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

Tyler D. Ellison1,2, Kohtaro Kato3,4, Zi-Wen Liu2, and Timothy H. Hsieh2

1University of Washington, Seattle, WA 98195, USA
2Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
3Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA
4Center for Quantum Information and Quantum Biology, Institute for Open and Transdisciplinary Research Initiatives, Osaka University, Osaka 560-8531, Japan

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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 $\mathbb{Z}_2 \times \mathbb{Z}_2$ SPT states (e.g. cluster state) have a symmetry-protected sign problem, and two-dimensional $\mathbb{Z}_2$ 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.

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[5] Nathanan Tantivasadakarn, Ryan Thorngren, Ashvin Vishwanath, and Ruben Verresen, "Building models of topological quantum criticality from pivot Hamiltonians", arXiv:2110.09512.

[6] Pureum Noh and Eun-Gook Moon, "Construction of Entangled Many-body States via the Higgs Mechanism", arXiv:2201.06585.

[7] Tyler D. Ellison, Yu-An Chen, Arpit Dua, Wilbur Shirley, Nathanan Tantivasadakarn, and Dominic J. Williamson, "Pauli stabilizer models of twisted quantum doubles", arXiv:2112.11394.

The above citations are from SAO/NASA ADS (last updated successfully 2022-01-23 15:03:05). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2022-01-23 15:03:03).