Towards Demonstrating Fault Tolerance in Small Circuits Using Bacon-Shor Codes

TL;DR

This paper demonstrates the potential for fault-tolerant quantum circuits using Bacon-Shor codes, showing pseudo-thresholds up to 3% error rate, and suggests feasible small-scale demonstrations with current hardware.

Contribution

It introduces a fault-tolerant implementation of quantum circuits with Bacon-Shor codes and computes pseudo-thresholds, advancing practical quantum error correction methods.

Findings

Pseudo-thresholds up to 3% for short circuits

Multiple stabilizer measurement rounds improve performance

Feasible small-scale fault-tolerant demonstrations with existing hardware

Abstract

Quantum error correction is necessary to perform large-scale quantum computations in the presence of noise and decoherence. As a result, several aspects of quantum error correction have already been explored. These have been primarily studies of quantum memory[1, 2], an important first step towards quantum computation, where the objective is to increase the lifetime of the encoded quantum information. Additionally, several works have explored the implementation of logical gates[3-5]. In this work we study a next step - fault-tolerantly implementing quantum circuits. We choose the $[[4, 1, 2]]$ Bacon-Shor subsystem code, which has a particularly simple error-detection circuit. Through both numerics and site-counting arguments, we compute pseudo-thresholds for the Pauli error rate $p$ in a depolarizing noise model, below which the encoded circuits outperform the unencoded circuits. These pseudo-threshold values are shown to be as high as $p=3\%$ for short circuits, and $p=0.6\%$ for circuits of moderate depth. Additionally, we see that multiple rounds of stabilizer measurements give an improvement over performing a single round at the end. This provides a concrete suggestion for a small-scale fault-tolerant demonstration of a quantum algorithm that could be accessible with existing hardware.