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PropHunt: Automated Optimization of Quantum Syndrome Measurement Circuits
Joshua Viszlai, Satvik Maurya, Swamit Tannu, Margaret Martonosi, Frederic T. Chong
TL;DR
SM circuits are critical to QEC performance, but existing optimization methods do not model how faults propagate through them. PropHunt optimizes CNOT-level SM circuits by minimizing ambiguity, while Hook-ZNE uses intermediate circuits for error mitigation; evaluations report lower logical error rates and improved ZNE performance.
Problem
Existing SM-circuit optimization methods provide limited generally applicable tooling and may rely on imperfect predictors that do not directly model error propagation.
Method
PropHunt optimizes SM circuits at individual-CNOT granularity by identifying and minimizing ambiguity, using MaxSAT to find minimum-weight logical errors.
Results
2.5x-4x lower logical error rates are reported for PropHunt-generated LP and RQT circuits, while Hook-ZNE reduces error by 3x-6x compared to DS-ZNE.
Takeaways & Limitations
PropHunt recovers hand-designed circuit performance and provides intermediate SM circuits that can improve Zero-Noise Extrapolation through Hook-ZNE.
Takeaways & Limitations
The connection between PropHunt and related SM-circuit problems and solutions is left for future work.
Abstract
from arXiv · showhide
Fault-Tolerant Quantum Computing (FTQC) relies on Quantum Error Correction (QEC) codes to reach error rates necessary for large scale quantum applications. At a physical level, QEC codes perform parity checks on data qubits, producing syndrome information, through Syndrome Measurement (SM) circuits. These circuits define a code's logical error rate and must be run repeatedly throughout the entire program. The performance of SM circuits is therefore critical to the success of a FTQC system. While ultimately implemented as physical circuits, SM circuits have challenges that are not addressed by existing circuit optimization tools. Importantly, inside SM circuits themselves errors are expected to occur, and how errors propagate through SM circuits directly impacts which errors are detectable and correctable, defining the code's logical error rate. This is not modeled in NISQ-era tools, which instead optimize for targets such as gate depth or gate count to mitigate the chance that any error occurs. This gap leaves key questions unanswered about the expected real-world effectiveness of QEC codes. In this work we address this gap and present PropHunt, an automated tool for optimizing SM circuits for CSS codes. We evaluate PropHunt on a suite of relevant QEC codes and demonstrate PropHunt's ability to iteratively improve performance and recover existing hand-designed circuits automatically. We also propose a near-term QEC application, Hook-ZNE, which leverages PropHunt's fine-grained control over logical error rate to improve Zero-Noise Extrapolation (ZNE), a promising error mitigation strategy.
1 Introduction
The paper motivates automated syndrome measurement circuit optimization as essential for evaluating QEC codes because existing approaches do not scale and common circuit objectives can misrepresent logical performance.
- SM circuits physically implement QEC parity checks, enabling error detection while requiring full compilation to evaluate a code.
- Hand-designed circuits, reduced-space brute-force searches, and imperfect predictor-based synthesis are insufficient for the growing space of QEC codes.
- 10x increase in logical error rate can result when circuit-depth optimization ignores error propagation through CNOTs in surface codes.
- PropHunt optimizes individual CNOT orderings, minimizes ambiguity, and automatically recovers hand-designed surface-code circuit performance.
- LP and RQT circuits achieve 2.5x-4x lower logical error rates than a standard coloration circuit at a physical error rate of 0.1%.
- Hook-ZNE uses intermediate PropHunt circuits as low-overhead error-mitigation gadgets for Zero-Noise Extrapolation.
2 Background
The background defines stabilizer-based syndrome measurement, circuit-level noise modeling, and hook errors, showing how CNOT ordering can alter effective code distance and logical performance.
- QEC codes encode logical qubits into physical qubits, with code distance defined by the smallest logical operator causing an undetectable logical error.
- Surface-code stabilizers are measured by ancilla-mediated CNOT circuits whose X- and Z-type directions and measurement bases differ.
- A decoder maps measured syndrome vectors to physical errors and logical observables, distinguishing correctable patterns from undetected logical errors.
- Circuit-level matrices H and L replace qubits with gate errors and capture how noisy SM circuits map faults to syndromes and logical effects.
- Hook errors arise when ancilla faults propagate through CNOTs to multiple data qubits, potentially reducing effective code distance.
- Poor CNOT ordering can reduce d_eff, whereas carefully chosen schedules can preserve d_eff = d.
3 Motivation
SM circuit design is difficult because errors are expected and their propagation shapes decodable syndrome patterns, while automated, generally applicable optimization remains limited.
- NISQ-era objectives such as gate count and depth do not directly address how SM-circuit errors propagate into decoded syndrome patterns.
- Hand-tailored circuits are difficult to generalize, and imperfect predictors complicate both circuit design and code-level comparisons.
- 3.1 Motivating Example: Surface Code: The surface code’s N-Z schedule orders CNOTs so worst-case hook errors are perpendicular to corresponding logical operators.
- 3.1 Motivating Example: Surface Code: A d=3 surface code’s logical performance changes notably between proper and poor CNOT schedules, with poor ordering potentially reducing d_eff.
- The sensitivity of logical error rate to SM-circuit choice is code-specific, motivating automated methods that can recover good schedules and explore other codes.
4 Error Ambiguity Explained
PropHunt frames circuit-level logical faults as ambiguity: distinct error patterns can share a syndrome while producing different logical effects, and CNOT rescheduling can remove such pathways.
- Ambiguous errors are error pairs with identical syndromes but different logical effects, formally satisfying He1 = He2 and Le1 ≠ Le2.
- Ambiguity can be detected from subgraphs when a logical vector is not in the row space of the corresponding check matrix.
- Changing the SM circuit changes H and L, allowing a modified circuit to make a syndrome subset decodable without ambiguity.
- In the d=7 example, poor CNOT ordering creates a reduced distance of 4 because equally likely hook-error explanations have different effects on Z_L.
- A modified CNOT ordering removes the ambiguity in the same decoding subgraph and eliminates the corresponding logical error pathway.
5 PropHunt
PropHunt iteratively optimizes CSS-code syndrome-measurement circuits by finding ambiguous logical errors and modifying CNOT error propagation to remove them. It searches a broader circuit-design space while preserving circuit validity and selecting among compatible verified changes.
- Optimization loop: PropHunt builds a circuit-level decoding graph, expands random connected subgraphs, finds minimum-weight logical errors with MaxSAT, and modifies corresponding subcircuits to remove ambiguity.The loop proceeds from graph construction through ambiguity detection, optimization, and circuit modification.
- Minimum-weight errors: MaxSAT enforces syndrome and logical-observable parity constraints while minimizing error assignments, producing a minimum-weight logical error for each ambiguous subgraph.Auxiliary variables and CNF conversion make the XOR-based formulation suitable for solving.
- Circuit changes: PropHunt enumerates reordering and rescheduling changes that alter which syndromes circuit-level errors flip by changing CNOT interaction order or relative scheduling.Reordering changes affect hook-error data qubits, while rescheduling changes alter detection timesteps and may require paired swaps for opposite-type stabilizers.
- Circuit representation: Its directed multi-edge graph represents shared data-qubit interactions between syndrome qubits, with edge directions encoding interaction order and same-round error propagation.The representation converts scheduling dependencies into CNOT schedules and supports tracking rescheduling changes.
- Validation and application: Candidate changes are pruned unless they preserve stabilizer commutation, keep all CNOTs schedulable, and resolve the ambiguity in the updated decoding matrices.When verified changes conflict, PropHunt applies the change yielding the shortest-depth syndrome-measurement circuit.
6 Evaluation
PropHunt is evaluated on surface, Lifted Product, and Random Quantum Tanner codes using coloration circuits as baselines. It improves logical error rates across the benchmark suite, recovers hand-designed surface-code performance, and remains effective across randomized inputs while balancing depth-related idle errors.
- Benchmark-code results: PropHunt improves logical error rates for all evaluated codes relative to the coloration baseline and matches well-known hand-designed surface-code circuits.For LP and RQT codes, where performant circuits are unknown, it identifies problematic CNOT ordering and scheduling.
- Benchmark-code results: 2.5x-4x lower logical error rates are achieved than the coloration circuit at a physical error rate of 0.1% for the evaluated LP and RQT codes.The comparison is against the selected coloration baseline.
- Robustness: PropHunt consistently improves three different random coloration circuits, despite variation in their starting and ending performance.This evaluation indicates robustness across randomized input schedules for the benchmark codes.
- Solver scaling: Ambiguous-subgraph MaxSAT models are more tractable than global formulations, and parallel sampling allows multiple circuit regions to be optimized simultaneously.Solve times increase with d_eff and can show substantial variability across codes and effective distances.
- Idle-error sensitivity: Across a broad range of relevant idle-error strengths, PropHunt’s small circuit-depth increases are outweighed by its logical-error-rate improvements.The study varies idle errors while holding gate error rate at 0.1% and leaves code-specific hardware evaluation for future work.
7 Fine-Grained Noise Scaling for QEC–ZNE
Hook-ZNE uses intermediate, suboptimal SM circuits from PropHunt to vary logical noise finely at fixed code distance, avoiding several costs of existing QEC–ZNE approaches. Compared with DS-ZNE, its finer scaling produces more stable extrapolation under the same shot budget.
- PropHunt and ZNE: PropHunt’s intermediate logical error rates can be tuned by reducing samples per iteration and increasing iteration count, making them suitable for ZNE.This slows convergence but provides more intermediate rates for extrapolation.
- Limits of existing approaches: Logical Circuit Folding amplifies noise by stretching circuit depth, increasing execution time on slow, resource-intensive error-corrected hardware.
- Limits of existing approaches: DS-ZNE varies logical error rates through code-distance scaling, but its coarse odd-integer distances, low-distance variance, and recompilation requirements hinder reliable fitting.Reducing algorithmic distance also provides little practical parallelism because magic-state distillation already uses low distance.
- Limits of existing approaches: Physical-level noise amplification requires fine-grained gate control and can distort noise scaling while increasing decoding pressure near threshold.
- Hook-ZNE: Hook-ZNE smoothly amplifies logical noise at fixed code distance using suboptimal PropHunt circuits, without increasing logical circuit depth or qubit count.This enables fine-grained bias–variance trade-offs for resource-constrained FTQC.
- Evaluation: Hook-ZNE consistently estimates expectation values closer to the true values than DS-ZNE across tested configurations.With 20,000 shots, finer noise scaling avoids very low distances where amplification and estimator variance grow rapidly, yielding more stable fits.
8 Related Work
Related work spans automated, hand-designed, flag-based, hardware-aware, adaptive, and defect-tolerant approaches to syndrome-measurement circuit construction. PropHunt is described as complementary to these problems while not using extra ancilla qubits for hook-error detection.
- Automated and hand-designed approaches: Parameterized brute-force searches over repetitive code structures have produced CNOT schedules for bivariate bicycle and color codes.The strategy identifies repetitive structures, constructs a smaller parameterized circuit, and searches that circuit exhaustively.
- Automated and hand-designed approaches: Hand-designed syndrome-measurement circuits have been successfully developed for surface codes and generalized bicycle codes.
- Flag fault-tolerant circuits: Flag-based methods use extra qubits to detect hook errors, with some producing distance-preserving circuits or circuits satisfying d_eff = d.Flag-Proxy Networks combines flag and proxy qubits while targeting low connectivity requirements.
- Flag fault-tolerant circuits: PropHunt does not use extra ancilla qubits to detect hook errors, though augmenting its output circuits with flag fault-tolerance is identified as future work.
- Related syndrome-measurement problems: Other related efforts address hardware incompatibility, reduced connectivity, adaptive measurement frequency, and defective components in syndrome-measurement circuits.These include QECC-synth, tangling schedules, adaptive strategies for concatenated codes, and high-weight superstabilizers built from low-weight gauge operators.
- Scope and complementary approaches: PropHunt is presented as complementary to these related syndrome-measurement problems, while thoroughly exploring the connection remains future work.Alternative syndrome-measurement methods such as Shor-, Steane-, and Knill-style schemes require extra logical ancilla qubits for state preparation.
9 Conclusion
The conclusion presents PropHunt as an optimizer for CSS-code syndrome measurement circuits and reports lower logical error rates for LP and RQT circuits, alongside Hook-ZNE improvements over DS-ZNE.
- PropHunt optimizes syndrome measurement circuits for CSS QEC codes by minimizing ambiguity and addressing logical error rate.
- Hook-ZNE reduces error by 3x-6x compared to DS-ZNE.Hook-ZNE uses intermediate syndrome measurement circuits produced by PropHunt for error mitigation with ZNE.
A.1 Abstract
The artifact provides the materials needed to reproduce the paper's results and includes PropHunt for optimizing CSS-code syndrome measurement circuits.
- The artifact includes source code, data, scripts, and PropHunt for optimizing syndrome measurement circuits of CSS codes.
A.2 Artifact check-list (meta-information)
The artifact is publicly available with documented software, hardware, storage, preparation, and experiment requirements, plus scripts for reproducing the paper's evaluations and figures.
- Requirements: The artifact uses an MIT license and requires Docker, Python, and a multi-core CPU.
- Requirements: Approximately 2GB of disk space, 5-10 minutes to build the Docker image, and 8-12 hours are required for experiments.
- Access: The code is publicly available and archived at Zenodo DOI 10.5281/zenodo.17945386.
- Workflow: The repository includes Docker and Makefile support for integrating the Loandra MaxSAT solver and simplifying evaluation.
- Experiments: Provided scripts reproduce motivational, benchmark, solver-timing, sensitivity, and DS-ZNE versus Hook-ZNE experiments.
- Configuration: PropHunt benchmarks can customize random samples, iterations, and CPU cores, including a d= 3 surface-code example with 100 samples, 5 iterations, and 48 cores.