Fermat's Library | q-2019-04-08-131 annotated/explained version.

Analysing correlated noise on the surface code using adaptive

decoding algorithms

Naomi H. Nickerson

and Benjamin J. Brown

2,3

Quantum Optics and Laser Science, Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2AZ,

United Kingdom

Niels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark

Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia

April 3, 2019

Laboratory hardware is rapidly pro-

gressing towards a state where quantum

error-correcting codes can be realised. As

such, we must learn how to deal with

the complex nature of the noise that may

occur in real physical systems. Single

qubit Pauli errors are commonly used to

study the behaviour of error-correcting

codes, but in general we might expect the

environment to introduce correlated er-

rors to a system. Given some knowledge

of structures that errors commonly take,

it may be possible to adapt the error-

correction procedure to compensate for

this noise, but performing full state to-

mography on a physical system to analyse

this structure quickly becomes impossible

as the size increases beyond a few qubits.

Here we develop and test new methods

to analyse blue a particular class of spa-

tially correlated errors by making use of

parametrised families of decoding algo-

rithms. We demonstrate our method nu-

merically using a diﬀusive noise model. We

show that information can be learnt about

the parameters of the noise model, and ad-

ditionally that the logical error rates can

be improved. We conclude by discussing

how our method could be utilised in a

practical setting blue and propose exten-

sions of our work to study more general

error models.

1 Introduction

The success of scalable quantum computation de-

pends on our ability to diagnose and repair errors

incident to the underlying physical qubits [1, 2,

3, 4, 5, 6, 7] of the system. To this end quantum

error-correcting codes [8, 9, 10, 11, 12, 13, 14]

have been developed to allow us to correct for

a ﬁnite density of physical errors that might

otherwise irreversibly disturb a quantum com-

putation. Of particular interest are topological

codes [10, 11], which are experimentally viable

due to their local structure and low-weight syn-

drome measurement operations [15, 16].

Quantum error-correcting codes are commonly

studied with a simple error model where it is

assumed that each physical qubit is subject to

some rate of local noise that is independent of the

state of other nearby qubits and its local environ-

ment. Identifying thresholds [11] and ﬁnding the

resource overheads [17, 18, 19] of diﬀerent codes

using simple error models can provide a proof of

principle that a code can withstand local noise,

and oﬀers a standardised benchmark to uniformly

compare diﬀerent error-correcting codes with one

another [20, 21]. However, when we consider

some realisation of a quantum error-correcting

code, where we take into account physical fea-

tures of its underlying hardware [22, 23, 24], an

independent noise model is unlikely to be real-

istic. Instead, we might anticipate that corre-

lated errors [25, 26, 27, 28, 29, 30, 31] could oc-

cur, where the probability of a qubit suﬀering an

error depends on its neighbouring qubits having

also suﬀered errors.

Correlated noise has been identiﬁed in a num-

ber of physical systems that might serve to re-

alise quantum error-correcting codes. For exam-

ple, external ﬁelds and proximity eﬀects in solid

state systems [30, 32] introduce correlated errors,

as do quantum codes that are coupled to a bo-

son environment [33, 34, 35, 36, 37, 13, 31, 38].

Correlated errors have also been identiﬁed when

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arXiv:1712.00502v4 [quant-ph] 1 Apr 2019

non-Markovian eﬀects are modelled [26, 39]. It

has also been seen that correlated errors can be

introduced by certain schemes of quantum error-

correction. This has been observed, for instance,

in Ref. [19]. Correlated errors also occur when

considering syndrome measurements under the

circuit-based noise model [40, 41, 42, 43, 44], and

while applying transversal logical gates to quan-

tum error-correcting codes that are not in their

code space [45]. See also recent work. [46].

The aim of this Manuscript is to develop nu-

merical methods to characterise correlated error

models. Speciﬁcally, we develop families of de-

coding algorithms, each member of which is de-

signed to correct for a particular correlated error

model. We show that we can use a decoder fam-

ily, which we will refer to as an adaptive decoder,

to analyse a source of noise with some unknown

parameter by comparing the logical error rates

of diﬀerent members of a decoder family. We

perform simulations to demonstrate our meth-

ods explicitly by adapting Edmonds’ minimum-

weight perfect matching algorithm [47, 48] to

identify string-like correlated errors introduced to

the toric code [10]. Gaining information on the

structure of correlations in noise can be used as a

diagnostic tool to minimize the eﬀect of correla-

tions at the hardware level. It may also be used

to improve classical post-processing. We show

that we can use our adaptive scheme to ﬁnd a

decoder with an improved threshold to deal with

the physical error model.

There are a number of methods and results

in the literature that look to determine correla-

tions that may emerge in fault-tolerant quantum

hardware. In Ref. [49] Combes et al. showed

that syndrome data can be used to discriminate

between diﬀerent correlated noise processes act-

ing on small quantum codes including the classi-

cal repetition code and the ﬁve-qubit code [50].

Our results build on this work by examining

diﬀerent codes and we give numerical evidence

that these ideas are applicable to degenerate

codes [51]. Moreover, the use of eﬃcient decoding

algorithms that we propose make our diagnostic

techniques scalable to systems of large size. We

point out that recently, during the preparation of

the present manuscript Ref. [52] showed that it is

possible to monitor time-dependent parameters

of uncorrelated noise models to improve thresh-

old error rates using methods similar to ours.

Other methods have also been found to de-

termine correlated errors that occur during the

error correction process. In Ref. [42] the au-

thors simulate a circuit-level error model to di-

rectly calculate the likelihood that correlated er-

rors occur while syndrome data is measured.

The information obtained with this procedure is

used to improve decoding schemes. Further, in

Ref. [53], drifts in experimental parameters of a

small repetition code are identiﬁed and corrected

by measuring changes in the rate at which error-

detection events occur. See also Ref. [54] that

recently became available. These works make

use of local syndrome data to determine diﬀer-

ent parameters in the error model. In contrast,

we make use of a decoding algorithm that uses

global syndrome data to distinguish more generic

correlated error models where error events may

correlate over a large number of adjacent qubits.

We also remark on previous work to specialise

decoders to account for known correlated errors.

It has been shown [55, 56, 57, 58, 59, 60, 61,

62, 63] that we can obtain improved thresholds

by using a decoder that speciﬁcally accounts for

known correlations that occur under depolarising

noise. Hutter and Loss have also identiﬁed that

specialised decoders can be used to correct for

known correlated errors introduced by a thermal

environment [29]. The present work extends these

ideas by demonstrating that we can use learned

knowledge to calibrate a decoder to give improved

thresholds without complete knowledge of the er-

ror model a priori.

The remainder of the Manuscript is structured

as follows. In Section 2 we introduce the toric

code and develop decoding algorithms for gen-

eral correlated errors. In Section 3 we show that

the generalised decoders we present can be used

to ﬁnd signatures that distinguish correlated er-

rors. Section 4 shows that we can use error model

data to obtain improved decoding schemes. In

Sec. 5 we give a discussion on how we might ap-

ply our scheme to the practical development of

fault-tolerant quantum hardware. We conclude

in Section 6 by discussing potential directions to

develop our methods in order to diagnose more

general forms of correlated error.

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Figure 1: The toric code with qubits arranged on the

edges of a sqaure lattice of size L = 7. (a) A plaquette

operator is shown in blue. (b) A star operator is shown

in orange. (c) and (d) show logical operators Z

and

, respectively.

2 The toric code and error models

We study correlated errors acting on the well-

studied toric code model. In this Section we

brieﬂy review the toric code, and introduce the

notation we use to describe correlated error

events. We go on to describe the minimum-weight

perfect-matching (MPWM) algorithm that we

will use to analyse correlated errors.

2.1 The toric code

The toric code [10] is a quantum error-correcting

code of n = 2L

qubits, arranged on the edges on

a square lattice of L × L vertices with periodic

boundary conditions, as shown in Fig. 1. The

code space of the toric code is described by its

stabilizer group, S [64]. The stabilizer group is

an Abelian subgroup of the Pauli group, P, where

the code states of the toric code, |ψ

i, are the

common +1 eigenspace of all elements S

∈ S

|ψ

i = (+1)|ψ

i, ∀j, k. (1)

Elements of the stabilizer group are commonly

known as stabilizers.

The stabilizer group of the toric code consists

of two types of stabilizers, star operators, A

, and

plaquette operators, B

, deﬁned as follows

∂j3v

, B

j∈∂f

, (2)

where j is an index over the qubits and X

and

operators are Pauli-X and Pauli-Z operators

acting on qubit j. The set ∂f are the qubits on

the boundary of face f and ∂j is the set of ver-

tices v at either end of the edge that supports

qubit j. The stabilizer group consists of one star

operator for each vertex, v, and one plaquette

operator for each face f. Colloquially, a star op-

erator is the product of Pauli-X operators acting

on all the qubits on edges incident to vertex v,

and a plaquette operator is the product of Pauli-

Z operators acting on all the edges that bound

faces of the lattice, f. We show pictures of a pla-

quette and a star operator in Figs. 1(a) and (b),

respectively.

The code space of the toric code encodes two

logical qubits that are acted upon by logical oper-

ators L, whose elements are Z

and X

with j =

1, 2. The logical operators are non-contractible

loops of Pauli-Z or Pauli-X operators acting on

the primal or dual edges of the lattice respec-

tively. One pair of these logical operators, X

and Z

, is shown in Figs. 1(c) and (d).

Quantum error-correcting codes are designed

to identify and correct errors suﬀered by the code.

We consider error operators E ∈ P. The proce-

dure for error correction involves measuring the

generators of the stabilizer group. Stabilizers

commute with the logical operators of the stabi-

lizer code, and as such we do not disturb encoded

logical information when stabilizer measurements

are made. The outcomes of the stabilizer mea-

surements can be used to identify the error as

the elements of the stabilizer group S

that do

not commute with E will return a −1 outcome.

We see this with the eigenvalue equation

E|ψ

i = −ES

|ψ

i = (−1)E|ψ

i, ∀j, k. (3)

We refer to elements of the generating set of S

that return −1 eigenvalues as defects. The col-

lection of measurement outcomes of the selected

stabilizer generators is known as the error syn-

drome. In the case of the toric code that we study

here we use the star and plaquette operators in

Eqn. 2 to represent the canonical set of stabilizer

generators.

We require a decoder, a classical post-

processing algorithm that interprets syndrome in-

formation, to estimate the most likely error that

might have occurred on the lattice. A decoder

will subsequently return a correction operator, C,

to attempt to reverse error E. Speciﬁcally, we

seek a decoder such that the value

P = prob(CE ∈ S), (4)

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is optimised. To do so, the decoder must make

use of some knowledge of the type of errors that

commonly occur in the system. In the remain-

der of this Section we describe general correlated

noise models acting on the toric code, the syn-

dromes they give rise to, and the decoder we use

to interpret the syndrome information.

2.2 Error events

In the previous Subsection we introduced error

operators E ∈ P. However, we wish to design

a decoder which takes into account more infor-

mation about the structure of an error operator.

Here we make rigorous the concept of a structured

error model by considering an error operator as a

series of error events.

To study the performance of quantum error-

correction procedures against correlated errors we

generate random Pauli errors of the form

E =

(1−σ

)/2

, (5)

where operators e

∈ P describe a set of error

events that may occur through the noise chan-

nel. Error event e

occurs with likelihood p

such

that σ

= −1 with probability p

and σ

= +1

otherwise.

Typically quantum error-correcting codes are

tested using an independent and identically dis-

tributed(i.i.d.) noise model. For instance, the

error events of the bit-ﬂip noise model are sin-

gle qubit Pauli errors acting on each qubit of the

code i.e. e

= X

for all j that index the qubits of

the system. Each error event in the model occurs

with uniform probability p

= p for all j. Al-

ternatively, the depolarising noise model is some-

times considered, where similarly, error events e

act uniformly on all qubits j of the system, i.e.

= p, and then error events can take values X

or Z

with uniform probability 1/3. In our

analysis that follows we will consider more gen-

eral bit-ﬂip error models with error events that

are supported on multiple qubits. The restriction

to bit-ﬂip errors simpliﬁes the problem of decod-

ing as all of the stabilizer defects are detected by

the plaquette operators.

2.3 Decoding with the minimum-weight per-

fect matching algorithm

In the next Section we generalise the well-studied

MWPM decoding algorithm [11]. For numerical

Figure 2: Defects on the lattice (left) are mapped onto

a graph with weighted edges (right) where the weights

correspond to the probabilities that its adjacent pair of

defects were created by an event that corresponds to

errors along that edge.

studies we make use of the Blossom-V implemen-

tation of Edmonds MWPM algorithm [47] due to

Kolmogorov [48]. This decoding algorithm is par-

ticularly well suited to the diﬀusive noise model

we will introduce in the next Section.

The algorithm due to Edmonds takes as input

a graph with weighted edges and returns a match-

ing graph, i.e. a graph where each vertex has one

and only one incident edge, such that the sum

of the weights of all the edges are minimal. We

denote edges, e, by pairs of vertices of the graph

such that e = (u, v) where u and v denote ver-

tices of the graph. We also denote by W (e) the

weight assigned to edge e.

To apply the MWPM algorithm to the prob-

lem of decoding we assign to each stabilizer de-

fect measured by a plaquette operator a vertex

of the matching graph. Ideally we then construct

a complete weighted graph where we weight each

edge with the function

W (e) = −log(prob(e)), (6)

where prob(e) is the probability that the error

model introduced an error event, or a combina-

tion of error events, such that defects u and v of

edge e occurred. In general evaluating W (e) may

be a complex summation of exponentially many

terms. We will typically approximate this value

by calculating only low-order terms of the sum-

mation. We summarise the construction of the

complete graph in Fig. 2.

To ﬁnd a correction operator we apply the Ed-

monds MWPM algorithm to the generated graph

to ﬁnd a matching graph where the edges corre-

spond to a collection of error events that is likely

to have caused the error syndrome. We use this

information to ﬁnd correction operator C.

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We remark that for the well-studied case of

i.i.d. noise we will typically have W (e) =

−log(prob(e)) ∝ d(u, v) where d(u, v) is the sep-

aration of vertices u and v of edge e on the lat-

tice [11]. Throughout this work we consider the

Manhattan distance where

d(u, v) = d

+ d

, (7)

where d

) is the separation between u and v

along the x(y) direction on the lattice.

We refer to the MWPM decoder where W (e) ∝

d(u, v) which is designed to deal with i.i.d. noise

model as the standard decoder. Other work has

also considered introducing an extra logarithmic

term to account for degeneracy in errors that

cause defects at vertices u and v [65, 59], but

for simplicity we do not use this extension here.

In this work we will generalise the construction of

the MWPM decoder by assigning diﬀerent func-

tions to W (e) depending on the anticipated cor-

related noise model.

The MWPM decoder will ﬁnd a single error

that was likely to have caused a given syndrome

with respect to an error model that is deter-

mined by W . We expect that our methods can

be improved using maximum-likelihood decod-

ing schemes that identify the most-likely equiv-

alence class of errors that caused a given syn-

drome [65, 55, 56, 66]. However, in general,

few eﬃcient maximum-likelihood decoding algo-

rithms are known [66]. Here we assume that iden-

tifying a single error that was likely to cause the

syndrome is a good approximation of maximum-

likelihood decoding. This has been demonstrated

to be the case for the independent and identically

distributed noise model in Ref. [11, 67].

3 Noise characterisation

In this Section we consider how to adapt the

MWPM decoding algorithm, such that it can

be parametrised to target diﬀerent noise models.

This parameterisation allows us to deﬁne a family

of decoders, which we call an adaptive decoder.

Speciﬁcally, we consider a MWPM decoder where

edges are assigned weights by the function W

(d)

where d is the distance between the two defects,

and the parameter λ denotes diﬀerent members

of the decoder family. In general, λ can represent

many variables, but here, for simplicity, we will

consider only integer values. We show that we

Figure 3: Error events of the ballistic noise model for

ξ = 3. The model introduces error strings along vertical

and horizontal lines that introduce pairs of stabilizer de-

fects that are separated by distance ξ. (a) An error event

acting on a horizontal edge marked by a square box. A

bit ﬂip is introduced to the edge in the square box, and

the ξ − 1 horizontal edges immediately below the edge

where the error event occurs. (b) An error event acting

on a vertical edge marked by a square box. The error

event introduces a bit ﬂip on the vertical edge and the

ξ − 1 vertical edges to the immediate right of the edge

where the event occurs.

can distinguish diﬀerent noise models by study-

ing logical error rates as a function of λ with an

adaptive decoder. We ﬁnd that, for certain noise

models with unknown parameters, the logical er-

ror rates, or thresholds of diﬀerent decoders can

allow us to deduce the noise parameters.

We ﬁrst consider a simple example noise model

to analytically justify the use of logical error rates

to learn noise parameters. We will then test two

diﬀerent adaptive decoders, which we will deﬁne

shortly, to analyse a diﬀusive noise model using

numerical experiments.

3.1 An illustrative example of the diagnostic

tools

We ﬁrst develop some intuition for our diagnos-

tic method using a simple correlated noise model,

that we call the ballistic noise model, by quoting

some results that are shown in Appendix A. We

will argue that improved logical error rates are

achieved with a targeted decoder that uses in-

formation about the noise model in the regime

where the error rate is small. We use this exam-

ple to motivate the simulations we conduct in the

remainder of the paper.

In the ballistic error model, error events form

‘straight lines’ of Pauli errors with a ﬁxed weight,

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Figure 4: An error caused by the ballistic noise model in

the path-counting regime. The noise model introduces

strings of length ξ = L/4. A standard decoder will have

diﬃculty dealing with this error conﬁguration as there

are two inequivalent correction operators that have the

same weight. Conversely the targeted decoder which is

designed to anticipate errors that introduce defects that

are separated by a distance ξ will correct errors from this

error model with high probability.

ξ. More speciﬁcally, the error event associated to

vertical(horizontal) qubit edge j introduces a bit-

ﬂip to edge j and the subsequent ξ −1 edges that

are located immediately to the right of edge j

(below edge j). Each error event occurs on edge

j with uniform probability p

= p. We show ex-

amples of error events of the ballistic noise model

where ξ = 3 in Fig. 3.

In Appendix A we show that for arbitrary ξ ≥ 3

in the regime where p is very low, otherwise

known as the path-counting regime [11, 18], the

logical error rates can be improved by using a tar-

geted decoder. It follows that we can construct

a family of two diﬀerent decoders to distinguish

i.i.d. noise from the ballistic noise model for ﬁxed

ξ ≥ 3 by comparing the logical error rates of

the two diﬀerent decoders. The two MWPM de-

coders of this family are the following, the ﬁrst is

the standard decoder, where the matching graph

is constructed with edges weighted according to

the separation of stabilizer defects. This decoder

seeks the lowest-weight correction operator. The

second decoder in the family is the targeted de-

coder, which anticipates the ballistic noise model

and assigns weight a to edges where stabilizer de-

fects are separated by distance aξ, and otherwise

assigns very high weights to edges of the match-

ing graph.

Consider the ballistic noise model with ξ = L/4

in the path-counting regime where 1/p  4ξL.

In this example the common error conﬁgurations

that will cause either the standard decoder or the

targeted decoder to fail are those where two non-

overlapping error events occur on the same row

or column. However, there are far fewer non-

overlapping error conﬁgurations that will cause

the targeted decoder to fail compared with the

standard decoder. We give an example error con-

ﬁguration in Fig. 4 caused by the ballistic noise

model that will be corrected using the targeted

decoder, but will not be corrected by the stan-

dard decoder.

For the case of the error conﬁguration shown

in Fig. 4, the standard decoder will construct

a graph that will assign equal probabilities to

two inequivalent correction operators, C

and C

independent of L, where C

repairs the error,

E ∈ S, and the other correction is such that

E 6∈ S. The standard decoder will struggle to

deal with this case as both C

and C

will have

weight ∼ L/2 and as such, the standard decoder

will not be able to identify the error events that

have caused the given syndrome. In contrast, the

targeted decoder that is designed to identify the

error events that introduce defects separated by

distance ξ, and will assign very low weights to

the edges of the set that correspond to the cor-

rect correction operator, and very high weights to

the other edges.

One can count the number of non-overlapping

error conﬁgurations using Eqn. (11). There are

exactly L(L − 2ξ + 1)/2 error conﬁgurations of

two error events where ξ = L/4 such that there

are are exactly L/2 bit ﬂips along a given row.

Each of these error conﬁgurations will cause the

standard decoder to fail with probability 1/2 in-

dependent of the size of the system. However,

of these non-overlapping conﬁgurations, there

are only 6ξ = 3L/2 error conﬁgurations where

the targeted decoder will fail with probability

1/2, other non-overlapping error conﬁgurations

of weight L/2 will be decoded successfully by the

targeted decoder.

Having shown that there are O(L

) low-weight

error conﬁgurations that will cause the standard

decoder to fail for the ballistic noise model, com-

pared with only O(L) low weight conﬁgurations

that will cause the targeted decoder to fail, we

can expect to be able to identify the ballistic

noise model by comparing logical error rates of

the noise source with respect to the two diﬀerent

decoding algorithms of the family by taking sys-

tem of suitably large size. In general, we show

in Appendix A that R, the ratio of the failure

Accepted in Quantum 2019-03-26, click title to verify 6

probability of the targeted decoder and the failure

probability of the standard decoder in the path

counting regime calculated with respect to the

ballistic noise model, vanishes exponentially with

L for constant ξ ≥ 3. Speciﬁcally, we ﬁnd

R ≤

ξ + 1





L/2ξ

. (8)

To this end we can expect to easily identify the

ballistic noise model in the path counting regime

by comparing the logical error rates of the two

decoders of the family.

3.2 Diﬀusive noise

We now consider a diﬀusive error model where

each error event introduces a continuous random

walk of bit-ﬂip errors that extends across a num-

ber of nearby qubits such that two defects appear

at its end points. We use this error model to nu-

merically demonstrate our diagnostic method.

We index error events e

with the faces of the

lattice f. Event e

will occur with probability

= p where p is uniform for all faces. Given

event e

occurs, a random walk is generated for a

ﬁxed number of steps, ξ  L, between adjacent

faces of the lattice. At a given step the walk di-

rection is chosen uniformly from the four adjacent

faces. Each edge, indexed j, that is crossed dur-

ing the walk an integer w times will suﬀer error

. An error event of this kind generates a sin-

gle error chain, that is, where only two syndrome

defects are created at the initial and ﬁnal face of

the walk.

Finally, we point out that while we observe the

behaviour in the case of odd and even values of

ξ to be qualitatively similar, we do not compare

odd and even values of ξ directly, as we observe

there to be a slight discrepancy in their perfor-

mance. We clearly see this eﬀect in Fig. 5. We

speculate that this is due the fact that, unlike

the odd case, for even ξ there is a non-zero prob-

ability that error events are members of the sta-

bilizer group, and therefore act trivially on the

code space.

3.3 Evaluating ξ for the diﬀusive noise model

using the single-weight adaptive decoder

Here we demonstrate numerically that using

adaptive decoding the unknown parameter ξ of

the diﬀusive noise model can be identiﬁed. We



2 4 6 8 10 12

0.01

0.02

0.05

0.10

a) Threshold dependence on decoder parameter

b) Example thresholds for 𝜉=3



0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.5

0.6

0.7

0.8

0.9

1.0

error

successrate

𝝀=2

𝝀=3

0 2 4 6 8 10

20 000

40 000

60 000

80 000

100 000

anyon%separa*on,%d

!="

W(d)

#="



0 5 10 15 20 25

0.5

0.6

0.7

0.8

0.9

1.0



Decoding success rate

L = 100, xi = 2, p = 0.11, efferr = 0.0770453

Decoder parameter, "



0 5 10 15 20

0.00

0.02

0.04

0.06

0.08

0.10



0 10 20 30 40

Correlation length, 

Optimal decoder parameter, 

ii)

Decoder weighting function

Error event rate, p



2 4 6 8 10

0.01

0.02

0.05

0.10

Defect separation, 𝑑

𝑊(𝑑)

𝝀

Figure 5: Threshold error rates as a function of the de-

coder parameter λ under a diﬀusive noise model that is

characterised by parameter ξ. (a) Solid lines show the

results for ξ = 3, 5, 7 and 9. Dashed line indicates the

threshold error rate for the noise model where ξ = 1.

Each value of ξ shows a unique behavior. Each thresh-

old value is obtained by repeated simulation of the noisy

lattice and decoding process to identify a crossing point.

Two threshold examples for a correlation parameter of

ξ = 3 are shown in (b), corresponding to the points

labelled (i) and (ii) in (a). Each point in these plots

represents the aggregated result of at least 20,000 indi-

vidual simulations.

Accepted in Quantum 2019-03-26, click title to verify 7

use an adaptive decoder based on the MWPM

decoding algorithm that we refer to as the single-

weight adaptive decoder. The weighting func-

tion for this decoder assigns low weights to edges

of the matching graph with length λ, and to

all other edges assigns high weights that scale

with the separation of pairs of stabilizer defects.

Speciﬁcally, it assigns edge weights with the func-

tion

(d) =

(

d d = λ,

d∆ otherwise,

(9)

where d is the Manhattan distance between two

defects, with ∆  L. The weighting function for

the single-weight adaptive decoder is shown in

the inset of Fig. 5(a). We will see that the logical

failure rate of the single-weight decoder depends

on the choice of λ relative to ξ.

We numerically simulate this scenario for in-

teger values of λ between 1 and 10 and values of

ξ = 1, 3, 5, 7, 9. To obtain the threshold error rate

we plot the logical error rate P

fail

as a function

of p for at least three large system sizes. To min-

imise the eﬀects of simulating ﬁnite size systems

we study systems where L 

√

ξ given that we

expect random walks to typically achieve walks

of length ∼

√

ξ. The value of p for which P

fail

coincides for all L determines p

. We ﬁnd the

crossing point by ﬁtting the data to the function

fail

= A

+ A

x + A

where x = (p −p

1/µ

and A

, p

and µ are constants to be evalu-

ated. We show an example threshold plot for

ξ = 3 using the single-weight adaptive decoder

with λ = 2, 3 in Fig. 5(b).

We remark on the clear trend in Fig. 5(a) where

the threshold presented as a function of p tend to

decay quickly with ξ. This decay is partly due to

the fact that the weight of the error increases with

ξ even while p remains constant. A comparison

of the eﬀective single qubit error rates are shown

In fact, given good knowledge of the value of ξ

the threshold remains relatively high even in the

presence of the spatial correlations. We discuss

this further in Sec. 4.

Having calculated the threshold for each error

model using a range of decoders from the family,

we go on to estimate the value of ξ by plotting

as a function of λ for each noise model, as

shown in Fig. 5(a). We observe that each value

of ξ generates a distinct plot as λ is varied. We

can therefore regard this as a signature of a given

noise model, as such, we refer to this plot as a

a) Parameter estimation

b) Continuous 𝝀* estimation

●

■

◆

▲

0 1 2 3 4 5 6 7

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

●

●●●

■

●

Discrete λ

■

Continuous λ

3 5 7 9 11

0.08

0.02

0.04

0.06

Figure 6: Estimating ξ values using signature plots in

Fig. 5(a). (a) Estimates of λ

∗

are shown. The orange

points show the discrete values of λ for which the adap-

tive decoder has the greatest threshold error rate. A

continuous estimation is made by ﬁtting a parabola to

the data points of the signature plot with odd λ close to

the peak λ value (red data points). This ﬁtting is shown

in (b). We observe that λ

∗

grows with ξ with the trend

∗

∼ 1.3 · ξ

0.39

. We show the ﬁtting with the dotted

blue line in (a).

Accepted in Quantum 2019-03-26, click title to verify 8

signature plot. We now investigate how we can

use the signature plot to estimate the unknown

parameter of the noise model, ξ and determine

other features of the noise model.

Studying Fig. 5(a) we ﬁrst remark on the dis-

crepancy between the data obtained for even and

odd values of λ. We have considered only odd

length error chains, which means single error

events will never introduce defects separated by

an even distance. As such, a decoder with even

λ will predominantly be correcting for second or-

der error events, which we would expect to have a

poor performance. This eﬀect is indeed observed

in the data. When the value of λ is odd, even

when it does not match ξ the decoder is prefer-

entially correcting for ﬁrst-order error events. It

is interesting to see that this feature of the noise

model, namely that the noise model only intro-

duces error strings of odd length, can be readily

identiﬁed from the signature plot.

In Fig. 6 we plot estimates for the value of λ

for which the threshold error rate is maximal. We

denote this value λ

∗

. We determine λ

∗

using two

rudimentary methods. First of all we estimate λ

∗

as the discrete value of λ for which the threshold

is maximal. We show these discrete data points

in orange in the Figure. Notably, for the rela-

tively small values of ξ we study, we observe that

∗

increases with ξ in one discrete step. This ob-

servation indicates that λ

∗

is sensitive to changes

in the value of ξ.

Ideally, we would like to determine ξ precisely

using the signature plot. However, using the

method for estimating λ

∗

we have suggested so

far, it is diﬃcult to resolve ξ from λ

∗

as there are

error models with diﬀerent values of ξ that return

the same value λ

∗

. To deal with this we reﬁne the

estimate of λ

∗

by using more data points from the

signature plot. In the Fig. 6(b) we ﬁt a quadratic

curve to the odd λ data points close to λ

∗

, which

are shown in Fig. 5. The maximal values of the

quadratic ﬁttings are shown in Fig. 6(a) with red

data points. We observe that the maximal values

increase like λ

∗

∼ 1.3 · ξ

0.39

which is plotted in

blue against the acquired data points. We might

expect a relationship λ

∗

∼

√

ξ that represents the

average distance covered by an error event that

performs a uniform random walk. We speculate

that this discrepancy in the ﬁtting is due to mi-

croscopic details and small size eﬀects. Nonethe-

less, our numerical results show that the value of

∗

, which is found using the signature plot, esti-

mates the unknown parameter ξ precisely.

3.4 Estimating ξ using the Gaussian decoder

One might argue that the adaptive decoder we

used in the previous section uses too much knowl-

edge of the error model to be applicable in any

real system, since we use the fact that the random

walk is of a ﬁxed length. In this section we show

that a similar result is achieved with a more ‘gen-

eral purpose’ adaptive decoder, which could ap-

ply to many more possible noise models. We call

this the Gaussian decoder, as it assigns weights

to the edges of the matching graph according to

a Gaussian function based on the separation of

two defects

(d) = d



− 9999 e

−

(d−µ)

2σ



(10)

where d is the separation of the two defects mea-

sured via the Manhattan distance on the lattice,

µ = λ and σ = λ/2. A schematic of the weight-

ing function of the Gaussian adaptive decoder is

shown in the inset of Fig. 7. The decoder is de-

signed to assign bias to identify common error

events that cause defects that are separated by a

distance that is on average d ∼ λ which we ex-

pect from the diﬀusive noise model. Errors that

are not consistent with a diﬀusive random walk

are assigned probabilities that are proportional to

their separation as we would using the standard

decoder.

In the previous section we produced signature

plots by computing the threshold error rate for

each decoder parameter λ, and then using the

result to estimate the correlation length ξ. In

practice it will be simpler to use logical failure

rates directly, rather than thresholds, as they can

be computed for a single code of ﬁxed size, and

a single value of physical error rate. As such,

for this case, we use logical error rates instead of

threshold error rates to ﬁnd a signature plot.

At the top of Fig. 7(a) we generate a signature

plot using the logical failure rate of the adaptive

decoder as a function of λ for diﬀerent odd values

of ξ ranging from ξ = 3 to ξ = 51, for a lattice

size of L = 100 such that

√

ξ  L. Only the

odd values of λ are shown. We see that, as in the

previous Subsection, the peak logical success rate

varies with ξ. In Fig. 7(b) we aim to estimate ξ

using the signature plot. Two diﬀerent estimates

Accepted in Quantum 2019-03-26, click title to verify 9

◆

0 5 10 15 20 25 30

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Decoding success rate

L = 100, xi = 3, p = 0.07, efferr = 0.0697621

a) Logical error rate dependence on decoder parameter 𝝀

b) Parameter estimation

0 2 4 6 8 10

20 000

40 000

60 000

80 000

100 000

anyon%separa*on,%d

!="

W(d)

#="



0 5 10 15 20 25

0.5

0.6

0.7

0.8

0.9

1.0



Decoding success rate

L = 100, xi = 2, p = 0.11, efferr = 0.0770453

Decoder parameter, "



0 5 10 15 20

0.00

0.02

0.04

0.06

0.08

0.10



0 10 20 30 40

Correlation length, 

Optimal decoder parameter, 

𝝈=𝝀/2

𝜇=𝝀

Defect separation, 𝑑

𝑊(𝑑)

0 2 4 6 8 10

20 000

40 000

60 000

80 000

100 000

anyon%separa*on,%d

!="

W(d)

#="



0 5 10 15 20 25

0.5

0.6

0.7

0.8

0.9

1.0



Decoding success rate

L = 100, xi = 2, p = 0.11, efferr = 0.0770453

Decoder parameter, "



0 5 10 15 20

0.00

0.02

0.04

0.06

0.08

0.10



0 10 20 30 40

Correlation length, 

Optimal decoder parameter, 

𝜉

●

●●●

●● ●

●●

●

■

●

Discrete estimation of λ

■

Continuous estimation of λ

0 10 20 30 40 50

Correlation length, ξ

Optimal decoding parameter, λ

Figure 7: (a) Logical error rates as a function of the

decoder parameter λ are shown for values of ξ =3,11,31

and 41 under a diﬀusive noise model. The inset shows a

schematic of the decoder weighting function. For each

value of ξ we see a peak in the logical error rate at

approximately λ ∼

√

ξ. (b) Estimation of the optimal

decoder parameter λ

∗

. We identify the discrete value of

λ for which the logical error rate is maximal (orange data

points), and make a continuous estimation by ﬁtting a

quadratic function around the peak (red data points).

We observe that λ

∗

∼

√

ξ as we expect from a diﬀusive

noise model where each error event will typically intro-

duce a pair of stabilizer defects that are separated by

distance

√

ξ.

of λ

∗

are shown, a discrete estimation by choos-

ing the value of λ for which the logical success

rate is the greatest, and an estimate based on a

quadratic ﬁt around the peak values. We observe

a trend λ

∗

∼

√

ξ as we might expect from the

diﬀusive noise model.

We ﬁnally remark that we chose a ﬁxed rela-

tionship between µ and σ in our weighting func-

tion, but more generally the functional form can

be parametrized by more variables to explore a

wider space of functions. This may be an inter-

esting route for further work. Using our choice of

adaptive decoder the results we have presented

show that the logical failure rate of the decoder

is sensitive to our choice of λ, and that we can

use this information to improve the failure rates

of our decoding scheme.

4 Improved decoding schemes

In the previous subsection we demonstrated that

we can establish features of noise models by use of

an adaptive decoder, and we showed this explic-

itly for the case of the diﬀusive noise model. Be-

ing able to gain a knowledge of the noise acting on

underlying hardware is a valuable tool for improv-

ing the performance of quantum error-correcting

codes. Particularly in the regime where the sys-

tem is too large to allow for tomography. Infor-

mation about the error processes involved may

make it easier to identify the source of the noise,

and it may be possible to improve the decoding

algorithm to account for these known errors in

classical post-processing. In this Section we show

that we can use signature plot data to improve

decoding algorithms to increase both threshold

error rates and logical success rates.

We will focus on the data obtained in Sub-

sec. 3.3 to design better decoding algorithms.

Our aim is to use signature plots to design a de-

coder that performs better than the best decoder

we had already identiﬁed by sweeping the param-

eter λ. We propose a specialised decoder, that

identiﬁes the two highest peaks on the signature

plot, and uses a weighting function that assigns

very low weight two both of these values of λ,

and high values elsewhere, otherwise the func-

tion assigns a large weight that scales with the

separation of the defects.

We compare the specialised decoder with both

the standard decoder, and the ﬁxed-weight de-

Accepted in Quantum 2019-03-26, click title to verify 10



 



(eff err)

, standard decoder



(eff err)

, fixed-weight decoder



(eff err)

, specialised decoder

(event)

, standard decoder

1 3 5 7 9

0.02

0.04

0.06

0.08

0.10

0.12

0.14

correlation length, ξ

Figure 8: The threshold under diﬀusive noise for varying

values of ξ for the standard decoder, the ﬁxed weight de-

coder with λ = λ

∗

and a specialised decoder where we

assign a very low weight to more than one value of the

λ. The threshold is shown as a function of the correla-

tion parameter ξ, for the three approaches to decoding.

We see that for ξ > 3, with an appropriate choice of a

non-monotonic weighting function we obtain improved

threshold error rates, thus indicating that we can design

better decoders using the data available in the signature

plot. The thresholds in the eﬀective single qubit error

rate are shown in the solid lines. The dashed blue line

gives a comparison to the event error rate.

coder which has been calibrated with λ = λ

∗

where we use the discrete value of λ

∗

found in the

previous section, i.e. the orange points shown in

Fig. 6(a). Fig. 8 compares the thresholds of the

diﬀerent decoders. We ﬁnd that the specialised

decoder outperforms the other decoders for suf-

ﬁciently high ξ. The calibrated ﬁxed-weight de-

coder also outperforms the standard decoder for

larger values of ξ, but does not perform as well as

the specialised decoder. Importantly, our results

show that updating the weighting function using

information from the signature plot improves the

logical failure rates of the surface code.

We remark that the specialised decoder we

have suggested here is by no means optimal, we

have only demonstrated that even using a very

simple method for using information from the sig-

nature plot the decoder can be improved. We ex-

pect that more sophisticated techniques will al-

low further improvements in the threshold, but

we leave a thorough exploration of the decoder

performance that can be achieved using signature

plots to future work.

It is common to think about noise only as a rate

of Pauli errors on single qubits, and it is useful to

consider how the correlations in the noise aﬀect

this eﬀective single qubit error threshold. Fig. 8

shows the thresholds in the single qubit error rate,

which is deﬁned as the probability with which any

individual qubit has suﬀered a Pauli error at the

end of the round of noise application. The eﬀec-

tive single qubit error thresholds for the standard

decoder (solid blue line) can be compared with

the error event thresholds (dashed blue line). The

latter decreases rapidly, since a smaller number of

events introduces more single qubit Pauli errors.

However, we see that even with the standard de-

coder, the eﬀective error threshold decreases only

gradually as correlations increase. We also plot

the eﬀective error threshold using the ﬁxed weight

decoder, and specialised decoder, as described in

the previous sections. Here, we see a more grad-

ual decline in the threshold.

It is often believed that any correlations in the

noise model are highly destructive, and should be

avoided. While this may sometimes be the case,

for example with very long range correlations in

which a single error event spans the code, the

eﬀect on the threshold depends strongly on the

exact structure of the correlations. These results

show that for a diﬀusive noise model resulting in

spatial correlations much smaller than the code

distance, the threshold is not strongly aﬀected,

and using a targeted decoding strategy the loss

of logical error rate can be reduced.

5 Practical considerations

It is worth noting the practicality of the scheme

we have discussed for analysis of a real exper-

imental setup. Given that some hardware has

been developed to implement an error correcting

code, we can perform the following operations:

prepare the code in a known logical state, per-

form stabilizer measurements, and measure out

the logical qubit with the intention of implement-

ing the identity gate. We can repeat this process

many times to build up a large data set of mea-

surement outcomes. A decoder can analyse the

measurement outcomes to determine if a logical

error occurred. We can then perform the analysis

we have described in classical post-processing on

this data set, using a large suite of decoders tar-

geted at errors over a large space of possible noise

Accepted in Quantum 2019-03-26, click title to verify 11

models. The fact that this analysis is a classical

and oﬄine makes it a very low cost way of study-

ing the performance of a logical qubit. Further-

more, since no direct interaction is needed with

the physical device, there need be no requirement

that the decoder is fast, in the sense that it can

keep up with the operating speed of the physi-

cal qubits. Given that we would expect to apply

our method oﬄine, we remark that this may be

a practical use case for slow decoders with high

thresholds, or other useful properties, even if such

decoders are impractical for online decoding.

The results of such an analysis, giving some

characterization of the noise model, could then

be used in multiple ways to improve a device. For

example, adjustments at the hardware level may

be possible to minimize sources of noise, and as

we have shown, we can also improve the classi-

cal processing accompanying the error correcting

code to account for the known structure of the

noise.

6 Conclusions

To summarise, we have developed diagnostic

tools to analyse an unknown parameter of a cor-

related error model acting on the toric code. Our

method generalises the MWPM algorithm to de-

ﬁne a parametrised family of decoders. We go on

to compare the logical error rates of the family

of decoders to hypothesise the types of correlated

errors a code suﬀers. We have shown that a code

can tolerate a higher error rate if we calibrate

a decoder under the error events that commonly

occur, and furthermore that an increased thresh-

old can be found for certain spatially correlated

noise models. We also discussed how our scheme

could be implemented in a practical setting.

In this work we have studied only a simple

noise model, but we expect that other correlated

error models where string-like errors are drawn

from some distribution, such as thermal noise,

will display a similar behaviour. Beyond spa-

tial correlations, the methods we have proposed

can be more broadly applied to other experi-

mentally relevant models of correlated noise, as

long as an appropriate family of decoders could

be found to characterize the error. For exam-

ple, errors correlated between X and Z bases,

higher probabilities of error in certain physical

locations, and temporal error correlations, per-

haps due to malfunctions in measurement appa-

ratus [68, 69, 70, 71], are all sources of concern.

Many known decoders exist which could poten-

tially be adapted to handle these types of corre-

lations [55, 56, 72, 73, 74, 75, 76, 63]. As a ﬁrst

step, it will be interesting to extend the ideas we

have presented here to identify multiple unknown

parameters of an error model to enable us to use

decoding algorithms to identify features of more

realistic noise models.

Beyond the toric code, similar techniques to

those we have demonstrated could potentially be

used to probe string-like correlated errors in other

error-correction procedures that use adaptations

of the MWPM algorithm [77, 78, 79]. This read-

ily extends to decoders for the toric code in the

setting where measurements are not ideal [80].

Moreover, it will be interesting to study local cor-

related error events that introduce many stabi-

lizer defects. We expect the Metropolis-based de-

coders presented in Refs. [56, 72], neural network

decoders [81], tensor-network approximants of

maximum likelihood decoders [66, 76] or perhaps

minimum-weight matching decoders that are sup-

plemented by belief propagation [57, 58, 59] may

be suitable for this purpose, but we leave this for

future work.

As hardware rapidly progresses towards the

ﬁrst implementations of small error correcting

codes, it is important to consider practical meth-

ods for analysing and optimising the performance

of these systems. The methods we have proposed

here oﬀer a simple way of studying and correct-

ing for correlated errors with oﬄine classical post

processing. With further development, we hope

these methods will prove a useful tool for evalu-

ating the performance of the ﬁrst generation of

fault-tolerant quantum processors.

Acknowledgements

The authors thank S. Benjamin, H. Bombín,

D. Browne, J. Combes, M. Kastoryano, D.

Poulin and J. Wootton for helpful and encour-

aging discussions. We are also grateful to C.

Chubb comments on an earlier draft of the

manuscript. We acknoweldge the use of the

Imperial College Research Computing Service,

DOI: 10.14469/hpc/2232. NHN is supported by

the Engineering and Physical Sciences Research

Council. BJB is supported by the Villum Foun-

Accepted in Quantum 2019-03-26, click title to verify 12

dation, the University of Sydney Fellowship Pro-

gramme and the Australian Research Council via

the Centre of Excellence in Engineered Quantum

Systems(EQUS) project number CE170100009.

A Logical Error Rates for the ballistic

error model in the path-counting regime

Here we show that a targeted decoder will have

better logical error rates in the path-counting

regime for the ballistic noise model for ξ ≥ 3. For

simplicity we assume only lattice sizes L = kξ for

even integers k.

In the path-counting regime the errors that

will most frequently cause the decoder to fail are

where k/2 = L/2ξ errors align along the same

row or column of the lattice in a suitable conﬁgu-

ration. The standard decoder will fail with prob-

ability independent of the size of the system if all

of these k/2 error events occur in a conﬁguration

where no errors overlap. However, as discussed in

the main text, we can design a targeted decoder

that makes use of value ξ to improve logical fail-

ure rates.

There are N

st.

conﬁgurations of k/2 non-

overlapping error events that will cause the stan-

dard decoder to fail where

st.

2ξ

ξ + 1

(k(ξ + 1)/2)!

(k/2)!(kξ/2)!

. (11)

In the case of the targeted decoder however there

are only N

sp.

error conﬁgurations that will cause

logical failure with high probability. This is be-

cause the targeted decoder is designed such that

in the path-counting regime the decoder will fail

if and only if all stabilizer defects of a given error

conﬁguration are separated from each other by

distances that are all factors of ξ. There are only

sp.

ξk!

(k/2)!(k/2)!

, (12)

of such conﬁgurations. We can use the expres-

sions given in Eqn. (11) and Eqn. (12) to bound

the ratio

R =

sp.

st.

≤

ξ + 1





L/2ξ

, (13)

which is proportional to the ratio of the decoder

failure probabilities for the targeted decoder and

the standard decoder for the ballistic noise model.

Clearly, for small ξ ≥ 3 the logical failure proba-

bility for the targeted decoder is exponentially

smaller than the logical failure probability of

the standard decoder when dealing with ballistic

noise.

The remainder of this Appendix will prove

Eqn (11). We consider two families of functions,

the ﬁrst, N

(t), denotes the number of conﬁgu-

rations that t error events of ﬁxed length ξ can

be aligned on a ring of l sites such that none of

the t error events overlap. The second, M

(t), is

the number of conﬁgurations that t error events

of size ξ can be arranged along a line of l sites

with open boundary conditions such that none of

the error events overlap.

We have that

st.

= N

l=L

(k/2), N

(0) = 1. (14)

We show Eqn. (11) recursively by ﬁnding a re-

lationship between N

(t) and N

(t −1). To do so,

we derive some facts relating the function N

(t)

to M

(t). Speciﬁcally, we have that

(t) =

l−ξ

(t − 1). (15)

We ﬁnd N

(t) of Eqn. (15) by ﬁrst considering

a single error event of size ξ that can be placed

on any of the l possible positions along a one-

dimensional lattice with periodic boundary con-

ditions. The remaining t − 1 non-overlapping er-

ror events can be placed on the vacant l − ξ lat-

tice sites that are not occupied by the ﬁrst er-

ror event. We thus count lM

l−ξ

(t − 1) diﬀer-

ent non-overlapping conﬁgurations. With this

expression, each error conﬁguration counted by

(t) is counted t times. The right-hand side of

the expression thus includes an additional factor

of 1/t to account for the degeneracy. We also

require the relationship

(t) = N

(t) − (ξ − 1)M

l−ξ

(t − 1). (16)

This expression is found by realising that the

number of conﬁgurations included in N

(t) that

are not included in M

(t) are those where a sin-

gle error event crosses one particular ﬁxed point

of the ring of l sites. There are ξ − 1 po-

sitions that one error event may lie to cross

this particular ﬁxed point, and the remaining

t − 1 non-overlapping error events may lie in

any of M

l−ξ

(t −1) conﬁgurations. We substitute

Accepted in Quantum 2019-03-26, click title to verify 13

Eqn. (16) into Eqn. (15) to ﬁnd

(t) =



1 −

t(ξ − 1)



(t). (17)

We ﬁnally combine Eqn. (15) and Eqn. (17) to

obtain

(t) =



1 −

(t − 1)(ξ − 1)

l − ξ



l−ξ

(t−1). (18)

We recursively apply Eqn. (18) to Eqn. (14) to

obtain the result of Eqn. (11).

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