Graphical CSS Code Transformation Using ZX Calculus

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 transfo...

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Bibliographic Details
Published in:arXiv.org 2023-09
Main Authors: Huang, Jiaxin, Sarah Meng Li, Yeh, Lia, Kissinger, Aleks, Mosca, Michele, Vasmer, Michael
Format: Article
Language:English
Subjects:
Coders
Codes
Equivalence
Fault tolerance
Morphing
Subsystems
Transformations
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.2307.02437