Fault-tolerant fidelity based on few-qubit codes: Parity-check circuits for biased error channels

TL;DR

This paper investigates fault-tolerant quantum error correction using few-qubit codes in the deep sub-threshold regime, demonstrating efficient circuits for biased error channels and estimating physical error rates needed for extremely low logical error rates.

Contribution

It introduces optimized error correction circuits for biased error channels and quantifies physical error thresholds for achieving very low logical error rates with small codes.

Findings

Efficient error correction circuits for biased channels identified

Logical error rate of 10^{-15} achievable with physical error rate of 10^{-5} for biased errors

Physical error rate thresholds are higher for biased channels compared to depolarising errors

Abstract

In the shallow sub-threshold regime, fault-tolerant quantum computation requires a tremendous amount of qubits. In this paper, we study the error correction in the deep sub-threshold regime. We estimate the physical error rate for achieving the logical error rates of $10^{-6} - 10^{-15}$ using few-qubit codes, i.e. short repetition codes, small surface codes and the Steane code. Error correction circuits that are efficient for biased error channels are identified. Using the Steane code, when error channels are biased with a ratio of $10^{-3}$, the logical error rate of $10^{-15}$ can be achieved with the physical error rate of $10^{-5}$, which is much higher than the physical error rate of $10^{-9}$ for depolarising errors.