Fault-tolerant conversion between adjacent Reed–Muller quantum codes based on gauge fixing

We design forward and backward fault-tolerant conversion circuits, which convert between the Steane code and the 15-qubit Reed–Muller quantum code so as to provide a universal transversal gate set. In our method, only seven out of a total 14 code stabilizers need to be measured, and we further enhance the circuit by simplifying some stabilizers; thus, we need only to measure eight weight-4 stabilizers for one round of forward conversion and seven weight-4 stabilizers for one round of backward conversion. For conversion, we treat random single-qubit errors and their influence on syndromes of gauge operators, and our novel single-step process enables more efficient fault-tolerant conversion between these two codes. We make our method quite general by showing how to convert between any two adjacent Reed–Muller quantum codes RM(1, m) and RM (1, m + 1), for which we need only measure stabilizers whose number scales linearly with m rather than exponentially with m obtained in previous work. We provide the explicit mathematical expression for the necessary stabilizers and the concomitant resources required. D-X Quan et al Fault-tolerant conversion between adjacent Reed–Muller quantum codes based on gauge fixing Printed in the UK 115305 JPHAC5 © 2018 IOP Publishing Ltd 51 J. Phys. A: Math. Theor.