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Fault-tolerant conversion between adjacent Reed-Muller quantum codes based on gauge fixing. (English) Zbl 1390.81125

Summary: 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 \(\overline{\mathsf{RM}}(1,m)\) and \(\overline{\mathsf{RM}}(1,m+1)\), 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.

MSC:

81P70 Quantum coding (general)
81P68 Quantum computation
94B60 Other types of codes
94C05 Analytic circuit theory
68M15 Reliability, testing and fault tolerance of networks and computer systems
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