Fault-tolerant conversion between adjacent Reed–Muller quantum codes based on gauge fixing. (19th February 2018)

• Record Type: Journal Article
• Authors: Quan, Dong-Xiao; Zhu, Li-Li; Pei, Chang-Xing; Sanders, Barry C
• Is Part Of: Journal of physics. Volume 51:Number 11(2018)
• Publication Date: 2018-02-19
• Subjects: quantum error correction -- Reed–Muller quantum code -- code conversion -- gauge fixing; Mathematical physics -- Periodicals; Statistical physics -- Periodicals; Quantum theory -- Periodicals; Matter -- Properties -- Periodicals; 530.105
• DOI: 10.1088/1751-8121/aaad13
• Languages: English
• ISSNs: 1751-8113

Abstract

Abstract: 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.