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Simulation and performance analysis of quantum error correction with a rotated

surface code under a realistic noise model

Mitsuki Katsuda,1, ∗Kosuke Mitarai,1, 2, 3, †and Keisuke Fujii1, 2, 4, 5, ‡

1Graduate School of Engineering Science, Osaka University,

1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan

2Center for Quantum Information and Quantum Biology,

Osaka University, 1-2 Machikaneyama, Toyonaka 560-0043, Japan

3JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan

4RIKEN Center for Quantum Computing (RQC), Hirosawa 2-1, Wako, Saitama 351-0198, Japan

5Fujitsu Quantum Computing Joint Research Division at QIQB,

Osaka University, 1-2 Machikaneyama, Toyonaka 560-0043, Japan

(Dated: April 26, 2022)

The demonstration of quantum error correction (QEC) is one of the most important milestones

in the realization of fully-ﬂedged quantum computers. Toward this, QEC experiments using the

surface codes have recently been actively conducted. However, it has not yet been realized to protect

logical quantum information beyond the physical coherence time. In this work, we performed a full

simulation of QEC for the rotated surface codes with a code distance 5, which employs 49 qubits

and is within reach of the current state-of-the-art quantum computers. In particular, we evaluate

the logical error probability in a realistic noise model that incorporates not only stochastic Pauli

errors but also coherent errors due to a systematic control error or unintended interactions. While

a straightforward simulation of 49 qubits is not tractable within a reasonable computational time,

we reduced the number of qubits required to 26 qubits by delaying the syndrome measurement in

simulation. This and a fast quantum computer simulator, Qulacs, implemented on GPU allows

us to simulate full QEC with an arbitrary local noise within reasonable simulation time. Based

on the numerical results, we also construct and verify an eﬀective model to incorporate the eﬀect

of the coherent error into a stochastic noise model. This allows us to understand what the eﬀect

coherent error has on the logical error probability on a large scale without full simulation based on

the detailed full simulation of a small scale. The present simulation framework and eﬀective model,

which can handle arbitrary local noise, will play a vital role in clarifying the physical parameters

that future experimental QEC should target.

I. INTRODUCTION

Quantum error correction (QEC) is essential for the re-

alization of quantum computers because physical qubits

suﬀer from errors due to decoherence caused by undesir-

able interactions with the environment. QEC protects

them from such errors by encoding logical information of

qubits on many physical qubits [1]. With the progress

of quantum hardware, it is becoming possible to pre-

cisely control tens to a hundred qubits, leading to ex-

perimental demonstrations of simple QEC codes on a va-

riety of physical systems [2–7]. Surface codes proposed

by Kitaev [8] has attracted much attention as a method

to implement an error correction with superconducting

qubits due to their relatively high threshold for local er-

rors and their implementability using a two-dimensional

lattice of qubits. By using the rotated surface code [9],

we can make one logical qubit with code distance dus-

ing only 2d2−1 qubits. This means that we can im-

plement d= 3 and d= 5 surface codes with 17 and

49 physical qubits, respectively. Currently available de-

vices such as the ones presented in [10,11] can handle

∗m.katsuda0729@gmail.com

†mitarai@qc.ee.es.osaka-u.ac.jp

‡fujii@qc.ee.es.osaka-u.ac.jp

qubits of this scale. Experimental eﬀorts are underway

to demonstrate QEC with the surface codes using the su-

perconducting qubits. Ref. [7] has achieved exponential

suppression of the logical error probability with the one-

dimensional repetitive code, which is a one-dimensional

substructure of the surface code. Moreover, Refs. [12–

14] have implemented all of the operations necessary for

the implementation of QEC in a two-dimensional surface

code with d= 3, while they do not achieve the break-even

point, that is, the logical error probability is smaller than

the physical error probability.

To demonstrate that QEC can actually achieve a logi-

cal error probability lower than the physical one, it is es-

sential to implement a surface code with a code distance

of d= 5 that can correct up to two errors. This is because

two-qubit operations inevitably introduce two-qubit cor-

related errors on physical qubits. However, successful

experimental realizations of surface codes [7,12–15] are

still up to d= 3 [12–14] possibly due to the limited ﬁ-

delity of gates and readouts in present quantum devices.

To clarify what ultimately limits the experimental im-

plementation of the error correction, detailed numerical

analyses are required; we need to determine how much

performance is necessary for which parameters, and how

much the logical error probability can be reduced if they

are achieved.

Numerical experiments of surface codes have been car-

arXiv:2204.11404v1 [quant-ph] 25 Apr 2022

2

ried out under various noise models. The simplest, albeit

not truly realistic, model is the stochastic Pauli errors

where Pauli operators act on the qubits probabilistically.

In this case, the Gottesman-Knill theorem [16] allows us

to simulate even large systems eﬃciently. In real exper-

imental systems, unfortunately, there is a noise that has

coherence and cannot be described by stochastic Pauli er-

rors originating from, for example, over-rotation with a

systematic control error, global external ﬁelds, cross-talk

and so on.

Simulating QEC with these noises is as diﬃcult as sim-

ulating a universal quantum computer in general. How-

ever, in certain limited cases, it is possible to numeri-

cally evaluate their performance. For example, a mixture

of coherent and incoherent noise on the one-dimensional

repetition code has been analysed in detail by making

use of its exact solvability mapping it to free-fermionic

dynamics [17]. The d= 3 rotated surface code with am-

plitude damping and dephasing [18] has been analysed

by exact simulation of the system, which is possible due

to its small number of qubits. A sophisticated technique

for contraction of 2D tensor networks has been used to

simulate surface codes with arbitrary local noise on the

data qubits, while the syndrome measurements are as-

sumed to be ideal [19]. It is not clear if this method can

be extended to simulate circuit-level noise, where each

elementary operation is subject to noise. More recently,

quasiprobability decomposition of non-Cliﬀord channels

into Cliﬀord channels is utilized to simulate surface codes

with a small coherent noise [20]. This method is, how-

ever, not applicable to arbitrary noise because the coher-

ence of the noise increases its sampling overhead expo-

nentially. Given the ﬁnite precision of control systems

in actual experiments, the existence of coherent errors

is inevitable. A detailed simulation of the surface code

with d= 5, which is the near-term milestone of QEC, is

still challenging for classical computers. A framework to

realize such simulations and to obtain knowledge on the

impact of a coherent error on QEC is highly demanded.

In this work, we fully simulate the QEC under a realis-

tic noise model, including incoherent and coherent noise,

in the d= 5 rotated surface code with 49 physical qubits,

and analyzed the eﬀect of coherent errors on the logical

error probability. The main obstacle to its analysis is

that a straightforward simulation of 49 qubits would re-

quire a complex vector of dimension 249. This prevents us

from simulating the dynamics with a realistic computa-

tional resource. We overcome this obstacle by exploiting

the structure of the syndrome measurement and reusing

the measured qubits in the simulation. This allows us

to achieve a full simulation of the d= 5 rotated surface

code by simulating only 26 qubits, thus making it feasi-

ble to analyze the eﬀects of arbitrary local noise models

on this QEC code. In particular, assuming its implemen-

tation on superconducting qubits, we use a realistic gate

set and noise model, such as coherent errors in one-qubit

operations and cross resonance gates in addition to naive

stochastic Pauli errors. Moreover, we develop an eﬀective

model of physical error probability for incorporating the

eﬀects of coherent errors in rotated surface codes com-

bining the simulation results and the previous analysis

of coherent errors in 1D repetitive codes [17]. Using this

model, we investigate the possible regime of coherence

time, gate time, coherent error ratio, etc., required for

maintaining the quantum information of a logical qubit

beyond the coherence time of a physical qubit. The re-

sults show that if the ratio of gate operation time to

coherence time is below 0.005, the lifetime of the logical

qubit exceeds that of the physical qubit, even if coherent

errors occur with the same magnitude as their incoherent

counterparts. On the other hand, it was also found that

if the magnitude of the coherent error can be reduced to

20% of the incoherent one, the ratio of gate operation

time to coherence time is acceptable up to 0.007. The

present simulation framework for QEC would provide an

important guideline for future experimental demonstra-

tions of QEC to extend the lifetime of logical qubits.

II. SIMULATION METHODS FOR d= 5

ROTATED SURFACE CODE

A. Circuits for syndrome measurements

The d= 5 rotated surface code, which is the target

in this work, is shown in FIG.1a. White circles and

black circles in Fig. 1a represent data qubits and ancilla

qubits for the sydrome measurements, respectively. The

list of stabilizers of this code is summarized in Table I.

In this table, Xi, Zimean Pauli operators acting on the

i-th data qubit. These stabilizers are also illustrated in

Fig. 1a.X-type stabilizers correspond to yellow plaque-

ttes, and Z-type stabilizers do to ocher ones. These sta-

bilizers are measured with the circuits shown in Figs. 1b

and 1c, respectively.

The order of CNOT gates is important to determine

the minimum depth of the syndrome measurement cir-

cuit. In this work, we choose the following order. In

the case of X-type stabilizers, they are applied clockwise

from the bottom right, while in the case of Z-type it is

TABLE I. List of stabilizers of the d= 5 rotated surface code

Index X-stabilizer Z-stabilizer

1X1X2Z16Z21

2X3X4Z6Z11

3X2X3X7X8Z11Z12 Z16Z17

4X4X5X9X10 Z1Z2Z6Z7

5X6X7X11X12 Z17 Z18Z22 Z23

6X8X9X13X14 Z7Z8Z12 Z13

7X12X13 X17X18 Z13 Z14 Z18Z19

8X14X15 X19X20 Z3Z4Z8Z9

9X16X17 X21X22 Z19 Z20 Z24Z25

10 X18X19 X23X24 Z9Z10 Z14 Z15

11 X22X23 Z15 Z20

12 X24X25 Z5Z10

3

123 4 5

6 7 8 9 10

11 12 13 14 15

16 17 18 19 20

21 22 23 24 25

12

16 320 4 24

14 518 6 22

15 719 8 23

13 917 10 21

11 12

(a)

|ψXi |0iH H

|ψai

|ψbi

|ψci

|ψdi

(b)

|ψZi |0i

|ψai

|ψbi

|ψci

|ψdi

(c)

FIG. 1. (a)The d= 5 rotated surface codes. Yellow faces( )

corresponds to X-type stabilizer and ocher ones( ) cor-

responds to Z-type stabilizer.White circles( ) means data

qubits with their index, and black circles( ) means ancilla

qubits used for measurement and the order in which the mea-

surement circuits are run. (b)Circuit for the X-type stabi-

lizer measurement (c)Circuit for the Z-type stabilizer mea-

surement.

clockwise from the top right. For example, if Z1Z2Z6Z7

is to be measured, we apply CNOT gates with the or-

der of 2,7,6,1. We have to make a special treatment for

the measurement qubits on the boundary as they have

fewer CNOT gates than the other measurement qubits.

In this work, we make them “wait” if their target data

qubit doesn’t exist. While “waiting“, one-qubit noise is

applied. For example, the sequence for measuring X1X2

is “2, 1, wait, wait”, and for measuring X22X23 is “wait,

wait, 22, 23”.

|0i

|0i

|0i

|0i

|0i

|0i

|0i

|0i

FIG. 2. A method to reduce the number of qubits in sim-

ulation in the case of the one-dimensional repetition code.

(left) The original syndrome measurement circuit. (right) A

time-shifted version of the original one.

B. Reducing the number of qubits for simulation

If we straightforwardly simulate 49 qubits, we need to

reserve a 249-dimensional complex vector on a classical

memory. It prohibits us to simulate the error correction

procedure with a practical computational resource. To

overcome this obstacle, we reduce the number of qubits

that have to be simulated by reusing the measured qubits.

As an illustrative example of this, we show the case for

the one-dimensional repetition code in FIG.2. While in

the actual experiment, the syndrome measurements are

done in parallel for each measurement qubit, we delay

them so that one syndrome measurement runs at a time.

This strategy allows us to reuse one measurement qubit

for multiple stabiliser measurements without changing

the system to be simulated. Note that the same eﬀect can

be obtained by analytically calculating a set of POVM

operators corresponding to each syndrome measurement

and applying them with appropriate probabilities. How-

ever, the circuit-based noise model with coherent error

considered here is somewhat complicated and hence we

avoid this approach.

When applying the proposed method to surface codes,

we have to take care of the order of the two-qubit gates

in the three-dimensional arrangement so that each syn-

drome measurement circuit can be delayed without any

collision. A three-dimensional unit block corresponding

to the syndrome measurements for X- and Z-type sta-

bilizers is shown in FIG. 3a. The unit block consists of

two types of rectangular blocks stacked on a time axis.

The rectangle block of 1 ×1×1 represents a one-qubit

gate applied to a measurement qubit and the rectangle

block of 2 ×1×1 represents a two-qubit gate between a

measurement qubits and an adjacent data qubit. Using

these 3D blocks, we construct a syndrome measurement

circuit for the whole surface code as in FIG. 3b. As you

can see, the blocks are assembled in such a way that they

can be delayed without any collision. This indicates that

the measurement qubits can be reused as same as the

one-dimensional case by delaying the syndrome measure-

ments appropriately. Speciﬁcally, the numbers written in

black circles in FIG. 1a correspond to the order in which

4

the measurements are to be executed.

H

H

|0 |0

(a)

(b)

FIG. 3. (a) Left and right blocks represent the syndrome mea-

surement circuits for X- and Z-stabilizers, respectively. (b)

Each syndrome measurement circuit is complied for parallel

X- and Z-type syndrome measurements on the surface codes.

Note that each syndrome measurement circuit can be delayed

without any collision following the order shown in each block

so that the syndrome measurements are done sequentially to

reduce the number of qubits in simulation.

III. NUMERICAL EXPERIMENT

A. Simulated system

The simulation method in Sec. II allows us to eﬀec-

tively perform the full-vector simulation of d= 5 rotated

surface code, and therefore to evaluate its performance

under any local noise model. We consider a situation

where two types of noise, incoherent and coherent, act

on physical qubits in a syndrome measurement circuit.

Here, we describe the concrete gateset and noise model

that are used in the simulation.

To conduct a realistic simulation, we consider a

hardware-native gateset common to the superconducting

devices and compile the syndrome measurement circuit

with those gates. More concretely, we use a gate-

set {Rx(π/2), R†

x(π/2), Rzx (π/2), R†

zx(π/2), Rz(π/2)},

where

Rx(π/2) = e−iπX/4,(1)

Rz(π/2) = e−iπZ/4,(2)

Rzx(π/2) = e−iπX ⊗Z/4,(3)

to perform the syndrome measurements. This gateset

is commonly used in the superconducting qubits with

cross resonance gate [21,22]. With these gate sets, the

syndrome measurement circuit can be rewritten as in

FIG. 4, where three changes are made from the circuit

shown in FIG.1b and FIG.1c. Firstly, we replace CNOT

gates in FIG.1c with Rzx(π/2) gates. Second, for the

two Hadamard gates in the circuit of FIG. 1b, we re-

placed the former one by the R†

x(π/2) gate and the lat-

ter one by the Rx(π/2) gate. Finally, we apply Rx(π/2)

gates to the data qubits after the circuit shown in FIG.1b

and Rz(π/2) gates to the data qubits after the circuit in

FIG.1c. These one-qubit rotation gates restore the errors

transformed to Yerror by Rzx(π/2) gates.

Next, let us deﬁne the noise model used in the simu-

lation and see how incoherent noise is incorporated. We

assume that the single-qubit depolarizing noise,

E1(ρ) = (1 −p)ρ+p

3X

A∈{X, Y, Z }

AρA, (4)

acts on every qubit at each step of the measurement cir-

cuit in FIG. 4. Same noise occurs in the “waiting” qubit

where the gate does not act. For the two qubits after the

cross-resonance gates, we apply the two-qubit depolariz-

ing noise:

E2(ρ) = (1 −p)ρ+p

15 X

A,B∈{I ,X,Y,Z }

(A⊗B)ρ(A⊗B).

(5)

We call the parameter pthe physical error probability.

In this paper, the probability of each Pauli error is set

to be equal, but it is not diﬃcult to reﬂect the actual

distribution of stochastic errors.

5

a

b

c

X

d

Z

e

f

FIG. 4. A syndrome measurement circuit for the rotated surface codes consisting of cross resonance gates and Virtual-Z gates.

The grey rectangle ( ) represents a cross resonance gate acting with another qubit. The rotation angle of the Rx, Rz, Rz x

gates is π/2 and that of the R†

xgate is −π/2.

Next, we describe coherent noise. The coherent noise

was modelled as an (unintentional) increase in the rota-

tion angle of each rotation gate. This increase is referred

to as an over-rotation error. Since this is associated with

the rotation gate, it does not act on the “waiting” data

qubit. In this study, we want to model the error probabil-

ity with the single parameter p. To this end, we introduce

a coherent error ratio cand set the over-rotation angle

to be θ= 2c√p. In other words, for a rotation gate gen-

erated by a Pauli operator A, we add the coherent noise

channel in the form of:

Ec(ρ) = e−ic√pAρeic√pA .(6)

The reason why we set θ= 2c√pis as follows. Consider

the expectation value of P1=|1ih1|when Rx(θ) = e−iθ

2X

is applied to |0i. It is calculated as,

h0|R†

x(θ)P1Rx(θ)|0i= sin2θ

2(7)

Since θ1, the bit-ﬂip probability pﬂip associated with

this over-rotation is,

pﬂip =c2p. (8)

Or equivalently, if the coherence of noise is destroyed

at each step, for example, by using the twirling opera-

tion, then such a decohered noise map corresponds to

a probabilistic Pauli error with probability c2p. If the

coherent errors experience constructive or destructive in-

terferences, then the eﬀect of the coherent error would

be increased or decreased against c2p. The parameter c

controls the magnitude of the coherent error compared

to the incoherent one.

B. Numerical simulation

We employ one of the fastest classical simulators of

quantum circuits, Qulacs [23]. In the simulation, all data

qubits are ﬁrst initialised to |0iand projected to the sur-

face code state by performing noise-free syndrome mea-

surements with zero outcomes. Next, we run syndrome

measurement circuits with circuit-level noise as explained

above for ﬁve rounds. After completing ﬁve rounds of

measurements, all data qubits were subjected to projec-

tive measurements in the Pauli Zbasis. The above pro-

cedure was repeated 10000 times. In decoding, as usual,

we took the XOR of the syndrome of the adjacent rounds

to determine the position where the syndrome is ﬂipped.

With this information, we estimated the error positions

using the minimum weight perfect matching (MWPM)

algorithm implemented in NetworkX [24]. After apply-

ing the recovery operation using the estimated errors,

we ﬁnally calculate the eigenvalue of the logical Pauli Z

operator ZL=Z1Z7Z13Z19Z25 from the ﬁnal projective

measurement. The logical error probability pLis esti-

mated by dividing the number of ZL= 1 occurrences by

10000. We varied pfrom 10−3to 7.0×10−2and cfrom

0.0 to 1.0 by 0.25. For each parameter pair (p, c), the cal-

culation took 18 hours in the absence of coherent errors

and up to 80 hours in their presence using an NVIDIA

A100 GPU.

The obtained pLis shown in FIG. 5. For psmaller than

about 3.0×10−3, we can see that pLdecreases if pis de-

creased. Furthermore, pLsatisﬁes pL=Apξat any value

of cwithin the statistical error. By ﬁtting the numerical

values of pLat c= 0 for p= 1.0×10−3to 3.0×10−3

with pL=Apξusing two parameters (A, ξ), we obtain

A= 6.5×105and ξ= 2.92. The value of ξis consistent

with the expectation that it should be d+1

2= 3. One

might think that the contribution of the coherent error

is small because such a small angle rotation can be frozen

by repetitive projections for the syndrome measurements.

However, even for a small ratio c= 0.25, it has a negli-

gible contribution to the logical error probability. This

result clearly shows that the eﬀect of the coherence error

is important to estimate the experimentally achievable

logical error probability accurately.

6

FIG. 5. The logical error probability is plotted as a function

of the physical error probability pwith coherent-noise param-

eters c= 0,0.25,0.5,0.75,1.0. The dotted lines correspond to

the results of the ﬁtting.

FIG. 6. The ratio peﬀ /p is plotted as a function of the coher-

ence parameter c. The dotted curve represents 1 + αc2with

the ﬁtted value α= 0.872.

C. Eﬀective model incorporating the coherent

error

In order to understand the eﬀect of coherent noise on

pL, we consider how to eﬀectively incorporate the eﬀect

of coherent error as a leading order correction to the in-

coherent error p. Since we expect that the leading-order

contribution of the coherent error to the eﬀective error

peﬀ is proportional to c2p, we model peﬀ as,

peﬀ =1 + αc2p. (9)

This is because while the probability amplitude for the

coherent error is ∼c√p, such an error becomes a de-

tectable event if and only if such an error occurs twice

on either ket or bra spaces in the density operator pic-

ture [17]. The coeﬃcient αtakes into account the fact

that the coherent errors can interfere with each other and

therefore their contribution to peﬀ can be not exactly c2p.

As mentioned before, if the coherence is destroyed at each

step, then αshould be a unit.

Then, we assume that the logical error probability un-

der the coherent error is given by replacing the physical

probability pin the case of c= 0 with peﬀ . More pre-

cisely, the logical error probability pLshould be obtained

by replacing pin Eq. (9) with peﬀ :

pL=A·1 + αc2·pξ,(10)

where Aand ξare thought to be the same as those with

c= 0. The validity of this model is conﬁrmed by detailed

numerical calculations carried out on the 1D repetition

codes in the previous study [17]. To test this assumption

for the two-dimensional case, we estimate the value of α

by the following procedure. First, we ﬁt pLobtained at

various c’s in the range of p= 1.0×10−3to 3.0×10−3

with pL=A(Bp)ξusing Bas the ﬁtting parameter while

using the ﬁxed Aand ξobtained at c= 0. Then, we ﬁt

Bwith 1 + αc2. As a result we obtain α= 0.872 and the

ratio peﬀ /p shown in FIG. 6implies that this ﬁtting goes

well supporting our assumption. Another interesting fact

is the coeﬃcient α= 0.872 is smaller than a unit meaning

the coherent error interferes in a destructive way. This

implies that the coherent error modelled in this work is

not so damaging for QEC, since the logical error prob-

ability is increased when the coherence is destroyed at

each step by twirling.

Since this behaviour of the leading order contribution

of the coherent error is a local property of the noise, the

eﬀective model obtained here is expected to be valid not

only when pis reduced for d= 5, but also when the code

distance dis further increased. If this is true, we can

estimate the logical error probability under the coherent

error by combining a limited size of full simulation and a

large size of simulation with stochastic Pauli noise.

D. Experimental consideration

As a concrete usage of this model, here we argue in

what situation an experimental QEC can achive a longer

lifetime tLof logical quantum information against a phys-

ical coherence time tcby calculating tL/tc. First, we

rewrite the error probability pusing the coherence time

tcand time tgrequired for each gate operation. For clar-

ity, we assume that the error probability is well approxi-

mated by:

p'1−e−tg

tc.(11)

This is the situation where a quantum gate is well-

calibrated and the coherence-time-limited ﬁdelity is

achieved. Second, we rewrite the logical lifetime tLusing

the logical error probability pL. Letting Nsteps = 11 be

the number of steps in one cycle of the syndrome mea-

surement (see Fig. 4), it takes physical time dNstepstgto

conduct dcycles of syndrome measurements. Since the

logical error occurs with probabilty pLin this time, the

logical lifetime is roughly given by

tL=dNstepstg

pL

.(12)

7

FIG. 7. The lifetime tLof a logical qubit, calculated from

the coherent noise parameter c, the gate operation time tg

and the coherent time tcof the physical qubit, divided by

tc. The upper graph is a three-dimensional representation of

tL/tc, while the lower graph shows the relationship between

tg/tcand cin more detail. The red area in the upper graph

and the shaded area in the lower graph show the area where

tL< tc.

Finally, combining Eqs. (10)-(12), we obtain the follow-

ing relation:

tL

tc

=d·Ngates

A·hβ(1 + αc2)·1−e−tg

tciξ

tg

tc

.(13)

A graph plotting tL/tcas a function of cand tg/tcis

shown in the FIG. 7. The part of the graph marked in

red is the parameter region where tL/tcis below 1. In

this region, the error correction procedure itself damages

the lifetime and therefore is meaningless for protecting

the quantum information. To achieve tL/tc>1, the gate

and coherence time ratio tg/tcdepends on the amount of

coherent error and ranges from 0.005 to 0.007. If the ra-

tio is reduced to tg/tc= 0.001, the lifetime of the logical

qubit is improved by a factor of tens against the coher-

ence time even in the presence of the coherence error with

c= 1. In this case, the over-rotation angle is θ∼0.06,

which would be experimentally detectable and hopefully

can be calibrated [25].

Finally, let us discuss the experiments on the rotated

surface code with code distance 3 carried out by diﬀerent

research groups in 2021 [12,13]. The main parameters

of each experimental system are given in TABLE II. For

p1Q, p2Qin the table, Krinner’s group evaluated it by

Interleaved Randomized Benchmarking (Interleaved RB)

and Zhao’s group evaluated it by Cross Entropy Bench-

marking (XEB). Note that while these are great progress

toward experimental QEC, neither group has succeeded

in QEC in a strict sense. Krinner’s group has imple-

mented a QEC protocol, but was unable to make the

lifetime of the logical qubit T1,L longer than the lifetime

of the physical one T1. Zhao’s group has only imple-

mented the error detection and postselection. The main

reason for the lack of successful error correction would

be the short code distance. The distance 3 code can-

not correct two-qubit errors that occur during two-qubit

gates.

Let us see whether or not a successful QEC is reach-

able if the code distance is increased to 5 and con-

sider what elements should be improved if it is not the

case. Since p2Qis 1.5 %,1.035 %, which is outside the

region where pL=Apd+1

2holds in Figure 5, QEC is ex-

pected to fail even if the code distance is increased to 5.

The ratios of the two-qubit gate and coherence time are

Tg,2/T1= 0.003 and = 0.001 for Krinners’ and Zhaos’

groups, respectively. This is suﬃciently small from our

analysis. However, p2Qare, respectively, given by 1.5 %

and 1.035 %, which is much higher than those limited

by the coherence time. This implies that this inﬁdelity

caused by a systematic control error or cross-talk result-

ing in coherent errors. Therefore, by improving their

control strategies, there would be a great possibility to

achieve a breakeven point by experimental QEC in the

near future; at least the coherence time does not become

the main obstacle for it.

TABLE II. Parameters of experiments which are implemented

by two diﬀerent groups.

Parameter Krinner [12] Zhao [13]

1Q gate duration Tg,1(ns) 40 25

2Q gate duration Tg,2(ns) 98 32

Lifetime T1(µs) 32.5 26.1

Coherence Time T∗

2(µs) 37.5 3.6

1Q gate error p1Q(%) 0.09 0.098

2Q gate error p2Q(%) 1.5 1.035

Measurement duration TM(ns) 300 1500

Measurement error pM(%) 0.9 4.752

Logical Lifetime T1,L(µs) 16.4 64.4

Logical Coherence Time T∗

2,L(µs) 18.2 69.0

8

IV. CONCLUSION

In this study, we constructed a framework to fully sim-

ulate QEC on the distance 5 rotated surface code un-

der an arbitrary local noise. Furthermore, we have con-

structed an eﬀective model that explains the behaviour

of the logical error probability under the coherent errors

within the stochastic Pauli noise model with an appro-

priate modiﬁcation. Therefore, combining our numerical

result and the eﬀective model, we can analyse the be-

haviour of the logical error probability with smaller a

physical error probability or larger code distance. While

we only modelled the over-rotation caused by a system-

atic control error, there are plenty of sources of coher-

ent errors such as unintended interactions in Hamilto-

nian, cross-talk, global ﬁelds and so on. These sources of

noise would be straightforwardly incorporated into our

framework. The performance analysis under a realistic

noise model is becoming increasingly important, and our

framework will provide a vital guideline for future im-

provements on experimental sides.

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