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The XZZX Surface Code

J. Pablo Bonilla Ataides, David K. Tuckett, Stephen D. Bartlett, Steven T. Flammia, Benjamin J. Brown

arXiv:2009.07851v3quant-ph

TL;DR

Fault-tolerant quantum computing needs practical codes that handle realistic noise with modest resources. The paper introduces the XZZX surface code and tailored decoders, finding hashing-level thresholds across single-qubit Pauli channels, favourable biased-noise scaling, and persistent benefits in fault-tolerant settings.

  • Problem

    Practical fault-tolerant quantum computing requires error-correcting architectures that handle realistic physical noise with modest overhead.

  • Method

    The paper uses XZZX stabilizers and noise-tailored decoders to exploit structure in biased Pauli noise, including unreliable syndrome measurements.

  • Results

    XZZX thresholds closely match the hashing bound for all single-qubit Pauli channels, with numerical evidence of exceeding it in some regimes and surpassing known fault-tolerant biased-noise thresholds.

  • Takeaways & Limitations

    Tailoring the code and decoder to structured noise can improve thresholds and reduce the physical-qubit overhead of fault-tolerant quantum computation.

  • Takeaways & Limitations

    The minimum-weight perfect-matching decoder can scale as O(V^3), with V=O(pd^3) when syndrome measurements are unreliable.

Abstract

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Performing large calculations with a quantum computer will likely require a fault-tolerant architecture based on quantum error-correcting codes. The challenge is to design practical quantum error-correcting codes that perform well against realistic noise using modest resources. Here we show that a variant of the surface code -- the XZZX code -- offers remarkable performance for fault-tolerant quantum computation. The error threshold of this code matches what can be achieved with random codes (hashing) for every single-qubit Pauli noise channel; it is the first explicit code shown to have this universal property. We present numerical evidence that the threshold even exceeds this hashing bound for an experimentally relevant range of noise parameters. Focusing on the common situation where qubit dephasing is the dominant noise, we show that this code has a practical, high-performance decoder and surpasses all previously known thresholds in the realistic setting where syndrome measurements are unreliable. We go on to demonstrate the favourable sub-threshold resource scaling that can be obtained by specialising a code to exploit structure in the noise. We show that it is possible to maintain all of these advantages when we perform fault-tolerant quantum computation.

INTRODUCTION

The paper develops the XZZX surface code as a practical fault-tolerant architecture tailored to structured noise, targeting high thresholds, efficient decoding, and lower resource overhead.

  • INTRODUCTION: Fault-tolerant quantum computation requires error correction that operates below threshold while using a modest physical-qubit and gate overhead.
  • INTRODUCTION: The XZZX code changes each surface-code stabilizer to the product XZZX, exploiting structure in non-depolarising noise.
  • INTRODUCTION: XZZX code-capacity thresholds closely match the hashing bound for every single-qubit Pauli noise channel and may exceed it for strongly X- or Z-biased noise.
  • INTRODUCTION: Efficient generalised matching decoders retain high thresholds for dominant dephasing noise and surpass previously known thresholds when syndrome measurements are unreliable.
  • INTRODUCTION: O((p/√η)d/2) logical-failure scaling under bias η improves resource requirements by approximately η^-d/4, while near-term devices can achieve O(p^d2/2) scaling.
  • INTRODUCTION: The advantages persist for fault-tolerant computation, including low-overhead Clifford gates implemented through measurement-based code deformations.

RESULTS

The XZZX code is locally equivalent to the conventional surface code but transforms biased Pauli errors into structured strings that support specialised decoding.

  • The XZZX surface code: The XZZX code differs from the conventional surface code by Hadamard rotation on alternate qubits while preserving its code parameters.
  • The XZZX surface code: A decoder maps syndrome defects to a correction, with failure probability decreasing rapidly with code distance below threshold.
  • The XZZX surface code: Pauli-Z errors form string segments whose defects occur at the endpoints, allowing multiple errors to combine into longer strings.
  • The XZZX surface code: Pauli-Z error strings align along one direction because diagonal products of face operators impose a parity-conservation symmetry, enabling one-dimensional decoding.
  • The XZZX surface code: 50% threshold error rates are reported for the relevant high-bias decoding regime.
  • The XZZX surface code: Pauli-X errors produce strings orthogonal to Pauli-Z strings, while finite bias couples errors in conjugate bases and requires generalised decoding.

Optimal Thresholds

The XZZX surface code achieves exceptional code-capacity thresholds across single-qubit Pauli channels, closely matching the zero-rate hashing bound and appearing to exceed it in some high-bias regimes.

  • Universal thresholds: The XZZX code’s code-capacity thresholds closely match the zero-rate hashing bound for all single-qubit Pauli noise channels and appear to exceed it in some regimes.These estimates use an efficient maximum-likelihood decoder that gives the optimal threshold attainable with the code for a given noise model.
  • Hashing-bound comparison: For high bias η ≥30, XZZX threshold estimates exceed the hashing bound, with the excess persisting across the investigated code-distance sets.The gap generally decreases with larger distances but appears to stabilise for η = 30, 100, and 1000.
  • Universal thresholds: 18.7(1)% is the global minimum XZZX threshold at standard depolarising noise, while peaks of ∼50% occur at pure X, Y, and Z noise.The thresholds were estimated over the stochastic vector r parameterising the general single-qubit Pauli channel.
  • Hashing-bound comparison: The apparent hashing-bound excess is associated with a possible superadditive coherent channel under concatenation with a finite-rate outer code.If both codes are LDPCs, the concatenation would provide an example of a superadditive LDPC code.
  • Practical decoding: A matching decoder produces thresholds that closely follow the hashing bound at high bias despite using a sub-optimal decoder that does not use all syndrome information.The data again appear to marginally exceed the bound at high bias.

Fault-tolerant thresholds

The XZZX code retains high thresholds with practical matching decoding when stabilizer measurements are unreliable. Its fault-tolerant decoding interprets data and measurement errors as strings in spacetime and outperforms conventional CSS decoding under biased noise.

  • Fault-tolerant decoding: A generalized matching decoder yields exceptionally high fault-tolerant thresholds for XZZX under biased phenomenological noise with unreliable stabilizer measurements.For unbiased noise, the decoder recovers the standard matching decoder.
  • Fault-tolerant decoding: Measurement errors are represented as strings along the temporal axis, enabling minimum-weight perfect matching to decode faulty syndrome measurements.Repeated measurements identify defects when a stabilizer outcome differs from the previous round.
  • Noise model: The phenomenological noise model assigns high-rate Pauli-Z errors probability p_h.r. and low-rate Pauli-X and Pauli-Y errors probability p_l.r. per unit time.Its noise bias is η = p_h.r./(2p_l.r.), and stabilizer measurements have error probability q = p_h.r. + p_l.r.
  • Threshold comparison: The XZZX matching decoder significantly outperforms the CSS matching decoder for Pauli-Z-biased noise across all noise biases shown.The comparison uses fault-tolerant threshold error rates as functions of noise bias and measurement error rate.
  • Threshold comparison: In the infinite-bias limit, the decoder treats the XZZX code as independent repetition codes and effectively decodes d decoupled copies of a two-dimensional surface code.This yields an expected minimum-weight perfect-matching threshold of ∼10.3%.
  • Threshold comparison: ∼10% is the observed large-bias fault-tolerant threshold, slightly below the expected ∼10.3% because of a suggested small-size effect.The decoder’s success depends on correctly decoding approximately d independent surface-code copies.

Overheads

The XZZX code improves sub-threshold logical-failure scaling under biased noise, reducing overhead and enabling rapid suppression in near-term, high-bias regimes. Its geometry can also reduce the physical-qubit array size while preserving protection against dominant errors.

  • Overheads: At high bias near threshold, logical failure rates fit P = AeBd2 rather than P = AeBd.The quadratic-in-distance scaling is especially relevant for finite systems operating near threshold.
  • Overheads: The decoder can correct approximately d/4 low-rate errors alongside many high-rate errors occurring simultaneously on the lattice.This behavior corresponds to the mixed-error regime where P lin. dominates P quad.
  • Overheads: O((p/√η)d/2) scaling improves the logical failure rate under biased noise compared with generic O(pd/2) scaling.This yields an overhead reduction by a factor of approximately η−d/4 at large bias.
  • Overheads: In the high-bias regime, logical failure can decay like ∼pd2/2 when systems tolerate ∼d2/2 dephasing errors.This rapid quadratic-distance decay is particularly relevant for near-term devices with few qubits operating near threshold.
  • Overheads: Choosing dX ≪ dZ reduces one lattice dimension by O(1/log η) because high-rate error strings align horizontally.Rectangular open-boundary geometries are argued to retain comparable large-system overhead scaling.

Low-overhead fault-tolerant quantum computation

The XZZX code’s noise-adapted advantages can be retained during fault-tolerant computation through code deformations and generalized lattice surgery. The proposed construction uses twist-defect surface codes, ancillary parity measurements, and decoders adapted to biased noise.

  • Low-overhead fault-tolerant quantum computation: Code deformations preserve the XZZX code’s high thresholds and reduced resource costs while implementing fault-tolerant logic gates.The paper focuses on a lattice-surgery example for this construction.
  • Low-overhead fault-tolerant quantum computation: Generalized lattice surgery entangles twist-defect surface-code qubits through parity measurements with an ancillary surface code.The construction includes a hexon surface code with six boundary twist defects.
  • Low-overhead fault-tolerant quantum computation: The proposal leaves detailed implementation questions and fault-tolerant quantum-computation threshold estimates for future work.It provides a high-level overview centered on one lattice-surgery example.
  • Low-overhead fault-tolerant quantum computation: Hexon initialization detects high-rate Pauli-Z errors on red vertices and low-rate Pauli-X errors on blue vertices using selected stabilizers.These initialization errors can be decoded with a minimum-weight perfect-matching decoder.
  • Low-overhead fault-tolerant quantum computation: In the infinite-bias limit, generalized lattice surgery reduces primarily to decoding one-dimensional repetition codes, with a single branching point at the twist.The twist introduced by a logical Pauli-Y measurement creates this exceptional branch.

DISCUSSION

The discussion presents the XZZX code as combining high thresholds with low overhead across biased-noise models, while emphasizing that the ultimate achievable thresholds remain unknown. It identifies tailored codes and decoders, including extensions to correlated noise, as future directions.

  • DISCUSSION: XZZX architectures yield remarkably high memory thresholds and low overhead compared with the conventional surface code.A generalized fault-tolerant decoder realizes these advantages across a broad range of experimentally relevant biased-noise models.
  • DISCUSSION: XZZX thresholds match random-coding hashing performance and numerically exceed the hashing bound for certain error models.The highest achievable thresholds for code capacity and fault-tolerant quantum computing are not yet known.
  • DISCUSSION: Tailoring the code and decoder to the relevant noise model can produce substantial threshold gains over separately decoding Pauli-X and Pauli-Z errors.The paper points toward extending these methods to correlated errors in more realistic fault-tolerant settings.
  • DISCUSSION: The decoder’s minimum-weight-matching basis presents no fundamental obstacle to adaptation from phenomenological noise to circuit-level noise.The authors expect the largest gains when gates preserve the noise structure, but this remains an expectation.
  • DISCUSSION: Further tailored codes and decoders may reduce the physical-qubit overhead required for fault-tolerant quantum computing.The paper identifies overhead scaling as a key direction for future research.

Optimal thresholds

The paper estimates optimal code-capacity thresholds using maximum-likelihood decoding and tensor-network approximations across single-qubit Pauli noise channels. It also performs compute-intensive simulations in regimes where XZZX thresholds exceed the hashing bound.

  • Optimal thresholds: Maximum-likelihood decoding selects the most probable logical coset consistent with the syndrome and is optimal by definition.Exact coset-probability evaluation is generally inefficient, motivating efficient approximations.
  • Optimal thresholds: Figure 2 estimates thresholds over single-qubit Pauli noise channels using four code distances d ∈ {13, 17, 21, 25}.The threshold surfaces use 211 values, with 111 estimated values per surface after exploiting X/Z symmetry.
  • Optimal thresholds: The threshold estimates use a tensor-network decoder approximation with χ = 16 across sampled single-qubit Pauli channels.Each code distance and physical error probability uses 30 000 simulations.
  • Optimal thresholds: For bias 30 ≤ η ≤ 1000, XZZX threshold estimates exceed the hashing bound and are tested with code distances up to d ∈ {65, 69, 73, 77}.Each threshold uses at least fifteen physical error probabilities and 60 000 simulations per distance and error probability.
  • Optimal thresholds: All threshold error rates are evaluated using the critical exponent method.This method follows the procedure of Ref. [45].

The minimum-weight perfect-matching decoder

The decoder uses minimum-weight perfect matching to pair syndrome defects according to error probabilities, with anisotropic weights adapted to biased noise and extensions for fault-tolerant decoding.

  • Decoder construction: Minimum-weight perfect matching pairs nearby defects using string operators likely to have caused each defect pair.The decoder represents defects as graph vertices and assigns edge weights based on the probability of candidate error strings.
  • Practical considerations: The matching algorithm can scale as O(V^3), with typical graph size V = O(pd^2) for reliable measurements and V = O(pd^3) for unreliable measurements.A look-up table and lattice translational invariance are used to reduce repeated weight computation and memory usage.
  • Biased-noise weighting: At infinite bias, decoding reduces to parallel one-dimensional matching problems along diagonal sets Dj, equivalent to repetition-code majority vote.This parallelised procedure substantially improves decoding speed in the infinite-bias limit.
  • Decoder construction: The decoder assigns each defect pair an edge weight proportional to −log prob(Eu,v), where Eu,v is the most probable string connecting them.The matching output identifies which defect pairs should be paired by the correction.
  • Biased-noise weighting: For biased noise, edge weights are anisotropic, scaling with Manhattan separations along code-aligned axes x′ and y′.On large lattices, the weight is chosen proportional to wh.r.lx′ + wl.r.ly′.
  • Fault-tolerant extension: The decoder checks for high-rate strings wrapping around the torus and extends naturally to 2 + 1-dimensional syndrome histories when measurements are unreliable.For unbiased noise at η = 1/2, the decoder is equivalent to the conventional phenomenological minimum-weight perfect-matching decoder.

Ansatz at low error rates

The paper tests a low-error-rate ansatz for logical failures by extracting gradients from code-distance scaling and extrapolating them to zero physical error.

  • Ansatz: The ansatz models logical failure as arising from approximately d/4 low-rate and d/4 high-rate errors along a weight-d logical operator.The low-rate and high-rate errors occur along the support of the logical operator.
  • Ansatz: The analysis approximates log P as G(p, η)d after neglecting the small n log(1 −p) term.Here G(p, η) is the gradient of logical-failure scaling with code distance.
  • Numerical test: For η = 3, gradients are extracted from logical-failure data plotted against code distance and then examined as a function of β = log[p/(1 −p)].The physical error rates analyzed are 0.0001, 0.0002, 0.0005, 0.001 and 0.002.
  • Numerical test: The gradient is approximately 0.5, consistent with the ansatz prediction of 1/2.The estimate is obtained from the distance dependence of the plotted logical-failure data.
  • Numerical test: The fitted intercept has gradient 0.22 ± 0.03, consistent with the expected value 1/4, while γ is estimated as 1.8 ± 0.06.The γ estimate is consistent with the expected range 3/2 ≤γ ≤2.

Code availability

Software for all simulations is publicly available under the OSI-approved BSD 3-Clause licence.

  • Availability: Software for all simulations is available at https://bitbucket.org/qecsim/qsdxzzx/ under the OSI-approved BSD 3-Clause licence.The software extends and uses qecsim and several scientific software packages.
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