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.

• Received 27 September 2016
• Corrected 15 February 2017

DOI:https://doi.org/10.1103/PhysRevA.95.022313