Flag fault-tolerant error correction with arbitrary distance codes

TL;DR

This paper introduces a versatile fault-tolerant quantum error correction protocol using flag circuits applicable to arbitrary distance codes, extending previous methods and reducing resource requirements for quantum error correction.

Contribution

It presents the first general flag error correction protocol for codes beyond distance three and broadens applicability to more distance-three codes, using fewer qubits.

Findings

Applicable to stabilizer codes of arbitrary distance satisfying certain conditions

Analyzed numerically for distance-three and -five codes

Potential for low-overhead fault-tolerant quantum computing

Abstract

In this paper we introduce a general fault-tolerant quantum error correction protocol using flag circuits for measuring stabilizers of arbitrary distance codes. In addition to extending flag error correction beyond distance-three codes for the first time, our protocol also applies to a broader class of distance-three codes than was previously known. Flag circuits use extra ancilla qubits to signal when errors resulting from $v$ faults in the circuit have weight greater than $v$. The flag error correction protocol is applicable to stabilizer codes of arbitrary distance which satisfy a set of conditions and uses fewer qubits than other schemes such as Shor, Steane and Knill error correction. We give examples of infinite code families which satisfy these conditions and analyze the behaviour of distance-three and -five examples numerically. Requiring fewer resources than Shor error correction, flag error correction could potentially be used in low-overhead fault-tolerant error correction protocols using low density parity check quantum codes of large code length.