Optimizing the Frequency of Quantum Error Correction using the [[7,1,3]] Steane Code

TL;DR

This paper explores how often to perform quantum error correction with the Steane code, showing that less frequent correction can be nearly as effective as correcting after every gate, especially when considering errors introduced during correction.

Contribution

It introduces a simplified error model to optimize the frequency of fault-tolerant quantum error correction, accounting for errors caused by the correction process itself.

Findings

Optimal correction frequency is insensitive to postselection failure probability.

Less frequent error correction can maintain low logical error rates.

Model aligns well with Monte Carlo simulation data.

Abstract

A common assumption in analyses of error thresholds and quantum computing in general is that one applies fault-tolerant quantum error correction (FTQEC) after every gate. This, however, is known not to always be optimal if the FTQEC procedure itself can introduce errors. We investigate the effect of varying the number of logical gates between FTQEC operations, and in particular the case where failure of a postselection condition in FTQEC may cause FTQEC to be skipped with high probability. By using a simplified model of errors induced in FTQEC, we derive an expression for the logical error rate as a function of error-correction frequency, and show that in this model the optimal frequency is relatively insensitive to postselection failure probability for a large range of such probabilities. We compare the model to data derived from Monte Carlo simulation for the $[[7,1,3]]$ Steane code.