Multivariate bicycle codes

Quantum error correction suppresses noise in quantum systems to allow for high-precision computations. In this Letter, we introduce multivariate bicycle quantum low-density parity-check (QLDPC) codes, via an extension of the framework developed by Bravyi [S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J. Yoder, ], 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 biplanar structure and often possess 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 four 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.