Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Self-dual bivariate bicycle codes with transversal Clifford gates

Zijian Liang, Yu-An Chen

TL;DR

This work enumerates weight-8 self-dual bivariate bicycle codes with up to $n \leq 200$ physical qubits, realized on twisted tori that enhance code distance and improve stabilizer locality.

Abstract

Bivariate bicycle codes are promising candidates for high-threshold, low-overhead fault-tolerant quantum memories. Meanwhile, color codes are the most prominent self-dual CSS codes, supporting transversal Clifford gates that have been demonstrated experimentally. In this work, we combine these advantages and introduce a broad family of self-dual bivariate bicycle codes. These codes achieve higher encoding rates than surface and color codes while admitting transversal CNOT, Hadamard, and $S$ gates. In particular, we enumerate weight-8 self-dual bivariate bicycle codes with up to $n \leq 200$ physical qubits, realized on twisted tori that enhance code distance and improve stabilizer locality. Representative examples include codes with parameters $[[n,k,d]]$: $[[16,4,4]]$, $[[40,6,6]]$, $[[56,6,8]]$, $[[64,8,8]]$, $[[120,8,12]]$, $[[152,6,16]]$, and $[[160,8,16]]$.

Self-dual bivariate bicycle codes with transversal Clifford gates

TL;DR

This work enumerates weight-8 self-dual bivariate bicycle codes with up to physical qubits, realized on twisted tori that enhance code distance and improve stabilizer locality.

Abstract

Bivariate bicycle codes are promising candidates for high-threshold, low-overhead fault-tolerant quantum memories. Meanwhile, color codes are the most prominent self-dual CSS codes, supporting transversal Clifford gates that have been demonstrated experimentally. In this work, we combine these advantages and introduce a broad family of self-dual bivariate bicycle codes. These codes achieve higher encoding rates than surface and color codes while admitting transversal CNOT, Hadamard, and gates. In particular, we enumerate weight-8 self-dual bivariate bicycle codes with up to physical qubits, realized on twisted tori that enhance code distance and improve stabilizer locality. Representative examples include codes with parameters : , , , , , , and .

Paper Structure

This paper contains 1 section, 1 theorem, 20 equations, 1 figure, 2 tables.

Table of Contents

  1. Acknowledgement

Key Result

Theorem 1

Let $a,b,c,d \in \mathbb{Z}$, and define the reciprocal trinomial pair These polynomials generate the stabilizer of a self-dual bivariate bicycle code. Setting $\Delta \coloneqq ad-bc$, the maximal number of logical qubits is Moreover, this maximum is realized on the twisted torus with periodicities specified by the basis vectors i.e., the coordinate $(n_x,n_y)$ is identified with both $(n_x+3a

Figures (1)

  • Figure 1: Stabilizer pattern of a weight-8 self-dual BB code on a twisted torus. Red and blue dots indicate vertices defined by $f(x,y)=1+x+y+y^{-1}$ and $g(x,y)=\overline{f(x,y)} = 1+x^{-1}+y^{-1}+y$, respectively. An $X$-stabilizer is the product of Pauli $X$ operators on all red and blue vertices (and similarly for $Z$). The stabilizer possesses inversion symmetry about the plaquette center, ensuring commutation between $X$- and $Z$-stabilizers. On a twisted torus with basis vectors $(0,8)$ and $(4,4)$, i.e., with boundary conditions $y^8=1$ and $x^4y^4=1$ (using the $x,y$ conventions of the top-left panel), this pattern realizes the $[[64,8,8]]$ code. The bottom-right panel shows the twisted torus with a longitudinal twist by a fraction of $2\pi$. The construction extends to arbitrary $f(x,y)$ and to twisted tori with basis vectors $\vec{a}_1$ and $\vec{a}_2$.

Theorems & Definitions (3)

  • Theorem 1: Maximal logical dimension for self-dual BB codes
  • Example 2
  • proof : Proof of Theorem \ref{['thm:bb-max-k']}