Thresholds for universal concatenated quantum codes

TL;DR

This paper analyzes the error thresholds of a concatenated quantum error-correcting code combining Steane and Reed-Muller codes, demonstrating a high asymptotic threshold suitable for universal quantum computation without magic states.

Contribution

It introduces a concatenated code scheme with a proven high asymptotic error threshold, enhancing fault-tolerance in quantum computing.

Findings

Lower bound of $1.28 imes 10^{-3}$ for the asymptotic threshold

Error suppression of logical CNOT gates improves threshold at higher levels

Threshold is competitive with existing models without magic state reliance

Abstract

Quantum error correction and fault-tolerance make it possible to perform quantum computations in the presence of imprecision and imperfections of realistic devices. An important question is to find the noise rate at which errors can be arbitrarily suppressed. By concatenating the 7-qubit Steane and 15-qubit Reed-Muller codes, the 105-qubit code enables a universal set of fault-tolerant gates despite not all of them being transversal. Importantly, the CNOT gate remains transversal in both codes, and as such has increased error protection relative to the other single qubit logical gates. We show that while the level-1 pseudo-threshold for the concatenated scheme is limited by the logical Hadamard, the error suppression of the logical CNOT gates allows for the asymptotic threshold to increase by orders of magnitude at higher levels. We establish a lower bound of $1.28~\times~10^{-3}$ for the asymptotic threshold of this code which is competitive with known concatenated models and does not rely on ancillary magic state preparation for universal computation.