Abstract
Surface codes are building blocks of quantum computing platforms based on 2D arrays of qubits responsible for detecting and correcting errors. The error suppression achieved by the surface code is usually estimated by simulating toy noise models describing random Pauli errors. However, Pauli noise models fail to capture coherent processes such as systematic unitary errors caused by imperfect control pulses. Here we report the first largescale simulation of quantum error correction protocols based on the surface code in the presence of coherent noise. We observe that the standard Pauli approximation provides an accurate estimate of the error threshold but underestimates the logical error rate in the subthreshold regime. We find that for large code size the logicallevel noise is well approximated by random Pauli errors even though the physicallevel noise is coherent. Our work demonstrates that coherent effects do not significantly change the error correcting threshold of surface codes. This gives more confidence in the viability of the faulttolerance architecture pursued by several experimental groups.
Introduction
Recent years have witnessed major progress towards the demonstration of quantum error correction and reliable logical qubits.^{1,2,3,4,5,6} Such qubits can be protected from noise by measuring syndromes of suitable parity check operators and applying syndromedependent recovery operations. Topological quantum codes such as the surface code^{7,8} are among the most attractive candidates for an experimental realization, as they can be implemented on a twodimensional grid of qubits with local check operators.
It is believed that such codes can tolerate a high level of noise^{9,10,11} which is comparable to what can be achieved in the latest experiments.^{6} The general confidence in the noiseresilience of topological codes primarily rests on considerations of Pauli noise—a simplified noise model where errors are Pauli operators X,Y,Z drawn at random from some distribution. An example is the case where each qubit j experiences noise described by the channel
with suitable probabilities \(\varepsilon _j^x,\varepsilon _j^y,\varepsilon _j^z\) and \(\varepsilon _j = 1  \varepsilon _j^x  \varepsilon _j^y  \varepsilon _j^z\). This kind of noise can be fully described by the stabilizer formalism.^{12} In pioneering work, Dennis et al.^{9} exploited this algebraic structure to establish the first analytical threshold estimates, see also.^{13} The effect of Pauli noise also is efficiently simulable thanks to the GottesmanKnill theorem, providing numerical evidence for high error thresholds of topological codes.^{14} The efficient simulability property has recently been extended beyond Pauli noise to random Cliffords and Paulitype projectors.^{15}
While such algebraically defined noise models are attractive from a theoretical viewpoint, they often do not correspond to noise encountered in realworld setups. They are—in a sense—not quantum enough: they model probabilistic processes where errors act randomly on subsets of qubits. Rather than being of such a probabilistic (or incoherent) nature, noise in a realistic device will often be coherent, i.e., unitary, and can involve small rotations acting everywhere. A typical situation where this arises is if e.g., frequencies of oscillator qubits are misaligned: this results in systematic unitary over or underrotations. On a singlequbit level, this means that (1) should be replaced by noise of the form
with a suitable unitary operator U_{j}∈SU(2). Since such errors generally cannot be described within the stabilizer formalism, understanding their effect on a given quantum faulttolerant scheme is a challenging problem.
Prior theoretical work indicates that the difference between coherent and incoherent errors could be significant. In particular, it was observed^{16,17,18,19,20} that coherent errors can lead to large differences between averagecase and worst case fidelity measures suggesting that a critical reassessment of commonly used benchmarking measures is necessary. This observation motivates the question of how much coherence is present in the effective logicallevel noise^{21,22} experienced by encoded qubits. Depending on whether or not the logical noise is coherent one may choose different metrics for quantifying performance of a given faulttolerant scheme. Significant progress has been made towards understanding the structure of the logical noise for concatenated codes.^{21,22,23} However, these studies are not directly applicable to large topological codes such as those considered here.
Bruteforce simulations of coherent noise in small codes were presented in^{24,25,26,27} for Steane codes and surface codes with up to 17 qubits. Simulating coherent errors by brute force clearly requires time (and memory) exponential in the number of qubits n. For the surface code, Darmawan and Poulin^{28,29} proposed an algorithm with a runtime exponential in n^{1/2} based on tensor networks, and simulated systems with up to 153 qubits. This algorithm can handle arbitrary noise (including, e.g., amplitude damping). Unfortunately, its formidable complexity prevents accurate estimation of error thresholds, e.g., for the systematic rotations considered here. In,^{30} threshold estimates for the 1D repetition code were obtained. To our knowledge, there are no analogous threshold estimates for topological codes subject to coherent noise.
Results
Coherent noise and quantum memory
We shall consider a particular version of the surface code proposed in Refs.^{31,32} A distanced surface code has one logical qubit and n = d^{2} physical qubits located at sites of a square lattice of size d × d with open boundary conditions, see Fig. 1.
We consider the situation where a logical state ψ_{L} initially encoded by the surface code undergoes a coherent error U = U_{1}⊗…⊗U_{n} which applies some (unknown) unitary operator U_{j} to each qubit j. To diagnose and correct the error without disturbing the encoded state we assume the standard protocol based on the syndrome measurement is applied. It works by measuring the eigenvalue (syndrome) s_{f} = ±1 of each stabilizer B_{f} on the corrupted state Uψ_{L}〉 and then applying a Paulitype correction operator C_{s} depending on the measured syndrome s = {s_{f}}_{f}. The correction C_{s} is computed by a classical decoding algorithm (for example, one may choose C_{s} as a minimumweight Pauli error consistent with s). We note that the syndrome s is a random variable with some probability distribution p(s) since the error U maps the initial logical state to a coherent superposition of states with different syndromes. We only consider noiseless syndrome measurements, and also assume that the correction operations C_{s} can be executed noiselessly. These assumptions (which can be relaxed by adapting our methods) are motivated by our focus on the effect of coherent errors. They also imply that we can assume that the correction C_{s} always returns the system to the logical subspace resulting in some final logical state ϕ_{s}〉. Thus the process of (noisy) storage with error U and subsequent error correction maps an initial encoded state ψ_{L}〉 to a certain final logical state ϕ_{s}〉 with probability p(s), while also providing the classical syndrome s.
The main question addressed here is how close the final state ϕ_{s}〉 and the initial state ψ_{L}〉 are as a function of the noise. Here we will assume that the error applied to qubit j is of the form U_{j} = exp(iη_{j}Z) for some (unknown) angle η_{j}. The restriction to Zrotations is dictated by the limitations of our simulation algorithm for storage. It provides a paradigmatic example of coherent noise. We also discuss a related problem—that of state preparation—and provide analogous results for general SU(2) coherent errors (see Supplementary Note 2).
Effective coherent logical noise
We show analytically that the syndrome probability distribution p(s) is independent of the initial logical state ψ_{L} whereas the final logical state has the form
for some logical rotation angle θ_{s}∈[0, π) depending on the syndrome s (see Supplementary Note 1). In other words, the error correction process converts physicallevel coherent noise U to effective coherent noise exp(iθ_{s}Z_{L}) on the logical qubit, with a strength θ_{s} depending on the (randomly distributed) syndrome s.
The parametrization (3) concisely captures the effect of coherent noise, providing us with a window into the nature of this conversion. We use the quantity
as a measure of the (average) logical error rate: this is the average (over syndromes) diamondnorm distance^{33} \(\left\ {\Lambda _s  {\mathrm{id}}} \right\_\diamondsuit = 2\left {{\mathrm{sin}}\theta _s} \right\) between the conditional logical channel
and the identity channel id. While the quantity P^{L} provides a meaningful measure for how well the initial state is preserved, the full information of the structure of residual logical errors is given by the distribution p(s) over logical rotation angles θ_{s}.
Polynomialtime classical simulation
We construct a polynomialtime classical algorithm which takes as input ψ_{L}〉 and the rotation angles η_{1},…,η_{n}, samples a syndrome s from the distribution p(s), and outputs s as well as the associated final state ϕ_{s}〉 (i.e., the logical rotation angle θ_{s}). The runtime of this algorithm scales as O(n^{2}), where we measure complexity in terms of the number of additions, multiplications, and divisions of complex numbers that are required. Strictly speaking, the simulation time scales as O(n^{2}) + t(n), where t(n) is the runtime of the decoding algorithm that computes the correction C_{s}. In our simulations the decoding time was negligible compared with the time required to sample the syndrome and compute the final logical state. By sampling sufficiently many syndromes, one can thus learn how frequently and in which ways error correction may fail in the presence of coherent noise. In particular, we may estimate the quantity P^{L}. By providing both the syndrome s and the the logical rotation angle θ_{s} conditioned on s, our algorithm also gives us a unique opportunity to investigate the full structure of the logicallevel noise.
Numerical results
Using our algorithm, we perform the first numerical study of large topological codes subject to coherent noise, performing simulations for surface codes with up to n = 2401 physical qubits, see Table 1 for a timing analysis.
This shows that efficient classical simulation of faulttolerance processes under coherent noise is possible, and allows us to extract key characteristics of these codes in the limit of large system size.
In more detail, in our simulations we consider translationinvariant coherent noise of the form (e^{iθZ})^{⊗n}, where θ∈[0, π) is the only noise parameter. The Pauli correction C_{s} was computed using the standard minimum weight matching decoder^{9,34} with constant weights independent of θ.
Our numerical results for the logical error rate P^{L} are presented in Fig. 2. Using the symmetries of the surface code one can easily check that P^{L} is invariant under flipping the sign of θ; accordingly, it suffices to simulate θ ≥ 0. The data suggests that the quantity P^{L} decays exponentially in the code distance d for θ<θ_{0}, where
can be viewed as an error correction threshold. We observe the exponential decay of P^{L} as a function of the code distance d in the subthreshold regime.
Surprisingly, the threshold estimate Eq. (6) agrees very well with the socalled Pauli twirl approximation^{35,36} where coherent noise of the form \({\cal N}(\rho ) = e^{i\theta Z}\rho e^{  i\theta Z}\) is replaced by its Paulitwirled version, i.e., dephasing noise of the form \({\cal N}_{{\mathrm{twirl}}}(\rho ) = (1  \varepsilon )\rho + \varepsilon Z\rho Z\) with \(\varepsilon = {\mathrm{sin}}^2\theta\). For the latter the threshold error rate is around ε_{0} ≈ 0.11, see Ref.^{9} Solving the equation \(\varepsilon _0 = \mathop {{sin}}\nolimits^2 (\theta _0)\) for θ_{0} yields θ_{0} ≈ 0.10π, in agreement with Eq. (6).
Applying the Pauli twirl at the physical level amounts to ignoring the coherent part of the noise. To assess the validity of this approximation, let us compare logical error rates \(P^L\) computed for coherent physical noise \({\cal N}\) and its Paulitwirled version \({\cal N}_{{\mathrm{twirl}}}\). Let \(P^L({\cal N}_{{\mathrm{twirl}}})\) be the logical error rate corresponding to \({\cal N}_{{\mathrm{twirl}}}\). The plot of \(P^L({\cal N}_{{\mathrm{twirl}}})\) and the ratio \(P^L/P^L({\cal N}_{{\mathrm{twirl}}})\) are shown on Fig. 3. It can be seen that applying the Pauli twirl approximation to the physical noise gives an accurate estimate of the error threshold but significantly underestimates the logical error probability in the subthreshold regime. We conclude that coherence of noise has a profound effect on the performance of large surface codes in the subthreshold regime which is particularly important for quantum faulttolerance. This phenomenon was previously observed in,^{23,28} but has not been studied for large topological codes prior to our work.
More information about the structure of the effective noise can be gained from Fig. 4, which shows the empirical probability distribution of the logical rotation angle θ_{s} obtained by sampling 10^{6} syndromes s for the physical Zrotation angle θ = 0.08π (which we expect to be slightly below the threshold). We compare the cases d = 9 and 25. In both cases the distribution has a sharp peak at θ_{s} = 0 (equivalent to θ_{s} = π). This peak indicates that error correction almost always succeeds in the considered regime. It can be seen that increasing the code distance has a dramatic effect on the distribution of θ_{s}. The distance9 code has a broad distribution of θ_{s} meaning that the logicallevel noise retains a strong coherence. On the other hand, the distance25 code has a sharply peaked distribution of θ_{s} with a peak at θ_{s} = π/2 which corresponds to the logical Pauli error Z_{L}. Such errors are likely to be caused by “ambiguous” syndromes s for which the minimum weight matching decoder makes a wrong choice of the Pauli correction C_{s}. We conclude that as the code distance increases, the logicallevel noise can be well approximated by random Pauli errors even though the physicallevel noise is coherent.
To get a deeper insight into this phenomenon, we introduce and numerically study associated measures of “incoherence”. To define a metric quantifying the degree of coherence present in the logicallevel noise, let us consider the twirled version of the logical channel Λ_{s},
and the corresponding logical error rate
Comparison of Eqs. (4,7) reveals that \(P^L \ge P_{{\mathrm{twirl}}}^L\) with equality iff the distribution of θ_{s} has all its weight on {0,π/2}, that is, when the logical noise is incoherent. It is therefore natural to measure coherence of the logical noise by the ratio \(P^L/P_{{\mathrm{twirl}}}^L\). This “coherence ratio” is plotted as a function of θ on Fig. 5(a). The data indicates that the coherence ratio decreases for increasing system size approaching one for large code distances. This further supports the conclusion that the logical noise has a negligible coherence. Finally, in Fig. 5(b), we show the analogous quantity for the average logical noise channel^{21} defined as \({\mathrm{\Lambda }} = \mathop {\sum}\nolimits_s p(s){\mathrm{\Lambda }}_s\). This average channel provides an appropriate model for the logicallevel noise if the environment has no access to the measured syndrome. This may be relevant, for instance, in the quantum communication settings where noise acts only during transmission of information. Thus one can alternatively define the coherence ratio as
where Λ^{twirl} is the Paulitwirled version of Λ. A straightforward computation gives \(P^L/P_{{\mathrm{twirl}}}^L = \frac{{\sqrt {\varepsilon ^2 + \delta ^2} }}{\varepsilon }\) where \(\varepsilon = \mathop {\sum}\nolimits_s p(s){\mathrm{sin}}^2(\theta _s)\) and \(\delta = \mathop {\sum}\nolimits_s p(s){\mathrm{sin}}(2\theta _s)/2\). The coherence ratio of the average logical channel is plotted as a function θ on Fig. 5(b). It provides particularly strong evidence that in the limit of large code distances, coherent physical noise gets converted into incoherent logical noise.
Discussion
Our work extends the range of noise models efficiently simulable on a classical computer. It allows—for the first time—to numerically investigate the effect of coherent errors in the regime of large code sizes which is important for reliable error threshold estimates. Our simulation algorithms make no assumptions about the particular decoder used. Hence the proposed approach should be universally applicable to benchmarking the performance of different faulttolerance strategies in the presence of coherent noise. Although we simulated only translationinvariant noise models, all our algorithms apply to more general qubitdependent noise. This enables numerical study of recently proposed state injection protocols,^{37} e.g., preparation of logical magic states, in the presence of coherent errors. Another possible application could be testing the socalled disorder assisted error correction method^{38,39,40} where artificial randomness introduced in the code parameters suppresses coherent propagation of errors due to the Anderson localization phenomenon. We leave as an open problem whether our algorithm can be extended to more general codes, such as as color and hyperbolic codes, as well as more general noise models such as those including systematic crosstalk errors.
Our numerical results spell good news for quantum engineers pursuing surface code realizations: thresholds for state preparation and storage are reasonably high, suggesting that coherent noise is not as detrimental as one could expect from the previous studies. The numerical investigation of the logicallevel noise gives rise to a conceptually appealing conjecture: error correction converts coherent physical noise to incoherent logical noise (for large code sizes). Whether this is an artifact of the considered error correction scheme or manifestation of a more general phenomenon is an interesting open question.
Methods
Our algorithm is based on a fermionic representation of the surface code proposed by Kitaev^{41} and Wen,^{31} see Fig. 6. It works by encoding each qubit of the surface code into four Majorana fermions in a way that simplifies the structure of the surface code stabilizers. The KitaevWen representation has previously been used by Terhal et al.^{42} to design fermionic Hamiltonians with topologically ordered ground states. Here we show that this representation is also wellsuited for the design of efficient simulation algorithms.
The fermionic version of the surface code is described by Majorana operators c_{1},…,c_{4n} that obey the standard commutation rules \(c_p^\dagger = c_p\), \(c_p^2 = I\), and c_{p}c_{q} = −c_{q}c_{p} for p ≠ q. Each vertex u of the surface code lattice contains a copy of a simple Majorana fermion code^{41} with four Majorana modes, a single logical qubit, and a stabilizer S_{u} = ±c_{p}c_{q}c_{r}c_{s}. It has logical Pauli operators \(\overline X _u = ic_pc_q\) and \(\overline Z _u = ic_qc_r\). We call this Majorana code the C4encoding of a qubit. Each Pauli operator P has a fermionic counterpart \(\overline P\) obtained by applying the C4encoding independently to each qubit of P.
The first step towards the construction of our algorithm is to translate notions from the surface code to the fermionic setting (see Supplementary Note 1). For example, we find that the fermionic counterpart \(\overline B _f\) of a surface code stabilizer B_{f} associated with a face f can be expressed as a product
of link operators (see Fig. 6) associated with edges bordering f. This implies that the syndrome measurement may effectively be replaced by the measurement of the commuting link operators by taking the product of the corresponding measurement outcomes. Similarly, we find that the logical operators X_{L} and Z_{L} (see Fig. 1) have fermionic counterparts of the form
for suitably chosen subsets LEFT and TOP of edges. Given the eigenvalues of the link operators, we may thus use these identities to compute final logical states from expectation values of the unpaired modes. Finally, as the problem of simulating storage requires considering arbitrary surface code states, we need to identify fermionic version \(\bar \psi _L\rangle\) of an encoded (logical) state \(\psi \in {\Bbb C}^2\). This takes the form
where \(\bar \psi\) is a C4encoded version of ψ on the unpaired modes, and ϕ_{link} is the unique state of 4n−4 Majoranas stabilized by all link operators. Eq. (11) implies that we may replace the initial logical state by a simpler state at the expense of measuring additional stabilizers.
Using these relationships, we show (see Supplementary Notes 2 and 3) that the error correction protocols considered in this paper can be decomposed into a sequence of O(n) elementary gates from a gate set known as a fermionic linear optics (FLO), see Refs.^{43,44} It includes the following operations:

1.
Initialize a pair of Majorana modes p,q in a basis state 0〉 satisfying ic_{p}c_{q}0〉 = 0〉.

2.
Apply the unitary operator U = exp(γc_{p}c_{q}). Here γ∈[0, π) is a rotation angle.

3.
Apply the projector Λ = (I + ic_{p}c_{q})/2. Compute the norm of the resulting state.
It is wellknown that quantum circuits composed of FLO gates can be efficiently simulated classically.^{43,44,45,46} The simulation runtime scales as O(n) for gates of type (1,2) and as O(n^{2}) for gates of type (3). By exploiting the geometrically local structure of the surface code we are able to reduce the number of modes such that at any given time step the simulator only needs to keep track of O(n^{1/2}) modes. Accordingly, each FLO gate can be simulated in time at most O(n). Since the total number of gates is O(n), the total simulation time scales as O(n^{2}).
Code availability
The code used in this study is available upon request to the corresponding author.
Data availability
Data used in this study is available upon request to the corresponding author.
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Acknowledgements
The authors thank Steven Flammia, Jay Gambetta, and David Gosset for helpful discussions. RK is supported by the Technical University of Munich—Institute for Advanced Study, funded by the German Excellence Initiative and the European Union Seventh Framework Programme under grant agreement no. 291763. He acknowledges support by the German Federal Ministry of Education through the funding program Photonics Research Germany, contract no. 13N14776 (QCDAQuantERA) and partial support by the National Science Foundation under Grant No. NSF PHY1125915, and thanks the Kavli Institute for Theoretical Physics for their hospitality. NP acknowledges support by TUMPREP. SB acknowledges support from the IBM Research Frontiers Institute and from Intelligence Advanced Research Projects Activity (IARPA) under contract W911NF160114.
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S.B., M.E., and R.K. undertook preliminary studies. S.B. and R.K. conceived the simulation algorithms. N.P., together with M.E. and R.K., implemented and tested the state preparation algorithm. S.B. implemented the storage simulation algorithm and collected and analyzed data. The manuscript was prepared by S.B. and R.K.
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Correspondence to Robert König.
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