Multivariate Bicycle Codes

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.