Multivariate Bicycle Codes

TL;DR

This paper introduces Multivariate Bicycle quantum error-correcting codes, especially Trivariate Bicycle codes, which offer higher encoding rates and compact layouts suitable for near-term quantum hardware, outperforming surface codes in efficiency.

Contribution

The paper develops new weight-5 Trivariate Bicycle QLDPC codes with bi-planar and toric structures, demonstrating improved encoding rates and practical suitability for near-term quantum devices.

Findings

Weight-5 TB-QLDPC codes encode 4 qubits with distance 5 using 60 physical qubits.

Codes have bi-planar and toric structures, facilitating implementation.

Compared to surface codes, these codes achieve higher encoding rates with fewer physical qubits.

Abstract

Quantum error correction suppresses noise in quantum systems to allow for high-precision computations. In this work, we introduce Multivariate Bicycle (MB) Quantum Low-Density Parity-Check (QLDPC) codes, via an extension of the framework developed by Bravyi et al. [Nature, 627, 778-782 (2024)] and particularly focus on Trivariate Bicycle (TB) codes. Unlike the weight-6 codes proposed in their study, we offer concrete examples of weight-5 TB-QLDPC codes which promise to be more amenable to near-term experimental setups. We show that TB-QLDPC codes up to weight-6 have a bi-planar structure and often posses a two-dimensional toric layout. Under circuit level noise, we find substantially better encoding rates than comparable surface codes while offering similar error suppression capabilities. For example, we can encode $4$ logical qubits with distance $5$ into $60$ physical qubits using weight-5 check measurements of circuit depth 7, while a surface code with these parameters requires $200$ physical qubits. The high encoding rate and compact layout make our codes highly suitable candidates for near-term hardware implementations, paving the way for a realizable quantum error correction protocol.