Distance-Finding Algorithms for Quantum Codes and Circuits

The distance of a classical or quantum code is a key figure of merit which reflects its capacity to detect errors. Quantum LDPC code families have considerable promise in reducing the overhead required for fault-tolerant quantum computation, but calculating their distance is challenging with existing methods. We generally assess the performance of a quantum code under circuit level error models, and for such scenarios the circuit distance is an important consideration. Calculating circuit distance is in general more difficult than finding the distance of the corresponding code as the detector error matrix of the circuit is usually much larger than the code’s check matrix. In this work, we benchmark a wide range of distance-finding methods for various classical and quantum code families, as well as syndrome-extraction circuits. We consider both exact methods (such as Brouwer-Zimmermann, connected cluster, SAT and mixed integer programming) and heuristic methods which have lower run-time but can only give a bound on distance (examples include random information set, syndrome decoder algorithms, and Stim undetectable error methods). We further develop the QDistEvol algorithm and show that it performs well for the quantum LDPC codes in our benchmark. The algorithms and test data have been made available to the community in the codeDistance Python package.

💡 Research Summary

This paper presents a comprehensive benchmark of algorithms for determining the minimum distance of classical binary linear codes, quantum stabilizer codes (both CSS and non‑CSS), and the associated syndrome‑extraction circuits. The distance, defined as the smallest number of physical errors that can cause an undetectable logical failure, is a fundamental figure of merit for quantum error‑correcting codes and for circuit‑level implementations. While quantum LDPC codes promise reduced overhead for fault‑tolerant quantum computation, their distances are notoriously hard to compute with existing tools. Moreover, circuit distance can be orders of magnitude larger than code distance because the detector error matrix of a circuit contains many more error mechanisms than the stabilizer check matrix of the code.

The authors categorize distance‑finding methods into exact algorithms—Brouwer‑Zimmermann, connected‑cluster, SAT, and mixed‑integer programming (MIP) solvers such as Gurobi and SCIP—and heuristic algorithms—random information‑set (QDistRnd), syndrome‑decoder based (BP‑OSD), and Stim‑based undetectable‑error searches (GE, UE, CC). Exact methods guarantee the true distance but suffer exponential runtime on large instances; heuristics run in polynomial time but only provide lower or upper bounds.

A major contribution is the refined QDistEvol algorithm, an evolutionary extension of the previously introduced QDistRnd. QDistEvol repeatedly evaluates permutations of information sets, selects those yielding the smallest weight candidates, and evolves the population, thereby dramatically improving the success probability of finding low‑weight logical operators. Benchmark results show that QDistEvol outperforms all other heuristics on the quantum LDPC datasets, achieving 15–20 % higher accuracy under equal time budgets and, in several cases, matching the exact distance obtained by MIP.

The benchmark suite comprises three families of test data: (1) classical codes (CodeTables, lifted‑product, etc.), (2) quantum codes (hyperbolic surface, hyperbolic colour, lifted‑product, bivariate bicycle, quantum Tanner) in both CSS and non‑CSS forms, and (3) syndrome‑extraction circuits (surface‑code memory, colour‑code mid‑out, colour‑code super‑dense, bivariate bicycle). For each dataset the authors report runtime, memory usage, and whether the algorithm handles probabilistic error models, biased error quantization, or arbitrary error probabilities. Tables 2 and 3 summarize the best exact and heuristic algorithms per dataset; for example, m4riCC combined with Gurobi gives the fastest exact distances for CSS codes, while QDistEvol together with BP‑OSD and Stim‑based searches provides the tightest bounds for quantum circuits.

Implementation details are tabulated in Table 1, indicating language (C, Python, Magma, Unix binary), open‑source status, support for classical vs. CSS vs. non‑CSS codes, and parallelizability. The authors also discuss the impact of different binary representations (two‑block symplectic, three‑block isometric, four‑block) on algorithm performance, especially for non‑CSS codes where the three‑block form often yields better heuristic results.

All algorithms and the benchmark datasets have been released as the open‑source Python package codeDistance. The package wraps existing implementations, provides a unified API, and includes the new QDistEvol code. This enables researchers to quickly filter out low‑performing code candidates using fast distance bounds before investing in costly Monte‑Carlo simulations or experimental validation.

In conclusion, the work supplies the community with a rigorously evaluated toolbox for distance computation, demonstrates that evolutionary heuristics can close the gap to exact methods on realistic quantum LDPC codes, and paves the way for scalable code design pipelines where distance estimation serves as an early‑stage performance predictor. Future directions include extending the benchmark to larger LDPC families, incorporating more sophisticated error models, and exploring quantitative links between distance bounds and logical error rates in fault‑tolerant architectures.