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Distance-Finding Algorithms for Quantum Codes and Circuits

Mark Webster, Abraham Jacob, Oscar Higgott

Abstract

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.

Distance-Finding Algorithms for Quantum Codes and Circuits

Abstract

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.
Paper Structure (45 sections, 17 equations, 28 figures, 31 tables, 13 algorithms)

This paper contains 45 sections, 17 equations, 28 figures, 31 tables, 13 algorithms.

Table of Contents

  1. Introduction
  2. Contribution of this Work
  3. Background
  4. Distance of Classical Linear Codes
  5. Distance of Quantum Stabiliser Codes
  6. Two-Block (Symplectic) Representation of Stabiliser Codes
  7. Three Block Representation Stabiliser Codes
  8. Four-Block Representation of Stabiliser Codes
  9. CSS Codes
  10. Distance of Quantum Circuits
  11. Direct Calculation of Detector Matrices for Classical and Quantum Codes
  12. Computational Complexity of Distance Finding
  13. Distance-Finding Algorithms
  14. Desiderata and Characteristics of Distance-Finding Algorithms
  15. Enumeration and Sampling of Codewords
  16. ...and 30 more sections

Figures (28)

  • Figure 1: Benchmarking of Meet-in-the-Middle and Brouwer-Zimmermann algorithms for hyperbolic surface codes. We benchmark against m4riCC which is the fastest exact algorithm for the dataset. Note that the QubitserfMM algorithm gave results for codes up to $n=96$ and QubitserfBZ up to $n=168$
  • Figure 2: QDistEvol Sensitivity Analysis for 756-qubit bivariate bicycle code. Here we plot the success rate for QDistEvol returning a distance of 34 when varying the number of generations (genCount), number of offspring per parent ($\mu$) and number of transpositions per mutation (pMut) parameters with iterCount=10000 RREF calculations per trial. We used 250 trials for this analysis with default settings genCount=100, $\mu$ = 10 and pMut=2 and swapPivot=TRUE.
  • Figure 3: Benchmark data by code length - codetables.de GF(2) codes
  • Figure 4: Benchmark data by code length - classical lifted product codes
  • Figure 5: Sensitivity of Exact Distance-Finding Methods to Choice of Block Encoding. We compare processing times for the non-CSS codetables data set for different block encoding choices for the Brouwer Zimmermann MW, m4riCC and Gurobi distance-finding algorithms. We include the processing time for Magma's two-block Brouwer-Zimmermann implementation as a base line.
  • ...and 23 more figures

Theorems & Definitions (3)

  • Example 3.1
  • Example 3.2
  • Example 4.1: Quantum MacWilliams Identity