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Surface code off-the-hook: diagonal syndrome-extraction scheduling

Gilad Kishony, Austin Fowler

arXiv:2602.09099v1quant-ph

TL;DR

Poorly chosen syndrome-extraction schedules can halve rotated-surface-code distance, and N/Z orientation planning becomes cumbersome in lattice-surgery geometries. The paper introduces a globally uniform diagonal schedule whose hook errors avoid logical-operator directions. Across memory and lattice-surgery primitives, it preserves full distance and yields equivalent or improved logical error rates while simplifying construction.

  • Problem

    Hook errors from poorly chosen extraction schedules can halve circuit-level code distance, while lattice-surgery geometries make traditional N/Z orientation planning cumbersome.

  • Method

    The diagonal schedule orders plaquette gates across opposite diagonal pairs, using one uniform schedule for each stabilizer type.

  • Results

    The diagonal schedule preserves full code distance and produces equivalent or improved logical error rates across memory experiments and lattice-surgery primitives.

  • Takeaways & Limitations

    A globally uniform diagonal circuit simplifies construction across varied lattice-surgery geometries and offers a compact period on hardware supporting parallel measurement and reset.

Abstract

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In the rotated surface code, hook errors (errors on auxiliary qubits midway through syndrome extraction that propagate to correlated two-qubit data errors) can reduce the circuit-level code distance by a factor of two if the extraction schedule is poorly chosen. The traditional approach uses N-shaped and Z-shaped schedules, selecting the orientation in each plaquette to avoid hook errors aligned with logical operators. However, this becomes increasingly complex within lattice surgery primitives with varied boundary geometries, and requires a 7-step schedule to avoid gate collisions. We propose the diagonal schedule, which orients hook errors along the diagonal of each plaquette. These diagonal errors crucially never align with logical operators regardless of boundary orientation, achieving full code distance. The diagonal schedule is globally uniform: all X-type plaquettes use one schedule and all Z-type plaquettes use another, eliminating geometry-dependent planning. On hardware supporting parallel measurement, reset, and gate operations, the schedule achieves a minimal period of 6 time steps, compared to 7 for the traditional approach. We demonstrate effectiveness for memory experiments, spatial junctions, spatial Hadamard gates, and patch rotation, showing equivalent or improved logical error rates while simplifying circuit construction.

I. INTRODUCTION

The rotated surface code protects quantum information through repeated stabilizer measurements, but syndrome-extraction schedules determine how auxiliary faults propagate. Poorly oriented hook errors can halve effective distance, while N/Z scheduling becomes cumbersome in lattice-surgery geometries.

  • I. INTRODUCTION: The surface code is attractive for fault-tolerant computation because of its high threshold, local connectivity, and compatibility with planar hardware.In the rotated layout, data qubits occupy square-lattice vertices and alternating plaquettes support repeated weight-4 X and Z stabilizer measurements.
  • I. INTRODUCTION: Hook errors arise when an auxiliary fault midway through stabilizer extraction propagates into correlated errors on two data qubits.The affected data qubits are those coupled to the auxiliary during the circuit’s second half, and the error Pauli type matches the measured stabilizer.
  • I. INTRODUCTION: Poorly designed schedules can halve effective code distance when hook-error orientations follow logical-operator paths between same-type boundaries.This reduces distance by a factor of two relative to the phenomenological-noise case.
  • I. INTRODUCTION: Traditional N-shaped and Z-shaped schedules produce vertical and horizontal hooks, respectively, so orientations are selected per plaquette from nearby boundary geometry.For a simple memory patch, Z-shaped X stabilizers and N-shaped Z stabilizers prevent short paths between same-type boundaries.
  • I. INTRODUCTION: Lattice surgery creates varying three-dimensional boundary geometries that make per-plaquette, per-time-step N/Z planning difficult or sometimes impossible.Mixing orientations across adjacent regions also requires a 7-step schedule to avoid simultaneous gates on shared data qubits.
  • I. INTRODUCTION: Alternating a schedule with its time-reversed version improves a fixed bad schedule’s effective distance to d − 1, but still sacrifices one distance unit.Only every other round has unfavorable hook propagation, compared with the ⌈d/2⌉ distance resulting from a fixed bad schedule.

II. THE DIAGONAL SCHEDULE

The diagonal schedule orders gates so hook errors follow plaquette diagonals, preventing alignment with logical operators for any boundary geometry. It uses uniform plaquette circuits and can reduce the cycle period under parallel measurement and reset hardware.

  • II. THE DIAGONAL SCHEDULE: The diagonal schedule applies the first two gates to one plaquette diagonal and the last two to the opposite diagonal.All X-type plaquettes use one schedule and all Z-type plaquettes use another.
  • II. THE DIAGONAL SCHEDULE: Diagonal hook errors never align with logical-operator directions, so they cannot create shortcuts between same-type boundaries regardless of boundary geometry.This preserves full code distance while avoiding geometry-dependent schedule planning.
  • II. THE DIAGONAL SCHEDULE: The diagonal circuits satisfy the even-order coupling constraint required when opposite-type stabilizers share data qubits.This constraint ensures the auxiliary measurement returns the intended stabilizer parity.
  • II. THE DIAGONAL SCHEDULE: On hardware with separate measurement/reset steps, diagonal extraction has period 9, reducible to 8 when its spatial-Hadamard-supporting delay is removed.Uniform N/Z scheduling can instead reach period 6 by removing its idling step.
  • II. THE DIAGONAL SCHEDULE: 6 time steps are sufficient for diagonal extraction when measurements and resets run in parallel with entangling gates, versus 7 for mixed-orientation N/Z scheduling.The diagonal schedule has no auxiliary idling in this setting.
  • II. THE DIAGONAL SCHEDULE: The schedule’s timing advantage is strongest when measurement/reset operations parallelize with gates or dominate cycle time, while idling noise is often smaller than gate and measurement/reset errors.These hardware-dependent tradeoffs qualify the practical benefit of shortening the period.

III. SIMULATIONS

The simulations benchmark diagonal scheduling across increasingly complex lattice-surgery settings under a uniform depolarizing noise model. Effective distance is inferred from the low-noise scaling of logical error rates, with decoder limitations able to lower observed values.

  • III. SIMULATIONS: The study verifies with integer linear programming that diagonal scheduling achieves the expected maximal circuit-level code distance across increasingly complex lattice-surgery settings.The benchmarks use a uniform depolarizing model and shortest periods available for hardware with parallel measurement and reset.
  • III. SIMULATIONS: Effective distance is extracted by fitting p_logical ∝ p^(t+1) at small p, where t = floor((d−1)/2) is the number of adversarial errors corrected.The definition d_eff = 2t + 1 is always odd, and observed distance can be reduced by decoder limitations.
  • III. SIMULATIONS: Diagonal hook errors trigger four neighboring stabilizer measurements rather than two, making them non-matchable for a matching decoder.They can nevertheless be decoded by decomposing the correlated error into independent single-qubit errors without reducing effective distance in the reported setting.

IV. MEMORY EXPERIMENT

In a single-patch memory experiment, diagonal and traditional schedules use the same period-6 timing and produce nearly identical logical error rates across tested physical error rates and code distances.

  • IV. MEMORY EXPERIMENT: Nearly identical logical error rates are obtained for traditional and diagonal schedules across all tested physical error rates and code distances.Both schedules use the same period-6 timing because each is spatially uniform in the memory geometry.

V. SPATIAL JUNCTIONS

The diagonal schedule handles spatial junctions with one globally uniform circuit, avoiding geometry-dependent orientation planning while maintaining full code distance. At an X-shaped junction, it also slightly improves logical error rates through a shorter period-6 circuit.

  • L-shaped junction: The diagonal schedule uses the same circuit throughout spatial junctions, automatically avoiding problematic hook-error configurations and achieving full code distance.Traditional N/Z scheduling requires orientation choices that vary with junction geometry.
  • X-shaped junction: At X-shaped junctions, traditional scheduling is difficult because nearby boundaries share the same Pauli type, complicating avoidance of all problematic hook paths.
  • X-shaped junction: The diagonal schedule slightly lowers logical error rates than the N/Z schedule at the X-shaped junction.The diagonal circuit has period 6, whereas the standard circuit requires period 7 to accommodate adjacent plaquette orientations.

VI. SPATIAL HADAMARD

Spatial Hadamard introduces stretched stabilizers that are vulnerable to short-axis hook errors, so flag measurements are added to preserve full distance. Partial or full flags achieve full effective distance with the Tesseract decoder, while matching-based decoders and Tesseract runtime remain important limitations.

  • Stretched stabilizers: Stretched stabilizers are vulnerable because their syndrome circuits communicate information through a three-qubit chain and can propagate hook errors along the rectangle’s short axis.
  • Flagged stretched stabilizers: Without flags, stretched-stabilizer hook errors reduce circuit-level distance from 2k + 1 to k + 1, whereas partial or full flags restore full distance.Partial flags preserve the diagonal schedule’s period-6 cycle.
  • Decoder results: The Tesseract decoder achieves deff = 2k + 1 with partial and full flags, and partial flags match memory logical error rates for equal volume.
  • Decoder limitations: Matching-based decoders achieve only deff = 2⌊k/2⌋ + 1 for these flagged circuits because they cannot fully exploit flag–hook correlations.
  • Decoder limitations: Tesseract is 10^3–10^4 times slower than matching-based decoders, motivating faster decoders that retain full effective distance.

VII. PATCH ROTATION

Patch rotation applies the diagonal schedule without changing its uniform spatial or temporal structure as the patch geometry evolves. The process shows no distance reduction, while its logical error rate is roughly six times that of a memory experiment because it spans six patch-time steps.

  • Patch rotation: The diagonal schedule remains uniform throughout patch rotation and shows no distance reduction despite changing boundary geometry.
  • Logical error rates: The patch-rotation logical error rate is roughly six times the memory rate, consistent with a sequence comprising six patch-time steps.

VIII. SUMMARY

The diagonal schedule avoids geometry-dependent N/Z orientation planning while preserving full code distance, simplifying syndrome extraction across lattice-surgery primitives. It achieves equivalent or improved logical error rates across demonstrated settings, while spatial Hadamard circuits remain limited by matching-decoder distance loss.

  • VIII. SUMMARY: The diagonal schedule preserves full code distance while replacing geometry-dependent N/Z planning with a globally uniform circuit across demonstrated lattice-surgery primitives.The paper demonstrates memory, L-junction, X-junction, spatial Hadamard, and patch-rotation settings.
  • VIII. SUMMARY: For spatial Hadamard gates, Tesseract achieves full circuit-level distance, whereas matching-based decoders achieve deff = 2⌊k/2⌋+ 1.The matching-decoder distance reduction is limited to the two-dimensional interface surface.
  • VIII. SUMMARY: The spatial Hadamard construction uses flag measurements to detect short-axis hook errors from stretched stabilizers without increasing circuit depth.Correlated matching improves logical error rates over vanilla matching, but both matching approaches share the same asymptotic distance limitation.
  • VIII. SUMMARY: Yoked and crosshairs surface codes are identified as promising applications because their complex boundaries suit geometry-independent hook-error handling.Both constructions store logical qubits more compactly for a given code distance.
  • VIII. SUMMARY: A broader open question is whether efficient schedule-selection algorithms can maximize circuit-level distance for arbitrary qLDPC codes while maintaining low circuit depth.The paper also suggests investigating schedules for other quantum error-correcting codes, including color codes.

1. Matching Decoders with Full Noise

Under full noise, matching-based decoders do not reproduce the logical error rates of an equal-volume memory experiment for the spatial Hadamard circuit. At k = 4, the Hadamard logical error rate is roughly seven times higher with PyMatching and four times higher with correlated PyMatching.

  • 1. Matching Decoders with Full Noise: At physical error rate 10^-3 and k = 4, the spatial Hadamard logical error rate is roughly 7 times the equal-volume memory rate with PyMatching and roughly 4 times higher with correlated PyMatching.The comparison uses the same decoder for each Hadamard and memory experiment.
  • 1. Matching Decoders with Full Noise: Effective distances are difficult to determine from the full-noise plots because both bulk errors and hook errors affect the logical error rates.The reference memory rates are shown using x-shaped markers at physical error rate 10^-3.

2. Interface-Only Noise Model

An interface-only noise model isolates stretched-stabilizer behavior and makes effective distance measurable without prohibitively low physical error rates. Matching decoders achieve a reduced distance, while Tesseract reaches full distance at substantially higher computational cost.

  • 2. Interface-Only Noise Model: The interface-only model isolates effective-distance behavior and avoids the prohibitively many shots required to measure it at very low physical error rates under full noise.Under full noise, bulk errors dominate at higher error rates and obscure the stretched-interface contribution.
  • 2. Interface-Only Noise Model: Tesseract is typically 10^3–10^4 times slower per cycle than matching-based decoders across physical error rates and code distances.This runtime tradeoff must be weighed against Tesseract’s distance advantage.
  • 2. Interface-Only Noise Model: With full noise, matching-based decoders fail to match the equal-volume memory logical error rates for the spatial Hadamard circuit, so effective distances remain hard to extract.The full-noise comparison uses PyMatching and correlated PyMatching with different flag settings.
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