Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Multivariate Multicycle Codes for Complete Single-Shot Decoding

Feroz Ahmed Mian, Owen Gwilliam, Stefan Krastanov

TL;DR

The paper introduces Multivariate Multicycle (MM) codes, a unifying framework that embeds BB, MB, GB, TT, and higher‑dimensional toric codes as Koszul complexes over quotient rings with cyclic relations. By placing qubits on a Koszul complex’s middle degree and defining X/Z parity checks from boundary maps, with metachecks provided by adjacent boundaries, MM codes achieve complete single‑shot decoding through high confinement and structured redundancy; the construction naturally extends to t≥4, enabling more robust metacheck sets and potential for logical non‑Clifford gates. The authors develop explicit boundary and metacheck matrices via the Koszul formalism, perform extensive numerical searches to produce high‑rate, high‑distance MM codes, and demonstrate confinement profiles that surpass known single‑shot CSS codes of comparable size. Their results indicate MM codes can offer scalable, fault‑tolerant quantum memories with strong single‑shot decoding and suggest promising avenues for transversal non‑Clifford operations, weight‑reduction techniques, and cohomological gate constructions. Overall, MM codes provide a principled, algebraically transparent path to constructing large, robust single‑shot quantum CSS codes with practical decoding advantages.

Abstract

We introduce multivariate multicycle (MM) codes, a new family of quantum error correcting codes that unifies and generalizes bivariate bicycle codes, multivariate bicycle codes, abelian two-block group algebra codes, generalized bicycle codes, trivariate tricycle codes, and n-dimensional toric codes. MM codes are Calderbank-Shor-Steane (CSS) codes defined from length-t chain complexes with $t \ge 4$. The chief advantage of these codes is that they possess metachecks and high confinement that permit complete single-shot decoding, while also having additional algebraic structure that might enable logical non-Clifford gates. We offer a framework that facilitates the construction of long-length chain complexes through the use of Koszul complex. In particular, obtaining explicit boundary maps (parity check and metacheck matrices) is particularly straightforward in our approach. This simple but very general parameterization of codes permitted us to efficiently perform a numerical search, where we identify several MM code candidates that demonstrate these capabilities at high rates and high code distances. Examples of new codes with parameters $[[n,k,d]]$ include $[[96, 12, 8]]$, $[[96, 44, 4]]$ $[[144, 40, 4]]$, $[[216, 12, 12]]$, $[[360, 30, 6]]$, $[[384, 80, 4]]$, $[[486, 24, 12]]$, $[[486, 66, 9]]$ and $[[648, 60, 9]]$. Notably, our codes achieve confinement profiles that surpass all known single-shot decodable quantum CSS codes of practical blocksize.

Multivariate Multicycle Codes for Complete Single-Shot Decoding

TL;DR

The paper introduces Multivariate Multicycle (MM) codes, a unifying framework that embeds BB, MB, GB, TT, and higher‑dimensional toric codes as Koszul complexes over quotient rings with cyclic relations. By placing qubits on a Koszul complex’s middle degree and defining X/Z parity checks from boundary maps, with metachecks provided by adjacent boundaries, MM codes achieve complete single‑shot decoding through high confinement and structured redundancy; the construction naturally extends to t≥4, enabling more robust metacheck sets and potential for logical non‑Clifford gates. The authors develop explicit boundary and metacheck matrices via the Koszul formalism, perform extensive numerical searches to produce high‑rate, high‑distance MM codes, and demonstrate confinement profiles that surpass known single‑shot CSS codes of comparable size. Their results indicate MM codes can offer scalable, fault‑tolerant quantum memories with strong single‑shot decoding and suggest promising avenues for transversal non‑Clifford operations, weight‑reduction techniques, and cohomological gate constructions. Overall, MM codes provide a principled, algebraically transparent path to constructing large, robust single‑shot quantum CSS codes with practical decoding advantages.

Abstract

We introduce multivariate multicycle (MM) codes, a new family of quantum error correcting codes that unifies and generalizes bivariate bicycle codes, multivariate bicycle codes, abelian two-block group algebra codes, generalized bicycle codes, trivariate tricycle codes, and n-dimensional toric codes. MM codes are Calderbank-Shor-Steane (CSS) codes defined from length-t chain complexes with . The chief advantage of these codes is that they possess metachecks and high confinement that permit complete single-shot decoding, while also having additional algebraic structure that might enable logical non-Clifford gates. We offer a framework that facilitates the construction of long-length chain complexes through the use of Koszul complex. In particular, obtaining explicit boundary maps (parity check and metacheck matrices) is particularly straightforward in our approach. This simple but very general parameterization of codes permitted us to efficiently perform a numerical search, where we identify several MM code candidates that demonstrate these capabilities at high rates and high code distances. Examples of new codes with parameters include , , , , , , and . Notably, our codes achieve confinement profiles that surpass all known single-shot decodable quantum CSS codes of practical blocksize.
Paper Structure (30 sections, 1 theorem, 84 equations, 1 figure, 2 tables, 3 algorithms)

This paper contains 30 sections, 1 theorem, 84 equations, 1 figure, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries from homological algebra
  3. Chain Complexes and Homology
  4. Codes from Chain Complexes
  5. Classical Codes
  6. Quantum CSS Codes
  7. Tensor Product of Complexes
  8. Single-Shot Error Correction
  9. Preliminaries from commutative algebra
  10. Group Rings
  11. Polynomial Rings
  12. Quotient Rings
  13. Tensor Algebra
  14. Exterior Algebra
  15. Previous Constructions
  16. ...and 15 more sections

Key Result

Proposition 1

Let $S$ be a commutative algebra over $\mathbb{F}_2$ such that $s:=\dim_{\mathbb{F}_2}(S)$ is finite. Let $t = 2q$ and $\underline{a} = (a_1, \ldots, a_t)$ be a sequence of elements of $S$. The middle 5-term segment of the Koszul complex $K_\bullet(\underline{a};S)$ is a balanced mCSS code.

Figures (1)

  • Figure 1: Tesseract decoder performance across quantum error correcting codes. Including our new MM codes $[[96,12,4]]$, $[[96,12,8]]$, $[[96,44,4]]$, single-shot decodable 4D topological codes, and 2D topological codes. The legend includes the encoding rate $r = K/N$. Our codes show high rate and high distance and are being single-shot decodable. That includes options with particularly high rates (close to half), at the cost of a lower pseudo-threshold.

Theorems & Definitions (29)

  • Remark 1
  • Remark 2
  • Remark 3
  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Definition 1: Tensor product construction sather2009homological
  • Definition 2: Exterior algebra construction sather2009homological
  • Example 5: BB codes Bravyi_2024 ($t=2$)
  • ...and 19 more