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AlphaSyndrome: Tackling the Syndrome Measurement Circuit Scheduling Problem for QEC Codes

Yuhao Liu, Shuohao Ping, Junyu Zhou, Ethan Decker, Justin Kalloor, Mathias Weiden, Kean Chen, Yunong Shi, Ali Javadi-Abhari, Costin Iancu, Gushu Li

arXiv:2601.12509v2cs.ETquant-ph

TL;DR

Syndrome-measurement scheduling is underexplored beyond surface codes, despite commuting stabilizers permitting many orderings whose error propagation produces different logical-error rates. AlphaSyndrome uses MCTS to optimize schedules against logical-operator proximity and heuristic-decoder correctability, reducing logical error rates by 80.6% on average versus depth-optimal baselines while matching Google's schedule and outperforming IBM's.

  • Problem

    Beyond surface codes, syndrome-measurement schedules are not well optimized, even though realistic error propagation makes different valid orderings produce significantly different logical-error rates.

  • Method

    AlphaSyndrome uses Monte Carlo Tree Search to synthesize schedules for commuting-stabilizer QEC codes under a noise model, optimizing error propagation relative to logical operators and heuristic-decoder capability.

  • Results

    80.6% average logical error-rate reduction, up to 96.2%, versus lowest-depth baselines across code families, sizes, and decoders; it matches Google's schedule and outperforms IBM's.

  • Takeaways & Limitations

    Schedule quality should be evaluated by decoder-conditioned logical-error performance, not circuit depth alone, across diverse QEC codes and decoders.

  • Takeaways & Limitations

    Analytically guiding error propagation is difficult because general QEC codes have exponentially many logical operations and decoder-compatible error patterns.

Abstract

from arXiv · show

Quantum error correction (QEC) is essential for scalable quantum computing, yet repeated syndrome-measurement cycles dominate its spacetime and hardware cost. Although stabilizers commute and admit many valid execution orders, different schedules induce distinct error-propagation paths under realistic noise, leading to large variations in logical error rate. Outside of surface codes, effective syndrome-measurement scheduling remains largely unexplored. We present AlphaSyndrome, an automated synthesis framework for scheduling syndrome-measurement circuits in general commuting-stabilizer codes under minimal assumptions: mutually commuting stabilizers and a heuristic decoder. AlphaSyndrome formulates scheduling as an optimization problem that shapes error propagation to (i) avoid patterns close to logical operators and (ii) remain within the decoder's correctable region. The framework uses Monte Carlo Tree Search (MCTS) to explore ordering and parallelism, guided by code structure and decoder feedback. Across diverse code families, sizes, and decoders, AlphaSyndrome reduces logical error rates by 80.6% on average (up to 96.2%) relative to depth-optimal baselines, matches Google's hand-crafted surface-code schedules, and outperforms IBM's schedule for the Bivariate Bicycle code.

1 Introduction

Syndrome-measurement schedules in QEC have many valid orderings, but realistic errors make their logical-error performance differ substantially. AlphaSyndrome automatically optimizes these schedules using code structure, decoder behavior, and noise-model feedback.

  • 1 Introduction: QEC repeatedly measures commuting stabilizer generators through ancilla-mediated Pauli checks to detect physical errors while preserving the logical state.These repeated syndrome-measurement cycles consume substantial spacetime and hardware resources.
  • 1 Introduction: Although stabilizer circuits largely commute, physical errors propagate through two-qubit gates, so gate ordering changes error paths and logical error rates.An ancilla error during a four-qubit measurement can reach different data qubits under different schedules.
  • 1 Introduction: Lowest-depth scheduling is not necessarily performance-optimal, while many non-surface QEC codes lack analytically optimized or hand-crafted schedules.Existing approaches include trivial qubit-index ordering, lowest-depth scheduling, and Google’s specialized surface-code design.
  • 1 Introduction: AlphaSyndrome uses Monte Carlo Tree Search to synthesize and optimize syndrome-measurement schedules under a given noise model.The framework targets general QEC codes with commuting stabilizers and heuristic decoders.
  • 1 Introduction: The optimization shapes propagated errors away from patterns close to logical operations and toward patterns within the heuristic decoder's correctable capability.These objectives connect schedule choices to the code structure and decoder rather than depth alone.
  • 1 Introduction: 80.6% average logical error-rate reduction, reaching 96.2%, over lowest-depth baseline schedules across QEC code families, sizes, and decoders.AlphaSyndrome also matches Google's rotated-surface-code schedule and outperforms IBM's bivariate-bicycle-code schedule.

2 Background

QECCs encode logical information redundantly and use syndrome measurements plus decoding to detect and correct physical errors. Stabilizer-code structure supports flexible measurement ordering, while MCTS provides a heuristic framework for searching finite scheduling states.

  • Quantum Error Correction: QECCs encode logical qubits in a redundant physical-qubit subspace and repeatedly detect errors through syndrome measurement before decoder-controlled correction.Syndrome measurement identifies errors without directly disturbing the encoded state; decoding then determines corrective operations.
  • Stabilizer Codes: Stabilizer codes are generated by commuting Pauli operators, with code parameters [[n,k,d]] describing physical qubits, logical qubits, and minimum undetectable-error weight.Errors anticommuting with stabilizers are detectable, whereas commuting errors outside the stabilizer group can act as logical errors.
  • Logical Operations and Errors: Logical operations have multiple equivalent physical realizations because multiplying by stabilizers preserves their action on the code space.Surface codes offer intuitive logical-operator chains, but that structural picture does not directly generalize to other QECCs.
  • Syndrome Measurement and Decoding: Syndrome measurement uses ancillas to perform partial Pauli checks for each stabilizer, followed by ancilla measurement and classical decoding.Under ideal conditions, compatible stabilizer and Pauli-check orders can produce identical outcomes, motivating simple lexical or lowest-depth schedules.
  • Monte Carlo Tree Search: MCTS represents scheduling as a finite-state search in which moves transform states, terminal states receive evaluations, and tree nodes track accumulated scores and visits.The method seeks an initial move likely to lead to the best terminal state while balancing exploration and exploitation.

3 Problem Formulation and Analysis

Syndrome-measurement schedules are equivalent without noise but diverge under hook-error propagation, so scheduling should favor error patterns distant from logical operators and correctly decoded by heuristic decoders.

  • 3 Problem Formulation and Analysis: Different syndrome-measurement schedules produce different logical error behavior because physical errors propagate through two-qubit gates along schedule-dependent paths.Hook errors can affect only data-qubit checks that occur after the error, making ordering consequential.
  • 3.1 Errors Propagation in Syndrome Measurement: A single Pauli check propagates the matching Pauli error from the ancilla to the data qubit, while the other Pauli error remains on the ancilla.For CNOT-based checks, Z checks propagate ancilla Z errors and X checks propagate ancilla X errors.
  • 3.1 Errors Propagation in Syndrome Measurement: In a multi-check stabilizer circuit, an ancilla hook error can propagate to multiple data qubits and substantially increase logical error rate.The affected data qubits depend on which checks remain after the error occurs.
  • 3.2 Consideration Regarding Logical Operation: Clockwise and anti-clockwise surface-code orders bias logical X and Z error rates in opposite directions.Figure 7 compares the two orders and reports their resulting logical X and Z error rates.
  • 3.2 Consideration Regarding Logical Operation: Late-check orientations should be chosen so propagated errors are orthogonal to corresponding logical operators rather than close to them.The surface-code example motivates the orientation used in Google’s schedule.
  • 3.2 Consideration Regarding Logical Operation: For a [[n,k,d]] code, determining proximity to logical operators is difficult because valid logical X_L or Z_L implementations form an exponentially large set.The passage gives k2^n−k+1 possible implementations over k logical qubits.
  • 3.3 Consideration Regarding Decoder: Schedules should also produce error patterns that a heuristic decoder can decode correctly, because approximate decoders succeed on some patterns and fail on others.For the [[19,1,5]] hexagonal color code example, BP-OSD succeeds where hypergraph union-find fails.
  • 3.4 Design Objective: AlphaSyndrome therefore minimizes expected logical error rate by shaping residual errors away from logical-operator supports and toward the decoder’s correctable region.The objective applies to commuting-stabilizer codes with a physical noise model and heuristic decoder.

4 Data-Driven Scheduling with MCTS

AlphaSyndrome represents valid syndrome-measurement schedules as tick assignments and uses decoder-in-the-loop noisy simulation to guide MCTS toward low-error circuits while controlling search cost through subtree reuse.

  • 4 Data-Driven Scheduling with MCTS: The scheduling problem is combinatorial and analytically intractable because order, code-dependent logical proximity, heuristic decoding, circuit size, and device noise all matter.These factors make a single rule-based schedule unlikely to perform well across non-uniform hardware.
  • 4 Data-Driven Scheduling with MCTS: AlphaSyndrome models a schedule as Pauli-check assignments to discrete ticks, with each assignment recording data qubit, ancilla, Pauli type, and tick.Idle gates fill ticks where a qubit participates in no check.
  • 4 Data-Driven Scheduling with MCTS: Concurrent checks must avoid sharing either data or ancilla qubits within a tick.The non-conflict condition requires distinct data and ancilla qubits for checks scheduled together.
  • 4 Data-Driven Scheduling with MCTS: The framework partitions stabilizers when anticommuting checks cannot be freely swapped, then runs MCTS within partitions and concatenates the resulting partial circuits.For CSS codes, X and Z stabilizers form separate partitions.
  • 4 Data-Driven Scheduling with MCTS: MCTS states contain complete or incomplete tick assignments satisfying non-conflict constraints, and transitions append an unscheduled check at its earliest feasible tick.Choosing t=t_max+1 minimizes idling and circuit depth under the stated transition rule.
  • 4 Data-Driven Scheduling with MCTS: Completed schedules receive a score equal to the inverse overall logical error rate estimated by noisy stim simulations with the decoder in the loop.The sampling circuit measures logical operators, applies the custom syndrome schedule with hook errors, and counts logical flips.
  • 4.5 Continuous Searching: Continuous MCTS reuses the subtree rooted at the selected best child instead of restarting searches at every scheduling step.This reduces redundant exploration compared with separately searching from each successive state.

5 Evaluation

AlphaSyndrome is evaluated across diverse codes, decoders, noise models, and industry schedules, showing that decoder-aware error-propagation optimization can improve reliability and system-level cost beyond depth minimization.

  • 5.3 Comparing with Lowest Depth Schedules: 80.6% average logical-error reduction across 32 code/decoder instances, with a peak reduction of 96.2%, compared with low-depth schedules.AlphaSyndrome usually uses greater depth, but improves reliability by steering hook errors away from logical operations and toward decoder-correctable patterns.
  • 5.3.2 Space-Time Resource Estimation and Reduction: 20-90% lower space–time volume across evaluated code families at comparable logical error rates.The main benefit comes from achieving target reliability at smaller code distances, reducing physical-qubit count while keeping execution time comparable or lower.
  • 5.4 Comparing with Industry Hand-Crafted Schedules: AlphaSyndrome outperforms low-depth schedules and matches Google’s schedules across code sizes and distances, including rectangular rotated surface codes with biased-error correction.These comparisons cover industry hand-crafted schedules for actively studied surface-code and bivariate-bicycle settings.
  • 5.4 Comparing with Industry Hand-Crafted Schedules: 44% lower overall logical error rate with BP-OSD and 10% lower with Unionfind on the [[72, 12, 6]] bivariate-bicycle code versus IBM’s schedule.Figure 13 reports both decoder-specific comparisons.
  • 5.5 Cross-Decoder Evaluation: 25.4% and 34.3% average cross-decoder reductions show that schedules compiled for BP-OSD or Unionfind perform best with their target decoder.BP-OSD-compiled schedules win in 7/8 instances, while Unionfind-compiled schedules win in 8/8; cross-testing generally degrades performance.
  • 5.4 Comparing with Industry Hand-Crafted Schedules: AlphaSyndrome can outperform Google’s schedule under non-uniform error models by adapting scheduling to uneven decoder weights and different noise conditions.Google’s schedule assumes uniform qubit error probabilities, whereas AlphaSyndrome adapts to error models and decoder weights.

6 Related Work and Discussion

Prior work largely uses code-specific or depth-focused scheduling, while AlphaSyndrome addresses the broader decoder-conditioned scheduling problem with MCTS because partial schedules lack reliable local performance estimates.

  • Syndrome Measurement Scheduling: Prior syndrome-scheduling work focuses on specific code families, manual orderings, or minimum depth rather than an automatically extensible general method.Examples include S/Z-shaped surface-code orderings, trivial orderings for color and hypergraph-product codes, and minimum-depth BB-code schedules.
  • QEC Mapping and Gadget Synthesis: QEC mapping and layout methods modify placement, routing, or embedding, but do not change the internal execution order of stabilizer measurements.These approaches therefore operate at a different layer from syndrome-measurement scheduling.
  • QEC Mapping and Gadget Synthesis: SAT-based state-preparation methods target limited stabilizer checks and repeat-until-success initialization, not repeated full syndrome-measurement rounds.Their execution setting differs from ongoing fault-tolerant computation.
  • Discussion: SAT or MaxSAT is poorly suited because full rounds have enormous ordering spaces and decoder-conditioned logical error rates are stochastic, noise-dependent, and difficult to encode symbolically.These constraints motivate a search method that can evaluate complete schedules under realistic noise and decoder behavior.
  • MCTS-Based Quantum Synthesis and Routing: Unlike structural circuit objectives, syndrome-schedule quality cannot be reliably assessed from partial constructions because logical-error performance emerges only after the full round is completed.The global interaction of Pauli-check ordering, fault propagation, and decoder behavior removes accurate local cost estimates.

A.1 Abstract

The artifact provides AlphaSyndrome’s scheduling algorithm, tested quantum error-correction codes, scheduling results, and evaluation scripts for reproducing key paper results.

  • A.1 Abstract: The artifact includes the AlphaSyndrome scheduling algorithm, tested QECCs, scheduling results, and scripts for reproducing key tables and figures.It provides instructions for preparation, evaluation, and execution.
  • A.1 Abstract: The software is implemented in Python and uses Stim-based numerical simulation with reproducibility materials for the reported experiments.The artifact covers the paper’s core evaluation outputs.
  • A.1 Abstract: The artifact is intended to support reproduction of the paper’s tables and figures rather than introduce a separate dataset contribution.Its contents center on code, tested QECCs, scheduling results, and evaluation scripts.

A.2 Artifact check-list (meta-information)

The artifact checklist identifies AlphaSyndrome as a Python MCTS program for QECC scheduling experiments, with reported runtime, hardware, metrics, outputs, and public-access information.

  • A.2 Artifact check-list (meta-information): AlphaSyndrome is listed as an MCTS program implemented in Python for producing syndrome-measurement schedules.The checklist identifies QECCs as the experiment dataset and logical error rate as the metric.
  • A.2 Artifact check-list (meta-information): The artifact reports experiments corresponding to Tables 2 and 4 and Figures 12–15, with Linux and x86 execution requirements.The listed environment requires more than 16 GiB RAM and 20 GiB disk space.
  • A.2 Artifact check-list (meta-information): The checklist estimates five minutes for preparation and less than one hour for experiments, and identifies a public GitHub repository.The artifact requires approximately 20 GiB of disk space.

A.3.2 Hardware dependencies.

The artifact requires a Linux x86 server with more than 16 GiB of RAM and more than 20 GiB of disk space.

  • RAM requirement: more than 16 GiB.
  • Storage requirement: more than 20 GiB of disk space.
  • A graphical interface is not required.

A.3.3 Software dependencies.

The artifact depends on Linux and Python above version 3.9, with QECC definitions stored in JSON files. It includes scripts for reproducing figures and tables or customizing schedules.

  • The artifact requires Linux and Python > 3.9, with additional packages listed in its folder.A Python virtual environment is recommended before installing requirements.txt.
  • QECC files use JSON fields for family, qubit counts, code distance, logical operators, and stabilizers.
  • artifact.py reproduces selected paper assets, including tables 2–3 and Figures 12–15.Outputs are saved in /result/, with tables in LaTeX format; randomization may cause numerical differences.
  • main.py lets users customize schedules through command-line options.
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