Graphical CSS Code Transformation Using ZX Calculus

• URL: http://arxiv.org/abs/2307.02437v2
• Date: Fri, 1 Sep 2023 06:07:27 GMT
• Title: Graphical CSS Code Transformation Using ZX Calculus
• Authors: Jiaxin Huang (Dept. of Computer Science, Hong Kong University of Science and Technology), Sarah Meng Li (Institute for Quantum Computing, Dept. of Combinatorics & Optimization, University of Waterloo), Lia Yeh (Dept. of Computer Science, University of Oxford, Quantinuum), Aleks Kissinger (Dept. of Computer Science, University of Oxford), Michele Mosca (Institute for Quantum Computing, Dept. of Combinatorics & Optimization, University of Waterloo, Perimeter Institute for Theoretical Physics), Michael Vasmer (Institute for Quantum Computing, University of Waterloo, Perimeter Institute for Theoretical Physics)
• Abstract summary: We present a generic approach to transform CSS codes by building upon their equivalence to phase-free ZX diagrams. We show how ZX and graphical encoder maps relate several equivalent perspectives on these code-transforming operations.
• Score: 0.6734802552703861
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In this work, we present a generic approach to transform CSS codes by building upon their equivalence to phase-free ZX diagrams. Using the ZX calculus, we demonstrate diagrammatic transformations between encoding maps associated with different codes. As a motivating example, we give explicit transformations between the Steane code and the quantum Reed-Muller code, since by switching between these two codes, one can obtain a fault-tolerant universal gate set. To this end, we propose a bidirectional rewrite rule to find a (not necessarily transversal) physical implementation for any logical ZX diagram in any CSS code. Then we focus on two code transformation techniques: code morphing, a procedure that transforms a code while retaining its fault-tolerant gates, and gauge fixing, where complimentary codes can be obtained from a common subsystem code (e.g., the Steane and the quantum Reed-Muller codes from the [[15,1,3,3]] code). We provide explicit graphical derivations for these techniques and show how ZX and graphical encoder maps relate several equivalent perspectives on these code-transforming operations.

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