Richard D.P. East
AKLT-States as ZX-Diagrams: Diagrammatic Reasoning for Quantum States
East, Richard D.P.; van de Wetering, John; Chancellor, Nicholas; Grushin, Adolfo G.
Authors
John van de Wetering
Dr Nicholas Chancellor nicholas.chancellor@durham.ac.uk
Teaching Fellow QO
Adolfo G. Grushin
Abstract
From Feynman diagrams to tensor networks, diagrammatic representations of computations in quantum mechanics have catalyzed progress in physics. These diagrams represent the underlying mathematical operations and aid physical interpretation, but cannot generally be computed with directly. In this paper we introduce the ZXH-calculus, a graphical language based on the ZX-calculus, that we use to represent and reason about many-body states entirely graphically. As a demonstration, we express the one-dimensional (1D) AKLT-state, a symmetry protected topological state, in the ZXH-calculus by developing a representation of spins higher than 1/2 within the calculus. By exploiting the simplifying power of the ZXH-calculus rules we show how this representation straightforwardly recovers the AKLT matrix-product state representation, the existence of topologically protected edge states, and the nonvanishing of a string-order parameter. Extending beyond these known properties, our diagrammatic approach also allows us to analytically derive that the Berry phase of any finite-length 1D AKLT chain is π. In addition, we provide an alternative proof that the two-dimensional (2D) AKLT-state on a hexagonal lattice can be reduced to a graph state, demonstrating that it is a universal quantum-computing resource. Lastly, we build 2D higher-order topological phases diagrammatically, which we use to illustrate a symmetry-breaking phase transition. Our results show that the ZXH-calculus is a powerful language for representing and computing with physical states entirely graphically, paving the way to develop more efficient many-body algorithms and giving a novel diagrammatic perspective on quantum phase transitions.
Citation
East, R. D., van de Wetering, J., Chancellor, N., & Grushin, A. G. (2022). AKLT-States as ZX-Diagrams: Diagrammatic Reasoning for Quantum States. PRX Quantum, 3(1), https://doi.org/10.1103/prxquantum.3.010302
Journal Article Type | Article |
---|---|
Acceptance Date | Dec 7, 2021 |
Online Publication Date | Jan 4, 2022 |
Publication Date | 2022 |
Deposit Date | Feb 22, 2022 |
Publicly Available Date | Feb 22, 2022 |
Journal | PRX Quantum |
Electronic ISSN | 2691-3399 |
Publisher | American Physical Society |
Peer Reviewed | Peer Reviewed |
Volume | 3 |
Issue | 1 |
DOI | https://doi.org/10.1103/prxquantum.3.010302 |
Files
Published Journal Article
(21.6 Mb)
PDF
Publisher Licence URL
http://creativecommons.org/licenses/by/4.0/
Copyright Statement
Published by the American Physical Society under the terms of
the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the
author(s) and the published article’s title, journal citation, and
DOI.
You might also like
Graphical structures for design and verification of quantum error correction
(2023)
Journal Article
Using copies can improve precision in continuous-time quantum computing
(2023)
Journal Article
Comparing the hardness of MAX 2-SAT problem instances for quantum and classical algorithms
(2023)
Journal Article
Controller-Based Energy-Aware Wireless Sensor Network Routing Using Quantum Algorithms
(2022)
Journal Article
Modernizing quantum annealing II: genetic algorithms with the inference primitive formalism
(2022)
Journal Article
Downloadable Citations
About Durham Research Online (DRO)
Administrator e-mail: dro.admin@durham.ac.uk
This application uses the following open-source libraries:
SheetJS Community Edition
Apache License Version 2.0 (http://www.apache.org/licenses/)
PDF.js
Apache License Version 2.0 (http://www.apache.org/licenses/)
Font Awesome
SIL OFL 1.1 (http://scripts.sil.org/OFL)
MIT License (http://opensource.org/licenses/mit-license.html)
CC BY 3.0 ( http://creativecommons.org/licenses/by/3.0/)
Powered by Worktribe © 2024
Advanced Search