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Tailoring surface codes for highly biased noise

David K. Tuckett, Andrew S. Darmawan, Christopher T. Chubb, Sergey Bravyi, Stephen D. Bartlett, Steven T. Flammia

arXiv:1812.08186v3quant-ph

TL;DR

Dephasing-biased noise raises the question of which surface-code features produce ultra-high thresholds and lower logical failure rates. The paper analyzes pure and biased noise using structural results and maximum-likelihood-based decoding, finding a 50% pure-noise threshold, hashing-bound agreement, and major geometry-dependent gains. These gains are strongest for coprime and rotated codes, although the discussion notes boundaries at high bias and across code families.

  • Problem

    The paper asks which surface-code features explain ultra-high thresholds and improved logical failure rates under dephasing-biased noise.

  • Method

    The paper combines a concatenated-code structural analysis with polynomial-time, exact, and approximate maximum-likelihood decoding for pure and biased noise.

  • Results

    The surface code reaches a 50% pure-Y threshold, tracks the hashing bound across biases, and rotated codes with approximately half the physical qubits outperform square codes over a wide biased-noise range.

  • Takeaways & Limitations

    Coprime and odd-dimensional rotated geometries can substantially reduce logical failure rates, achieving target rates with quadratically fewer physical qubits in the pure-noise regime.

  • Takeaways & Limitations

    At high bias, other surface-code geometries may be more robust because combined X and Z errors can create lower-weight logical operators, and performance may not generalize across topological codes.

Abstract

from arXiv · show

The surface code, with a simple modification, exhibits ultra-high error correction thresholds when the noise is biased towards dephasing. Here, we identify features of the surface code responsible for these ultra-high thresholds. We provide strong evidence that the threshold error rate of the surface code tracks the hashing bound exactly for all biases, and show how to exploit these features to achieve significant improvement in logical failure rate. First, we consider the infinite bias limit, meaning pure dephasing. We prove that the error threshold of the modified surface code for pure dephasing noise is $50\%$, i.e., that all qubits are fully dephased, and this threshold can be achieved by a polynomial time decoding algorithm. We demonstrate that the sub-threshold behavior of the code depends critically on the precise shape and boundary conditions of the code. That is, for rectangular surface codes with standard rough/smooth open boundaries, it is controlled by the parameter $g=\gcd(j,k)$, where $j$ and $k$ are dimensions of the surface code lattice. We demonstrate a significant improvement in logical failure rate with pure dephasing for co-prime codes that have $g=1$, and closely-related rotated codes, which have a modified boundary. The effect is dramatic: the same logical failure rate achievable with a square surface code and $n$ physical qubits can be obtained with a co-prime or rotated surface code using only $O(\sqrt{n})$ physical qubits. Finally, we use approximate maximum likelihood decoding to demonstrate that this improvement persists for a general Pauli noise biased towards dephasing. In particular, comparing with a square surface code, we observe a significant improvement in logical failure rate against biased noise using a rotated surface code with approximately half the number of physical qubits.

I. INTRODUCTION

The paper explains why surface codes perform exceptionally under dephasing-biased noise and shows how code geometry and boundaries improve logical failure rates. It proves a 50% pure-noise threshold, links thresholds to the hashing bound, and demonstrates substantial savings with coprime and rotated codes.

  • Thresholds: The threshold error rate of the tailored surface code closely tracks the hashing bound across noise biases.Improved tensor-network decoding strengthens evidence at high and infinite bias, where earlier estimates fell short.
  • Pure dephasing: 50% is the proven error threshold for pure Y noise, achieved by an efficient decoder through a hidden concatenation of cycle and repetition codes.The concatenated structure is governed by g = gcd(j, k), with coprime and square codes representing contrasting cases.
  • Code geometry: Coprime codes have one Y-type logical operator of weight O(n), giving pure-Y distance O(n) versus O(√n) for square codes.This produces exponentially better subthreshold behavior for coprime geometries.
  • Rotated codes: Odd-dimensional rotated codes share coprime codes’ favorable structure and achieve the same distance as standard codes with approximately half the physical qubits.They are described as optimal because their Y-distance equals n.
  • Scope: The improvement is not universal across topological codes: the triangular 6.6.6 color code instead shows a decreasing threshold with bias.This contrast bounds the scope of the surface-code result.

III. FEATURES OF SURFACE CODES WITH PURE Y NOISE

For pure Y noise, the surface code exposes syndrome and structural features that support efficient decoding and explain its high threshold. Y errors generate more syndrome information than X or Z errors, while the code admits a concatenated classical-code description.

  • Concatenated structure: Pure-Y error correction can be represented as a concatenation of a bottom-level repetition code and a top-level cycle code.The paper presents this structure as the basis for its analytical results.
  • Syndrome structure: Y errors anticommute with both X- and Z-type stabilizers, producing more syndrome locations than comparable X- or Z-error strings.The additional syndrome information is a distinctive feature of pure Y noise on the surface code.
  • Syndrome structure: The displacement between X- and Z-type stabilizers matters because colocated stabilizers, as in the 6.6.6 color code, do not increase distinct syndrome locations for Y errors.Thus, syndrome advantages depend on stabilizer geometry rather than on Y errors alone.

B. Structure of the standard surface code with pure Y noise

Under pure Y noise, the surface code has a hidden concatenated structure: a cycle code at the top level and repetition codes at the bottom level. The decomposition is governed by g = gcd(j,k), explaining why coprime codes behave like large repetition codes while square codes retain a cycle-code component.

  • Concatenated structure: Under pure Y noise, surface-code error correction is equivalent to correcting classical bit-flip errors in the Y-code.The Y-code is defined by parity checks associated with vertices and plaquettes of the surface-code lattice.
  • Concatenated structure: The top-level cycle code C_m encodes m −1 bits and has parity checks corresponding to triangles in the complete graph K_m.Its checks are redundant, while a connected graph with m vertices has e−m+1 independent cycles.
  • Concatenated structure: The Y-code is a concatenation of the cycle code C_g+1 and g(g +1)/2 repetition codes with lengths determined by jk/g^2.The repetition-code multiplicities are 1, 2(g −1), and g(g +1)/2 −2g +1 for lengths jk/g^2, 2jk/g^2, and 4jk/g^2, respectively.
  • Concatenated structure: Coprime codes have g = 1 and are essentially a single repetition code whose size grows linearly with n, unlike square codes with a linearly growing cycle-code component.This structural difference yields better subthreshold behavior for the coprime case under pure Y noise.
  • Concatenated structure: For square codes, reflection orbits produce repetition codes REP(1), REP(2), and REP(4), with orbit representatives forming the top-level code.The symmetry group has one orbit of size 1, 2(j −1) orbits of size 2, and all remaining orbits of size 4.
  • Concatenated structure: For general rectangular codes, tiling into g × g blocks replaces these repetition lengths by REP(t), REP(2t), and REP(4t), where t = jk/g^2.Extended diagonals alternate between reflected diagonal patterns across the tiles and have no support on boundary edges.

2. Decoding the cycle code

The cycle-code decoder estimates each edge error from syndrome bits supplied by triangles containing that edge, then applies the procedure across the complete graph. It runs in polynomial time and achieves a 50% error threshold, but its subthreshold logical-error scaling is weak.

  • Threshold: The decoder achieves a 50% error threshold for IID bit-flip errors on the cycle code.The analysis uses a bias ϵ > 0 below the 50% error rate and concentration bounds for the syndrome observations.
  • Decoder construction: The cycle-code decoder estimates each edge error by thresholding the syndrome bits from triangles containing that edge.For edge (1,2), the subroutine outputs 0 when the relevant syndrome sum is at most m/2 and 1 otherwise.
  • Decoder construction: O(m^3) runtime results from applying the single-edge subroutine independently to all edges of K_m.A union bound gives a total misidentification probability bounded by 2m^2 exp(−2ϵ^2m).
  • Subthreshold behavior: Its logical-error probability decays exponentially with √n because the cycle code has distance O(m) while n scales as m^2/2.The resulting scaling is unavoidable for this code family.
  • Subthreshold behavior: Reducing the error rate from 49% to 1% would require an estimated cycle-code length of about 10^17.The repetition code REP(n), by contrast, has logical-error probability that decays exponentially with n.

C. Threshold of the standard surface code with pure Y noise

For pure Y noise, the standard surface code has a 50% error-correction threshold achievable with polynomial-time decoding, consistent with its concatenated classical-code structure.

  • The pure-Y surface code is equivalent to a concatenation of two classical codes, each with a 50% threshold.This concatenated structure underlies the threshold result.
  • 50% is the error-correction threshold for the surface code with pure Y noise.The threshold can be achieved by a polynomial-time decoding algorithm.
  • The 50% threshold also applies to rotated surface codes with odd linear dimensions.

D. Y-type logical operators of the standard surface code

Under pure Y noise, the minimum-weight Y-type logical operators are governed by g = gcd(j, k), making coprime and rotated codes substantially more favorable than square codes.

  • For square j×j codes, the minimum-weight Y-type logical operator is a full diagonal with weight dY = 2j −1.
  • For a standard j×k surface code, the Y-distance is determined by g = gcd(j, k).The code can be viewed as jk/g^2 tiled copies of a square g×g code under pure Y noise.
  • Square codes have dY = O(√n), whereas coprime and rotated codes have dY = O(n).The larger Y-distance contributes to lower logical failure rates for coprime and rotated codes.

2. Logical operator count

Y-type logical operators are much fewer than X- or Z-type operators, and odd rotated codes achieve a single full-weight Y-type logical operator.

  • A standard j×k surface code has a g-dependent number of Y-type logical operators, with g = gcd(j, k).The corresponding number of Y-type stabilizers is also cY.
  • For rotated codes with odd linear dimensions, cY = 1.This is much smaller than the number of X- or Z-type logical operators.
  • For coprime and rotated codes, cY = O(1), contributing to their improved logical failure rates over square codes.The comparison is made for pure Y and Y-biased noise.
  • Odd rotated codes have Y-distance dY = O(n), while even rotated codes have dY = O(√n) under pure Y noise.Odd rotated codes admit a single Y-type logical operator of weight O(n).
  • For odd rotated codes, Y⊗n is the only nontrivial Y-type logical operator and the code has dY = n.The odd-dimensional boundary checks enforce uniform action across all qubits.
  • The rotated code is optimal in Y-distance, whereas the coprime code has dY = O(n) with n = 2jk −j −k + 1 physical qubits.
  • Odd rotated codes are equivalent to the repetition code under pure Y noise and therefore have a 50% threshold.

IV. PERFORMANCE OF SURFACE CODES WITH PURE Y NOISE

Numerical results confirm a 50% pure-Y threshold and show that coprime and rotated codes achieve target logical failure rates with quadratically fewer physical qubits than square codes; the improvement persists for biased Pauli noise.

  • Pure-Y performance: The observed pure-Y improvement in coprime and rotated codes is predicted by their larger Y-distance and lower logical-operator count.
  • Pure-Y performance: 50% is observed as the pure-Y threshold for coprime and rotated codes, while square-code evidence is less definitive.The simulations use exact maximum-likelihood decoding across square, coprime, and rotated families.
  • Pure-Y performance: Logical failure rate decays exponentially with Y-distance, f ∼ exp(−αdY), at and below the threshold.The decay is observed for square, coprime, and rotated code families.
  • Pure-Y performance: Quadratically fewer physical qubits achieve a target logical failure rate with coprime or rotated codes than with square codes.This follows from dY = O(√n) for square codes and dY = O(n) for coprime and rotated codes.
  • Biased-noise performance: The improvement persists for general Pauli noise biased toward dephasing, with rotated codes significantly outperforming square codes using approximately half the physical qubits.
  • Biased-noise performance: Threshold estimates for biased noise are reported as very close to the hashing bound across biases, with residual differences attributed likely to finite-size effects.The estimates use an adapted tensor-network decoder with stronger convergence under biased noise.

B. Advantage of coprime and rotated surface codes with biased noise

Rotated surface codes substantially reduce logical failure rates under dephasing-biased noise while using roughly half as many physical qubits as comparable square codes. Their advantage is explained by larger Y-distance and fewer Y-type logical operators.

  • A rotated j×j code with n = j2 physical qubits significantly reduces logical failure rate against biased noise compared with a square j×j code.The comparison uses strongly converged approximate maximum-likelihood decoding across physical error probabilities.
  • 81 physical qubits versus 145 gives the rotated 9×9 code a substantial resource advantage over the square 9×9 code.At standard depolarizing noise and low bias, performance is similar; the improvement becomes very large in the pure-Y limit.
  • dY = 81 versus dY = 17 and cY = 1 versus cY = 28 explain the rotated code’s lower sensitivity to Y noise at equal dX = dZ = 9.The rotated and square 9×9 codes have the same X- and Z-distance, but differ strongly in Y-distance and the number of Y-type logical operators.

VI. IMPROVED TENSOR-NETWORK DECODING OF ROTATED CODES WITH BIASED NOISE

The rotated layout improves tensor-network decoding for biased noise by removing correlations that make standard-layout contraction difficult. This enables exact pure-Y decoding with a minimal bond dimension and improves convergence for finite bias.

  • The rotated layout removes tensor-network correlations, enabling efficient and optimal decoding for pure Y noise and improving efficiency when X and Z errors are rare.The standard layout retains correlations that make strongly Y-biased noise challenging.
  • Exact tensor-network contraction is not known for general local Pauli noise, so approximate contraction uses an MPS updated column by column.The approximation truncates the MPS to the χ largest Schmidt values; without truncation, parameters grow exponentially with the number of columns.
  • Exact pure-Y decoding is achieved with χ = 1 on the rotated layout, whereas the standard layout requires χ ∼48 for a reasonable 21 × 21 approximation.The rotated-code boundary MPS is unentangled for pure Y noise, so no truncation beyond χ = 1 is needed.
  • The rotated-layout MPS decoder extends to finite bias and shows substantially improved performance over the standard method.The extension applies when X and Z errors have nonzero probability alongside predominantly Y noise.
  • A related rotated-layout tensor-network decoder reaches exact pure-Y decoding with χ = 4, outperforming the standard-layout MPS decoder in efficiency.This is less efficient than the improved χ = 1 MPS decoder but still substantially more efficient than the standard layout.
  • Changing square to coprime dimensions on the standard layout provides little MPS-decoder improvement because early column contractions produce identical boundary states.The truncation error for a 21 × 22 code is therefore expected to be at least as large as for a 21 × 21 code.

A. Boundary entanglement in MPS decoder

For pure Y noise, the rotated layout makes the boundary MPS unentangled because compatible boundary configurations factor into independent row contributions. This removes the long-range correlations present in the standard layout and yields exact χ = 1 decoding.

  • The boundary state is formed by contracting columns up to j while leaving right-going boundary indices uncontracted.The boundary configuration records checks acting across the current column boundary, while bulk variables describe checks contained in the contracted region.
  • Under pure Y noise, each boundary choice has at most one compatible bulk configuration, otherwise its boundary amplitude is zero.Bulk checks are fixed sequentially by requiring each qubit to be satisfied, beginning near the two-qubit boundary check.
  • For the trivial coset, compatible configurations have constant check values along rows, with only rows terminated by two-qubit X checks allowed to vary.The remaining checks are fixed to zero, producing a restricted family of compatible configurations.
  • Flipping an odd boundary check introduces 2j Y errors and changes its probability by (pY /pI)2j, independently of other row flips.Because row-flip weights do not depend on which other rows are flipped, the boundary variables are independent.
  • The rotated boundary state is a product state, so the tensor network contracts exactly with χ = 1 for pure Y noise.This contrasts with the standard layout, where three-qubit boundary checks create long-range correlations requiring paired boundary flips.
  • The χ = 1 exactness extends to any coset and the improved contraction efficiency carries over to finite bias.The finite-bias extension includes nonzero pX and pZ and is supported by numerical convergence results.

VII. DISCUSSION

The paper links ultra-high biased-noise performance to the surface code’s structural features and shows that coprime and rotated geometries can substantially reduce logical failure rates. It also provides evidence that the threshold follows the hashing bound across biases, while identifying open questions for fault-tolerant decoding and alternative geometries.

  • Coprime and rotated codes significantly outperform square codes under pure Y noise, with target logical failure rates achievable using quadratically fewer physical qubits.Rotated codes with odd linear dimensions have a single Y-type logical operator of optimal weight n.
  • Under Y-biased noise, a rotated code with approximately half as many physical qubits outperforms a square code across a wide range of physical error probabilities for biases as low as η = 100.The reported advantage increases with code size until low-rate errors dominate logical failures.
  • The threshold error rate of surface codes tracks the hashing bound exactly for all biases, supported by a tensor-network decoder adapted for biased noise.The decoder achieves exact maximum-likelihood decoding for pure Y noise and converges more strongly with Y-biased noise than the prior decoder.
  • The paper identifies alternative geometries as a future direction because well-placed X and Z errors may combine with Y strings to create more common, lower-weight logical operators.This caveat applies particularly in the high-bias regime.
  • Whether high biased-noise performance persists under fault-tolerant quantum computing remains an open question, alongside optimal fault-tolerant thresholds and below-threshold performance.A forthcoming study reports fast but suboptimal fault-tolerant thresholds above 5% with biased noise.

Appendix A: Color-code thresholds with biased noise

The appendix evaluates biased-noise thresholds for triangular color codes and develops tensor-network and pure-Y decoding procedures. It contrasts the color code’s declining threshold under bias with the surface code’s increasing threshold and details the approximations used for numerical decoding.

  • Color-code thresholds with biased noise: The triangular 6.6.6 color-code threshold decreases as noise becomes more biased, contrasting with the surface code’s increasing threshold.Figure 19 compares color-code estimates with surface-code estimates and the hashing bound across bias values.
  • Decoder: The color-code decoder accounts for correlations between X- and Z-type stabilizer syndromes using a tensor-network approximate maximum-likelihood procedure.The construction represents qubits and stabilizers as tensors whose exact contraction yields coset probabilities.
  • Decoder: Exact tensor-network contraction is exponential in the number of qubits, while lattice transformation and bond-dimension truncation enable efficient approximation controlled by χ.Neighboring qubit tensors are merged into a square lattice before approximate contraction.
  • Color-code thresholds with biased noise: Threshold estimates use triangular color codes of distances d = 7, 11, 15, and 19 across biases η = 0.5 to ∞, with η = ∞ representing pure Y noise.The simulations impose pX = pZ and approximate maximum-likelihood decoding with χ = 36.
  • Pure-Y decoding: For pure Y noise, the surface-code decoder maximizes Y-type coset probabilities over candidate recovery operators and logical sectors.For a j×k code, the Y-type stabilizer group has size 2^g−1, where g = gcd(j, k), enabling efficient decoding for small g when the required operators are constructible.

1. Constructing Y-type stabilizers and logical operators

The construction of Y-type stabilizers and logical operators uses diagonal paths that reflect at code boundaries, with the recovery procedure depending on whether the lattice is coprime, square, or neither. These constructions determine how syndrome information is converted into candidate recovery operators.

  • Constructing Y-type stabilizers and logical operators: A minimum-weight Y-type logical operator is generated by applying Y operators along a diagonal path from the top-left corner until another corner is reached.Y-type stabilizers are constructed similarly from the next gcd(j, k)−1 top-row qubits.
  • Constructing candidate Y-type recovery operators: The candidate Y-type recovery construction differs for coprime, square, and non-coprime nonsquare surface codes.Each geometry requires a distinct treatment of residual syndrome locations.
  • Constructing candidate Y-type recovery operators: For coprime codes, Y-type destabilizers anticommute with individual syndrome locations, so multiplying the destabilizer for each location yields a candidate recovery operator.Destabilizers combine partial and residual recovery operators along reflected diagonal paths.
  • Constructing candidate Y-type recovery operators: For square codes, residual boundary syndrome locations cancel when partial recovery operators for all syndrome locations are multiplied.Figure 24 illustrates the original error, partial recoveries, residual boundary syndromes, and their cancellation.
  • Constructing candidate Y-type recovery operators: For neither coprime nor square codes, the lattice is divided into a coprime region and square regions, with stabilizers moving residual syndromes off the lattice.Partial recoveries leave residual syndrome locations only between regions.
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