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 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.
Highlights
Proposition 1 Any symmetryprotected topological (SPT) state belonging to a nontrivial group cohomology phase in D ≥ 2 dimensions protected by a G = Zm q symmetry has symmetry protected magic, if it is defined on q-dimensional qudits and the symmetry is represented by tensor products of Pauli operators
We have introduced the concepts of symmetryprotected magic and a symmetry-protected sign problem to facilitate the study of many-body magic and the sign structure of wave functions
Using the universal properties of certain nontrivial group cohomology phases in D ≥ 2 dimensions, we showed that the corresponding SPT states have symmetry-protected magic, assuming the symmetry is represented by products of Pauli operators
Summary
We introduce symmetry-protected magic and a symmetry-protected sign problem These simplified diagnostics of the complexity of a state allow us to make analytical statements about the structure of quantum information in quantum phases of matter. We contribute to the understanding of the quantum complexity of SPT states, by showing that certain SPT states have symmetry-protected magic and that some possess a symmetry-protected sign problem. The symmetry-protected sign problem, in contrast, informs us about the sign structure of SPT states and poses an obstruction to finding a non-negative representation through local symmetry-respecting basis changes. We state a number of conjectures, and in particular, we conjecture that states defined on qubits and belonging to the double semion phase have magic that is robust to arbitrary unitary local operations
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