Threshold Estimate for Fault Tolerant Quantum Computation

TL;DR

This paper provides a practical estimate of the accuracy threshold for fault-tolerant quantum computation using concatenated codes and Shor’s fault-tolerant error correction with the 7-qubit code under a depolarizing error model. By combining a simple Monte Carlo simulation with analytical hand calculations, the author obtains a gate-error threshold on the order of $\epsilon \approx 10^{-3}$ and a memory-error threshold about an order of magnitude smaller, around $\epsilon \approx 10^{-4}$. The work argues that higher-level FTEC improvements and concatenation can drive error rates down despite imperfect gates, and discusses the Toffoli gate’s role and potential improvements. While not executable on current hardware and relying on simplifying assumptions, the results offer a concrete guideline for the required gate accuracies and illuminate the trade-offs in concatenated fault-tolerant schemes. The study highlights the importance of efficient fault-tolerant ancilla preparation, syndrome strategies, and higher-level error correction to realize scalable quantum computation.

Abstract

I make a rough estimate of the accuracy threshold for fault tolerant quantum computing with concatenated codes. First I consider only gate errors and use the depolarizing channel error model. I will follow P.Shor (quant-ph/9505011) for fault tolerant error correction (FTEC) and the fault tolerant implementation of elementary operations on states encoded by the 7-qubit code. A simple computer simulation suggests a threshold for gate errors of the order ε\approx 10^{-3} or better. I also give a simple argument that the threshold for memory errors is about 10 times smaller, thus ε\approx 10^{-4}.