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Independent Trivariate Bicycle Codes

Aygul Azatovna Galimova

Abstract

We introduce six independent trivariate bicycle (ITB) codes, which extend the bivariate bicycle framework of Bravyi et al.\ to three cyclic dimensions. Using asymmetric polynomial pairs on three-dimensional tori, we construct four codes including a $[[140,6,14]]$ code with $kd^2/n = 8.40$. In the code-capacity setting, the $[[140,6,14]]$ code achieves a pseudothreshold of $8.0\%$ and $kd^2/n = 8.40$, exceeding the best multivariate bicycle code of Voss et al.\ ($7.9\%$, $kd^2/n = 2.67$). With circuit-level depolarizing noise, pseudothresholds reach $0.59\%$ for $[[140,6,14]]$ and $0.53\%$ for $[[84,6,10]]$. On the SI1000 superconducting noise model, the $[[140,6,14]]$ code achieves a per-round per-observable rate of $5.6 \times 10^{-5}$ at $p = 0.20\%$. We additionally present two self-dual codes with weight-8 stabilizers: $[[54,14,5]]$ ($kd^2/n = 6.48$) and $[[128,20,8]]$ ($kd^2/n = 10.0$). These results expand the design space of algebraic quantum LDPC codes and demonstrate that the third cyclic dimension yields competitive candidates for practical fault-tolerant implementations.

Independent Trivariate Bicycle Codes

Abstract

We introduce six independent trivariate bicycle (ITB) codes, which extend the bivariate bicycle framework of Bravyi et al.\ to three cyclic dimensions. Using asymmetric polynomial pairs on three-dimensional tori, we construct four codes including a code with . In the code-capacity setting, the code achieves a pseudothreshold of and , exceeding the best multivariate bicycle code of Voss et al.\ (, ). With circuit-level depolarizing noise, pseudothresholds reach for and for . On the SI1000 superconducting noise model, the code achieves a per-round per-observable rate of at . We additionally present two self-dual codes with weight-8 stabilizers: () and (). These results expand the design space of algebraic quantum LDPC codes and demonstrate that the third cyclic dimension yields competitive candidates for practical fault-tolerant implementations.
Paper Structure (18 sections, 5 equations, 1 figure, 7 tables)

This paper contains 18 sections, 5 equations, 1 figure, 7 tables.

Table of Contents

  1. Introduction
  2. The Quantum Error Correction Challenge
  3. Bicycle Codes and Their Extensions
  4. Our Contribution
  5. Mathematical Background
  6. Stabilizer codes and CSS construction
  7. Code construction
  8. Code instances
  9. Noise simulations
  10. Code-capacity noise results
  11. Circuit-level depolarizing noise results
  12. SI1000 noise results
  13. Comparison with existing codes
  14. Code parameters
  15. Code-capacity comparison
  16. ...and 3 more sections

Figures (1)

  • Figure 1: Per-round logical error rate $p_L$ versus physical error rate $p$ under circuit-based depolarizing noise bravyi2024high. Solid lines: BB codes; dashed: ours. Curves are fits $p_L = p^{d/2}\exp(c_0 + c_1 p + c_2 p^2)$; markers: Monte Carlo estimates ($10{,}000$--$60{,}000$ trials, $N_c = 12$ rounds); bars: $95\%$ Wilson intervals. BB data reproduces the published values of bravyi2024high (Table 1).