Modeling noise and error correction for Majorana-based quan-
tum computing
Christina Knapp
1
, Michael Beverland
2
, Dmitry I. Pikulin
3
, and Torsten Karzig
3
1
Department of Physics, University of California, Santa Barbara, California 93106 USA
2
Station Q Quantum Architectures and Computation Group, Microsoft Research, Redmond, Washington 98052 USA
3
Station Q, Microsoft Research, Santa Barbara, California 93106-6105 USA
August 27, 2018
Majorana-based quantum computing seeks to
use the non-local nature of Majorana zero modes
to store and manipulate quantum information
in a topologically protected way. While noise
is anticipated to be significantly suppressed in
such systems, finite temperature and system
size result in residual errors. In this work, we
connect the underlying physical error processes
in Majorana-based systems to the noise mod-
els used in a fault tolerance analysis. Standard
qubit-based noise models built from Pauli opera-
tors do not capture leading order noise processes
arising from quasiparticle poisoning events, thus
it is not obvious a priori that such noise mod-
els can be usefully applied to a Majorana-based
system. We develop stochastic Majorana noise
models that are generalizations of the standard
qubit-based models and connect the error prob-
abilities defining these models to parameters of
the physical system. Using these models, we
compute pseudo-thresholds for the d = 5 Bacon-
Shor subsystem code. Our results emphasize
the importance of correlated errors induced in
multi-qubit measurements. Moreover, we find
that for sufficiently fast quasiparticle relaxation
the errors are well described by Pauli operators.
This work bridges the divide between physical
errors in Majorana-based quantum computing
architectures and the significance of these errors
in a quantum error correcting code.
1 Introduction
Topological phases of matter provide an attractive platform
for quantum computation due to the possibility of manipu-
lating information stored in the non-local degenerate state
space of non-Abelian anyons. This gives rise to the idea of
topological quantum computation [1, 2]. Any local probe
of the system does not provide information about the qubit
state; this topological protection of the encoded informa-
tion is anticipated to suppress decoherence from the envi-
ronment, thereby leading to exceptionally long qubit life-
times. At present, the most promising approach for topo-
logical quantum computing uses the non-Abelian properties
of Majorana zero modes (MZMs) that can be localized at the
ends of topological superconducting wires [314]
1
.
Each MZM is described by a Hermitian operator,
γ
a
= γ
a
, which obeys fermionic anticommutation rela-
tions {γ
a
, γ
b
} = 2δ
a,b
. A pair of MZMs corresponds
to a single fermionic state with occupation number
c
c = (1 +
a
γ
b
) /2, where c = (γ
a
+
b
) /2 is a Dirac
fermion operator. Quantum information can be encoded
through the occupation of the fermionic state that can be un-
occupied (even parity) or occupied (odd parity). Application
of one of the Majorana operators to the state flips the fermion
parity. When using a modular approach where each qubit is
encoded in a separate superconducting island, at least four
MZMs are required to store a qubit [15]. If the total fermion
parity of the superconducting island is fixed to be even, then
the qubit states correspond to both pairs of MZMs having
even parity or both pairs having odd parity. It is a choice
of basis how to pair the MZMs (e.g., pairing γ
1
with γ
2
and γ
3
with γ
4
as opposed to γ
1
with γ
3
and γ
2
with γ
4
);
which means that unlike other qubit schemes, phase errors
and bit-flip errors result from the same physical processes.
That is, for one encoding choice, the operator γ
1
γ
3
flips the
qubit state, while for the other, it results in a phase error.
Physically, errors described by Majorana operators are due
to, for instance, thermally excited quasiparticles, finite hy-
bridization of MZMs, or external quasiparticle poisoning of
the island. In the absence of non-equilibrium noise sources,
these errors will generally be exponentially suppressed in ra-
tios involving the physical parameters of the system that are
expected to be large, but cannot be made arbitrarily small for
practical reasons [1623].
The small error rates discussed above set an upper bound
1
While strictly speaking MZMs in a superconductor are not
dynamical and therefore are defects rather than quasiparticles,
they can be utilized for topological quantum computing in much
the same way as non-Abelian anyons.
Accepted in Quantum 2018-08-21, click title to verify 1
arXiv:1806.01275v3 [quant-ph] 24 Aug 2018
on the qubit lifetime. In order to store information for a
longer time, it is necessary to perform quantum error cor-
rection on the system. For a given quantum error correcting
code, if the error rate is below a particular value, known as
the code’s pseudo-threshold, the qubit’s lifetime is increased.
Studying the effectiveness of quantum error correcting codes
for Majorana-based systems is a relatively recent and impor-
tant development in the field of topological quantum com-
putation [15, 2436]. Thus far, the majority of studies (with
the exceptions of Refs. [24, 26, 34, 35]) have assumed qubit-
based noise models, which approximate noise in the system
by Pauli errors and measurement bit-flips. For a MZM sys-
tem, errors are naturally modeled by products of Majorana
operators. Some of the most important types of errors (e.g.,
quasiparticle poisoning [1618]) involve an odd number of
Majorana operators, which take the system out of the com-
putational subspace and are therefore not described by Pauli
operators. Thus, unless the effect of these errors can be cap-
tured with measurement bit-flips, the noise affecting a MZM
system is outside the scope of a qubit-based noise model.
The purpose of this paper is to connect the underlying
physical error processes in a Majorana-based quantum com-
puting architecture to a noise model that can be used to an-
alyze fault tolerance of the system. To this end, we develop
stochastic Majorana noise models from physical considera-
tions of proposed MZM systems and discuss the parameters
that control the probabilities of applying different products
of Majorana operators and the probability of measurement
bit-flips. These noise models reduce to qubit-based noise
models when the probabilities of an odd number of Majorana
operators being applied is set to zero. By analyzing these
noise models for a small Bacon-Shor subsystem code [37
39], we find that correlated errors induced through multi-
qubit measurements are most problematic for fault tolerance,
and therefore, would be most important to minimize in a
Majorana-based quantum computing architecture. We find
that for charging-energy-protected MZM qubits [40, 41], the
pseudo-threshold values calculated with a Majorana noise
model are well approximated by a qubit-based noise model
for finite, but sufficiently small, odd-Majorana error proba-
bilities. More generally, our work provides and exemplifies
a framework for analyzing Majorana-based error correction
that can be extended to other physical MZM architectures.
1.1 Guide to the Reader
The remainder of this paper is organized as follows. In Sec-
tion 2, we present four stochastic Majorana noise models in
order of simplest to most realistic. We define exactly what is
meant by a stochastic Majorana noise model and justify why
stochastic noise is an appropriate approximation of the errors
occurring in a Majorana-based system. We further discuss in
what limit our models reduce to analogous qubit-based noise
models. In Section 3, we use physical considerations to mo-
tivate the errors included in the noise models. We focus on a
particular Majorana-based qubit proposal [41] and estimate
the dependence of error rates on physical parameters in this
system. A summary of how these physical considerations
relate to the parameters of the noise models is given in Ta-
ble 2. In Section 4, we apply the Bacon-Shor subsystem
code to our noise models to estimate pseudo-thresholds and
the relative importance of the different error processes from
an error correction viewpoint. One motivation for using the
Bacon-Shor subsystem code is that it can be implemented
using only two-qubit measurements, which we anticipate to
be easier to perform experimentally on a MZM system than
the higher-weight measurements required for more standard
error-correcting codes (e.g., the surface code)
2
. We discuss
experimental implications of our analysis at the end of this
section. In Section 5, we discuss extensions of our analysis
to other MZM qubit proposals and error correcting codes.
Finally, in Section 6, we conclude and identify future direc-
tions. Details of the various discussions are relegated to the
appendices.
This paper is intended for both a condensed matter and a
fault tolerance audience; as such it contains review material
that readers from either community may find superfluous.
Readers from a fault tolerance background may wish to skip
Sections 2.1, 4.1, and 4.2. For those already familiar with
topological superconductivity, Sections 3.1 and 3.2 may be
skimmed. Additionally, those not interested in the technical
aspects of fault tolerance might wish to skim Sections 4.1
and 4.3, while those who are not invested in the physical
noise processes affecting a MZM system should skim Sec-
tions 3.1 and 3.2. The main results of this paper are con-
tained in Sections 2.2 and 4.4.
2 Stochastic Majorana Noise Mo dels
In this section, we develop stochastic Majorana noise models
analogous to the standard qubit-based noise models. There
are several motivations for tailoring a noise model to a sys-
tem of MZMs. (1) In general, the physical sources of errors
are best understood in terms of interactions of the environ-
ment with the MZMs; a Majorana noise model is therefore
more transparently connected to the physical system, affords
a more precise description of the noise that can lead to more
realistic quantum error correction simulations, and can be
applied independently of the encoding of quantum informa-
tion. (2) Majorana-based quantum computing architectures
do not necessarily group MZMs into qubits; as such, noise
models that describe environmental effects as qubit errors
are not applicable to all MZM systems. For instance, a Ma-
jorana fermion code [24, 34, 35] could not be fully analyzed
2
Higher-weight measurements could alternatively be built
from sequences of two-qubit measurements, but this is likely to
introduce additional errors.
Accepted in Quantum 2018-08-21, click title to verify 2
with a qubit noise model. (3) Even when MZMs are arranged
into qubits, some of the most common types of errors take
the system out of the computational subspace (e.g., quasi-
particle poisoning [1618]), and are therefore not captured
by the probabilistic application of Pauli errors.
Throughout this paper, we consider a set of 2n Majo-
rana zero modes (MZMs), with corresponding operators
γ
1
, γ
2
, . . . γ
2n
. Noise models discretize time into time
steps. In a stochastic Majorana noise model, after a time
step τ, a probabilistically generated string of Majorana op-
erators, γ
a
1
1
. . . γ
a
2n
2n
, for ~a {0, 1}
2n
, is applied to the state
ρ
M
of the MZM subsystem. Additionally, the noise models
allow for measurement errors that modify the binary vector
~m = (m
1
, . . . , m
N
) of the time step’s N measurement out-
comes by bitwise addition of the probabilistically generated
vector
~
b = (b
1
, . . . , b
N
), for ~m,
~
b {0, 1}
N
. The noise
model only tracks operators applied to either the MZM sub-
system or the measurement outcomes. Considering opera-
tors acting on the full system (i.e., MZM subsystem and its
environment) enables us to identify which operators to in-
clude in the noise model.
More explicitly, when the full system (MZMs plus envi-
ronment) begins in a product state ρ
M
|e
0
ihe
0
|, the time-
evolved projected density matrix can be written as [42]
X
j
he
j
|U [ρ
M
|e
0
ihe
0
|] U
|e
j
i =
X
j
ε
j
ρ
M
ε
j
, (1)
where U denotes the unitary evolution and ε
j
he
j
|U|e
0
iis
the projection of the environmental noise processes onto the
MZM subsystem. The operator ε
j
is therefore some combi-
nation of Majorana operators:
ε
j
=
X
~a
O
j
~a
=
X
~a
o
j
a
1
...a
2n
(γ
1
)
a
1
. . . (γ
2n
)
a
2n
. (2)
Instead of considering the density matrix ρ
M
, Eq. (1) can
equivalently be seen as a quantum trajectory where during
the time step the pure state |ψi
M
transforms as
|ψi
M
τ
X
~a
O
j
~a
|ψi
M
(3)
with some probability P
j
. Noise described by Eq. (3) de-
pends on the 2
n
coefficients o
l
a
1
...a
2n
, which renders numer-
ical simulations of large systems intractable.
Fortunately, Eq. (3) can be greatly simplified by noting
that decoherence processes such as energy relaxation and
phonons will destroy the coherence between different prod-
ucts of Majorana operators. In other words, local noise pro-
cesses do not result in superpositions of products of Majo-
rana operators (as opposed to non-local operations on the
computational state that allow coherent superpositions of
Majorana operators to be maintained over long time peri-
ods). Moreover, error correction itself separates many of the
linear combinations in Eq. (3) [42]. Given these considera-
tions, we can replace the intractable model of Eq. (3) with
a simpler, stochastic Majorana noise model. Then each time
step gives:
|ψi
M
τ
(γ
1
)
a
1
. . . (γ
2n
)
a
2n
|ψi
M
(4)
~m
τ
~m
~
b, (5)
with some probability Pr(~a,
~
b). The order of Majorana oper-
ators in Eq. (4) is unimportant as it only contributes to the
overall phase of the error operator.
The noise described by Eq. (4) is unitary, and thus does
not include an amplitude damping channel or an erasure
channel. This is not necessarily a crucial limitation since
a pessimistic estimate can be obtained by sufficiently strong
noise that randomly flips the qubits. This is the standard
approach for the qubit-based noise models reviewed in the
next section. As a consequence, however, the latter fails
to take into account the relaxation time T
1
characterizing
the timescale during which the system relaxes to the lower-
energy qubit state. In a MZM system, there is no such time
scale, since in practice the temperature is larger than the de-
generacy splitting of the qubit states. Thus, the assumption
that noise has unit amplitude provides a more accurate de-
scription of the physical noise processes for Majorana-based
qubits.
The content of different stochastic models is contained en-
tirely in the probability distribution {Pr(~a,
~
b)}. Given this
distribution, the errors in the system propagate classically
and can be efficiently simulated using standard Monte Carlo
techniques by tracking the net Majorana operators applied at
any given time. For this reason, when a model of the type
given in Eqs. (4) and (5) mimics the actual noise in a physi-
cal system, it is extremely useful for studying quantum error
correction.
In the following, a noise event refers to the application
of one of the operations of the right hand side of Eqs. (4) or
(5). For simplicity of relating the probabilities defining our
noise models to physical processes, we include noise events
that apply the same Majorana operator twice (therefore not
causing an error). The following presentation of the noise
models is tailored for conceptual ease; in Section 4 and Ap-
pendix C we give a more explicit description of how one can
simulate these noise models.
2.1 Qubit-based stochastic noise models
We first review three well-known qubit-based stochastic
noise models, all of which are built from Pauli operators
and measurement bit-flips. In each case, we consider a sce-
nario consisting of a sequence of time steps, where a set of
single- and multi-qubit measurements and/or gates are ap-
plied in each step.
Throughout the paper, the error probabilities of the Pauli
noise models are slightly reweighted compared to their
Accepted in Quantum 2018-08-21, click title to verify 3
standard presentation by also including the identity operator
as a possible error. This allows for an easier comparison
with Majorana noise models where noise events can lead to
the application of γ
2
a
= 1.
Pauli noise (or code capacity noise). For a given time
step and for each qubit:
1. Apply one single-qubit operator (either 1, X, Y , or
Z, chosen uniformly) with probability p; otherwise do
nothing.
2. Apply all measurement projectors perfectly.
More explicitly, Pauli operators X, Y , or Z are applied
with probability p/4 and identity is applied with probability
1 3p/4. As noted above, the non-standard normalization
is chosen for ease of comparison with the Majorana noise
models introduced in the following section. For the remain-
ing noise models, we will not explicitly write “otherwise do
nothing.
Pauli noise is defined by the single parameter p. While
the model is too simple to provide realistic estimates of an
error-correcting code’s performance, it serves as a quick
first test of any code.
Pauli noise with bit-flip measurement (or phe-
nomenological noise). For a given time step, for each qubit
apply step 1 of Pauli noise, then:
2. Apply all measurement projectors perfectly, then flip
each measurement outcome with probability p
mst
.
Note that flipping the measurement outcome does not change
the state of the system, only our information about the sys-
tem.
Pauli noise with bit-flip measurement is defined by two
parameters, {p, p
mst
}, which may be taken to be equal,
p = p
mst
, for a simple estimate of a code’s pseudo-threshold.
We emphasize that the measurement projections are still
applied exactly, but the classical bit which stores the mea-
surement outcome can be flipped. This model is motivated
on the grounds that qualitatively different error correction
approaches are required to handle faulty measurements
in addition to errors on the encoded information alone,
making this minimal addition to Pauli noise useful for
discriminating between codes.
Pauli circuit noise (or circuit-level noise) extends the
previous two models to account for the different noise pro-
cesses affecting a qubit during an operation (unitary gate or
measurement). For a given time step, each qubit is involved
in a k-qubit operation, where k = 0 for an idle qubit. For all
sets of qubits involved in the same k-qubit operation:
1. Do the following:
(a) For each qubit in the set, apply one single-qubit
operator (either 1, X, Y , or Z, chosen uniformly)
with probability p
(k)
.
(b) Apply a k-qubit Pauli operator with probability
p
(k)
cor
. For k = 2, this is any element of the set of
16 operators {Z X, 1 Y, . . . }. For j 1,
p
(j)
cor
= 0.
2. Apply the measurement projector perfectly, then flip the
k-qubit measurement outcome with probability p
(k)
mst
.
For an idle qubit, do nothing.
For step 1a, X, Y , or Z are applied with probability p
(k)
/4
and identity is applied with probability 1 3p
(k)
/4. For
step 1b, any given non-trivial Pauli operator is applied with
probability p
(k)
cor
/16 and identity is applied with probabil-
ity 1 15p
(k)
cor
/16. Again, the non-standard normalization
is chosen to simplify comparison with the Majorana noise
models in the following section.
Pauli circuit noise is defined by the set of probabilities
{p
(0)
, p
(k)
, p
(2)
cor
, p
(k)
mst
} for k {1, 2}. It is for this noise
model (slightly renormalized
3
) with the probability of a
single-qubit, two-qubit, and measurement bit-flip error
equally likely, that the well-known result is found that
the qubit surface code has an error threshold value of
p
th
1% [43, 44]. This means that a quantum state can be
reliably stored in an (arbitrarily large) surface code for an
indefinite period of time for qubits subjected to circuit-level
noise with p < p
th
.
2.2 Stochastic Majorana noise models
We now present four stochastic Majorana noise models in
order of increasing complexity. The first three are analogous
to the qubit-based models reviewed above. The fourth is
motivated by a particular physical implementation and mea-
surement protocol of a Majorana-based quantum computing
architecture [40, 41].
Naively, the simplest stochastic Majorana noise model to
consider would simply apply the Majorana operator γ
i
with
probability p for each i {1, 2, . . . , 2n} in each time step,
followed by perfect measurements. However, such a model
would on average spend an equal amount of time in an even
total MZM parity state as in an odd total MZM parity state.
In the case of superconducting islands with charging energy,
e.g., for the system described in Section 3.1, the energy sep-
aration of these states is large (on the order of the super-
conducting gap or the charging energy of the island) and
thus physically we would expect the system to spend much
3
Note that because of the renormalization of probabilities to
include the identity operation, this result is found for p = p
(k)
mst
=
3/4p
(0)
= 3/4p
(1)
= 15/16p
(2)
cor
.
Accepted in Quantum 2018-08-21, click title to verify 4
MZM
measurement
islands
Figure 1: Schematic of a quantum computing architecture with
2n MZMs (red stars) equally divided among n/m islands (blue
boxes). The inset zooms in on the jth island. As an example,
we indicate a two-MZM measurement and the possible noise
events for the Majorana noise model QpBf when the island
begins a time step in the even parity state.
more time in the lower-energy state corresponding to a spe-
cific MZM parity. To more accurately describe this situation,
we go beyond the naive model and introduce the concept of
MZM islands.
We assume that the MZMs are naturally split into n/m
subsets, each of which contains 2m MZMs belonging to the
same superconducting island (see Fig. 1). We assume that
the initial 2m-MZM parity on each island is even so that, for
a given time step, an island has even parity if an even number
of Majorana operators have been applied in its history, and
has odd parity otherwise. By keeping track of the parity of
the islands, the probability of applying an odd number of
Majorana operators can be adjusted depending on whether
this would relax the system back to the ground state or lead
to an excited state. This is captured by an additional step 0
in Majorana noise models that is not required for the qubit
ones.
Each noise model contains up to four types of noise
events:
Quasiparticle event: application of a single Majorana
operator: |ψi
M
γ
j,a
|ψi
M
.
Pair-wise dephasing event: application of a pair
of Majorana operators belonging to the same island:
|ψi
M
γ
j,a
γ
j,b
|ψi
M
.
Correlated event: application of Majorana operators
from multiple islands involved in the same measure-
ment (see later discussion or Table 1 for examples).
Measurement bit-flip: flipping of the classical bit stor-
ing the outcome of a 2k-MZM parity measurement:
e.g., ~m ~m (0, 1, 0, . . . , 0).
The naming of the noise events will become clear in Sec-
tion 3 when we describe the physical processes contributing
to each error-type. To simplify combinatorial prefactors, we
allow the same Majorana operator to be used multiple times
in a given noise event (e.g., pair-wise dephasing includes the
identity operator γ
2
a
). We also keep track of ordering, so that
applying the pairs γ
a
γ
b
and γ
b
γ
a
are considered different
noise events (multiple noise events contribute to the same
type of error). Unless otherwise noted, we assume that the
Majorana operators corresponding to a given noise event are
chosen uniformly over all MZMs on the island.
We note that even if subsets of MZMs are combined into
physical qubits, errors involving an odd number of Majo-
rana operators on any given island cannot be described by
Pauli operators, motivating the consideration of stochastic
Majorana noise models.
Quasiparticle noise (Qp). In a given time step, imple-
ment the following sequence for each island:
0. If the island begins the time step with odd parity, apply
a quasiparticle event with probability p
odd
.
1. Apply one single-island noise event: either a quasipar-
ticle event with probability p
qp
or a pair-wise dephasing
event with probability p
pair
.
2. Apply all measurement projectors perfectly.
We count 2m × 2m different pairs of Majorana operators
per island. In step 1, the operator γ
j,a
is applied with prob-
ability p
qp
/2m, the operator γ
j,a
γ
j,b
(a 6= b) is applied with
probability p
pair
/(2m)
2
, and identity is applied with prob-
ability 1 p
qp
3p
pair
/4 (because of pair-wise dephasing
events with γ
2
j,a
= 1).
The set of probabilities {p
odd
, p
qp
, p
pair
} defines this
model. In an encoding where four MZMs define a physi-
cal qubit, Qp reduces to the qubit-based model Pauli noise
when p
qp
= 0 (with p p
pair
).
As mentioned above, step 0 accounts for the energy
difference between an island with odd parity and even
parity: when p
qp
p
odd
, each island in the system spends
on average very little time in the odd MZM parity state. We
will return to this discussion in Section 3.1.
Quasiparticle noise and bit-flip measurement
(QpBf). In a given time step, implement steps 0 and 1 from
model (Qp) for each island, then:
2. Apply all measurement projectors perfectly, then inde-
pendently flip each classical bit storing a measurement
outcome with probability p
mst
.
This model is defined by the set of probabilities
{p
odd
, p
qp
, p
pair
, p
mst
}. In an encoding where four MZMs
define a physical qubit, when p
qp
= 0, QpBf reduces to
the qubit-based model Pauli noise and bit-flip measurement
(with p p
pair
and the same p
mst
). Example noise events
Accepted in Quantum 2018-08-21, click title to verify 5
MZM
measurement
Event Operator Qp QpBf MC PMC
qp γ
1,1
p
qp
/(2m) p
qp
/(2m) p
(2)
qp
/(2m) p
(0)
qp
/(2m)
γ
1,2m
p
qp
/(2m) p
qp
/(2m) p
(2)
qp
/(2m) p
(2)
qp
/(2m)
pair γ
1,1
γ
1,2
p
pair
/(2m)
2
p
pair
/(2m)
2
p
(2)
pair
/(2m)
2
p
(0)
pair
/(2m)
2
γ
1,1
γ
1,2m
p
pair
/(2m)
2
p
pair
/(2m)
2
p
(2)
pair
/(2m)
2
p
(2)
pair
/(2m)
2
cor γ
1,1
γ
1,2m
γ
2,2m
p
pair
p
qp
/(2m)
3
p
pair
p
qp
/(2m)
3
p
cor,odd
/N
odd
p
(2)
cor,odd
/(8m)
γ
1,1
γ
1,2m
γ
2,2m
γ
2,2
p
2
pair
/(2m)
4
p
2
pair
/(2m)
4
p
cor,even
/N
even
p
(2)
cor,even
/(16m
2
)
mst b
(2)
N/A p
mst
p
(2)
mst
p
(2)
mst
Table 1: Left panel: MZMs (red stars) are grouped into sets of 2m on an island (blue box). The operator of ath MZM on the jth
island is γ
j,a
. For the time step considered here, the two islands are involved in a four-MZM measurement. Right panel: Example
noise event probabilities when both islands begin the time step with even parity. The table could equivalently be understood as
noise event probabilities after the initial step 0 of each noise model accounting for the asymmetry between even and odd parity
islands. The left-most column labels the error type, with the abbreviations meaning quasiparticle, pair-wise dephasing, correlated,
and measurement bit-flip, respectively. The operators for correlated noise events for models MC and PMC could be applied from
two independent single-island noise events, analogously to models Qp and QpBf, or from a single correlated event; for simplicity
we only write the probability of the latter. The parameters N
even
=
(2m)
2
(2m + 1)
m2
2
and N
odd
= N
even
/2m in the fifth
column are defined to be the number of different odd correlated events and even correlated events, respectively, in the model
MC. Note that in model PMC, correlated events must involve a pair of MZMs connected by the measurement, which reduces the
combinatorial factors in the denominators. For instance, an odd correlated event has 4m possibilities for the initial excitation (γ
1,1
in the table) and only two choices for the pair of MZMs that transfers the excitation to the second island (γ
1,2m
γ
2,2m
in the table).
and their corresponding probabilities are schematically de-
picted in Fig. 1 and listed in Table 1 for an island beginning
a time step in the even parity state.
Correlated events. Models Qp and QpBf do not dis-
tinguish the probabilities of noise events involving MZMs
on idle islands from those on measured islands. We would
now like to account for these differences. Define γ
j,a
to be
the operator corresponding to the ath MZM on the jth is-
land. We consider two types of correlated events possible in
a multiple-island measurement:
Odd correlated event: application of a string of three
or more Majorana operators involving at least two is-
lands, such that an odd number (up to m 1) of Ma-
jorana operators are applied to one of the islands in-
volved in a k-island measurement. There is an even
number (up to m) of Majorana operators applied to
the remaining k 1 islands involved in the measure-
ment. For example, an odd correlated event in a
two-island measurement of islands i and j results in
|ψi
M
γ
i,a
γ
j,b
γ
j,c
|ψi
M
.
Even correlated event: application of a string of four
or more Majorana operators involving at least two is-
lands, such that an even number (up to m) of Majorana
operators are applied to all the islands involved in the k-
island measurement. For example, an even correlated
event in a two-island measurement of islands i and j
results in |ψi
M
γ
i,a
γ
i,b
γ
j,c
γ
j,d
|ψi
M
.
In the following, we assume that during a time step each
island is either idle or involved in a single measurement. The
spread of correlated events can be mitigated by restricting
the number of islands involved in a measurement. For
instance, for the Bacon-Shor code studied in Section 4,
correlated events only involve nearest neighbor islands.
Majorana circuit noise (MC). In a given time step,
for a set of islands involved in the same k-island measure-
ment (k = 0 for an idle island), implement the following
sequence:
0. For each island that begins the time step with odd parity,
apply a quasiparticle event with probability p
(k)
odd
.
1. For the set of islands involved in the same k-island mea-
surement, do the following:
(a) For each island in the set, apply one single-island
noise event: either a quasiparticle event with
probability p
(k)
qp
or a pair-wise dephasing event
with probability p
(k)
pair
.
(b) Apply a correlated event to the set: either an
odd correlated event with probability p
(k)
cor,odd
or
an even correlated event with probability p
(k)
cor,even
.
For j 1, p
(j)
cor,odd
= p
(j)
cor,even
= 0.
2. Apply the measurement projector perfectly, then flip
the classical bit storing the measurement outcome with
probability p
(k)
mst
. For an idle island, do nothing.
MC is defined by the probability set
{p
(k)
odd
, p
(k)
qp
, p
(k)
pair
, p
(k)
mst
, p
(k)
cor,odd
, p
(k)
cor,even
} for k k
max
,
Accepted in Quantum 2018-08-21, click title to verify 6
where k
max
is the maximum number of islands in-
volved in a measurement. MC has the same action
on an idle island (k = 0) as Qp, with the probability
set {p
odd
, p
qp
, p
pair
} {p
(0)
odd
, p
(0)
qp
, p
(0)
pair
}. MC has the
same action on an island involved in a single-island
measurement (k = 1) as QpBf, with the probability set
{p
odd
, p
qp
, p
pair
, p
mst
} {p
(1)
odd
, p
(1)
qp
, p
(1)
pair
, p
(1)
mst
}. In an
encoding where four MZMs define a physical qubit and
if the probability of any error involving an odd number
of Majorana operators on a given island is set to zero
(i.e., p
(k)
qp
= p
(k)
cor,odd
= 0), MC reduces to the qubit-based
model Pauli circuit noise, with the probabilities related by
{p
(k)
, p
(2)
cor
, p
(k)
mst
} {p
(k)
pair
, p
(2)
cor,even
, p
(k)
mst
}.
In Table 1, we compare the probabilities of noise events
in step 1 of models Qp, QpBf, and MC. We see that MC
allows for the possibility that noise events affecting multiple
islands connected by a measurement happen with greater
probability than independent noise events on the islands
(e.g., when p
(2)
cor,even
> p
2
pair
).
Physical Majorana circuit noise (PMC). This model
refines MC by considering a specific physical implementa-
tion and measurement protocol of the MZM system [40, 41].
Focusing on a specific measurement protocol allows us to
drop many of the correlated events included in MC, as well
as to separate noise events involving measured MZMs from
those only involving unmeasured MZMs. These two modi-
fications enable a more accurate description of this physical
system, see Section 3.1 for a description of the underlying
causes of errors in such a system.
We assume that our measurement protocol allows parity
measurements of two MZMs belonging to the same island
and joint parity measurements of a set of four MZMs on two
islands. During a given time step, each island is now either
idle (k = 0), involved in a two-MZM measurement (k = 1),
or involved in a four-MZM measurement (k = 2).
The model follows the same steps as MC, with slight mod-
ifications of steps 0 and 1(a) for islands that are involved in
a measurement, to account for whether particular MZMs are
connected to the measurement apparatus or not:
0. For each island that begins the time step with odd parity,
apply a quasiparticle event corresponding to:
A MZM not involved in the measurement with
probability
2m2
2m
p
(0)
odd
.
A MZM involved in the measurement with prob-
ability
2
2m
p
(k)
odd
.
1. (a) For islands in the set not involved in a measure-
ment (k = 0) apply either a quasiparticle of pair-wise
dephasing event with respective probabilities p
(0)
qp
and
p
(0)
pair
.
For each island in the set with k 1, apply either a
quasiparticle event corresponding to:
A MZM not involved in the measurement with
probability
2m2
2m
p
(0)
qp
.
A MZM involved in the measurement with prob-
ability
2
2m
p
(k)
qp
.
or a pair-wise dephasing event corresponding to:
A pair of MZMs not involved in the measurement
with probability
(2m2)
2
(2m)
2
p
(0)
pair
.
A pair of MZMs, with at least one of them
involved in the measurement, with probability
4(2m1)
(2m)
2
p
(k)
pair
.
Furthermore, we consider a restricted set of correlated events
in step 1(b) so that odd correlated events are strings of three
Majorana operators γ
i,a
γ
i,b
γ
j,b
and even correlated events
are strings of four Majorana operators γ
i,a
γ
i,b
γ
j,b
γ
j,c
, such
that i 6= j and the indices are chosen such that γ
i,b
and γ
j,b
are involved in a measurement.
PMC is defined by the same set of probabilities as MC,
with k
max
= 2. While MC treated all MZMs on a given is-
land identically, PMC distinguishes between the measured
and unmeasured MZMs within the island for single-qubit
noise events. Furthermore, correlated events in PMC always
involve a pair of MZMs directly coupled by the measure-
ment. In Table 1, we compare noise event probabilities for
all four models. Only a restricted set of correlated events are
considered in PMC, which changes the combinatorial pref-
actors of the probabilities of these events between MC and
PMC.
3 Physical System
In the following, we use a recently proposed implementation
of a measurement-based topological quantum computer [41]
as an example to illustrate the different types of errors and
to connect them to our noise models. The system consists
of an array of four-MZM islands, tetrons, depicted in Fig. 2.
In this proposal, the MZM islands have a finite charging en-
ergy E
C
. Each tetron corresponds to a physical qubit, whose
states are stored in the two nearly-degenerate ground states
within a total parity subspace. In the absence of quasipar-
ticle poisoning, the latter is fixed to be either even or odd.
The inset of Fig. 2 shows the details of a tetron implemen-
tation based on a single superconducting island composed
of two topological superconducting wires of length L (gray
lines), connected by a conventional superconducting bridge
(blue line). There are four MZMs (red stars), localized at
the end points of the topological superconductors. Multiple
tetrons are connected to each other by semiconducting wires
(orange lines), which may be gated (not shown) to allow for
Accepted in Quantum 2018-08-21, click title to verify 7
MZM
top. supercond. supercond.
semicond.
Figure 2: Example physical system: an array of charging
energy-protected islands with four-MZMs (tetrons) connected
by a network of semiconducting wires (orange lines). The jth
tetron, shown in the inset, is a superconducting island formed
out of two topological superconducting wires (gray lines) con-
nected by a regular superconducting bridge (blue line). A MZM
(red star) is localized at the ends of each topological supercon-
ductor. In terms of the noise models of the previous section,
each tetron corresponds to a single island with 2m = 4 MZMs.
measurement by quantum dots [41]. The Hilbert space of a
single tetron contains many of the features applicable to a
large class of MZM systems, however the measurement pro-
tocol is specific to this qubit proposal.
3.0.1 Spectrum and single MZM excitations
In this section, we identify the types of errors that appear
in a Majorana-based quantum computing architecture and
discuss how the probabilities of these errors depend on the
physical parameters of the system. For ease of explanation,
we focus on a particular Majorana-based qubit proposal, re-
viewed in Section 3.1. In Section 3.2, we discuss the dif-
ferent error processes and corresponding transition rates. In
Section 3.3, we connect the parameters of the noise mod-
els with a small set of physical parameters that can be ex-
pressed in terms of the transition rates. When possible, we
distinguish between the features that are generic to systems
of MZMs and those that are specific to the example system
we consider. The former will motivate the errors included
in MC, while the latter will explain the refinement of MC to
PMC.
3.1 Example system: Tetron array
The jth tetron can be described by the Hamiltonian
4
H = E
C
(ˆn
s
n
g
)
2
+
X
k
E
k
ˆn
,k
+
X
a6=b
δE
ab
j,a
γ
j,b
,
(6)
4
In Eq. (6) we assumed that “mutual charging energy” terms
that could couple pairs of MZMs belonging to different tetrons
are perfectly quenched. We will comment more these terms in
Section 3.2.4.
Figure 3: Cartoon of the energy levels of a single tetron against
dimensionless gate voltage for E
C
= 2∆ (left panel) and
E
C
= /2 (right panel), where E
C
is the charging energy
of the island and is the superconducting gap. For fixed gate
volgate 0.5 < n
g
< 0.5, the system has a nearly-degenerate
ground state with an even four-MZM parity. The degeneracy
is broken by the MZM hybridization δE
ab
, leading to distinct
states |gi|ψ
i
i
M
denoted by solid blue and purple curves. The
energy bands bordered by solid orange and red curves corre-
spond to the states |e
i|
¯
ψ
i
i
M
that have total fermion parity
even, and odd four-MZM parity. The shaded regions indicate
that the bands contain many discrete energy levels, with level
spacing δ
is
. The dashed blue and purple curves correspond
to the states |e
C,±
i|
¯
ψ
i
i
M
that have total fermion parity odd,
and odd four-MZM parity. Comparing the right and left panels
near n
g
0, we see that when E
C
is larger (smaller) than ,
the quasiparticle-poisoned states with one extra or one fewer
electron on the superconducting island, |e
C,±
i|
¯
ψ
i
i
M
, are higher
(lower) in energy than the lowest energy states with a thermally
excited quasiparticle, |e
i|
¯
ψ
i
i
M
.
where a, b = 1, 2, 3, 4. The last two terms (BCS Hamilto-
nian and MZM hybridization) are present in all MZM sys-
tems, while the first term (charging energy Hamiltonian) is
present in systems for which the superconducting island is
Coulomb-blockaded (i.e., not grounded).
In the charging energy Hamiltonian, the operator ˆn
s
, with
integer eigenvalues n
s
, is the number operator for electrons
on the superconducting island. Generally the energy will be
minimized when n
s
is the integer closest to n
g
, the dimen-
sionless gate voltage applied to the superconducting island.
When n
g
is an integer, adding an electron to or subtract-
ing an electron from the island costs a charging energy E
C
,
which is set by the capacitance of the superconducting is-
land.
The BCS Hamiltonian is written in terms of the quasipar-
ticle number operator ˆn
,k
, with integer eigenvalues n
,k
.
More specifically, ˆn
,k
counts the number of above-gap
quasiparticles on the superconducting island with crystal
momentum k, occupying the state with energy
E
k
=
s
k
2
2m
µ
2
+
2
, (7)
where is the superconducting gap and µ is the chemical
potential. Due to the finite size of the island, only a dis-
Accepted in Quantum 2018-08-21, click title to verify 8
crete set of momenta, {k
i
}, is allowed. The level spacing of
the island, δ
is
, is the separation between adjacent energies:
δ
is
= E
k
E
kδk
.
The last term of Eq. (6) describes the MZM hybridiza-
tion energy, δE
ab
, between MZMs γ
j,a
and γ
j,b
. Gener-
ally, δE
ab
is set by the wavefunction overlap, resulting in
a length dependence δE
ab
e
L
ab
, where L
ab
is the dis-
tance separating γ
j,a
and γ
j,b
and ξ is the superconducting
coherence length. The topological protection of the system
is manifested as an exponentially small ground state degen-
eracy splitting and requires L
ab
ξ.
We denote the eigenstates of Eq. (6) using the basis
|n
s
, {n
,k
}
i|
j,1
γ
j,2
,
j,3
γ
j,4
i. The first set of quantum
numbers describes the non-topological degrees of freedom:
charge or quasiparticle excitations. The second set denotes
the (almost) degenerate MZM subspace. The pairing of Ma-
jorana operators into
j,1
γ
j,2
and
j,3
γ
j,4
is an arbitrary
choice. Note that the total fermion parity of the island,
2(n
s
mod 2) 1, equals the product of the quasiparticle and
four-MZM parity.
To discuss the leading excitations, we compare the energy
levels of the system for E
C
= 2∆ and for E
C
= ∆/2 in the
left and right panels of Fig. 3. Solid curves correspond to an
even number of particles n
s
on the superconducting island,
dashed curves to odd n
s
. The tetron is operated at an even
integer value for the dimensionless gate voltage to maximize
protection from quasiparticle poisoning by maximizing the
energy separation between solid and dashed curves. Without
loss of generality we assume n
g
0. In this regime, there
are two nearly degenerate ground states:
|gi|ψ
i
i
M
= |0, {0}
i|ψ
i
i
M
, (8)
with i = 0, 1 and corresponding energy given by the solid
blue and purple curves centered at n
g
= 0. We write
{n
,k
}
= {0}
to denote that there are no quasiparti-
cles occupying energy levels above the superconducting gap.
The qubit is stored in the two MZM states |ψ
i
i
M
, which are
orthogonal linear combinations of the even four-MZM par-
ity basis states |
j,1
γ
j,2
= ±1,
j,3
γ
j,4
= ±1i. When the
only non-vanishing hybridizations are δE
12
and δE
34
, |ψ
i
i
M
is simply | ± 1, ±1i.
There are two bands of lowest excited states with even
n
s
, that correspond to a single thermally excited quasipar-
ticle. Their energies are shown in Fig. 3 by the overlap-
ping shaded regions bordered by solid orange and red curves.
Two-quasiparticle excitations require energies of at least 2∆
and are therefore much less likely; in the following, we re-
strict our attention to n
,k
{0, 1}. We denote a single
excitation with energy larger than the gap as e
. The two
bands are denoted by the states
|e
i|
¯
ψ
i
i
M
= |0, {n
,k
= 1}
i|
¯
ψ
i
i
M
, (9)
with i = 0, 1. In the above, e
may denote different k states;
this does not matter for our discussion as long as we focus
on states with energy . Here, the MZM states |
¯
ψ
i
i
M
are orthogonal linear superpositions of the odd four-MZM
fermion parity states |
j,1
γ
j,2
= ±1,
j,3
γ
j,4
= 1i.
Depending on whether n
g
is positive or negative, the two
lowest excited states with odd total fermion parity contain
either an extra electron or one fewer electron, respectively.
Such states are quasiparticle-poisoned. The states with one
extra (one fewer) electron are written as
|e
C,±
i|
¯
ψ
i
i
M
= | ± 1, {0}
i|
¯
ψ
i
i
M
, (10)
with energy levels shown in the dashed blue and purple
parabolas centered about n
g
= ±1.
The state of the MZMs is very similar for the two ex-
cited states discussed above. In both cases, an excitation
process exchanges an electron between the MZMs and other
fermionic modes represented by either the excited quasi-
particles or by an external environment. The excitation
acts by applying a single Majorana operator to the state
|ψ
i
i
M
. The ground states |gi|ψ
i
i
M
and the excited states
|e
i|
¯
ψ
i
i
M
, |e
C,±
i|
¯
ψ
i
i
M
are therefore distinguished by their
four-MZM parity, with odd parity states separated from the
even parity ground states by an energy min (∆, E
C
). In our
stochastic Majorana noise models, we focus only on these
four lowest energy states and can therefore use the four-
MZM parity as a measure of whether an island is in a ground
or an excited state. In order to include higher excited states,
we would need to separately track the four-MZM parity and
the quantum numbers n
s
and {n
,k
}.
3.1.1 Measurements
For an array of tetrons, parity measurements can be imple-
mented by coupling the MZMs to quantum dots [41] or by
a conductance-based readout scheme [15, 40]. Most of the
measurement concepts can be used interchangeably between
the two approaches. In the following, we focus on quantum-
dot based measurements. The most common examples are
measuring the parity of a pair of MZMs on a single island, or
measuring a four-MZM parity composed out of two MZM-
pairs from separate islands. The latter can induce corre-
lated excitations of different islands. Other Majorana-based
quantum computing architectures employ different measure-
ment schemes, which in general will change the correlated
events between islands being measured. In Section 5, we
comment on how the different measurement schemes of
other Majorana-based quantum computing architectures af-
fect which types of correlated events the noise model should
include.
To measure the parity of a pair of MZMs γ
1,m
and γ
1, ¯m
belonging to the same tetron, electrostatic gates in the semi-
conducting wire adjacent to the MZMs are tuned to form
a quantum dot tunnel-coupled to γ
1,m
and γ
1, ¯m
, as shown
in the left panel of Fig. 4. To measure the parity of four
MZMs (two pairs on two different superconducting islands),
Accepted in Quantum 2018-08-21, click title to verify 9
MZM
top. supercond. supercond.
semicond.
quantum dot tunneling amp.
Figure 4: MZM parity measurements using quantum dots. Left
panel: parity measurement of
1,m
γ
1, ¯m
by tunnel coupling
(yellow line) the corresponding MZMs to a quantum dot. Right
panel: parity measurement of γ
2,m
γ
2, ¯m
γ
3,m
γ
3, ¯m
by tunnel
coupling the corresponding MZMs to two quantum dots. For
both panels, we show only half of the tetron(s) involved in the
measurement. The tunnel couplings enable the transfer of elec-
trons between the quantum dots and the MZMs connected to
it. Because the MZM islands have charging energy, an electron
originating from a quantum dot returns to the quantum dot
with high probability. Importantly, an electron can return ei-
ther by (1) traveling through a subset of the measured MZMs
then backtracking its way back to the quantum dot or (2)
traversing the full loop formed by the tunnel couplings, the
quantum dot(s), and the superconducting island(s). In (1), an
even number of each Majorana operator is applied, while in
(2) a product of all Majorana operators involved in the mea-
surement loop are applied. The processes of type (2) result in
a measurement through a shift of the ground state properties
of the quantum dot depending on the parity of the involved
MZMs.
the electrostatic gates are tuned to form two quantum dots,
each of which is tunnel-coupled to a MZM on each island,
as shown in the right panel of Fig. 4.
An isolated quantum dot is described by a charging-
energy Hamiltonian
H
D
= E
QD
C
d
d N
g
2
, (11)
where d is the fermionic annihilation operator for the quan-
tum dot and N
g
is the dimensionless gate voltage on the dot.
Writing the eigenvalues of the number operator for quantum
dot levels as N
d
, the quantum dot Hamiltonian is spanned by
the occupation basis |N
d
i
d
. We assume that the quantum dot
is in the spin-polarized regime such that the available states
are |0i
d
, |1i
d
. When performing a measurement, a tunneling
Hamiltonian couples Eqs. (6) and (11). For instance, for the
two-MZM measurement, the tunneling term takes the form
H
t
= a
t
m
d
γ
1,m
+ t
¯m
d
γ
1, ¯m
+ H.c., (12)
where t
m
is the tunneling amplitude between MZM γ
1,m
and the quantum dot, and a
is a bosonic operator re-
moving a single electron charge from the island. Equa-
tion (12) hybridizes the two quantum dot states |0i
d
and |1i
d
in a two-MZM parity-dependent manner. By measuring the
energy levels, charge occupation, or quantum capacitance
Figure 5: Quasiparticle events for a single MZM island. The
corners of the triangle correspond to the three energy states we
consider: the ground state with even MZM parity (correspond-
ing to the computational states) |gi = |0, {0}
i, the thermal
excited state with an above-gap quasiparticle |e
i = |0, {1}
i,
and the quasiparticle-poisoned states |e
C,±
i = | ± 1, {0}
i.
The edges of the triangle denote the quasiparticle event that
transitions the system between the given states (thermal exci-
tation of an above-gap quasiparticle or extrinsic quasiparticle
poisoning), and are labeled by the corresponding transition rate
for that process (in blue) and the operators that act on the sys-
tem for that given process (in red). Arrows can be reversed by
conjugating the operators, but the rates for the opposite pro-
cesses can be drastically different.
of the quantum dot, one can extract the two-MZM parity
1,m
γ
1, ¯m
of the system. Such a two-MZM measurement
can be used to infer whether the tetron is in computational
state |ψ
0
i
M
or |ψ
1
i
M
. For a more-detailed discussion, see
Ref. [41].
In the remainder of this section, we make the follow-
ing gauge choice: all fermion operators are neutral and the
charge on the system is accounted for by the bosonic op-
erator ˆn
s
. We define the neutral creation operator of an
above-gap quasiparticle with crystal momentum k as c
k
(i.e., c
k
|n
s
, {n
,k
= 0}
i = |n
s
, {n
,k
= 1}
i). When
an electron enters or leaves the superconducting island, the
eigenvalue n
s
changes by 1. We account for this change with
the ladder operators a
±
, which raise or lower n
s
.
3.2 Error processes
In this section, we discuss the main physical processes con-
tributing to errors in Majorana-based quantum computing
architectures. The corresponding error rates will be inde-
pendent of which computational (MZM) state the system is
in, as such we simplify notation by dropping the MZM la-
bels |ψi
M
, |
¯
ψi
M
and only writing the energy state |gi, |e
i,
or |e
C,±
i. When an effect is independent of which excited
state the system is in, we will simply write |ei.
There are higher energy states, for instance, the state with
both an extra electron and an above-gap quasiparticle on the
island, that we have not discussed. Throughout this work,
Accepted in Quantum 2018-08-21, click title to verify 10
our aim will be to identify the lowest order errors: the
most prominent errors in the system. If the probability of
a given error is less than or equal to the product of the prob-
abilities of other errors, and the effect on the Hilbert space
of the MZMs is the same for that particular error as for the
combination of the other errors, then we call such an error
higher order. Processes involving excited states other than
|e
i and |e
C,±
i can be described as higher-order errors, see
Appendix A. Note that an error is classified as higher or-
der solely from its probability; it is possible for the physical
process causing a particular higher order error to be distinct
from the physical processes causing lower order errors.
Figure 5 schematically illustrates single island error pro-
cesses involving quasiparticles. A transition between the
ground and an excited state corresponds to the application
of a single Majorana operator and thus sets the probability
of a quasiparticle event. The combination of an excitation
and a relaxation, or a transition between excited states, cor-
responds to the application of an even number of Majorana
operators. These processes therefore contribute to the prob-
ability of a pair-wise dephasing event. We first quantify the
transition rates involving quasiparticles in a single island, be-
fore discussing correlated events between islands connected
during a measurement and error processes that do not in-
volve quasiparticles. In the rates given below, we only quote
the parametric dependence and ignore prefactors of O(1).
3.2.1 Thermally excited quasiparticles
Thermal excitation of an above-gap quasiparticle, the top left
process in Fig. 5 |gi |e
i, is present in all MZM systems.
This process occurs, for instance, when a Cooper pair breaks
into two electrons, one of which occupies one of the non-
local fermionic states formed by the MZMs while the other
occupies a state in the continuum above the superconducting
gap. Thermal excitation of a quasiparticle preserves the total
fermion parity of the island and can therefore occur in an
isolated system.
5
Crucially, in equilibrium, thermal excitation of a single
quasiparticle to the state |e
i is exponentially suppressed in
the ratio /T , where T is the temperature of the system.
5
There are also thermal excitations that conserve the parity
of the MZMs by exciting a pair of quasiparticles to the contin-
uum. These processes have an energy cost greater than twice
the superconducting gap and are therefore less likely to occur.
In practice, there will be a competition between a power-law en-
hancement for creating a particle-hole pair since it can happen
anywhere in the system, and the additional exponential suppres-
sion due to the higher energy cost. Thermally excited particle-
hole pairs do not cause errors by themselves, but increase the
chance of subsequently transitioning to states that flip the MZM
parity. Furthermore, pair-excitation can cause correlated events
when nearby superconducting islands are coupled together for a
measurement. For now we assume that these effects can be qual-
itatively captured by the transition rate of a single quasiparticle
excitation and the rates describing correlated events.
The excitation and relaxation rates take the form
Γ
ge
= τ
1
0
exp(/T ) (13)
Γ
e
g
= τ
1
0
, (14)
where τ
0
is a characteristic time scale describing the
electron-phonon coupling of the system. For InAs wires with
an Al half-shell, τ
0
50ns [23]. In the presence of non-
equilibrium quasiparticles, the factor exp(/T ) is essen-
tially replaced by the number of quasiparticles in the vicinity
of the MZMs.
A single thermal excitation event takes the computational
(MZM) state of the island from an even parity state to an
odd parity state, while a relaxation event does the reverse.
Both thermal excitation and relaxation apply a single Majo-
rana operator γ
a
to the computational state, although as can
be seen from Eqs. (13) and (14) the corresponding rates are
significantly different. This is the underlying reason why all
of the noise models of Section 2 account for quasiparticle
events with two different probabilities, p
(j)
odd
and p
(j)
qp
: if the
island begins a time step in an odd MZM parity state, a re-
laxation event is much more likely than an excitation event
if the island has even MZM parity, indicating a separation
of scales between p
(j)
odd
and p
(j)
qp
. Furthermore, the applica-
tion of a single quasiparticle event significantly changes the
probability of future quasiparticle events, which is why our
noise models only allow for a single noise event to occur in
step 0, and a single noise event to occur in step 1(a) (this
approximation is justified if the errors are sufficiently rare).
A pair-wise dephasing event can result from an excitation
and subsequent relaxation of a thermal quasiparticle. The
relaxation event is equally likely for all Majorana operators
on the island, regardless of which Majorana operator was
applied for the excitation event: this is the underlying reason
why we allow for all pairs of Majorana operators, including
the same Majorana operator twice, when considering pair-
wise dephasing in our noise models. Finally, notice that the
rates of thermal excitation and relaxation are unaffected by
whether or not the superconducting island is involved in a
measurement.
3.2.2 Extrinsic quasiparticle poisoning
Extrinsic quasiparticle poisoning, the top right process in
Fig. 5 |gi |e
C,±
i, is when a quasiparticle tunnels onto or
off of the superconducting island, thereby changing the to-
tal fermion parity and charge of the island. We focus on the
lowest-energy poisoning events where quasiparticles tunnel
onto (off of) the island and into (out of) one of the fermion
states provided by the MZMs. In the following discussion,
we will assume n
g
= 0 so that adding an electron to, or
taking an electron off of the superconducting island costs an
energy E
C
; the following expressions could be made more
general by replacing E
C
with E
C
(1 2n
g
). For an exter-
nal quasiparticle to enter the island, it has to overcome the
Accepted in Quantum 2018-08-21, click title to verify 11
energy barrier E
C
+δµ, where δµ = µ
is
µ
res
denotes a pos-
sible difference in the chemical potentials (a voltage bias) of
the reservoir and the island.
In the case of tetrons, the most likely source of quasiparti-
cles is the quantum dot explicitly coupled to the MZMs dur-
ing a parity measurement. When the quantum dot is tuned
to resonance
6
, the energy difference δµ = 0. Transitions
between |gi and |e
C,±
i are then described by the rates
Γ
(k)
ge
C,±
= g
T
(k)δ exp {−E
C
/T } (15)
Γ
(k)
e
C,±
g
= g
T
(k)δ (16)
where g
T
(k) is the dimensionless conductance between the
quantum dot and MZM island. The parameter δ is either
the level spacing of the quantum dot δ
dot
when an electron
transitions from the quantum dot to the island, or the ef-
fective MZM level spacing when an electron transitions
from the island to the quantum dot. Equations (15)-(16)
can also be generalized to conductance-based measurement
schemes [15, 40]. In that case, δ
dot
is replaced by the tem-
perature T and g
T
(k) is the dimensionless conductance be-
tween the lead and the MZM island. In the case of a quantum
dot connected to the superconducting island, only a single
channel with transmission T
(k)
contributes to the island-dot
coupling, thus g
T
(k) = |T
(k)
|
2
. The transmission T
(k)
is
essentially zero for k = 0 (i.e., no measurement), and be-
comes appreciable for k = 1, 2. Notice also that during the
measurement, the quantum dot is tuned close to the charge
degeneracy point, thus we do not have to pay extra energy to
remove a particle from it. If the quantum dot is then tuned
to the bottom of its charging parabola when a measurement
is not being performed, then E
C
E
C
+ E
QD
C
in Eq. (15),
further protecting the system from quasiparticle poisoning.
It thus follows that quasiparticle poisoning from the quan-
tum dot is significantly more likely during a measurement.
This motivates why the noise models MC and PMC allow for
different probabilities of quasiparticle events and pair-wise
dephasing when an island is involved in a measurement.
Alternatively, quasiparticles could come from nearby
metallic gates, for example those used to tune the MZM is-
land into the topological regime. The transition rates are now
independent of whether a measurement is being performed,
as this is not expected to change the coupling between the
island and the gate. If the metallic gates are kept at a large
voltage, the energy |δµ| may become much larger than E
C
so that it could be favorable for an extra quasiparticle to
be on the superconducting island. In this case, it is essen-
tial that the dimensionless conductance between the metallic
gate and the island g
env
is sufficiently small so that the island
is not constantly being poisoned by quasiparticles. Since an
insulating barrier to the gates suppresses g
env
exponentially
in the thickness of the barrier, sufficiently small values of
6
Tuning the quantum dot to resonance maximizes the mea-
surement visibility, but is not a necessary condition.
g
env
are possible in practice. In the following, we assume
that extrinsic quasiparticle poisoning rates are well approxi-
mated by Eqs. (15) and (16).
From Eq. (15), we see the benefit of the island having
a large charging energy is to suppress the rate of extrin-
sic quasiparticle poisoning events. Majorana-based quantum
computing proposals for which the superconducting island
is grounded (E
C
0) are likely to be more susceptible to
quasiparticle and pair-wise dephasing events.
A comment is in order about the fast relaxation of
Eq. (16). A crucial requirement is that the environment
can accept or provide an electron at no energy cost. This
is always the case for conductance-based measurements in
which the island is connected to leads with no charging en-
ergy. For quantum dot-based measurements, it is important
to properly initialize and decouple the quantum dots. An ex-
ample procedure that always allows for fast relaxation is a
double-dot measurement (see Fig. 4 right panel). By initial-
izing the quantum dots in the state |0i
d
|1i
d
and tuning them
so that each dot is at its charge degenerate point, the island
can relax its charge state by transitioning to the |1i
d
|1i
d
or
|0i
d
|0i
d
state. Alternatively, we can use the quantum dots to
check if a poisoning event has occured during the measure-
ment by performing an additional charge measurement of the
quantum dot(s) after the island is decoupled. This additional
measurement could allow immediate correction of extrinsic
quasiparticle poisoning events, in which case we would not
need to include such events in the noise models.
In terms of the computational state of the island, extrin-
sic quasiparticle poisoning has the same effect as the corre-
sponding thermal quasiparticle event: an excitation, |gi
|ei, takes the MZMs from an even parity state to an odd par-
ity state by applying a single Majorana operator γ
a
; relax-
ation, |ei |gi, does the reverse; and the combination of an
excitation and subsequent relaxation, |gi |ei |gi, ap-
plies the pair of Majorana operators γ
a
γ
c
where a can equal
c. Transitions between the excited states |e
C,±
i and |e
i
have the same effect on the computational state as a com-
bined excitation and relaxation. Moreover, a transition be-
tween excited states requires two unlikely processes to oc-
cur: first, an island must be excited and second, the island
must transition to the other excited state before relaxing to
the ground state. It follows that such transitions only con-
tribute a small correction to the probability of pair-wise de-
phasing events and can therefore be neglected.
3.2.3 Correlated events
Quasiparticle poisoning and thermal excitations can lead to
correlated events that transfer excitations between two (or
more) islands connected during a measurement. An excited
charge state can be transferred by a quasiparticle tunneling
between two islands through the low-energy subspace pro-
vided by the MZMs. A thermal excitation can be trans-
Accepted in Quantum 2018-08-21, click title to verify 12
ferred without changing the net particle number by an above-
gap quasiparticle tunneling between two islands and a cor-
responding reverse tunneling of an electron through the low
energy subspace. Each time an island transitions between the
ground and an excited state, a Majorana operator is applied
to the computational state of that island. Thus, a process
in which two islands swap being in the ground and excited
state,
|gi |ei |ei |gi, (17)
results in a Majorana operator being applied to each island.
The corresponding MZMs are tunnel-coupled to the same
quantum dot, as shown in the right panel of Fig. 4 (either
γ
2,m
γ
3,m
or γ
2, ¯m
γ
3, ¯m
). When this process combines with
excitations and relaxations on the two islands, it results in
correlated events involving three or four Majorana opera-
tors between two islands, described, e.g., by the probabilities
p
(2)
cor,odd
and p
(2)
cor,even
, respectively, in PMC.
The most prominent correlated events come from the pro-
cesses
|gi |gi |e
x
i |gi |gi |e
x
i |gi |gi (18)
|gi |gi |e
x
i |gi |gi |e
x
i (19)
|e
x
i |gi |gi |e
x
i |gi |gi (20)
where the excited states |e
x
i {|e
C,±
i, |e
i}. Equa-
tion (18) applies an even number of Majorana operators and
is described by the probability p
(2)
cor,even
in PMC. If relaxation
processes are fast compared to the time step τ, this is the
most probable type of correlated event. Equations (19) and
(20) describe odd correlated events captured by p
(2)
cor,odd
in
PMC and can be seen as being part of an even correlated
event split up over two time steps. Both odd correlated
events have the same effect on the computational subspace
and are equally likely. To simplify our noise models, we only
take into account the process described by Eq. (19). When
connecting the probabilities defining PMC to physical pro-
cesses, we therefore have to overestimate the probability of
the event described by Eq. (19) by at least a factor of two
in order to avoid undercounting odd correlated events. Simi-
larly to the previous discussion of single-island errors, we ig-
nore higher order correlated events between the two islands,
e.g., |e
C,±
i |gi |e
i |e
i, see Appendix A.
Assuming the islands are identical, there is no energy dif-
ference between the states |e
x
i |gi and |gi |e
x
i. As
such, the corresponding transition rates have no exponential
suppression:
Γ
e
e
C
,±
,gg,e
e
C
,±
= g
is-is
(21)
Γ
e
e
,gg,e
e
= g
2
is-is
δ
is
/E
C
, (22)
where g
is-is
is the dimensionless conductance between the
two islands. The second line originates from the tunneling of
an above-gap quasiparticle (level spacing δ
is
) and a tunnel-
ing through the MZMs (effective level spacing ) within the
time 1/E
C
of the virtual charge and thermal excited state.
Note that for weak coupling to the dots, the interisland con-
ductance g
is-is
g
(k)
T
, and grows quadratically with the tun-
neling amplitude to the dots. Despite the smaller prefactor
of Eqs. (21) and (22), the corresponding rates are expected
to be much larger than those given in Eqs. (13) and (15) due
to the absence of exponential suppression. It follows that
an excitation on one island moves to the other with higher
probability than an independent relaxation of one island and
excitation of the other; this is what is meant by a correlated
event.
Processes in which |e
x
i |gi |gi |e
y
i with
x 6= y are a combination of transitions between excited
states |e
C,±
i |e
i and the same-state correlated events of
Eqs. (21) and (22). We thus expect the corresponding tran-
sition rates to only contribute a small correction to the corre-
lated event probabilities in PMC, for the same reason that
transitions between excited states only contribute a small
correction to the probability of a pair-wise dephasing event.
3.2.4 Other errors
Processes that do not involve quasiparticles can also con-
tribute to the probabilities of pair-wise dephasing events and
measurement bit-flips. Finite MZM hybridization, δE
ab
in
Eq. (6), leads to dephasing of the MZMs associated with
operators γ
a
γ
b
. The MZM hybridization is caused by fi-
nite overlap of the MZM wavefunctions when the correlation
length of the topological superconductor is not sufficiently
smaller than the distance separating the MZMs. The degen-
eracy splitting can have both a constant and fluctuating piece.
The former would be problematic for topological gates that
rely on degeneracy of the computational subspace, but will
not be explicitly considered here. The latter is problematic
even for memory storage. Charge noise in the electrostatic
gates and substrate on which the MZM island is sitting can
cause fluctuations of the electric field at the topological wire,
which in turn results in a noisy time-dependence of the MZM
hybridization δE
ab
. We expect the rate of such errors to
scale as
Γ
EE
s
Z
S
EE
(ω)e
L/ξ
, (23)
where S
EE
(ω) is the spectral function of electric field fluc-
tuations in the system and L is the typical separation of
MZMs in the island. Any computing approach that involves
tuning a superconductor into the topological regime with
electrostatic gates will likely experience noise described by
Γ
EE
, however, as discussed in Ref. [23], the time scales
associated with this dephasing process are predicted to be
quite long for reasonably-sized systems (order of minutes for
L/ξ 30). At these system sizes, the thermal excitations of
Eq. (13) become the most important source of pairwise de-
phasing events. For concreteness, we use this limit and ne-
Accepted in Quantum 2018-08-21, click title to verify 13
glect hybridization-based errors for the probability estimates
in the next section.
Another source of error occurs when the classical bit stor-
ing the outcome of a 2k-MZM parity measurement does not
agree with the actual parity of the measured MZMs at the
end of the time step. Physically, the measurement is imple-
mented by adding a term to the Hamiltonian with the 2k Ma-
jorana operators to be measured. This term splits the energy
of the two eigenstates of 2k-MZM parity and the environ-
ment then quickly dephases these two states, collapsing the
system into one of the eigenstates. The measurement out-
come is obtained by gathering statistics throughout the time
step as to which eigenstate the system has collapsed.
There are two ways for measurement outcomes to dis-
agree with the state of the system at the end of the time step:
(1) the classical bit storing the measurement is flipped, or (2)
the state of the system changes between the measurement
projection and the end of the time step. Errors from case
(1) are described by the probability p
(k)
mst
in MC and PMC
(probability p
mst
in QpBf). This case either results from sta-
tistical error (the integration time of the measurement was
too short), or from classical noise in the measurement flip-
ping the bit’s outcome. Classical noise is strongly dependent
on the physical implementation of the measurement, as such,
we do not further analyze these processes other than to note
that a measurement error will be more probable for smaller
measurement signals, likely to occur when multiple islands
are involved in a single measurement.
Due to our convention that measurement projections are
performed at the end of a time step, our models do not
describe case (2). In reality, a measurement is performed
by collecting statistics throughout the time interval, thus
the measurement result will likely reflect a noise event that
occurs near the beginning of the time step, but not one that
occurs at the end. The latter are instead reflected in the next
measurement that occurs. In an error correction protocol in
which almost every island is measured in each time step,
it is only a small effect (for a large number of time steps)
whether we apply projectors at the beginning or the end of
a time step, as for one convention the noise events will be
reflected in the measurement associated with that time step,
and in the other convention the noise events will be reflected
in the measurements associated with the subsequent time
step.
3.3 Error probabilities
We now use the transition rates discussed in the previous sec-
tion to estimate how the error probabilities in the stochastic
Majorana noise model PMC depend on physical parameters
of the system. These estimates serve the double purpose of
connecting the noise models of the previous section to the
underlying physical system, and of simplifying the simula-
Figure 6: Probability tree for the different error types involving
quasiparticles for Majorana noise model PMC: p
k
is the prob-
ability that a MZM on an island in a k-island measurement is
excited, q
k
is the probability that this excitation is transferred to
a different island in the measurement, and r
k
is the probability
that this excitation does not relax. The probabilities p
(k)
qp
and
p
(k)
pair
are for single islands, while p
(k)
cor,odd
and p
(k)
cor,even
are for both
islands in a measurement. In the text, we relate these small,
dimensionless parameters to the transition rates discussed in
the previous section. When m = 2, corresponding to a tetron
architecture, and r
k
= 0, corresponding to no odd Majorana
operators, the model PMC reduces to the qubit-based model
Pauli circuit noise.
tions of pseudo-thresholds in the next section by reducing
the number of independent parameters in PMC. At the end
of this section, we discuss the other three Majorana noise
models Qp, QpBf, and MC, and the limit in which they re-
duce to the analogous qubit-based noise models.
In Fig. 6, we schematically show that for a k-island mea-
surement, all errors involving quasiparticles are related by
three small dimensionless parameters: p
k
, the probability
that a quasiparticle is excited on one of the islands involved
in the measurement; q
k
, the probability that the energy from
this excitation is transferred to a different island involved in
the measurement; and r
k
, the probability that an excitation
created during the time step does not relax. These dimen-
sionless parameters are related to the three types of transition
rates involving quasiparticles discussed in the previous sec-
tions: excitations Γ
(k)
ge
x
, energy transfers between islands
Γ
(2)
e
x
,gg,e
x
, and relaxations Γ
(k)
e
x
g
, respectively. More ex-
plicitly, denoting the length of the time step by τ, we find
p
k
= 2m
X
x
Γ
(k)
ge
x
τ (24)
r
k
= max
x
τΓ
(k)
e
x
g
1
= τ
(k)
r
(25)
q
2
= max
x
exp
n
Γ
(2)
e
x
,gg,e
x
τ
(2)
x
o
1
. (26)
The factor of 2m in Eq. (24) comes from any of the 2m
MZMs on an island initially being excited. We assume that
excitations are rare, so that Γ
(k)
ge
x
τ 1 and p
k
is a small
parameter. For an idle MZM, we expect extrinsic quasipar-
Accepted in Quantum 2018-08-21, click title to verify 14
ticle poisoning to be negligible, thus only Γ
(0)
ge
will con-
tribute to p
0
. Generally, we expect Γ
(0)
ge
Γ
(1)
ge
Γ
(2)
ge
and Γ
(1)
ge
C
. Γ
(2)
ge
C
, so that p
0
< p
1
. p
2
.
In Eq. (25), τ
(k)
r
max
x
h
1/Γ
(k)
e
x
g
i
is the typical time
scale over which the longest living excited state relaxes to
the ground state. We assume τ
(k)
r
τ based on physical
considerations. The parameter τ is bounded from below by
the longest measurement time, a four-MZM parity measure-
ment in the current discussion. We expect this measurement
time to be 1 10 µs for high-fidelity measurements, while
typical time scales for the electron-phonon coupling facili-
tating quasiparticle relaxation are 0.1 µs [45]. Physically,
we expect 0 < r
k
< 1/10 and r
0
r
1
r
2
. The probabil-
ity of an excitation, created at the beginning of the time step,
to not relax is exponentially small in 1/r
k
and will be ne-
glected through the remainder of the paper. Equation (25)
then follows from the probability of a given excitation to
happen withing the window τ
(k)
r
just before the end of the
time step. When r
k
= 0, an island never ends (or begins) a
time step in an excited state; for m = 2, the noise model then
reduces to the qubit-based model Pauli circuit noise. If the
assumption of r
k
1 is violated, excitations become long-
lived. In this case, excitations can travel long distances be-
fore relaxing, thereby significantly degrading the effective-
ness of error correction. We discuss noise models describing
long-lived excitations in Appendix B.
For simplicity, and to obtain an upper bound on the cor-
related errors in Eq. (26), we maximize the probability for
transferring energy over the excited states. Since q
2
is not
exponentially suppressed and can therefore be O(1) when
islands involved in a measurement are well-coupled, we did
not use the approximation for small transition rates. Phys-
ically, we expect the probability of an energy transfer be-
tween the two islands to be smaller than the probability that
the excitation remains on the same island, so we anticipate
0 < q
2
< 1/2. For an idle island or single-island measure-
ment, q
0
= q
1
= 0.
In the following, we describe the noise model PMC using
the parameters introduced above.
Quasiparticle events. The probability p
(k)
qp
that a single Ma-
jorana operator is applied to an island involved in a 2k-MZM
measurement can be obtained from considering an initial ex-
citation (p
k
) of a quasiparticle that does not relax (r
k
) and
does not transition to another island (1 q
k
). We therefore
find
p
(k)
qp
p
k
(1 q
k
) r
k
. (27)
The probability p
(k)
odd
of an initially excited state to relax
during the time step is given by
p
(k)
odd
1 e
1/r
k
(28)
p
(k)
odd
> 1 p
(k)
qp
. (29)
The bound in Eq. (29) reflects the assumption of fast
relaxation. In that case, it is more likely for an excitation to
relax and get re-excited, than for the excitation to remain
over the entire time step. Within the fast relaxation limit,
we can use the upper bound for 1 p
(k)
odd
= p
(k)
qp
. We discuss
how the noise models can be generalized beyond the fast
relaxation limit in Appendix B.
Pair-wise dephasing events. Pair-wise dephasing events
can occur from quasiparticle excitations and relaxations, or
from non-quasiparticle processes such as MZM hybridiza-
tion discussed in Section 3.2.4:
p
(k)
pair
p
k
(1 q
k
) (1 r
k
) + m Γ
EE
τ.
(30)
Here, (1 q
k
)(1 r
k
) is the probability of an excited
quasiparticle to relax within the same island and thus con-
tribute to pair-wise dephasing. The second term arises from
noise in the hybridization, see Eq. (23). The combinatoric
prefactor of m counts the number pairs of MZMs whose
wavefunction overlap contributes to pair-wise dephasing. In
the following, we will assume the MZMs are sufficiently
separated that we can drop the second term.
Correlated events. An odd correlated event occurs when
one island involved in a measurement is excited and trans-
fers its excess energy to a second island, which remains in
the excited state for the rest of the time step. Similar to quasi-
particle events, only excitations within a fraction of r
k
of the
time step contribute significantly to odd correlated events.
More explicitly, the probability of an odd correlated event is
p
(2)
cor,odd
2 p
2
q
2
r
2
.
(31)
The factor of two is a result of the initial quasiparticle exci-
tation occurring on either of the two islands involved in the
measurement.
An even correlated event occurs when a quasiparticle ex-
citation in one island is transferred to the other island, which
subsequently relaxes:
p
(2)
cor,even
2p
2
q
2
(1 r
2
).
(32)
The factor of two again comes from either island initially
being excited. When hybridization errors are subleading
(i.e., the MZMs are sufficiently well-separated), an even
correlated event is essentially a pair-wise dephasing event
interrupted by an energy transfer.
Accepted in Quantum 2018-08-21, click title to verify 15
Model Probabilities
Qp p
pair
= p (1 r), p
qp
= p r, p
odd
= 1 p
qp
QpBf p
pair
= p (1 r), p
qp
= p r, p
odd
= 1 p
qp
, p
mst
MC p
(k)
pair
= p
k
(1 q
k
) (1 r), p
(k)
qp
= p
k
(1 q
k
) r
& p
(k)
odd
= 1 p
(k)
qp
, p
(2)
cor,even
= 2p
2
q
2
(1 r)
PMC p
(2)
cor,odd
= 2p
2
q
2
r, p
mst
Table 2: Probabilities for pair-wise dephasing, quasiparticle,
and correlated events in the four stochastic Majorana noise
models can be written in terms of an excitation parameter p
k
(p), a relaxation parameter r, and a correlation parameter q
k
(with q
0
= q
1
= 0), see Eqs. (24-26). The expressions above
neglect the contribution of hybridization to pair-wise dephasing
events, and assume that quasiparticle relaxation times are fast,
i.e, p
(k)
qp
> exp(1/r) and similar regardless of whether an
island is involved in a measurement. Model PMC is physically
motivated when 0 r . 1/10 and 0 q
2
1/2. When
m = 2, the limit of r = 0 corresponds to the qubit model
Pauli circuit noise. The expressions in this table are used in the
pseudo-threshold calculations in Section 4.
Measurement bit-flip. A measurement bit-flip depends only
on classical and statistical noise, and is thus independent of
the other probabilities in the noise model. The signal-to-
noise ratio of a measurement typically increases as a square
root of the integration time [46]. We assume that τ is essen-
tially given by the integration time of the measurement. The
confidence of the measurement outcome therefore behaves
as
p
(k)
mst
e
τ
(k)
mst
, (33)
where τ
mst
is a characteristic time scale that gives a signal-
to-noise ratio of 1. Four-MZM measurements are expected
to have lower visibility and thus be more susceptible to
statistical and classical error, i.e., p
(1)
mst
< p
(2)
mst
.
We have shown that the probabilities defining model PMC
can be defined in terms of excitation parameters p
k
, relax-
ation parameter r
k
, and a correlation parameter q
2
. Varying
these parameters over the appropriate ranges and studying
the effect on the pseudo-threshold for the system should in-
form us about the relative importance of single-island vs.
two-island errors, as well as errors with an odd vs. even
number of Majorana operators. A natural choice for the three
other stochastic Majorana noise models is to assume a sim-
ilar dependence of the probabilities on these parameters. In
the next section, for ease of comparing the effects of differ-
ent errors, we will set r
k
= r and use the lower bound of
the relaxation probability p
(k)
odd
= 1 p
(k)
qp
. We summarize
the scaling relations between error probabilities for the four
models in Table. 2.
Finally, note that if we set p
0
= p
2
and restrict our at-
tention to tetrons (m = 2), the noise models MC and PMC
differ only in that some of the odd correlated events in MC
do not occur in PMC. For both MC and PMC, there are only
16 computationally distinct
7
even correlated events, each oc-
curring with equal probability. In contrast, there are 16 com-
putationally distinct odd correlated events that can occur in
PMC with equal probability, and 32 computationally distinct
odd correlated events that can occur in MC with equal prob-
ability. For larger values of m, we expect MC and PMC to
differ more substantially.
4 Application of noise models
We now apply the noise models presented in Section 2 to
analyze the error correction performance of the Bacon-Shor
code [3739] on a small system of tetrons. As explained
in Section 3, a tetron consists of four MZMs, and there-
fore stores a single qubit of information in the overall even
parity state. Although our noise models can be applied to
any quantum error correction scheme, there are several mo-
tivations for considering this small subsystem code rather
than (1) a Majorana fermion code specifically designed to
correct for quasiparticle events [24, 34, 35], or (2) a qubit-
based stabilizer code (e.g., the surface code). (1) A Majo-
rana fermion code either requires the ability to measure the
total parity of all MZMs on an island, or the ability to dy-
namically adjust the number of MZMs on a given island. We
discuss in Section 5 why it is experimentally challenging to
satisfy these conditions. Furthermore, the codes discussed
in Ref. [35] are categorized with a notion of code distance
in which the most probable noise events are quasiparticle
events, which is not expected to be the case for a tetron ar-
chitecture. Rather, for low temperatures (T E
C
, ) and
fast relaxation (r
k
1), quasiparticle events in a system of
tetrons are converted by the environment into Pauli errors,
which in turn are correctable by qubit-based codes. As dis-
cussed in Section 3.1, the odd total-MZM parity state of an
island is highly excited above the (even MZM parity) ground
state; as such, the environment relaxes the system back to
the ground state. In the language of Refs. [34, 35], the en-
vironment measures the missing stabilizers of the Majorana
fermion code needed to detect errors involving odd numbers
of Majorana operators.
Regarding point (2), in the Bacon-Shor code, error cor-
rection is built out of two-qubit measurements, which can be
simply implemented for tetrons with four-MZM parity mea-
surements. In contrast, typical stabilizer codes involve mea-
surements of at least four-qubit (eight Majorana) operators,
which are expected to be more difficult to implement ex-
perimentally. While in principle it is possible to implement
higher-weight measurements from a sequence of smaller-
weight measurements, the extra operations can significantly
7
Because the total MZM parity of an island is fixed, some
Majorana operators are computationally equivalent, e.g. γ
j,1
γ
j,2
,
γ
j,2
γ
j,1
, γ
j,3
γ
j,4
, and γ
j,4
γ
j,3
.
Accepted in Quantum 2018-08-21, click title to verify 16
increase the noise. For example, a six-MZM measurement
for a system of tetrons could be performed using two an-
cilla tetrons with six four-MZM and four two-MZM mea-
surements [41], or alternatively by preparing ancilla tetrons
in a cat state, then doing a sequence of four-MZM measure-
ments [32]. Adding ancilla tetrons and additional measure-
ments provides new locations and opportunities for noise
to occur before it can be corrected by the code. Addition-
ally, as is further discussed in Section 6, higher-weight mea-
surements can result in a higher probability of correlated
errors, with negative effects on the pseudo-threshold. Cir-
cumventing higher-weight measurements using subsystem
codes is therefore a natural starting point for error correction
in Majorana-based quantum computing architectures. De-
termining optimal error correction procedures by weighing
the advantages and disadvantages of higher-weight measure-
ments is an interesting direction for future research.
We review error correction with subsystem codes, focus-
ing on the Bacon-Shor code and how it applies to a system
of tetrons. We then analyze the conditions for fault toler-
ance (i.e., compute pseudo-thresholds) for each of the noise
models presented in Section 2. The analysis of this section
is restricted to quantum memory error correction; analyzing
the fault tolerance of logical gates is an important subject
that we relegate to future studies. The choice of error cor-
recting code and error correction protocol have not been op-
timized to minimize the number of resources [47, 48] or to
maximize the fault tolerance threshold [49]. As such, the
pseudo-threshold values reported here are more informative
in their relative magnitude (within a given noise model for
different parameter values) than in their absolute value. At
the end of this section, we separately discuss what experi-
mental implications can be drawn from our analysis.
4.1 Subsystem error correcting codes
As subsystem codes are less commonly studied compared to
stabilizer codes [42], we first review the standard applica-
tion of a subsystem code to a qubit noise model (i.e., errors
involving an even number of Majorana operators in a tetron
architecture). For a more extensive discussion of subsystem
codes, see Ref. [50]. We then discuss how a subsystem code
can also correct for errors involving an odd number of Ma-
jorana operators in a tetron architecture. We address mea-
surement bit-flips later when discussing the application of
the Bacon-Shor code to the noise models of Section 2. An
example illustrating the formal concepts introduced below is
given in Section 4.2.
Subsystem codes are generalizations of stabilizer codes in
which information is encoded in a subsystem of a subspace
of the Hilbert space, rather than a subspace. More explicitly,
the Hilbert space of physical qubits H can be decomposed
into a code space H
C
and the perpendicular subspace of er-
Figure 7: The d = 3 Bacon-Shor code storing a single logical
qubit is implemented by applying X-type (red solid ovals) and
Z-type (blue dashed ovals) Pauli operators on a grid of 9 qubits
(black dots). (a) Example of gauge generators. X-type (Z-
type) gauge generators act on pairs of nearest neighbor qubits
in the horizontal (vertical) direction. There are six different X
and Z gauge generators. (b) Example of an X-type stabilizer.
X stabilizers consist of pairs of X-type Pauli operators applied
to an even number of columns. Z stabilizers are pairs of Z-
type Pauli operators applied to an even number of rows (not
shown). (c) The Z-type bare logical operator is applied to an
odd number of rows. (d) The X-type bare logical operator is
applied to an odd number of columns.
rors, H
E
= H
C
:
H = H
C
H
E
. (34)
The code space can be further decomposed into a logical
subsystem H
L
, the Hilbert space of the logical qubits, and a
gauge subsystem H
G
:
H
C
= H
L
H
G
. (35)
In a stabilizer code, H
G
is trivial. The benefit of having the
gauge subsystem is that error correction only needs to cor-
rect an error modulo the gauge subsystem structure; a non-
trivial action on the gauge subsystem does not affect the en-
coded quantum information. As we detail below, this extra
degree of freedom can allow the error correction procedure
to be implemented directly with smaller measurements, but
comes at the cost of a reduction of the number of inequiva-
lent code states.
More formally, we specify a subsystem code using sub-
groups of the Pauli group for n physical qubits, P
n
:
The gauge group, G: a non-Abelian subgroup of P
n
which defines the subsystem code.
The stabilizer group, S(G): the largest subgroup of G,
excluding 1, consisting of elements which commute
with every element of G.
The group of bare logical operators, L
B
: Pauli opera-
tors that commute with, but do not belong to, the gauge
group G.
The group of dressed logical operators, L
D
=
hL
B
, Gi.
Accepted in Quantum 2018-08-21, click title to verify 17
The code space, H
C
(G), of a subsystem code with gauge
group G is the +1 eigenspace of all stabilizer operators. The
code distance, d, is defined to be the minimum support of
any logically non-trivial element of L
D
. We use the standard
qubit definition of code distance, not to be confused with the
Majorana fermion code distance used in Refs. [24, 34, 35].
Error correction with a subsystem code involves using the
measurement outcomes of the generators of the stabilizer
group (stabilizers) to infer if any unwanted Pauli noise op-
erators have been applied to the system
8
. One of the most
appealing features of subsystem codes is that the eigenvalue
of each stabilizer generator can be determined by multiply-
ing together the eigenvalues of some of the gauge generators,
which are often easier to measure. For an example of gauge
generators and stabilizers, see Fig 7.
To model memory error correction, the system is prepared
in an eigenstate of one of the logical operators and is periodi-
cally measured to check that the state remains unchanged. If
at least one, but fewer than d, Pauli errors occur in an error
correction round (measurement of all stabilizers), the sys-
tem will no longer be in H
C
(G) and some of the stabilizer
measurements will have outcome 1. The syndrome of a
given Pauli error is the corresponding set of stabilizer mea-
surement outcomes. The error correction protocol uses the
syndrome to infer which Pauli operators have been applied
(under the assumption that the minimal number of errors cor-
responding to that syndrome has occurred). The errors can
then be corrected by appropriate application of Pauli opera-
tors, returning the system to its intended state. However, if
more than (d 1)/2 Pauli errors occur in an error correction
round, the correction procedure might incorrectly diagnose
the error; in this case, the correction procedure might change
the logical state of the encoded quantum information, result-
ing in a failure of the error correction protocol.
4.2 Bacon-Shor codes
One particular subsystem code, the distance-d Bacon-Shor
code, is implemented on a d × d grid of qubits, with d
odd [37, 38]. Figure 7 shows the relevant operators of the
d = 3 Bacon-Shor code. The generators of the gauge group
G are XX acting on horizontal nearest neighbors and ZZ
acting on vertical nearest neighbors, as depicted in Fig. 7(a).
The stabilizer group S(G) has d1 X-type and d1 Z-type
generators, where each X-type generator consists of X ap-
plied to two columns (2d qubits) and each Z-type generator
consists of Z applied to two rows (2d qubits), see Fig. 7(b).
A Z-type bare logical operator is a string of Z operators ap-
plied to a row of qubits [Fig. 7(c)], and an X-type logical
operator is a string of X operators applied to a column of
qubits [Fig. 7(d)]. Applying a stabilizer operator to a bare
8
As stabilizer generators commute, they can be measured si-
multaneously. This is not the case for gauge generators, which do
not necessarily commute.
logical operator results in an equivalent bare logical opera-
tor. Application of a gauge operator to a bare logical operator
results in an equivalent dressed logical operator.
For the distance-d Bacon-Shor code (d = 3 in Fig. 7), a
single X error in a given row will anti-commute with any
Z-type stabilizer with support in that row and will therefore
have the same syndrome as all other X errors in that row.
This ambiguity does not cause any problem for error correc-
tion as X errors belonging to the same row differ from each
other by a gauge generator, and thus are corrected by the
same procedure: applying a single X operator to one qubit
in the row. In order to find each stabilizer measurement out-
come, it is enough to measure the d two-qubit gauge gener-
ators and multiply the outcomes (e.g., three two-qubit mea-
surements for the d = 3 code shown in Fig. 7). Single-qubit
Z errors can similarly be identified by finding the eigenval-
ues of the stabilizer generators, and corrected by applying a
single Z operator to any qubit in the same column.
In the following, we imagine a distance-d Bacon-Shor
code implemented in a tetron-like architecture (see Fig. 2)
using 4d
2
MZMs. The jth qubit hosts four MZMs,
γ
j,1
, γ
j,2
, γ
j,3
, γ
j,4
, and we identify the corresponding
Pauli operators as
X
j
γ
j,2
γ
j,3
, Y
j
γ
j,1
γ
j,3
, Z
j
γ
j,1
γ
j,2
. (36)
(When analyzing noise model PMC, we will alter this defi-
nition to take into account the geometric arrangment of the
MZMs on the island, see Fig. 13. Specifically, we use the
parity conservation of the ground state so that X
j
= γ
j,1
γ
j,4
or γ
j,2
γ
j,3
depending on whether a tetron is in a joint mea-
surement with its left or right neighbor; similarly Z
j
=
γ
j,1
γ
j,2
or γ
j,3
γ
j,4
depending on whether a tetron is in a
joint measurement with its top or bottom neighbor.) Using
Eq. (36), we can map each Pauli operator to a MZM-parity
measurement. Furthermore, we assume that corrections are
applied as Pauli frame updates: rather than actually apply-
ing an operator to correct an error, we classically track mea-
surement outcomes and appropriately reinterpret subsequent
measurement outcomes. As such, we can assume corrections
are perfect, since faulty measurements are already taken into
account.
We can choose to apply all Z corrections (Pauli frame
changes) to the qubit in the appropriate column in the top
row and all X corrections to the qubit in the appropriate row
in the left column. The stabilizer measurements are decoded
by assuming that the minimal number of Pauli errors cor-
responding to a given syndrome have occurred (we will ex-
plain how to treat measurement bit-flips in the next section).
For the d = 5 Bacon-Shor code, any two-qubit Pauli error
can be corrected in this way [37], see Fig. 8(a) for an exam-
ple. However, some three-qubit Pauli errors will be misdi-
agnosed by this decoding scheme, and can result in a logical
operator being applied to the system after the correction step,
see Fig. 8(b). In this case, the quantum information stored in
Accepted in Quantum 2018-08-21, click title to verify 18
Figure 8: Two examples of Pauli errors (red and blue stars) and
corresponding error correction (solid red and dashed blue cir-
cles). All X errors in the same row and all Z errors in the same
column have the same syndrome. (a) Example correction of a
single X and a single Z error. The net Pauli operator applied
to the system (errors and correction) is in the gauge group and
thus does not change the quantum information stored in the
code. (b) Example failure of the code for three X errors. They
syndrome of the three X errors is the same as the syndrome for
two X errors in the third and fifth rows. The correction thus
applies X operators to these rows, so that the net operator
(error and correction) applied to the system is a dressed logical
operator, which changes the quantum information.
the code has changed, resulting in a failure of the protocol.
Note that the stabilizer measurements do not distinguish
between the even and odd parity subspaces of the qubits: this
can be seen with the mapping in Eq. (36) by noting that γ
j,4
does not change any of the stabilizer outcomes. Therefore,
every error involving an odd number of Majorana operators
has the same syndrome as some error involving an even num-
ber of Majorana operators. Using the decoder described in
the previous paragraph, all syndromes will be interpreted as
corresponding to an error involving an even number of Ma-
jorana operators, thus the correction step (applying an X op-
erator to a qubit in the left column or a Z operator to a qubit
in the top row) will not return the system to the even par-
ity subspace. However, when the system relaxes back to the
ground state in a later time step, it either does this through
application of a Majorana operator that does not change the
stabilizer measurement outcomes (in which case the envi-
ronment has self-corrected), or the stabilizer measurements
are altered and the error correction procedure now identifies
the Pauli error resulting from the combined initial excitation-
intermediate correction steps-relaxation processes. Note that
when the excitation-relaxation process extends over more
than one time step the outcome of subsequent stabilizer mea-
surements might disagree. For example, an initial γ
j,2
exci-
tation would be interpreted as a Y
j
error. If there is a sub-
sequent relaxation to, say, γ
j,3
the combined process corre-
sponds to an X
j
error. The above process can lead to mis-
interpretation of the error and can therefore be thought as
contributing to measurement errors, which we discuss more
in Section 4.4. We say that the protocol works if the cor-
rection step after the system has returned to the even parity
subspace (possibly in a later time step) has applied a sta-
bilizer operator. Conversely, if the correction step after the
qubit returns to the even parity subspace results in a logical
error, the protocol fails.
4.3 Fault tolerant error correction
A distance-d error correcting code could be used to protect
against up to (d 1)/2 errors with a single round of perfect
measurements. To account for the fact that some noise mod-
els can have imperfect measurements, a framework known
as fault-tolerant error correction has been developed. This
framework distinguishes between faults and errors:
A fault is any noise event that adversely disturbs the
system or the measurements.
An error is any non-trivial operator applied to a qubit.
For a qubit-based model, a fault corresponds to any non-
trivial noise event (including measurement bit-flips) and an
error is some single-qubit Pauli operator (we will general-
ize the discussion to Majorana noise models at the end of
this section and clarify what is meant by a noise event that
“adversely disturbs the system”). Faults are defined with
respect to a given noise model; we illustrate this point by
considering the qubit-based noise models Pauli noise with
bit-flip measurement and Pauli circuit noise. For the for-
mer, a single fault is either a single-qubit Pauli operator or a
measurement bit-flip. For the latter, a single fault can be a
single-qubit Pauli operator, a two-qubit Pauli operator, or a
measurement bit-flip. Notice that if a noise model includes
correlated events, a single fault can result in multiple errors.
For a given qubit-based noise model, the requirement that
the error correction procedure is fault-tolerant amounts to
satisfying the following conditions [51]:
1. (EC A): For any initial state (irrespective of whether
or not it is a code state), a single fault that can occur
anywhere during the error correction procedure results
in an output which differs from a code state by at most
(d 1)/2 errors.
2. (EC A’): For an initial error-free code state, a single
fault that can occur anywhere during the error correc-
tion procedure outputs the error-free code state.
3. (EC B): If the error correction proceeds without addi-
tional faults, any code state with (d 1)/2 errors will
be corrected.
If the noise is sufficiently weak, information stored in a fault-
tolerant error correcting code is better-protected than infor-
mation stored directly in a physical qubit. The performance
Accepted in Quantum 2018-08-21, click title to verify 19
of a fault-tolerant error correction scheme can be quantified
by the pseudo-threshold, defined for a noise model charac-
terized by a single parameter p by
p
th
max{p|p
err
(p) p}, (37)
where p
err
is the logical error rate, i.e., the probability of an
uncorrectable error remaining in the system after the appli-
cation of strength-p noise and error correction. The pseudo-
threshold will depend on the noise model, the error cor-
recting code, and the error-correction protocol. A pseudo-
threshold is defined for a particular error correcting code on
a fixed number of qubits, in contrast to a threshold which
is defined for a family of error correcting codes, each for
different system size (i.e., a different number of physical
qubits). The threshold is the limit of the pseudo-threshold
for infinitely large system size. The Bacon-Shor code family
does not have a finite threshold.
For a Majorana-based system, any non-trivial string of
Majorana operators applied to a single island results in an
error. A fault is more subtle. For instance, some noise
events are a result of the system relaxing back to the even
parity ground state. This relaxation does not necessarily oc-
cur within the same time step as excitation, which is why
our models include noise events involving an odd number
of Majorana operators (i.e., quasiparticle and odd correlated
events). When using a qubit-based code (i.e., a code that
only corrects for Pauli errors on the qubits) to error correct
a Majorana-based system, it is essential for the environment
to relax the system to the even parity ground state so that the
net operator applied to any island involves an even number
of Majorana operators (and therefore corresponds to some
Pauli operator). This motivates a careful specification of
what is considered “adversely disturbing the system” in the
definition of a fault. If an island begins a time step in the odd
parity state and a quasiparticle event relaxes the island back
to the even parity state, this noise event is beneficial to the
error correction protocol and is thus not considered a fault.
Conversely, if an island begins or ends a time step in the odd
parity state, the lack of a relaxation is a fault. For all the Ma-
jorana noise models presented in Section 2, any non-trivial
noise event applied during steps 1 or 2 correspond to a fault;
a quasiparticle event applied in step 0 is not a fault; and not
applying a quasiparticle event in step 0 is a fault.
With the above understanding of what constitutes a fault
and what constitutes an error, fault tolerance conditions
(EC A)-(EC B) apply to a Majorana-based system. Note
that “a single fault” in conditions (EC A) and (EC A’) im-
plies that if the fault corresponds to a quasiparticle or odd
correlated event in a time step, the excitation is immediately
relaxed in the subsequent time step. Simiarly, “without ad-
ditional faults” in condition (EC B) implies that if the code
state with (d 1)/2 errors is in an odd parity state, then the
system is relaxed to the even parity state in the subsequent
time step.
Figure 9: Measurement steps for a single error correction round
of the d = 5 Bacon-Shor code. Same legend as Fig. 7, black
dots correspond to qubits, red ovals denote X measurements,
blue dashed ovals denote Z measurements. Each step involves
the measurement of two stabilizers built out of five gauge gen-
erator measurements. Four measurement steps are necessary
(for arbitrary d) for a single error correction round. As dis-
cussed in the main text, faulty measurements can be corrected
by repeating each error correction round at least four times.
One might, in principle, be able to do multiple measurements
on a single qubit and thereby reduce the number of measure-
ment steps required; however, this is inadvisable as it allows
correlated errors to spread over longer distances.
4.4 Numerical results
In this section, we analyze fault-tolerant error correction cir-
cuits for storing a qubit using the d = 5 Bacon-Shor code
under the Majorana noise models presented in Section 2. We
define the parameters of each noise model according to Ta-
ble 2, so that we can easily scale the probability of a logical
error p
err
as a function of the probability p
0
of an excita-
tion of an idle island. The analogous qubit noise models are
found by setting the probability of all errors involving an
odd number of Majorana operators to zero (equivalently, by
setting the relaxation parameter r from Section 3.3 to zero).
An exhaustive simulation of all possible errors to find the
exact pseudo-threshold p
A
th
of a noise model A quickly be-
comes intractable. We therefore follow a Monte Carlo ap-
proach of simulating n
trials
runs for a single error correction
round to estimate p
err
(p
0
) from the fraction of trials which
result in a logical error we extract
p
err
(p
0
) =
n
fail
n
trials
. (38)
The uncertainty of p
err
(p
0
) for a given p
0
is of order
1/
n
trials
.
The Majorana noise models Qp, QpBf, and MC are de-
signed without respect to a specific physical measurement
protocol. Thus, when simulating pseudo-thresholds for
these three models, we define measurements according to
Eq. (36), i.e., without considering the geometric arrange-
ment of MZMs within a tetron. In contrast, model PMC is
Accepted in Quantum 2018-08-21, click title to verify 20
designed with the measurement protocol of Section 3.1 in
mind; as such when analyzing this model we redefine the
measurements to account for a convenient physical imple-
mentation of the Bacon-Shor code.
There are three main differences between our simulations
of models MC and PMC and the most realistic approach
to estimating optimal pseudo-threshold values. (The sim-
ple models Qp and QpBf will clearly not estimate realistic
pseudo-threshold values as they do not include all lowest or-
der noise events in a Majorana-based system.) Firstly, to
help the simulations run faster, we do many trial runs of a
single error correction round, rather than simulating many
rounds and analyzing the frequency of logical errors being
applied.
9
The latter approach is a more accurate description
of noise in an actual system because it feeds the errors from
a previous error correction round into the next. However,
feeding errors into the next time step is only important when
the system has the unlucky combination of an error going
undetected (e.g., from occuring in one of the later time steps
of an error correction round and thus being identified as a
measurement bit-flip) followed by enough additional errors
occuring in the next error correction round to result in a log-
ical error. When the noise event rate is sufficiently small, the
difference between simulating many error correction rounds
and repeatedly simulating a single round will only result in
a small correction to the pseudo-threshold estimates. Details
of the code used for simulating the pseudo-thresholds are in
Appendix C.
Secondly, to correct for faulty measurements, we use the
Shor error correction approach [39] of repeating stabilizer
measurements multiple times in a single error correction
round. Generally, Steane error correction [52], which uses
entangling gates (e.g., CNOTs) of the logical data qubit with
logical ancilla qubits to locate errors, results in higher thresh-
old estimates (in part because the effort of error correction is
shifted from operations on the data qubits to preparation of
ancillas [51]). Steane error correction is only known to apply
to stabilizer codes, which means that in order to do Steane er-
ror correction on the Bacon-Shor code we must fix the gauge
subsystem and thereby lose the ability to build stabilizer
measurements out of two-qubit measurements. Furthermore,
the logical ancillas (tripling the number of tetrons) and mul-
tiple CNOT gates (each using an additional ancilla tetron and
multiple measurements [41]) needed for Steane error correc-
tion quickly complicate the pseudo-threshold simulations.
Finally, we include correlated events, which reduces the
number of faults that can be corrected for models MC and
PMC because two faults can result in a logical error (e.g., a
correlated event plus a pair-wise dephasing event can imple-
ment Pauli errors on three islands). We could have avoided
correlated events reducing the code distance by using mea-
9
This approach allows us to speed up the run time by using
importance sampling for the first error.
0.00 0.02 0.04 0.06 0.08 0.10 0.12
p
0
0.00
0.05
0.10
0.15
p
err
p
err
p
err
= p
0
Figure 10: Logical error rate p
err
(solid red curve) as a func-
tion of the noise event rate p
0
for the Qp noise model (see
Table 2). The dot-dashed blue line indicates no error cor-
rection. The pseudo-threshold is the intersection of the two
curves: p
Qp
th
0.09. This plot was generated for 10
5
Monte
Carlo trials per data point, see Appendix C for simulation de-
tails.
surement gadgets [51], which use CNOT gates and ancilla
tetrons to ensure that any correlated event only affects one
data tetron. However, measurement gadgets introduce more
opportunities for a fault to occur within an error correction
round due to both the introduction of ancilla tetrons and be-
cause CNOT gates require three measurements for a tetron
architecture [41]. It is therefore not clear whether measure-
ment gadgets would improve the error correction for tetron-
based architectures.
Given the above considerations, we believe the pseudo-
threshold estimates for models MC and PMC in this section
are below their optimal values and should not be taken as
experimental targets. Our analysis should rather be taken as
indication of the relative importance of the different noise
events (quasiparticle, measurement, correlated, and pair-
wise dephasing) on the pseudo-threshold. In Section 6, we
identify future directions of study that could improve simu-
lations of the noise models presented in this paper and result
in more accurate pseudo-threshold estimates. We explicitly
discuss experimental implications of our analysis at the end
of this section.
4.4.1 Quasiparticle noise with perfect measurement (Qp)
Model Qp is the simplest stochastic Majorana noise model
given in Section 2, in which only pair-wise dephasing events
or quasiparticle events can occur. We define measurements
according to Eq. (36). The relative distribution of quasiparti-
cle events and pair-wise dephasing events can be adjusted by
varying the relaxation parameter r, see Table 2. In the limit
r = 0, Qp reduces to the qubit model Pauli noise (or code
capacity noise).
Accepted in Quantum 2018-08-21, click title to verify 21
The standard procedure for calculating pseudo-thresholds
for Pauli noise is to consider a noisy time step, followed by
perfect application of all stabilizer measurements (i.e., er-
rors cannot occur between the four different measurement
steps displayed in Fig. 9). We implement the same proce-
dure for Qp, however, since the syndrome of a quasiparticle
event after a single time step is identical to that of some pair-
wise dephasing event, a single time step of Qp is not able
to distinguish quasiparticle events from pair-wise dephasing
events. The r-dependence of the model will therefore only
become apparent after simulating multiple time steps, for in-
stance by considering several error correction rounds or by
modeling application of a logical gate (not included in this
paper). We do not simulate multiple time steps for Qp, as
the simulations of QpBf and MC clearly indicate that the
pseudo-threshold only has weak r dependence. The pseudo-
threshold estimate under Qp (and Pauli noise) for a single
time step is p
Qp
th
0.09 (see Fig 10).
4.4.2 Quasiparticle noise with bit-flip measurement
(QpBf)
The model QpBf builds on the previous model Qp by allow-
ing the classical bit storing the measurement outcome to be
flipped. We define the error probabilities in Table 2 and the
measurements according to Eq. (36). For this noise model,
we repeat the four measurement steps comprising a single
round of error correction in order to avoid introducing errors
into the code state from a single measurement bit-flip (i.e.,
we use a Shor error correction scheme). Our procedure is
to consider a single (noisy) time step preceding each error
correction round, repeated four times. We again apply all
stabilizer measurements simultaneously, as in the Qp simu-
lation. If the repetition rounds result in different syndromes,
we accept the last syndrome which repeats for two consec-
utive rounds. When multiple errors occur, it is possible that
no two consecutive rounds will have the same syndrome. In
this case, we assume the syndrome for the fourth round.
The pseudo-threshold dependence on r and p
mst
is shown
in Fig. 11. In the top panel, we see weak dependence on r,
with p
QpBf
th
8 × 10
3
for 0 r 1/3 and p
mst
= 10
4
.
It follows that under this noise model, the d = 5 Bacon-
Shor code is insensitive to the relative distribution of quasi-
particle and pair-wise dephasing events. Intuitively, we can
understand this result as follows: in the parameter regime
considered, the environment relaxes each island with high
probability, thereby converting quasiparticle events in one
time step to pair-wise dephasing events in the subsequent
time step. A pair-wise dephasing event in one time step does
not have a significantly different effect than a pair-wise de-
phasing event broken into two time steps. As noted in Sec-
tion 3.3, larger r is not captured by model QpBf and would
be highly problematic (see Section 5 and Appendix B), re-
quiring a Majorana fermion code to efficiently correct a sin-
0.0000 0.0025 0.0050 0.0075 0.0100 0.0125
p
0
0.000
0.005
0.010
0.015
0.020
0.025
p
err
p
mst
= 10
4
r = 0
r = 1/10
r = 1/3
p
err
= p
0
0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014
p
0
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
p
err
r = 1/10
p
mst
= 0.0001
p
mst
= 0.001
p
mst
= 0.01
p
mst
= 0.08
p
mst
= 0.1
p
err
= p
0
Figure 11: Pseudo-threshold simulation results for the QpBf
noise model varying r (top) and p
mst
(bottom) (see Table 2).
Solid curves are the logical error rate as a function of the noise
event rate p
0
, dot-dashed blue lines indicate no error correc-
tion, and the pseudo-threshold for each choice of parameters r
and p
mst
is their intersection. These plots were generated for
1.5 × 10
5
Monte Carlo trials per data point, see Appendix C
for simulation details. See text for discussion of results.
gle excitation [24, 34, 35].
The bottom panel of Fig. 11 shows the pseudo-threshold
dependence on p
mst
for r = 1/10. The logical error rate,
and therefore the pseudo-threshold, have much weaker de-
pendence on p
mst
than on p
0
. Intuitively, this insensi-
tivity to measurement bit-flip can be understood as a di-
rect consequence of the repetition of stabilizer measure-
ments: a measurement bit-flip must be repeated in order to
show up in the accepted syndrome, whereas a quasiparti-
cle or pair-wise dephasing event occurring in the first time
step will affect all subsequent syndromes. We expect p
mst
to contribute significantly to p
err
when p
mst
& O
p
0
10
.
This estimate is consistent with the approximately con-
10
One example scenario in which measurement bit-flips con-
tribute to a failure of the error correction protocol is if two single
qubit errors occur in the first time step, e.g., x errors on qubits in
different columns, and a measurement bit-flip occurs for a qubit
in a different column in both the second and third time step (or
Accepted in Quantum 2018-08-21, click title to verify 22
0.0000 0.0002 0.0004 0.0006
p
0
0.0000
0.0002
0.0004
0.0006
0.0008
0.0010
p
err
q = 1/5, p
mst
= 10
4
r = 0
r = 1/10
r = 1/3
p
err
= p
0
0.0000 0.0002 0.0004 0.0006
p
0
0.0000
0.0002
0.0004
0.0006
0.0008
0.0010
p
err
r = 1/10, p
mst
= 10
4
q = 0
q = 1/10
q = 1/5
q = 1/3
q = 1/2
q = 2/3
q = 1
p
err
= p
0
0.0000 0.0002 0.0004
(4p
2
+ p
0
)/5
0.0000
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
p
err
r = 1/10, q = 1/5, p
mst
= 10
4
p
0
/p
2
= 2
p
0
/p
2
= 1
p
0
/p
2
= 1/2
p
pair
= p
err
0.0000 0.0002 0.0004
p
0
0.0000
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
0.0007
p
err
r = 1/10, q = 1/5
p
mst
= 0.05
p
mst
= 0.01
p
mst
= 0.001
p
mst
= 0.0001
p
pair
= p
err
Figure 12: Pseudo-threshold simulation results for the MC noise model varying r (top left), q (top right), p
2
(bottom left), and p
mst
(bottom right) (see Table 2). Solid curves are the logical error rate as a function of the noise event rate per qubit (averaged over
the 20 measured qubits and 5 idle qubits in the system), dot-dashed lines indicate no error correction, and their intersections denote
the pseudo-threshold for those parameter choices. In the top right panel, the pseudo-threshold for q = 1/10 is 6.5 × 10
4
, and
for q = 0 is 9.8 × 10
4
. See text for discussion of the results. The first three plots were generated for 10
8
effective Monte Carlo
trials per data point and the bottom right plot was generated for 10
7
effective Monte Carlo trials per data point, see Appendix C
for simulation details.
stant value p
QpBf
th
8 × 10
3
for p
mst
= 10
4
10
2
q
p
QpBf
th
, and the sharp pseudo-threshold decrease when
p
mst
= 0.08
q
p
QpBf
th
. The insensitivity to p
mst
is inde-
pendent of whether or not there are MZMs in the system
and should apply generally to any error correction scheme
that relies on repetition to correct for measurement bit-flips.
A similar result was found in the context of the surface
code built from qubits subjected to Pauli noise with bit-flip
measurement in Ref. [53]: for small Pauli error probability
(p
0
< 0.01), the probability of a stabilizer measurement bit-
flip could approach 50% without strongly affecting the log-
ical error rate. In comparison, the probability of a stabilizer
both the third and fourth time steps). The same logical error
could instead occur if three x errors affecting qubits in different
columns occur in the first, second, or third time steps. The two
scenarios are similarly probable when p
2
mst
O (p
0
).
measurement bit-flip here is 1 (1 p
mst
)
10
, which equals
50% for p
mst
0.07.
Finally, we note that in the regime of weak p
mst
depen-
dence, the pseudo-threshold estimate for QpBf is approxi-
mately consistent with the simulation of Qp if the logical
error rate is plotted as a function of p
0
/4. If we ignore mea-
surement bit-flips, the simulation for QpBf mainly differs
from Qp in that there are four time steps before applying
the decoder, thus the QpBf simulation roughly quadruples
the probability of a noise event compared to the Qp simula-
tion
11
.
11
This argument is not exact as only errors occuring during
the first three time steps will be actively corrected by the error
correction protocol.
Accepted in Quantum 2018-08-21, click title to verify 23
4.4.3 Majorana circuit noise (MC)
The error probabilities for MC are given in Table 2 (here we
write q
2
as q). Because we are restricting our analysis to the
operations necessary for error correction of quantum mem-
ory, during any given time step every island is either idle or
involved in a four-MZM measurement with another island.
Measurements are still defined according to Eq. (36). It is
now important to keep track of which islands are involved
in a given measurement, therefore we can no longer assume
that all stabilizers are applied simultaneously. Rather, each
set of stabilizer measurements (see Fig. 9) is preceded by
a noisy time step, so that a single error correction round is
comprised of four time steps and four measurement steps.
As for QpBf, faults in the measurement process require rep-
etition and we use the same protocol as before: we accept
the last repeated syndrome, or if no syndrome repeats, we
assume the syndrome for the last round. We show in Ap-
pendix D that this repetition scheme satisfies the conditions
for fault tolerance.
The simulation results are plotted in Fig. 12. The top left
panel shows the dependence on r for q = 1/5 (single-qubit
errors four times more likely than two-qubit errors), p
0
= p
2
,
and p
mst
= 10
4
. The pseudo-threshold dependence on r
is even weaker than was the case for QpBf, and is within
noise of the simulation for realistic choices of r (including
unphysical values of r = 1/2 and 1 does reveal a weak r-
dependence). This indicates that the code is insensitive to a
redistribution of errors involving an even or odd number of
Majorana operators. We stress that the insensitivity of the
pseudo-threshold to changes in r is highly dependent on the
assumption e
1/r
k
< p
k
(see Section 3.3) that went into
the construction of the circuit noise models. If this were
not the case, a single excitation in one island could spread
throughout the system through correlated events occurring
with probability q O(1). We comment further on archi-
tectures where this inequality is not satisfied in Section 5 and
Appendix B.
The pseudo-threshold’s dependence on q is shown in the
top right panel of Fig. 12 for r = 1/10, p
0
= p
2
, and
p
mst
= 10
4
. As is expected from similar studies using
qubit noise models, we see that a higher weight of corre-
lated events significantly lowers the pseudo-threshold [54].
When all initial excitations in an island lead to correlated
events, i.e., q = 1, the pseudo-threshold p
MC
th
1.2 ×10
4
.
When half of the errors on a measured island are from cor-
related event processes, i.e., q = 1/2, the pseudo-threshold
increases to the p
MC
th
4.5 × 10
4
. Finally, in the limit of
no correlated events, i.e., q = 0, the pseudo-threshold is
p
MC
th
9.8 ×10
4
. This last limit agrees with the simulation
results for Qp and QpBf by scaling the x-axes of Figs. 10
and 11 by factors of 1/16 and 1/4, respectively, to account
for the factor of four increase in the number of time steps
that can contribute to a logical error compared to the QpBf
measurement step (1)
measurement step (3)
Figure 13: Measurement steps (1) (top panel) and (3) (bottom
panel), see Fig. 9, for the d = 3 Bacon-Shor code implemented
for a system of tetrons. Measured MZMs are highlighted in
yellow. Tetrons involved in XX measurements between hori-
zontal nearest neighbors are highlighted in red, tetrons involved
in ZZ measurements between vertical nearest neighbors are
highlighted in blue. The PMC simulation takes into account
the geometrical arrangement of MZMs on a tetron, so that dif-
ferent MZMs on the left and right islands are measured in an
XX measurement, and different MZMs on the top and bottom
islands are measured in a ZZ measurement, see Eqs. (39) and
(40).
simulation.
The bottom left panel of Fig. 12 plots the pseudo-
threshold for p
2
= p
0
/2, p
0
, and 2p
0
with r = 1/10, q =
1/5, and p
mst
= 10
4
. In order to keep the total probability
of an error fixed, the x-axis is now given by (p
0
+ 4p
2
) /5,
the average rate of creating an excitation in the d = 5 system.
The pseudo-threshold is lower for p
2
> p
0
, which is consis-
tent with our expectation that a higher percentage of corre-
lated events should decrease the pseudo-threshold. This de-
pendence is not especially strong, which can be understood
as resulting from the small ratio (1/5) of islands that are idle
in any given time step.
Finally, we show the pseudo-threshold dependence on
p
mst
in the bottom right panel of Fig. 12 for r = 1/10,
q = 1/5, and p
0
= p
2
. The curves for p
mst
= 10
4
and
p
mst
= 10
3
are essentially within noise of each other, and
only for p
mst
& 10
2
O
p
0
does the pseudo-threshold
depend noticeably on p
mst
. The intuition is the same as for
model QpBf: measurement bit-flips must be repeated over
successive stabilizer measurements in order to affect the ac-
cepted syndrome and influence the correction procedure.
4.4.4 Physical Majorana circuit noise (PMC)
For PMC, we follow the same protocol as for MC: every
time step is followed by a stabilizer measurement and we re-
peat the error correction round four times (for a total of 16
time steps, 16 stabilizer measurements, and four sets of syn-
dromes). We assume the last syndrome which repeats twice
Accepted in Quantum 2018-08-21, click title to verify 24
0.0000 0.0002 0.0004 0.0006
p
0
0.0000
0.0002
0.0004
0.0006
0.0008
0.0010
p
err
q = 1/5, p
mst
= 10
4
r = 0
r = 1/10
r = 1/3
p
err
= p
0
0.0000 0.0002 0.0004 0.0006
p
0
0.0000
0.0002
0.0004
0.0006
0.0008
0.0010
p
err
r = 1/10, p
mst
= 10
4
q = 0
q = 1/10
q = 1/5
q = 1/3
q = 1/2
q = 2/3
q = 1
p
err
= p
0
0.0000 0.0002 0.0004 0.0006
(4p
2
+ p
0
)/5
0.0000
0.0001
0.0002
0.0003
0.0004
0.0005
p
err
r = 1/10, q = 1/5, p
mst
= 10
4
p
0
/p
2
= 2
p
0
/p
2
= 1
p
0
/p
2
= 1/2
p
pair
= p
err
0.0000 0.0002 0.0004
p
0
0.0000
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
0.0007
p
err
r = 1/10, q = 1/5
p
mst
= 0.05
p
mst
= 0.01
p
mst
= 0.001
p
mst
= 0.0001
p
pair
= p
err
Figure 14: Pseudo-threshold simulation results for the PMC noise model varying r (top left), q (top right), p
2
(bottom left), and
p
mst
(bottom right) see Table 2). Solid curves are the logical error rate as a function of the noise event rate per qubit (averaged
over the 20 measured qubits and 5 idle qubits in the system), dot-dashed lines indicate no error correction, and their intersections
denote the pseudo-threshold for those parameter choices. See text for discussion of the results. The first three plots were generated
for 10
8
effective Monte Carlo trials per data point and the bottom right plot was generated for 10
7
effective Monte Carlo trials
per data point, see Appendix C for simulation details.
is correct; if no two syndromes are repeated, we assume the
last syndrome.
As noted at the end of Section 3.3, models PMC and MC
applied to a system of tetrons with p
0
= p
2
only differ in a
slight redistribution of the odd correlated events. However,
since PMC is defined with a particular physical implemen-
tation in mind and keeps track of which MZMs are involved
in a measurement, we now take into account the geometric
arrangement of MZMs on an island to define the measure-
ments. We replace Eq. (36) with the following definitions of
an XX measurement of horizontal nearest neighbor tetrons
and a ZZ measurement of vertical nearest neighbor tetrons
(see Fig. 13):
X
j
X
j+1
= γ
j,2
γ
j,3
γ
j+1,1
γ
j+1,4
(39)
Z
j
Z
j+5
= γ
j,3
γ
j,4
γ
j+5,1
γ
j+5,2
. (40)
Note that now, for every tetron not located at a corner, each
MZM on that tetron is involved in at least one measurement.
This is a departure from the case where one MZM (e.g., γ
j,4
)
is not touched by any measurement and prevents a one-to-
one mapping of quasiparticle events and Pauli errors. In par-
ticular, a single-MZM error can lead to measurement syn-
dromes that are not described by a local Pauli error. Equa-
tions (39) and (40) are more convenient measurements to
implement physically; however, there is no fundamental lim-
itation to implementing measurements described by Eq. (36)
if the tetron architecture includes additional ancilla islands to
facilitate longer-range measurements [41].
The PMC simulation results are shown in Fig. 14. The r-
dependence, q-dependence, and p
mst
-dependence of the logi-
cal error rate are shown in the top-right, top-left, and bottom-
right panels, respectively. These plots are not noticeably dif-
ferent from those in Fig. 12 for the MC noise model, indicat-
ing that the redistribution of odd correlated errors (due to the
difference between models MC and PMC) and the measure-
ment redefinition have little effect on the pseudo-thresholds.
Accepted in Quantum 2018-08-21, click title to verify 25
The bottom-left panel of Fig. 14 plots the logical error
for p
0
/p
2
= 2, 1, 1/2 with r = 1/10, q = 1/5, and
p
mst
= 10
4
. The x-axis, (4p
2
+ p
0
)/5, is chosen for
easy comparison to model MC but is no longer the aver-
age rate of creating an excitation (the probability some of
the noise events in PMC on a measured island are indepen-
dent of p
2
). As expected, increasing p
2
worsens the logical
error rate because it increases the relative distribution of cor-
related events, but it does so to a smaller degree than for MC
since fewer of the noise events depend on p
2
.
4.5 Experimental Implications
The pseudo-threshold calculations in the previous section
have important implications for experimental realizations of
a tetron-based quantum computing architecture. In the limit
of a hard superconducting gap and large charging energy,
i.e., T min(∆, E
C
), quasiparticle events are rare and the
noise model PMC is a reasonable description of the most-
likely error sources in the system. The weak dependence of
the pseudo-threshold on r in this regime indicates that once
quasiparticle events are sufficiently rare and the relaxation
time is less than one tenth of the four-MZM measurement
time, it is not essential to further suppress the quasiparticle
poisoning rate.
A second important experimental implication regards the
tradeoff between correlated and measurement errors. Statis-
tical errors in the measurement contribute to the probability
p
mst
in our noise models. For the same length of measure-
ment time, larger measurement visibility, which is controlled
by the tunneling amplitude between the quantum dot and
MZMs (see Section 3.1), results in smaller statistical error
and hence smaller p
mst
. Conversely, the parameter q control-
ling the relative distribution of single-qubit and two-qubit
errors increases with tunneling amplitude [see Eqs. (21),
(22) and (26)]. Thus, we see a tradeoff in which reduc-
ing the tunneling amplitude can suppress correlated events
at the expense of increasing measurement errors. How-
ever, the pseudo-threshold results indicate that, at least for
Shor error correction of quantum memory, correlated events
have a much stronger effect on the pseudo-threshold than
do measurement bit-flips. Therefore, it is desirable to work
with a smaller tunneling amplitude from the point of view
of the pseudo-threshold. Reference [54] reached a similar
conclusion about the relative importance of two-qubit errors
compared to measurement errors for small-distance surface
codes. The tradeoff of measurement and correlated errors
is particularly interesting when also considering different er-
ror correcting codes that might provide certain advantages
but introduce stronger correlated errors due to higher-weight
measurements.
As noted earlier, the pseudo-threshold estimates in this pa-
per prioritize simplicity of simulation over optimal magni-
tude, and should therefore not be interpreted as quantitative
experimental targets. However, if the correlation parameter
q can be reasonably estimated for a given system, then pre-
vious studies using the qubit-based model Pauli circuit noise
are applicable. For instance, if q 1/15, then Ref. [55]
suggests a pseudo-threshold for the d = 5 Bacon-Shor code
O
10
3
(with Steane error correction).
5 Extensions
We now discuss how the noise models can be modified to
describe other Majorana-based systems. The physical error
sources we considered (thermal excitation, extrinsic quasi-
particle poisoning, fluctuations in the MZM hybridization
energies, error in the measurements) are ubiquitous to all
Majorana-based quantum computing architectures. There-
fore, quasiparticle events, pair-wise dephasing events, and
measurement bit-flips, as well as the corresponding proba-
bilities p
(k)
qp
, p
(k)
pair
, and p
(k)
mst
, respectively, should be present
in any realistic stochastic Majorana noise model. The maxi-
mal number of MZMs involved in a parity measurement will
vary depending on the proposal, therefore the allowed in-
teger values of k will be system-dependent. Furthermore,
the degree to which a particular proposal is susceptible to
these errors will depend on experimental parameters of the
system, including the size and gap of the superconducting
islands, whether or not the island has a charging energy, and
what potential quasiparticle reservoirs (e.g., metal or super-
conducting leads, quantum dots) are present.
The main assumption of all four Majorana noise models
introduced in Section 2 regards the energy separation be-
tween even and odd parity states of the underlying quantum
computing architecture. More specifically, we assume the
system is divided into superconducting islands and that the
environment relaxes islands with odd Majorana parity to the
even parity subspace within a time step with high probabil-
ity. When this is not the case, e.g., for grounded supercon-
ducting islands [5658], step 0 of the Majorana noise models
needs to be modified. In such systems, a single quasiparti-
cle excitation can be long-lived, and multi-island stabilizer
measurements can propagate the excitation throughout the
system, see Appendix B.
Qubit-based error correcting codes are less effective in a
MZM system with long-lived quasiparticle excitations. If the
error correcting measurements introduce connected paths of
islands spanning the system, a single quasiparticle excitation
can cause a logical error within a time step. Therefore, in
this scenario, the measurements must be carefully designed
to minimize the spread of high-weight errors through quasi-
particle excitations. Provided there is still a finite relaxation
rate, the probability for an excitation not to relax over a
full time step is exp {−1/r}. Thus, if measurements are
designed so that excitations can only travel to neighboring
Accepted in Quantum 2018-08-21, click title to verify 26
islands in a time step
12
, the probability for a single excita-
tion to travel through d (the code distance) islands (possi-
bly resulting in a logical error) is exp {−(d 1)/2r}. For
d-large, this probability becomes vanishingly small and the
qubit-based code retains some error correcting ability.
Alternatively, one could use a Majorana fermion code [24,
34, 35] to distinguish odd and even parity states, thereby
correcting quasiparticle excitations before they can spread
throughout the system. Such codes require the ability to ei-
ther (1) measure the total parity of the MZMs on an island,
or (2) dynamically adjust the number of MZMs on an is-
land, both of which are experimentally challenging for the
following reasons. (1) Proposals to date are not able to dis-
tinguish the total fermion parity of an island from the total
parity associated with the MZMs. The measurements pro-
posed in Refs. [5659] measure the total charge on the is-
land, and are thus unable to identify the presence of a ther-
mally excited quasiparticle. Charge measurements of the is-
lands are therefore helpful only if the charge excitations are
long-lived while thermal excitations relax sufficiently fast.
(2) Dynamically adjusting the number of MZMs on an is-
land requires sufficient tunability of certain experimental pa-
rameters (e.g., Josephson energy) to transition from the fully
disconnected regime to the fully connected regime: residual
coupling in the former can lead to increased probability of
correlated events; non-fully connected regions within an is-
land in the latter can result in mutual capacitances leading to
higher weight errors, see Appendix A. Furthermore, the tun-
ing procedure must be done sufficiently slowly and smoothly
to avoid introducing diabatic errors into the system [6064].
Model PMC was built on the further assumption of single
or two-island quantum dot-based measurements involving at
most two MZMs per island [40, 41]. Additional correlated
events can arise for alternate measurement schemes [32, 35,
5658] or if more than two islands are connected during a
measurement. For instance, surface code implementations
that rely on eight-MZM stabilizer measurements connecting
four islands [15] could result in a correlated event involving
any subset of the four islands.
Finally, the pseudo-threshold calculations in Section 4 are
specific to Shor error correction of quantum memory for a
system with four MZMs per island. The pseudo-threshold
estimates would likely change for a Steane error correction
analysis, if logical operations on the system are included, or
if islands have more than four MZMs (e.g., a hexon archi-
tecture [41]). The benefit of the Bacon-Shor code is lost for
Steane error correction, since the stabilizers could no longer
be implemented from two-island measurements. Generally,
for a stabilizer code, Steane error correction will estimate
higher pseudo-thresholds, however we do not anticipate the
change from Shor to Steane error correction to affect the
12
Note that the Bacon-Shor code already utilizes this clever
design by only using two-qubit measurements.
relative importance of quasiparticle and correlated events
(r and q dependence). Including logical operations would
probably result in a greater dependence on the relative mag-
nitudes of p
0
, p
1
, and p
2
, as the majority of islands would
no longer be involved in two-island measurements for each
time step. With hexons (m = 3), the percentage of corre-
lated events that do not commute with all stabilizer measure-
ments is greater than for the case of tetrons; therefore we ex-
pect the pseudo-threshold to have a stronger q dependence.
Furthermore, since physical considerations can constrain the
number of allowed correlated events in PMC relative to MC,
more substantial differences in the pseudo-threshold esti-
mates are possible than were found for the tetron case con-
sidered here.
6 Conclusions and Outlook
The primary goal of this paper is to connect the underly-
ing physical processes causing errors in a Majorana-based
quantum computing architecture to the noise models used
for fault tolerance analysis of the system. We developed
stochastic Majorana noise models in close analogy to the
standard qubit-based noise models generally used in thresh-
old calculations. These Majorana noise models allow for
errors involving an odd number of Majorana operators to
occur, and have different probability distributions depend-
ing on whether a MZM island begins a time step in an
even or odd parity state. The result of our analysis is that
for quasiparticle-poisoning-protected qubits, the pseudo-
threshold estimate for the d = 5 Bacon-Shor code under
each of the Majorana noise models is well-approximated by
the estimate using the analogous qubit-based model. Essen-
tially, when the relaxation parameter r . 1/5, a quasipar-
ticle event in one time step relaxes with high probability in
the subsequent time step; the cumulative effect is to apply
a Pauli error broken-up over two time steps, which does not
dramatically alter the error correcting code’s performance.
This result does not depend on any particular feature of the
Bacon-Shor code, thus we expect it to apply more generally
to larger error correcting codes, as well as to error correction
of logical gates. It is a positive result that MZM systems with
short-lived quasiparticle excitations (e.g., charging-energy-
protected qubits) can be analyzed with the simpler qubit-
based noise models, as this simplifies numerical simulations
and allows for studies of larger error-correction schemes.
Conversely, a MZM system with long-lived quasiparti-
cles cannot be accurately described by a qubit-based noise
model. In such a system, an excitation could travel between
islands connected by measurements over several time steps
with O(1) probability, thereby spreading throughout the sys-
tem. In order to prevent excitations traveling long distances,
it is therefore beneficial when performing measurements in
parallel to avoid creating connected paths of islands span-
Accepted in Quantum 2018-08-21, click title to verify 27
ning the system. The Majorana noise models developed here
could be extended to systems with long-lived excitations (see
Appendix B). The latter make qubit-based codes less effi-
cient and might require a Majorana fermion code for error
correction [24, 34, 35] (see discussion in Section 5).
A useful observation is that in a Shor error correction
scheme, in which stabilizer measurements are repeated to
protect against a single measurement bit-flip, the error cor-
recting code can sustain high probabilities of measurement
errors without affecting the pseudo-threshold. This result
applies equally well to conventional qubit systems. While
generally Steane error correction results in higher pseudo-
thresholds, if measurement error is a limiting obstacle, Shor
error correction could be an attractive alternative.
Our pseudo-threshold analysis further demonstrates that
there is a strong dependence on the relative distribution of
single-island and correlated events. It is therefore essen-
tial that any realistic noise analysis of a given system care-
fully estimate the rate of transferring excitations between is-
lands so as to choose an appropriate distribution of single-
island and correlated events. Furthermore, proposals involv-
ing higher-weight measurements (e.g., four-island stabilizer
measurements in the surface code) will be subject to higher-
weight correlated events that could significantly affect the
threshold estimate. The relative importance of correlated er-
rors compared to measurement errors also indicates that it is
worth optimizing the measurement processes so that corre-
lated errors are suppressed even if this reduces the measure-
ment fidelity.
This work is a first step towards comprehensive analysis
of the fault tolerance of Majorana-based quantum computing
architectures. Future directions of study include understand-
ing to what extent our results hold for both larger error cor-
recting codes and logical gate error correction. Additionally,
an important open question is to determine the optimal error
correction procedure for Majorana-based quantum compu-
tation. This analysis should weigh the experimental feasi-
bility of the quantum error correcting codes under consider-
ation (e.g., ability to perform stabilizer measurements and
whether the underlying system needs to be charging-energy-
protected) and the physical noise sources affecting the un-
derlying architecture in addition to the usual fault tolerance
criteria (e.g., threshold values and ratio of physical to logical
qubits). We believe the connections elucidated here between
physical error processes in MZM systems and noise models
would aid in such an analysis.
Acknowledgments
We are grateful to Bela Bauer, Chris Chamberland, Nicolas
Delfosse, Daniel Litinski, Tom O’Brien, and Krysta Svore
for helpful discussions and comments on a draft of this pa-
per. C.K. acknowledges support from the NSF GRFP under
Grant No. DGE 114085.
A Higher order errors
The probabilities defining PMC correspond to the lowest or-
der error processes occuring in a tetron architecture. These
lowest order processes include measurement bit-flips and all
noise events that involve only a single excitation (one factor
of p
k
in the language of Fig. 6 and Table 2): quasiparticle,
pair-wise dephasing, and correlated events. All other errors
in the model occur at higher order, that is, with a probabil-
ity that is the product of probabilities explicitly defined in
the model (i.e., a probaiblity O
p
2
k
). We now justify why
other error processes (transitions into higher excited states,
other correlated events, or mutual capacitance terms) that are
physically present in the system can be absorbed into these
higher order terms.
Throughout our discussion, we have only consid-
ered the excited states |e
i and |e
C,±
i. In particu-
lar, we have neglected an extrinsic quasiparticle tunnel-
ing into an above-gap state, states with multiple above-
gap quasiparticles, or states with charge 2e or higher.
Let Γ
ge
= max
Γ
ge
, Γ
ge
C,±
. Transitions into all
higher energy states are additionally suppressed compared to
Γ
ge
by a factor exp{−δε/T }, where δε is the energy dif-
ference between the higher excited state and max (∆, E
C
).
As δε min (∆, E
C
), this additional exponential factor
makes such a transition a higher order process. For
the same reason, a correlated event involving the process
|e
x
i |gi |e
x
i |e
x
i (|e
x
i {|e
i, |e
C,±
i}), is also a
higher order process, as it requires two excitations.
For a noise model that gives conservative threshold esti-
mates, it is sometimes beneficial, for the sake of keeping the
model simple, to overestimate the probability of lowest or-
der errors, so that multiple lowest order errors capture the
effects of additional error processes. As an example, we
note that the presence of interactions in the MZM system
can lead to an additional error source that we have not ex-
plicitly included in our model. So far we have neglected
mutual charging energies within a superconducting island,
e.g., between the two MZM nanowires comprising a tetron.
In the case of tetrons, mutual capacitances are exponentially
suppressed in the number of channels in the superconduct-
ing backbone (blue line in the left panel of Fig. 2) connecting
the two MZM nanowires. The remaining small but finite mu-
tual capacitances, however, would add four-(or more)-MZM
terms to Eq. (6) that involve pairs of MZMs belonging to
different tetrons. This capacitance can, depending on the
screening properties of the system, fall off as a powerlaw
with distance, and could therefore give rise to high-weight
errors (errors involving products of 2n MZMs with n 2)
in the MZM system. To justify ignoring these high-weight
errors at lowest order in the noise model, we could overes-
Accepted in Quantum 2018-08-21, click title to verify 28
Figure 15: Decision tree for the generalized step 0 for models
MC and PMC in the case of long-lived excitations. If an island
involved in a k-island measurement begins in the odd parity
state, it can either relax (left black line) with probability p
odd
,
hop to another island involved in the measurement (center red
line) with probability p
hop
, or remain excited throughout the
time step (right blue line) with probability p
stay
. In the first or
the last case, the noise model proceeds as described in the main
text to allow for single island noise events, correlated events, or
measurement bit-flips. In the latter case, if at least one of the
islands involved in the measurement is still in the odd parity
state, it again proceeds to the next step of the decision tree
(relax, hop, or stay excited). Even for q
k
1, the tree can be
truncated after k
2
steps where all k island have been visited
by the excitation. In the final step the excitation can then be
assumed to either relax or stays excited. The probabilities for
p
odd
and p
stay
are modified for the final step of the decision tree
as shown.
timate the probability p
(k)
pair
of pair-wise dephasing events so
that the probability of an error involving 2n MZMs is given
by
p
(k)
pair
n
.
B Long-lived excitations
The Majorana noise models presented in Section 2 need to be
modified when the system has long-lived excitations (e.g., in
the case of islands with small charging energies) to include
an additional type of correlated event:
Hopping event: application of two Majorana operators
involving two islands connected by a measurement. A
hopping event for islands i and j is of the form γ
i,a
γ
j,b
.
A hopping event can occcur when an island involved in a
k-island measurement begins a time step in the odd parity
state and transfers its excitation to another island involved
in the same measurement. Such a process is more likely
to occur when an excitation is long-lived. Models MC and
PMC can be generalized to account for long-lived excitations
by modifying step 0 as follows:
0. For the set of islands involved in the same k-island mea-
surement, if at least one of the islands begins the time
step with odd parity, do one of the following:
(a) Apply a hopping event involving an odd parity is-
land with probability p
hop
, and return to the begin-
ning of step 0.
(b) Apply a quasiparticle event to an island with odd
parity with probability p
odd
, and proceed to step 1.
(c) Do not apply any noise events with probablity
p
stay
= 1 p
hop
p
odd
, and proceed to step 1.
The generalized step 0 results in the decision tree depicted
in Fig. 15. The chance for a large number n
hop
of hopping
events is exponentially suppressed by q
n
hop
k
. Moreover, even
for large q
k
1, the tree can be truncated, with the final step
modified to not include step 0a. For instance, if every island
is only directly connected to at most two other islands (as
in a tetron measurement), then a hopping event is a random
walk through the connected path of islands. After k
2
steps,
the average path includes the path involving all islands in the
measurement; therefore all possible noise events occuring
after k
2
+1 steps can be captured by a more-probable process
in a smaller number of steps.
In the language of Section 3, the probabilities for the mod-
ified step 0 are given by
p
hop
= q
k
, (41)
p
odd
= (1 q
k
)
1 e
1/r
, (42)
p
stay
= (1 q
k
) e
1/r
. (43)
(If truncating the modified step 0, the probabilities for the fi-
nal step do not involve q
k
, see Fig. 15.) Recall that q
k
is the
probability of an energy transfer between islands connected
by a measurement, while r is the probability that an excita-
tion occuring during a time step does not relax before the end
of the time step. The probability that an island beginning a
time step in the odd parity state does not relax is e
1/r
.
In the main text, we set p
hop
= 0. The justification for this
approximation is that we assume r is small enough (quasi-
particle excitations are short-lived enough) that noise events
resulting from a sequence of hopping events are already cap-
tured by a sequence of correlated and single qubit noise
events. More concretely, one might worry that not includ-
ing hopping events neglects the possibility that a single ex-
citation can result in a high-weight error spread over several
time steps from a quasiparticle traveling between different
islands connected by subsequent measurements. Consider,
for instance, the following two scenarios leading to a weight-
three error: (1) An idle island is initially excited in the first
Accepted in Quantum 2018-08-21, click title to verify 29
time step, the excitation is transferred to a second island in
the second time step and does not relax, and the excitation is
transferred to a third island in the third time step and then re-
laxes. This process applies a pair of Majorana operators on
all three islands involved. In an alternative scenario (2) an
idle island experiences no noise events in the first time step,
the first and second islands are involved in an even correlated
error in the second time step, and the third island undergoes
a pair-wise dephasing event in the third time step. Process
(1) only requires a single excitations to cause a weight-three
error, while process (2) is the most probable implementation
of a weight-three error over three time steps that does not
involve hopping events. Process (1) occurs with probability
p
qp
(p
hop
p
stay
) (p
hop
p
odd
) (44)
and process (2) occurs with probability
1 p
qp
3
4
p
pair
p
(2)
cor,even
p
(2)
pair
. (45)
(We assume the first island is idle in the first time step for
simplicity, the argument naturally extends if the island is in-
stead involved in a measurement.) Processes (1) and (2) re-
sult in the same computational effect for the MC simulation
discussed in Section 4.4, because the net application of Ma-
jorana operators at the end of the third time step are equiva-
lent for (1) and (2), and intermediate steps are interpreted as
measurement errors. We are therefore justified in neglecting
process (1) when the probability (44) is much less than (45).
This is the case provided
p
0
q
1 q
r
(1 r)
2
e
1/r
1 e
1/r
qre
1/r
.
(46)
The simplifying approximation holds for small q and r. For
q = 1/5 and p
0
= 10
3
condition (46) is valid provided
r < 1/5, and for q = 1/5 and p
0
= 10
4
is valid provided
r = 1/7. Note that the r values in Fig. 12 were chosen to
exagerate the r dependence of the noise model and do not
strictly fall into this regime. Including additional time steps
does not result in a stronger condition on the relative size of
p
0
and r.
When r is too large to satisfy Eq. (46), we need to use the
more complicated step 0. In this regime, a single excitation
can result in a high-weight error by continually spreading to
new islands with each multi-island measurement.
In scenarios with long-lived excitations it becomes crucial
to devise a careful measurement protocol to avoid introduc-
ing paths of connected islands spanning the system. If errors
can only hop to O(k) nearby islands in a single time step, an
excitation will eventually relax before traversing the system
if the system size becomes sufficiently large. This implies
that, while being less efficient, large qubit-based codes will
still be able to correct for long-lived excitations. At some
Algorithm 1 Perfect Measurement Monte Carlo
1: nFail = 0
2: for i = 1 to i = N do
3: MList = ZeroArray[n]
4: err = Noise[MList]
5: MList = MList + err
6: gaugeOutcome = Dot[gaugeMatrix,MList]
7: stabOutcome = Dot[stabGaugeMatrix,gaugeOutcome]
8: corr = LookupBaconShor[stabOutcome]
9: MList = MList + corr
10: logicalErr = Dot[logicalMatrix,MList]
11: if Total[logicalErr] > 0 then
12: nFail = nFail +1
13: end if
14: end for
15: return nFail/N
point it might, however, be more practical to use Majorana
fermion codes to be able to detect whether an island is in the
odd parity state before the excitation can propagate.
Finally, if the quantum computing architecture is built
from grounded superconducting islands, an island can spend
an equal amount of time in the even and odd parity states. In
this case, the probabilities of different noise events are the
same regardless of in which parity state the system begins.
In this regime, excitations are dominated by extrinsic quasi-
particle poisoning events and Eq. (24) is modified to
p
k
= 2m
exp
n
Γ
(k)
ge
C
τ
o
1
. (47)
In particular, p
k
is no longer exponentially suppressed and
all noise events become much more likely.
C Pseudo-threshold code
In this appendix, we give further details of the code used
to simulate the pseudo-thresholds reported in the main text
Section 4.
Perfect measurement Monte Carlo algorithm: In
both the qubit-based model Pauli noise and the Majorana
model Qp, the measurements are perfect and the algorithm
to model error correction is as in Algorithm 1.
The algorithm estimates p
err
by sampling simulated
trials of faulty quantum error correction and counting the
fraction of trials which result in a logical error. In each trial,
errors occur according to the noise model (for a given set
of noise parameters), and are recorded in a binary vector
‘err’ of length 100 = 4 × 25, i.e., one for each MZM. A
bit value of 1 indicates that the corresponding Majorana
operator was applied to the system, otherwise the bit value
is 0. This error is added bitwise to the length 100 binary
vector ‘MList’ (initially trivial), which tracks the net set of
Majorana operators which have been applied to the system.
Accepted in Quantum 2018-08-21, click title to verify 30
Algorithm 2 Perfect Measurement Importance
Sampling Monte Carlo
1: pErr = 0
2: for v = 2 to v = vTrunc do
3: nSample = 0
4: nFail = 0
5: while nFail < N do
6: w = RandomSubset[W,v]
7: nSample = nSample + 1
8: MList = ZeroArray[n]
9: err = PostSelectedNoise[MList,w]
10: MList = MList + err
11: gaugeOutcome = Dot[gaugeMatrix,MList]
12: stabOutcome = Dot[stabGaugeMatrix,gaugeOutcome]
13: corr = LookupBaconShor[stabOutcome]
14: MList = MList + corr
15: logicalErr = Dot[logicalMatrix,MList]
16: if Total[logicalErr] > 0 then
17: nFail = nFail +1
18: end if
19: end while
20: pErr = pErr + Pr(v) nFail/nSample
21: end for
22: return pErr
After the noise has been applied, the gauge measurement
outcomes are inferred, in practice by storing each gauge
operator as a row of a binary array ‘gaugeMatrix’ (with
a bit value of 1 for each Majorana operator appearing in
the gauge operator), and multiplying it by ‘MList’. The
results are stored as a binary vector ‘gaugeOutcome’. The
stabilizers ‘stabOutcome’ are found by multiplying the
appropriate subset of gauge measurement outcomes and
a correction ‘corr’ is inferred from a lookup table for the
Bacon-Shor code, which would apply the appropriate Pauli
correction for two or fewer single qubit Pauli errors (again
encoded in a bit string) as described in Section 4.2. The final
result of the error correction attempt is obtained by adding
‘corr’ to ‘MList’. If the error correction was successful
‘MList’ is a binary representation of a trivial operator. This
is checked by taking a binary dot product of the list of
all possible non-trivial logical operators ‘logicalMatrix’
with ‘MList’ which yields 0 (1) when ‘MList’ shares an
even (odd) number of Majorana operators with a certain
logical operator and thus (anti)commutes. Anticommutation
with any local operator is then counted as a failure. In our
simulations, the number of samples used to generate each
data point was N = 10
5
.
Importance sampling with perfect measurement:
The above Monte Carlo procedure would become slow for
small error rates, since the number of repetitions becomes
very large, requiring huge computational resources. This
is largely because, for very low error rates, almost every
run has only zero or one faults. In practice, the error rates
required for our purposes were sufficiently large for Pauli
noise and the Majorana model Qp to allow us to use Monte
Carlo, but it is useful to describe this case for illustrative
purposes. Since the protocol is fault tolerant (as proven in
Appendix D), zero or one faults will never cause a logical
failure. Therefore, we can save time by only generating sam-
ples in which at least two faults occur. In order to do this,
we must be able to both calculate the probability that at least
one fault occurs, and efficiently sample from the distribution
formed by post selecting on there being at least one fault.
This procedure is known as importance sampling. Here, we
can apply importance sampling relatively simply by consid-
ering each tetron independently, and by separately consider-
ing errors on all subsets of v tetrons, for v = 2, 3, 4 . . . . We
can truncate v at v
trunc
when the probability of v
trunc
tetrons
having an error becomes negligible.
Let W be the set of all tetrons, and let Pr(v) be the
probability of precisely v of the tetrons having an error.
Then, the importance sampling Monte Carlo algorithm is
implemented as in Algorithm 2. In each run, a subset w
containing v tetrons is randomly selected; noise is drawn
from the post-selected distribution where errors occur on
each tetron in the subset w and nowhere else.
Imperfect measurement Monte Carlo algorithm:
We now consider the noise models for which the measure-
ments can be imperfect (i.e., QpBf, MC, PMC, and their
analogous qubit-based models). For these models, our quan-
tum error correction protocol repeats measurements four
times to achieve fault-tolerance, essentially by increasing the
reliability of the inferred stabilizer outcomes. Algorithm 3
therefore includes four error correction rounds t, and the cor-
rection protocol is implemented as normal, but by using the
last repeated set of stabilizer outcomes (or the last stabilizer
outcome if none are repeated). We also apply an additional
round of perfect error correction to ensure that the system is
returned to the code space.
We have used S to represent the number of stabilizer
generators. The function ‘Noise[MList]’ here is very
general in the MC noise model for example, it actually
entails a sequence of four time steps, during each of which
a different set of gauge generators is (noisily) measured.
Importance sampling with imperfect mea-
surement: A key feature of the importance sampling
in the perfect measurement case described above is that
each possible fault is independent. However, now that
we have multiple time steps, and since in some of our
noise models, the noise can depend on the current state
of the system, we can no longer treat the faults as being
independent. Moreover, importance sampling really is
necessary for these cases since the error rates in the region
of interest are very small rendering Monte Carlo extremely
time consuming. The number of samples that would be
Accepted in Quantum 2018-08-21, click title to verify 31
Algorithm 3 Imperfect Measurement Monte Carlo
1: nSample = 0
2: nFail = 0
3: while nFail < N do
4: nSample = nSample + 1
5: MList = ZeroArray[n]
6: for t = 1 to 4 do
7: err = Noise[MList]
8: MList = MList + err
9: gaugeOutcome = Dot[gaugeMatrix,MList]
10: stabOutcome
[t] = Dot[stabGaugeMatrix,gaugeOutcome]
11: end for
12: t=4;
13: while t>1 and stabOutcome
[t] 6= stabOutcome
[t-1] do
14: t = t-1
15: stabOutcome = stabOutcome
[t]
16: end while
17: if t == 1 then
18: stabOutcome = stabOutcome
[4]
19: end if
20: corr = LookupBaconShor[stabOutcome]
21: MList = MList + corr
22: GaugeOutcome = Dot[gaugeMatrix,MList]
23: resStabOutcome = Dot[stabGaugeMatrix,GaugeOutcome]
24: resCorr = LookupBaconShor[resStabOutcome]
25: resMList = MList + resCorr
26: logicalErr = Dot[logicalMatrix,resMList]
27: if Total[logicalErr] > 0 then
28: nFail = nFail +1
29: end if
30: end while
31: return nFail/nSample
needed for the regime of interest for MC and PMC would
require extremely large sample numbers without any kind
of importance sampling. Fortunately, we can implement a
partial importance sampling: we run importance sampling
for the first fault that occurs, and then after that we run
the system forward in time with usual Monte Carlo. The
algorithm is a straightforward combination of Algorithms 2
and 3. The effective number of total samples used per data
point for imperfect measurements was 10
7
for the plots in
which p
mst
was varied, and and 10
8
samples for the rest of
the simulations (although many of these were not actually
run due to the importance sampling step).
Single island error probabilities: Lastly, in Table 3
we explicitly write the eight computationally distinct bit
strings for a single island and the corresponding probability
of applying that bit string to ‘MList’ in a time step. This table
is used to generate single-island error strings in models Qp,
QpBf, and MC. For the jth tetron, the bit string (1, 0, 1, 0)
corresponds to γ
j,1
γ
j,3
, or equivalently to γ
j,2
γ
j,4
(because
the total parity of an island is fixed). For simplicity, we write
the probabilities for models Qp and QpBf. Model MC is
straightforwardly generalized by adding the appropriate su-
perscript to each of the probabilities: (0) for an idle island
or (k) for a k-island measurement. Model PMC is slightly
more complicated for a measured island as we need to distin-
guish which MZMs are being measured. Note that since the
parity of a tetron is fixed, bit strings that differ by (1, 1, 1, 1)
are computationally equivalent. Models MC and PMC will
additionally have two-island errors, whose probabilities de-
pend on p
cor,even
and p
cor,odd
.
D Fault tolerant conditions
In this appendix, we detail how fault tolerance conditions
(EC A)-(EC B) are satisfied for noise models Qp, QpBf,
MC, and PMC.
Qp: We simulate model Qp with a noisy time step fol-
lowed by perfect application of all stabilizer measurements
followed. Measurements are defined according to Eq. (36).
We use the minimum weight perfect matching decoder for
the resulting syndrome to choose which correction operator
to apply.
Model Qp assumes perfect measurements, thus (EC A)
and (EC A’) are automatically satisfied. To check that
(EC B) holds, we know that the d = 5 Bacon-Shor code
can correct all single- and two-qubit Pauli errors, and that
in a tetron system all pair-wise dephasing events can be
expressed as Pauli errors. Therefore, we just need to check
that quasiparticle errrors are similarly correctable. A single
fault corresponds to either a quasiparticle excitation in one
time step followed by relaxation in the next time step, or a
pair-wise dephasing event. The former process has the same
net effect as the latter, and is thus correctable. To see this
more explicitly, consider the sequence of the quasiparticle
event γ
j,a
occuring in one time step and the island relaxing
back to the even parity subspace in the subsequent time
step with application of γ
j,b
. The first error has the same
syndrome as the Pauli error γ
j,a
γ
j,4
(because γ
j,4
is not
involved in any of the stabilizer measurements), thus the
error correction will inadvertently apply the operator γ
j,4
.
The net operator applied to the island at the end of the
next time step is γ
j,b
γ
j,4
, which is simply a Pauli error.
More generally, once the island has returned to the even
parity subspace, the net operator applied to the island is a
Pauli error, which can be corrected in the standard way.
Intermediate corrections (while the island remains in the
odd parity subspace) do not worsen the error.
QpBf: Model QpBf is simulated with a noisy time step
followed by application of all stabilizer measurements, re-
peated four times (i.e., one time step per error correction
round). Measurements are defined according to Eq. (36).
We apply the minimum weight perfect matching decoder to
the last syndrome to be repeated. If no syndrome is repeated,
we select the last syndrome.
To check that conditions (EC A) and (EC A’) hold, first
consider a single measurement bit-flip occuring, which flips
one of the four syndromes. Our protocol ensures that we se-
lect the correct syndrome, thus we correctly ignore the mea-
Accepted in Quantum 2018-08-21, click title to verify 32
Bit Strings Even Probabilities Odd Probabilities
(0, 0, 0, 0) 1 p
qp
3
4
p
pair
(1 p
odd
)
1 p
qp
3
4
p
pair
+
1
4
p
odd
p
qp
(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)
1
4
p
qp
1
4
(1 p
odd
) p
qp
+
1
4
p
odd
1 p
qp
1
4
p
pair
+
3
16
p
odd
p
pair
(1, 1, 0, 0), (1, 0, 1, 0), (1, 0, 0, 1)
1
4
p
pair
1
4
(1 p
odd
) p
pair
+
1
4
p
odd
p
qp
Table 3: The eight computationally distinct single-island bit strings (left column) and their corresponding probabilities when the
island begins the time step with even parity (middle column) and odd parity (right column). The probabilities are written for
models Qp and QpBf, but are easily generalized to MC by including the superscript (k) when the island in question is involved in
a k-island measurement. The first row corresponds to no error, the second row to errors involving an odd number of MZMs, and
the last row to errors involving two MZMs.
surement bit-flip. Now, suppose a pair-wise dephasing event
occurs in the first, second, or third time step. The third and
fourth syndromes will then agree, and correctly reflect the er-
ror applied to the system, thus the error correction step at the
end of the protocol will return the system to the error-free
code state. If the pair-wise dephasing event occurs in the
fourth time step, we will select the syndrome indicating no
error has occurred: this will neither correct nor worsen the
error, so that in the subsequent error correction round the er-
ror can be corrected. Finally, a quasiparticle event followed
by a relaxation has the net effect of applying a Pauli opera-
tor and is thus correctable (either in the same error correction
round or the subsequent round depending during which time
step the initial excitation occurs); the intermediate syndrome
will be interpreted as a measurement error and thus does not
worsen the error.
For condition (EC B), we need to show that any two faults
occuring within an error correction round can be corrected
either in the same error correction round or in the next. If
neither of the faults are measurement bit-flips, then they
are correctable for the same reasons given for model Qp
(although possibly not until the next error correction round
if they occur in the fourth time step). If only one fault is
a measurement bit-flip, it will not appear in the selected
syndrome and will not affect the correction protocol, and
the remaining fault is correctable for the same reasons given
in Qp. Finally, if both faults are measurement bit-flips, they
will only affect the correction protocol if they result in the
same syndrome in the second and third, or the third and
fourth measurement repetitions. The correction step will
then at most apply two Pauli operators to the system, which
can be corrected in the subsequent error correction round if
no additional faults occur.
MC: Model MC is simulated with the error correction round
shown in Fig. 9, with a noisy time step preceding each stabi-
lizer measurement. The entire error correction round is then
repeated four times (total of 16 time steps). Measurements
are defined according to Eq. (36). We apply the minimum
weight perfect matching decoder to the last syndrome to be
repeated. If no syndrome is repeated, we select the last syn-
drome. We use the d = 5 Bacon-Shor code as a d = 3 code
when including correlated events.
Conditions (EC A)-(EC B) are satisfied for the same
reasons given for QpBf. Even correlated events map to
two-qubit Pauli errors and can thus be corrected for the
same reasons as in the qubit system. Odd correlated events
γ
i,a
γ
i,b
γ
j,c
result in the same syndrome as the even cor-
related event γ
i,a
γ
i,b
γ
j,c
γ
j,4
, with i 6= j. After relaxation
back to the even parity subspace, the net Majorana operator
string applied to islands i and j is equivalent to a two-qubit
Pauli operator applied in a single time step. Because there
are more time steps for each error correction round, it is
even less likely for measurement bit-flips to be repeated in a
way that affects the accepted error syndrome.
PMC: Model PMC is simulated according to the same pro-
tocol as MC, with the only difference being that measure-
ments are now defined according to Eqs. (39) and (40).
The fault tolerance conditions (EC A)-(EC B) are satisfied
for the same reasons given in the other noise models, with
only a slight modification. Here, for all non-corner islands,
every MZM on the island is involved in some stabilizer mea-
surement, and the syndrome corresponding to a quasiparticle
event on that island will match the syndrome corresponding
to multiple pair-wise dephasing events. However, once the
island relaxes back to the even parity ground state, the net
operator applied to the island will be equivalent to a Pauli
operator. Thus, the syndrome will change, and the interme-
diate step at which the island was excited will be interpreted
as a measurement bit-flip.
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