Single-shot quantum error correction with the three-dimensional subsystem toric code

Fault-tolerant protocols and quantum error correction (QEC) are essential to building reliable quantum computers from imperfect components that are vulnerable to errors. Optimizing the resource and time overheads needed to implement QEC is one of the most pressing challenges. Here, we introduce a new topological quantum error-correcting code, the three-dimensional subsystem toric code (3D STC). The 3D STC can be realized with geometrically-local parity checks of weight at most three on the cubic lattice with open boundary conditions. We prove that one round of parity-check measurements suffices to perform reliable QEC with the 3D STC even in the presence of measurement errors. We also propose an efficient single-shot QEC decoding strategy for the 3D STC and numerically estimate the resulting storage threshold against independent bit-flip, phase-flip and measurement errors to be p(STC) approximate to 1.045%. Such a high threshold together with local parity-check measurements make the 3D STC particularly appealing for realizing fault-tolerant quantum computing. Topological quantum error correction is a promising approach towards fault-tolerant quantum computing, but suffers from large time overhead. Here, the authors generalise the stabiliser toric code to a single-shot 3D subsystem toric code, featuring good performance and resilience to measurement errors.