Distance-four quantum codes with combined postselection and error correction*

When storing encoded qubits, if single faults can be corrected and double faults postselected against, logical errors only occur due to at least three faults.  At current noise rates, having to restart when two errors are detected prevents very long-term storage, but this should not be an issue for low-depth computations.  We consider distance-four, efficient encodings of multiple qubits into a modified planar patch of the 16-qubit surface code.  We simulate postselected error correction for up to 12000 rounds of parallel stabilizer measurements, and subsequently estimate the cumulative probability of logical error for up to twelve encoded qubits.

Our results demonstrate a combination of low logical error rate and low physical overhead. For example, the distance-four surface code, using postselection, accumulates 25 times less error than its distance-five counterpart.  For six encoded qubits, a distance-four code using 25 qubits protects as well as the distance-five surface code using 246 qubits.

Hence distance-four codes, using postselection and in a planar geometry, are qubit-efficient candidates for fault-tolerant, moderate-depth computations.