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  • Open Access

Generalized Toric Codes on Twisted Tori for Quantum Error Correction

Zijian Liang1, Ke Liu2,3, Hao Song4, and Yu-An Chen1,*

  • *Contact author: yuanchen@pku.edu.cn
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PRX Quantum 6, 020357 – Published 23 June, 2025

DOI: https://doi.org/10.1103/rmy6-9n89

Abstract

The Kitaev toric code is widely considered one of the leading candidates for error correction in fault-tolerant quantum computation. However, direct methods to increase its logical dimensions, such as lattice surgery or introducing punctures, often incur prohibitive overheads. In this work, we introduce a ring-theoretic approach for efficiently analyzing topological CSS codes in two dimensions, enabling the exploration of generalized toric codes with larger logical dimensions on twisted tori. Using Gröbner bases, we simplify stabilizer syndromes to efficiently identify anyon excitations and their geometric periodicities, even under twisted periodic boundary conditions. Since the properties of the codes are determined by the anyons, this approach allows us to directly compute the logical dimensions without constructing large parity-check matrices. Our approach provides a unified method for finding new quantum error-correcting codes and exhibiting their underlying topological orders via the Laurent polynomial ring. This framework naturally applies to bivariate bicycle codes. For example, we construct optimal weight-6 generalized toric codes on twisted tori with parameters [[n,k,d]] for n≤400, yielding novel codes such as [[120,8,12]], [[186,10,14]], [[210,10,16]], [[248,10,18]], [[254,14,16]], [[294,10,20]], [[310,10,≤22]], and [[340,16,18]]. Moreover, we present a new realization of the [[360,12,≤24]] quantum code using the (3,3)-bivariate bicycle code on a twisted torus defined by the basis vectors (0,30) and (6,6), improving stabilizer locality relative to the previous construction. These results highlight the power of the topological order perspective in advancing the design and theoretical understanding of quantum low-density parity-check (LDPC) codes.

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Popular Summary

Quantum computers rely on error correction to protect fragile quantum states from noise. Among various error-correcting codes, bivariate bicycle codes, a specific type of quantum low-density parity-check (qLDPC) code, has recently emerged as a promising approach due to its high error thresholds and low overhead. However, finding optimal qLDPC codes remains a significant challenge.

This work introduces a ring-theoretic framework to systematically construct and analyze qLDPC codes using mathematical tools from algebraic topology and Grobner bases. Our method efficiently identifies fundamental anyon excitations in these codes and determines their properties, even under twisted boundary conditions. This enables us to discover new qLDPC code generalizations of the Kitaev toric code with the best-known parameters to date, surpassing previous designs.

Our findings have important implications for fault-tolerant quantum computing. The new qLDPC codes we construct advance the theoretical understanding of bivariate bicycle codes and provide practical guidance for experimental implementations of robust quantum error-correction codes.

Article Text

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Outline

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PRX Quantum 6, 020357– Published 23 June, 2025
  • Received 10 February 2025
  • Revised 2 May 2025
  • Accepted 13 May 2025
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