Distance-four quantum codes with combined postselection and error correction

TL;DR

This paper demonstrates that distance-four surface codes with postselection can significantly reduce logical error rates and physical qubit overhead, making them efficient for fault-tolerant quantum computing in low-depth scenarios.

Contribution

It introduces a distance-four, planar surface code encoding multiple qubits with combined postselection and error correction, showing improved efficiency over higher-distance codes.

Findings

Distance-four codes with postselection have 25 times lower error than distance-five codes.

Six qubits encoded in a distance-four code use only 25 qubits, comparable to larger codes.

Simulations of up to 12000 rounds show low logical error accumulation.

Abstract

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.