Overhead analysis of universal concatenated quantum codes

TL;DR

This paper evaluates the resource overhead and noise thresholds of concatenated quantum codes combining the Steane and Reed-Muller codes, comparing their efficiency to surface codes in fault-tolerant quantum computation.

Contribution

It provides new lower bounds on noise thresholds and resource requirements for concatenated codes that do not need magic state distillation, with comparisons to surface code implementations.

Findings

49-qubit code has a depolarizing noise threshold of 9.69×10⁻⁴

105-qubit code has an adversarial noise threshold of 8.33×10⁻⁶

Surface codes outperform concatenated codes in resource overhead for the tested parameters

Abstract

We analyze the resource overhead of recently proposed methods for universal fault-tolerant quantum computation using concatenated codes. Namely, we examine the concatenation of the 7-qubit Steane code with the 15-qubit Reed-Muller code, which allows for the construction of the 49 and 105-qubit codes that do not require the need for magic state distillation for universality. We compute a lower bound for the adversarial noise threshold of the 105-qubit code and find it to be $8.33\times 10^{-6}.$ We obtain a depolarizing noise threshold for the 49-qubit code of $9.69\times 10^{-4}$ which is competitive with the 105-qubit threshold result of $1.28\times 10^{-3}$. We then provide lower bounds on the resource requirements of the 49 and 105-qubit codes and compare them with the surface code implementation of a logical $T$ gate using magic state distillation. For the sampled input error rates and noise model, we find that the surface code achieves a smaller overhead compared to our concatenated schemes.