Fault-tolerant gates via homological product codes
Tomas Jochym-O’Connor
Walter Burke Institute for Theoretical Physics, Institute for Quantum Information & Matter, California Institute of Technology,
Pasadena, CA 91125, USA
January 30, 2019
A method for the implementation of a univer-
sal set of fault-tolerant logical gates is presented
using homological product codes. In particular,
it is shown that one can fault-tolerantly map
between different encoded representations of a
given logical state, enabling the application of
different classes of transversal gates belonging
to the underlying quantum codes. This allows
for the circumvention of no-go results pertaining
to universal sets of transversal gates and pro-
vides a general scheme for fault-tolerant compu-
tation while keeping the stabilizer generators of
the code sparse.
1 Introduction
Quantum error correction extends qubit coherence
times through error mitigation and will be a require-
ment for any large-scale quantum computation. In this
vein, tremendous research efforts have been placed on
finding quantum error correcting codes that may be re-
alizable in both the near and distant future. Among
the leading candidates for experimental implementa-
tion are 2D topological stabilizer codes, such as the
toric code [1, 2], which allow for the correction of errors
by measuring small-weight local checks while protecting
logical information in highly non-local degrees of free-
dom [37]. These codes are experimentally appealing
due to their stabilizers being low-weight, thus minimiz-
ing the effect of noise during measurement. They can be
generalized to higher spatial dimensions, again with the
stabilizer generators being relatively low-weight, and
provide theoretically simple implementations of differ-
ent classes of fault-tolerant logic [813]. The motivation
for quantum error correcting codes to have geometri-
cally local stabilizers (in a given dimension) is to sim-
plify experimental architectures, yet this may not nec-
essarily be a hard requirement. However, the need for
low-weight stabilizer checks is much stronger, as larger
weight checks lead to more noise propagation, and gen-
erally lower threshold error rates. The theory of quan-
Tomas Jochym-O’Connor: tjoc@caltech.edu
tum low-density parity check (LDPC) codes has been
developed to address such concerns and finding good
codes with low-weight checks remains a very active area
of research [1419]. Moreover, LDPC constructions can
be used to construct codes with very little overhead for
fault-tolerant computation, relying on the preparation
of logical ancillary states [2023]. This work will not
focus on the development of such codes, but rather will
center on how to generally use such codes for the pur-
poses of fault-tolerant computation.
Obtaining a universal set of fault-tolerant gates is
complicated by the existence of no-go results for such
constructions using only transversal gates [24, 25].
However, many alternative schemes have been devel-
oped to circumvent this restriction. They rely on the
preparation of special ancillary states and gate tele-
portation [2628], or tailored fault-tolerant construc-
tions for certain classes of codes [8, 2936]. This work
extends the set of fault-tolerant alternatives, present-
ing a scheme for fault-tolerant logic on any CSS code
while keeping the underlying stabilizer measurements
low-weight.
We present a method for implementing fault-tolerant
logical gates in a homological product code [16].
Namely, given the homological product of two quantum
codes, we show how to map in a fault-tolerant man-
ner between the encoded homological product logical
state to a logical state specified by one of the two codes.
Then, if the underlying codes have a set of transversal
gates, such logical gates can be applied and the state
can be re-encoded back into the full codespace fault-
tolerantly. There are no restrictions on the underlying
codes, other than having to be defined by a boundary
operator δ : C C such that δ
2
= 0 in the linear
space C. In particular, by using versions of the 2D and
3D color codes [9, 37] as the underlying codes in the con-
struction, a universal set of fault-tolerant operations can
be implemented. The mapping between different rep-
resentations of the code follows a similar construction
to Ref. [30], where one of the two codes is unencoded
while the other provides the protection. The key dif-
ference between the methods is that the stabilizers of
the homological product can remain low-weight, unlike
Accepted in Quantum 2019-01-18, click title to verify 1
arXiv:1807.09783v2 [quant-ph] 29 Jan 2019
those in a concatenated model. Moreover, we present
a decoding method to address errors that may occur
during the unencoding of one of the two codes. Given
certain properties of the underlying codes, this will re-
sult in a finite probability error threshold along the lines
of Ref. [38] as well as potential protection against mea-
surement errors.
The article is organized as follows: In Sec. 2 we review
the theory of CSS codes defined by chain complexes and
the construction of the homological product codes, care-
fully considering their underlying structure. In Sec. 3 we
present the main result, a fault-tolerant method to im-
plement a logical gate using homological product codes
as well as discuss a simple decoding procedure. In Sec. 4
we present examples of codes that exhibit a set of uni-
versal fault-tolerant gates, expanding on the notions of
code padding and doubling. Finally, we conclude with
some remarks and open questions in Sec. 5.
2 Homological Product Codes
2.1 Single sector theory
We begin by reviewing the connection between CSS
codes and homology. Namely, as in Ref. [16], we focus
on single sector theory
1
in Z
2
, that is a chain complex
defined by a linear space C and a linear boundary oper-
ator δ : C C, such that δ
2
= 0. We can then use such
a boundary operator to define a CSS code [39, 40], that
is a stabilizer code whose generators can be expressed
as either X-type or Z-type.
Let C be a n-dimensional binary space Z
n
2
, then δ
will be a n × n binary matrix. The (perhaps over-
complete) generating set of X stabilizers will be given
by the rows of δ, that is for a given row, a generator S
X
i
will have X support on the qubits with a 1 in the given
row. Similarly, the Z stabilizers will be defined by the
columns of the matrix. Given δ
2
= 0, we are thus as-
sured commutativity of the stabilizers. The number of
independent generators of both type will be rank(δ),
and as such the number of logical qubits of the code
will be k = n 2rank(δ).
For the remainder of this section we shall focus on the
reverse implication. That is, given a CSS code whose X
and Z stabilizer spaces are of the same dimension, one
can construct a boundary operator δ of a single-sector
theory. The results presented are a fairly straightfor-
ward corollary of Lemma 3 from Ref. [16], yet we in-
clude them here for completeness and to review some
1
In general, CSS codes can be constructed from chain com-
plexes where δ
i
: C
i
C
i1
are linear operator satisfying:
δ
i1
δ
i
= 0.
important concepts, namely the canonical boundary op-
erator.
Lemma 1. Given a CSS code on n qubits, whose X and
Z stabilizer spaces are each of cardinality 2
l
, therefore
encoding k = n 2l qubits. Then, there exists a in-
vertible matrix W , and canonical boundary operator δ
0
defined as:
δ
0
=
0
k
0 0
0 0
l
1
l
0 0
l
0
l
, (1)
such that δ = W δ
0
W
1
, where the rows (columns) of δ
contain a set of generators of the X (Z) stabilizer group.
Proof. Given the existence of a CSS code, by the
Gottesman-Knill theorem [41] there exists a unitary
operator U composed solely of CNOT gates that
maps |ψi |0i
l
|+i
l
to the encoded stabilizer code,
where l = (n k)/2 and |ψi is a k-qubit state. This
statement can be expressed in terms of matrix manip-
ulation on Z
2
, where δ
0
will represent the initial state
of the stabilizers before the application of the encoding
circuit, that is the rows of δ
0
represent the initial |+i
states, and the columns the initial |0i states. A CNOT
gate with qubit i as control, and qubit j as target can
then be expressed according to the invertible matrix:
W
i,j
= 1 + w
i
w
T
j
, (2)
where w
k
is the standard basis column vector one non-
zero entry at position k. The action of W
i,j
by conjuga-
tion maps column c
j
to the sum of columns c
i
c
j
, and
row r
i
to the sum of rows r
i
r
j
. This is the exact ac-
tion required from a CNOT, as it maps X
i
to X
i
X
j
and
Z
j
to Z
i
Z
j
. Note, as required for a valid representation
of a CNOT gate, W
2
i,j
= 1 W
i,j
= W
1
i,j
, where again
we are working modulo 2. Then, the encoding opera-
tion W can be broken into its CNOT components and
can be expressed as W = W
i
N
,j
N
· · · W
i
2
,j
2
W
i
1
,j
1
. As
such,
δ = (W
i
N
,j
N
· · · W
i
1
,j
1
)δ
0
(W
i
N
,j
N
· · · W
i
1
,j
1
)
1
(3)
= (W
i
N
,j
N
· · · W
i
1
,j
1
)δ
0
(W
i
1
,j
1
· · · W
i
N
,j
N
), (4)
will be a valid representation of the stabilizers of the
code. Since W is a valid representation of the encoding
circuit of the code, the rows (columns) of δ will remain a
valid representation of the X (Z) stabilizers since they
were so for δ
0
.
Perhaps as importantly for the purposes of this arti-
cle, if the given CSS code has a generating set of stabi-
lizers that are sparse, then the resulting constructed δ
will be sparse. We define the sparsity of a code to be the
Accepted in Quantum 2019-01-18, click title to verify 2
smallest integer w such that there exists a set of genera-
tors of the code whose weights are at most w while any
given qubit participates in at most w stabilizer checks.
Corollary 2. Given a CSS code on n qubits with an
equal number of X and Z stabilizers and sparsity t, then
a boundary operator δ can be constructed such that no
row or column will have more than t
2
non-zero entries.
Proof. Given some sparse representative set of stabiliz-
ers {S
X
i
, S
Z
i
}, as in the proof of Lemma 1, a unitary
circuit W can be chosen that maps Z
k+i
S
Z
i
and
X
k+l+i
S
X
i
. Consider the right action of W
1
in
terms of its action on the canonical boundary operator:
δ
0
W
1
=
0
k×n
s
X
1
.
.
.
s
X
l
0
l×n
, (5)
where on the right side of the equality we have a matrix
whose rows are either all-zero or a binary representa-
tion s
X
i
of the stabilizer S
X
i
. This follows from the
fact that the right application of W
1
results in the
propagation of the initial X stabilizers to their final
generator form. Then, by the sparsity of the stabilizer
generators, each row will be of weight at most t and
each column will have weight at most t. Now consider
the left application of W applied to δ
0
W
1
, thus com-
pleting the conjugation, the resulting matrix W δ
0
W
1
will have rows that will be sums of the different rows
of δ
0
W
1
. Moreover, each row of W δ
0
W
1
will be a
sum of at most t rows s
X
i
, and will as such be of weight
at most t
2
. Finally, since a given row can map to at
most t other rows, each non-zero entry within a column
of δ
0
W
1
can map to at most t entries within that col-
umn. Therefore, since there were at most w non-zeros
entries in a column of δ
0
W
1
, there can be at most t
2
non-zeros in each column of W δ
0
W
1
2.2 Homological Product Construction
Given two complexes (C
1
, δ
1
), (C
2
, δ
2
) with their asso-
ciated spaces and single sector boundary operators, we
define a new operator as in Ref. [16],
= δ
1
1 + 1 δ
2
, (6)
acting on C
1
C
2
. It follows from δ
2
i
= 0 that
2
= 0,
again since we are working in Z
2
. Therefore, (C
1
C
2
, ) is a valid single sector complex, defining its own
quantum CSS code.
We now restate some important properties of the ho-
mological product.
Lemma 3 ([16]). Let (C
1
, δ
1
), (C
2
, δ
2
) be complexes
defining codes with k
1
, and k
2
logical operators, respec-
tively. Let = δ
1
1 + 1 δ
2
, then the resulting com-
plex (C
1
C
2
, ) will encode k = k
1
k
2
logical qubits and
ker = ker δ
1
ker δ
2
+ im . (7)
Suppose that w
a
is the sparsity of δ
a
. Moreover, let
d
X
a
, d
Z
a
be the X and Z distances for the correspond-
ing codes. Then, the weight and distances of the new
code can be bounded according to the parameters of the
original code.
Lemma 4 ([16]). The sparsity of is upper bounded
by w
1
+ w
2
. The X and Z distance of the new code
satisfy the following bounds:
max {d
α
1
, d
α
2
} d
α
d
α
1
d
α
2
, α = X, Z. (8)
2.3 Encoding the homological product code
In this subsection, we review some facts about the
encoding circuit for [16]. As eluded to in Sec-
tion 2.1, there are invertible matrices W
a
such that δ
a
=
W
a
δ
a,0
W
1
a
, where δ
a,0
are the canonical boundary op-
erators for δ
a
. The matrices W
a
are binary represen-
tatives of the encoding circuit for the given code, and
as such, by taking their tensor product we obtain the
encoding operation for . That is:
= (W
1
W
2
)(δ
1,0
1 + 1 δ
2,0
)(W
1
1
W
1
2
) (9)
:= (W
1
W
2
)
0
(W
1
W
2
)
1
, (10)
where we have defined
0
to be the canonical boundary
operator for . We can express
0
in matrix form as
follows:
0
= (δ
1,0
1 + 1 δ
2,0
) (11)
=
1
k
1
δ
2,0
0 0
0 1
l
1
δ
2,0
1
l
1
1
n
2
0 0 1
l
1
δ
2,0
, (12)
where k
i
are the number of logical qubits and l
i
= (n
i
k
i
)/2 is the number of X/Z stabilizers of the given code
code.
It is worth further exploring the form of
0
, as this
will be informative of how the logical information is
encoded in the code. Each row and column of
0
will be
of weight at most 2. Moreover, if a given row has 2 non-
zeros entries, say at positions q
i
and q
j
, then any column
with a non-zero entry at q
i
will also have a non-zero
entry at position q
j
in order to satisfy commutativity.
As such, these rows and columns represent an initial
entangled Bell pair between qubits q
i
and q
j
since they
will be stabilized by the operators X
q
i
X
q
j
and Z
q
i
Z
q
j
.
Accepted in Quantum 2019-01-18, click title to verify 3
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l
1
l
1
l
2
l
2
k
2
k
1
n
1
n
2
Figure 1: Initial state of the homological product code prior to
encoding [16]. Each circle represents a qubit, with the black
qubits representing those holding the logical information to be
encoded. Blue qubits are prepared in the |0i state, while red
qubits are prepared in |+i. The yellow qubits joined by an
oscillating edge are prepared in a Bell pair (|00i + |11i)/
2.
The initial state can be pictorially represented by
Fig. 1, where along a fixed row and column, the states
in C
1
and C
2
are fixed, respectively. Then, the ten-
sor product binary operators W
1
1 and 1 W
2
will
have geometric meaning in this picture. Note for the
remainder of this work, we will denote U
i
as the physi-
cal encoding unitaries composed of CNOT gates acting
on the quantum states whose binary representation is
given by W
i
. Thus U
1
will couple qubits within verti-
cal bands, while U
2
will couple qubits within horizontal
bands, as represented in Fig. 2.
3 Fault-tolerant logical gates
3.1 Partial decoding of the homological product
code
The key idea for expanding the set of available fault-
tolerant logical gates will be for the two underlying
codes composing the homological product to have dif-
ferent complimentary sets of transversal gates. Then,
we can achieve the application of these logical gates by
only decoding one of the two underlying codes, while re-
maining protected by the other. This is reminiscent of
the scheme for implementing fault-tolerant gates using
two concatenated codes [30], with the added advantage
that the stabilizers remain low-weight in the case of the
homological product.
The main result is that, while we decode one of the
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1
encoder
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(b) C
2
encoder
Figure 2: Schematic representation of the support of the en-
coding circuits for each of the codes underlying the homo-
logical product code, defined by the boundary operator =
δ
1
1 + 1 δ
2
. The overall encoding circuit has a binary
representation of the operator W
1
W
2
, which acts on the ini-
tial state represented in Fig. 1. That is, the physical encoding
unitary U
1
for the code C
1
will act on every column on qubits
from Fig. 1, and conversely the physical encoding unitary U
2
will act on every row.
codes, we still remain fully protected by the other code.
While errors may potentially propagate during the ap-
plication of the decoding process, they can still be cor-
rected as long as the number of faults is less than half
the distance of the underlying code protecting the in-
formation (that is the code that remains encoded at all
times). The main Theorem is a variant of Lemma 4
stated above (originally Lemma 2 from Ref. [16]), yet is
proved using the concept of error bands.
Recall the homological product is defined by the com-
plex (C
1
C
2
, ), where C
i
are binary spaces. The com-
plex defines a code with n
1
n
2
physical qubits, which
we label (i, j) where 1 i n
1
and 1 j n
2
.
Then, a Pauli operator P is supported only on (i, j)
if Tr
(i
0
,j
0
)6=(i,j)
(P ) = P
i,j
6= 0, we will therefore denote
such Pauli operators P
i,j
. We will call the error band E
1
a
to be all possible Pauli operators of the form
Q
j
P
a,j
,
that is a product of any Pauli operators P
i,j
with i = a.
Conversely, the error band E
2
a
will be all possible Pauli
operators of the form
Q
j
P
j,a
. Moreover, we will say
that a Pauli error E is supported on the set {e
1
, · · · e
l
} of
error bands E
1
a
if the operator is supported by the qubits
composing those error bands, that is E E
1
e
1
· · ·E
1
e
l
.
A Pauli operator supported on a set of error bands E
2
a
is
defined similarly and whether the operator is supported
on error bands E
1
a
or E
2
a
should be clear from context.
Note that the unitary U
1
will only couple qubits within
a fixed error bands E
1
a
, while conversely U
2
will only
couple qubits within fixed error bands E
2
a
.
Theorem 5. Let (C
1
, δ
1
), (C
2
, δ
2
) be complexes and let
(C
1
C
2
, ) be the homological product code constructed
Accepted in Quantum 2019-01-18, click title to verify 4
from these codes with = δ
1
1 + 1 δ
2
. Then any er-
ror E supported on fewer than d
2
error bands E
1
a
cannot
support a logical operator. Similarly, any error E sup-
ported on fewer than d
1
error bands E
2
a
cannot support
a logical operator.
Proof. Consider an error E supported on fewer than d
2
error bands E
1
a
, let the affected bands be denoted by the
set {e
1
, · · · , e
l
}, that is E E
1
e
1
· · ·E
1
e
l
, where l < d
2
.
Since any error can be expressed in the Pauli basis, if
we show any Pauli error supported on the above set
cannot support a logical operator, E cannot support
a logical operator. As such, without loss of general-
ity, suppose E is a Pauli error. Consider the modified
error E
0
= U
1
EU
1
, where U
1
is the encoding unitary
for the code C
1
. Then, if we can show that E
0
cannot
support a logical operator on the new codespace after
applying U
1
, then by unitary equivalence it cannot on
the homological product codespace.
Any logical operator must commute with all of the
stabilizers of the code. The modified codespace is given
by the boundary operator
1,0
= (W
1
1
1)(W
1
1) =
δ
1,0
1 + 1 δ
2
, that is it will correspond to k
1
logical
codeblocks encoded in the code (C
2
, δ
2
) along with ac-
companying encoded ancilla state
2
. This can be viewed
visually by considering the initial unencoded state in
Fig. 1 followed by the encoding operation of Fig. 2b.
Therefore, in order for E
0
to support a logical error, it
will have to support a logical operator on one of the
first k
1
error bands E
2
a
. Without loss of generality, con-
sider the first error band E
2
1
, that is the first row of
qubits in Figs. 1-2, and the support of E
0
on that band,
E
0
1
= E
0
E
2
1
(E
1
e
1
· · · E
1
e
l
) E
2
1
. Therefore, the
weight of the Pauli operator given by E
0
1
is limited by
the number of initial error bands E
1
a
on which the error
was supported, that is: wt(E
0
1
) l < d
2
, and as such
since any logical operator supported on the band E
2
1
must be of weight at least d
2
, the error E
0
1
cannot sup-
port a logical error. Since this will be true for all en-
coded logical bands supported on E
2
a
, with a k
1
, the
error E
0
cannot support a non-trivial logical error.
To conclude, since E
0
cannot support a logical error
on the code specified by the boundary operator
1,0
,
then E = U
1
E
0
U
1
cannot support a logical operator on
the code specified by = (W
1
1)
1,0
(W
1
1
1).
Equipped with Theorem 5, we propose the following
scheme to implement a fault-tolerant logical gate. Sup-
pose the logical gate G
1
can be implemented transver-
sally on the single-qubit partition of the code induced
by the complex (C
1
, δ
1
), that is it can be implemented
by applying gates that are each individually supported
2
An encoded ancillary state is a fixed state that contains no
non-trivial logical information, yet still may be partially encoded.
on single qubits of the code. Then, in order to apply
the logical gate G
1
on the logical state of the homo-
logical code (C
1
C
2
, ), we begin by unencoding the
code (C
2
, δ
2
), that is we apply the unitary 1U
2
. At this
point, the first k
2
blocks of n
1
qubits remain encoded
in the code (C
1
, δ
1
), while the the remaining blocks are
in an encoded ancillary state. Therefore, we can apply
the transversal implementation of the logical gate G
1
on any of the k
2
logical states we desire. We complete
the logical gate application by then reencoding into the
code (C
1
C
2
, ) by applying 1 U
2
.
The proposed scheme is fault-tolerant in that, it will
be able to correct against up to b(d
1
1)/2c faults
throughout the process. Any error P
a,b
that occurs dur-
ing the application of either 1 U
1
2
, 1 U
2
, or the
transversal gate can spread to a high-weight error, yet
such an error will remain within a single error band E
2
a
.
This follows as the application of 1 U
2
only ever cou-
ples qubits within the same error band E
2
a
. Similarly, a
transversal gate with respect to the code (C
1
, δ
1
) will not
couple different error bands E
2
a
, by definition. There-
fore, any single fault P
a,b
can result in an error that
is contained within the error band E
2
a
. Since any log-
ical error must be supported on at least d
1
such error
bands, any error affecting less than half of such error
bands must remain correctible by the Knill-Laflamme
condition [42].
By symmetry, given a transversal gate G
2
on the
single-qubit partition of the code (C
2
, δ
2
), a fault-
tolerant implementation of G
2
can be achieve by ap-
plying U
1
1, followed by the transversal gate, and a
reencoding U
1
1. Such a fault-tolerant gate will be
able to correct against up to b(d
2
1)/2c faults.
3.2 Correcting errors
In the last section, we showed how even when de-
coding one of the two codes, we can always protect
against at least b(d
i
1)/2c faults. However, the en-
coding/decoding operations may generally have O(n
c
i
)
time steps
3
, and as such, errors will accumulate within
a given error band (assuming an independent non-
Markovian error processes), resulting in an error with
probability: p
e
O(n
c
i
), where p
e
is the physical error
rate. This is undesirable from the perspective of fault
tolerance, as we hope that for a given family of codes,
by growing the distance, the probability of incurring a
logical error decreases exponentially below some thresh-
old value. Yet, if the underlying error rate is growing
3
Decoding of stabilizer codes amounts to Gaussian elimina-
tion on the binary-symplectic matrices representing the stabiliz-
ers. This can be done using a polynomial number (in the input
size) of elementary matrix row manipulations which correspond
to Clifford gates.
Accepted in Quantum 2019-01-18, click title to verify 5
polynomially with the distance, this yields a pseudo-
threshold for each code, rather than a global threshold
for a code family.
In this section, we present a simple decoding algo-
rithm for the homological product code, based on the
decoding algorithm of the individual codes composing
the homological product. While the presented scheme
will likely be far from ideal in many settings, it will
serve as a proof of principle decoder as well as provide
a means to correct against errors during the implemen-
tation of the fault-tolerant logical gates. This will help
alleviate the concern errors accumulating within an er-
ror band due to the encoding/decoding having a macro-
scopic number of individual time steps. As will be dis-
cussed at the end of this subsection, in some cases this
will guarantee a fault-tolerance threshold against inde-
pendent noise. However, the existence and value of such
a threshold would have to be studied on a case-by-case
basis.
Consider the homological product code as specified in
the previous subsection, with a boundary operator =
δ
1
1 + 1 δ
2
. Moreover, suppose we have recovery
operators R
i
for each code that returns the code to the
codespace and corrects with certainty when the weight
of the error is below half the distance of the respective
code. We present the following Corollary to Theorem 5,
which follows directly from the proof of that result.
Corollary 6 (of Theorem 5). Let (C
1
, δ
1
), (C
2
, δ
2
) be
complexes and let (C
1
C
2
, ) be the homological product
code constructed from these codes with = δ
1
1 +
1 δ
2
. Then, for any error E supported on fewer than
b(d
j
1)/2c error bands E
i
a
, the error E
0
= U
i
EU
i
is
correctible using the recovery operator R
j
, where i, j
{1, 2} such that i 6= j.
The above Corollary states that given a