Overhead analysis of universal concatenated quantum codes

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 × 10(−6). We obtain a depolarizing noise threshold for the 49-qubit code of 9.69 × 10(−4) which is competitive with the 105-qubit threshold result of 1.28 × 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