Generalized toric codes on twisted tori for quantum error correction

TL;DR

The paper tackles the challenge of designing two-dimensional quantum LDPC codes with larger logical spaces beyond the Kitaev toric code by introducing a ring-theoretic framework that leverages Laurent polynomial rings and Gröbner bases to classify anyons and compute code parameters. By formulating Pauli operators as a module over $R=\mathbb{Z}_2[x,x^{-1},y,y^{-1}]$ and using the stabilizer pair $f(x,y),g(x,y)$, the authors derive $k_{\max}=2\dim_{\mathbb{F}_2}(R/\langle f,g\rangle)$ and connect logical operators to Wilson lines, enabling efficient code analysis without large parity-check matrices. They apply this approach to generalized toric codes on twisted tori, producing numerous novel weight-6 $[[n,k,d]]$ codes for $n\le 400$, with improved stabilizer locality and practical advantages, plus a clear link to one-dimensional generalized bicycle codes. The results demonstrate how topological order and algebraic geometry can guide the systematic discovery of high-performance qLDPC codes with potential practical impact on quantum hardware.

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 $n \leq 400$, yielding novel codes such as $[[120,8,12]]$, $[[186,10,14]]$, $[[210,10,16]]$, $[[248, 10, 18]]$, $[[254, 14, 16]]$, $[[294, 10, 20]]$, $[[310, 10, \leq 22]]$, and $[[340, 16, 18]]$. Moreover, we present a new realization of the $[[360, 12, \leq 24]]$ 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.

Figures (3)

• Figure 1: Polynomial representation of Pauli operators haah2016algebraic. We choose two edges $e_1$ and $e_2$ at the original as the unit cell and label their Pauli $X$ and $Z$ operators by 4-dimensional vectors, as in Eq. \ref{['eq: X1 Z1 X2 Z2 definition']}. The translation group $\mathbb{Z}^2$, generated by $x$ and $y$, acts by sending an operator at the origin to the site $(m,n)$ via multiplying its vector by the monomial $x^n y^m$. As an illustration, the $A^\mathrm{TC}_v$ and $B^\mathrm{TC}_p$ stabilizers of the Kitaev toric code are shown, with monomials such as $x^2y^2$ and $x^3y^3$ indicating their positions relative to the origin. In this way, all Pauli operators form a module over the Laurent polynomial ring $R=\mathbb{Z}_2[x,x^{-1},y,y^{-1}]$.
• Figure 3: (a) A twisted torus embedded in three-dimensional space. The torus undergoes a twist along its longitudinal direction by an angle that is a fraction of $2\pi$, as indicated by the red curve tracing the large cycle. (b) A two-dimensional projection of the twisted torus, where points related by the lattice vectors $\vec{a}_1$ and $\vec{a}_2$ are identified. The parallelogram's opposite edges are identified, forming the twisted torus. The twisted torus was previously employed in Ref. Hastings2021Fiber to construct fiber bundle codes.
• Figure 4: (a) Alternative representation of the twisted torus from Fig. \ref{['fig: twisted torus']}, with lattice vectors $\vec{a}_1=(0,\alpha)$ and $\vec{a}_2=(\beta,\gamma)$ (here $\alpha=6$, $\beta=\gamma=3$). Green labels on the left and right edges (and top and bottom) indicate identified boundaries: for example, the horizontal edge in the red $A_v$ terms between labels 2 and 3 on the left reappears at the same location between 2 and 3 on the right, and similarly for the vertical edge in the blue $B_p$ terms at position 1. (b) Sequential numbering of qubits from $0$ to $2\alpha\beta-1$. (c) This twisted torus is equivalent to the rotated toric code, illustrating that a lattice rotation is a special case of the twisted‐torus construction.

Theorems & Definitions (14)

• Theorem 1
• Example 1
• Example 2
• Example 3
• Example 4
• Theorem 2
• proof
• Corollary 2.1
• Corollary 2.2
• Example 5