Logical Operators and Fold-Transversal Gates of Bivariate Bicycle Codes

TL;DR

This paper introduces high-rate bivariate bicycle quantum codes with explicit logical operator bases and fold-transversal gates, advancing fault-tolerant quantum computation with improved symmetry and gate implementation.

Contribution

It provides new high-rate BB codes with explicit logical bases and fold-transversal Clifford gates, enhancing fault-tolerance in quantum error correction.

Findings

Constructed $[[98,6,12]]$ and $[[162,8,12]]$ BB codes with fault-tolerant gates

Codes exhibit explicit logical operator bases similar to toric codes

Lays mathematical foundations for logical operators in quantum group algebra codes

Abstract

Quantum low-density parity-check (qLDPC) codes offer a promising route to scalable fault-tolerant quantum computation with constant overhead. Recent advancements have shown that qLDPC codes can outperform the quantum memory capability of surface codes even with near-term hardware. The question of how to implement logical gates fault-tolerantly for these codes is still open. We present new examples of high-rate bivariate bicycle (BB) codes with enhanced symmetry properties. These codes feature explicit nice bases of logical operators (similar to toric codes) and support fold-transversal Clifford gates without overhead. As examples, we construct $[[98,6,12]]$ and $[[162, 8, 12]]$ BB codes which admit interesting fault-tolerant Clifford gates. Our work also lays the mathematical foundations for explicit bases of logical operators and fold-transversal gates in quantum two-block and group algebra codes, which might be of independent interest.