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Symmetry-protected sign problem and magic in quantum phases of matter

Ellison, Tyler D. and Kato, Kohtaro and Liu, Zi-Wen and Hsieh, Timothy H. (2021) Symmetry-protected sign problem and magic in quantum phases of matter. Quantum, 5 . Art. No. 612. ISSN 2521-327X. doi:10.22331/q-2021-12-28-612.

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We introduce the concepts of a symmetry-protected sign problem and symmetry-protected magic to study the complexity of symmetry-protected topological (SPT) phases of matter. In particular, we say a state has a symmetry-protected sign problem or symmetry-protected magic, if finite-depth quantum circuits composed of symmetric gates are unable to transform the state into a non-negative real wave function or stabilizer state, respectively. We prove that states belonging to certain SPT phases have these properties, as a result of their anomalous symmetry action at a boundary. For example, we find that one-dimensional ℤ₂× ℤ₂ SPT states (e.g. cluster state) have a symmetry-protected sign problem, and two-dimensional ℤ₂ SPT states (e.g. Levin-Gu state) have symmetry-protected magic. Furthermore, we comment on the relation between a symmetry-protected sign problem and the computational wire property of one-dimensional SPT states. In an appendix, we also introduce explicit decorated domain wall models of SPT phases, which may be of independent interest.

Item Type:Article
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URLURL TypeDescription Paper ItemVideo
Ellison, Tyler D.0000-0002-1740-6889
Kato, Kohtaro0000-0003-3317-2004
Liu, Zi-Wen0000-0002-3402-9763
Hsieh, Timothy H.0000-0001-8187-7266
Additional Information:© 2021. This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. Published: 2021-12-28. We would like to acknowledge Tomotaka Kuwahara for helpful conversations related to Proposition 3, and we thank Sergey Bravyi for insights that led us to Lemma 2. TDE thanks Yu-An Chen, Kyle Kawagoe, Alex Kubica, and Beni Yoshida for clarifying discussions, and he is especially grateful to Lukasz Fidkowski and Nathanan Tantivasadakarn for providing feedback on the manuscript. TDE is also appreciative of the hospitality of Perimeter Institute, where this work was initiated. ZWL is supported by Perimeter Institute for Theoretical Physics. Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Economic Development, Job Creation and Trade. KK is supported by MEXT Quantum Leap Flagship Program (MEXT Q-LEAP) Grant Number JPMXS0120319794. TH acknowledges support from the Natural Sciences and Engineering Research Council of Canada (NSERC) through a Discovery Grant.
Group:Institute for Quantum Information and Matter
Funding AgencyGrant Number
Perimeter Institute for Theoretical PhysicsUNSPECIFIED
Department of Innovation, Science and Economic Development (Canada)UNSPECIFIED
Ontario Ministry of Economic Development, Job Creation and TradeUNSPECIFIED
Ministry of Education, Culture, Sports, Science and Technology (MEXT)JPMXS0120319794
Natural Sciences and Engineering Research Council of Canada (NSERC)UNSPECIFIED
Record Number:CaltechAUTHORS:20220202-382557100
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:113210
Deposited By: Tony Diaz
Deposited On:02 Feb 2022 16:40
Last Modified:02 Feb 2022 16:40

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