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Graphical structures for design and verification of quantum error correction

Chancellor, Nicholas; Kissinger, Aleks; Zohren, Stefan; Roffe, Joschka; Horsman, Dominic

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Authors

Aleks Kissinger

Stefan Zohren

Joschka Roffe

Dominic Horsman



Abstract

We introduce a high-level graphical framework for designing and analysing quantum error correcting codes, centred on what we term the coherent parity check (CPC). The graphical formulation is based on the diagrammatic tools of the ZX-calculus of quantum observables. The resulting framework leads to a construction for stabilizer codes that allows us to design and verify a broad range of quantum codes based on classical ones, and that gives a means of discovering large classes of codes using both analytical and numerical methods. We focus in particular on the smaller codes that will be the first used by near-term devices. We show how CSS codes form a subset of CPC codes and, more generally, how to compute stabilizers for a CPC code. As an explicit example of this framework, we give a method for turning almost any pair of classical [n,k,3] codes into a [[2n − k + 2,k,3]] CPC code. Further, we give a simple technique for machine search which yields thousands of potential codes, and demonstrate its operation for distance 3 and 5 codes. Finally, we use the graphical tools to demonstrate how Clifford computation can be performed within CPC codes. As our framework gives a new tool for constructing small- to medium-sized codes with relatively high code rates, it provides a new source for codes that could be suitable for emerging devices, while its ZX-calculus foundations enable natural integration of error correction with graphical compiler toolchains. It also provides a powerful framework for reasoning about all stabilizer quantum error correction codes of any size.

Citation

Chancellor, N., Kissinger, A., Zohren, S., Roffe, J., & Horsman, D. (2023). Graphical structures for design and verification of quantum error correction. Quantum Science and Technology, 8(4), Article 045028. https://doi.org/10.1088/2058-9565/acf157

Journal Article Type Article
Acceptance Date Aug 17, 2023
Online Publication Date Sep 22, 2023
Publication Date Oct 1, 2023
Deposit Date Jan 25, 2024
Publicly Available Date Jan 25, 2024
Journal Quantum Science and Technology
Electronic ISSN 2058-9565
Publisher IOP Publishing
Peer Reviewed Peer Reviewed
Volume 8
Issue 4
Article Number 045028
DOI https://doi.org/10.1088/2058-9565/acf157
Public URL https://durham-repository.worktribe.com/output/2164041

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