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

Tyler D. Ellison; Kohtaro Kato; Zi-Wen Liu; Timothy H. Hsieh

doi:10.22331/q-2021-12-28-612

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

Tyler D. Ellison^1,2, Kohtaro Kato^3,4, 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

Published:	2021-12-28, volume 5, page 612
Eprint:	arXiv:2010.13803v4
Doi:	https://doi.org/10.22331/q-2021-12-28-612
Citation:	Quantum 5, 612 (2021).

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Abstract

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.

► BibTeX data

@article{Ellison2021symmetryprotected,
  doi = {10.22331/q-2021-12-28-612},
  url = {https://doi.org/10.22331/q-2021-12-28-612},
  title = {Symmetry-protected sign problem and magic in quantum phases of matter},
  author = {Ellison, Tyler D. and Kato, Kohtaro and Liu, Zi-Wen and Hsieh, Timothy H.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {5},
  pages = {612},
  month = dec,
  year = {2021}
}

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