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

Ellison, Tyler D.; Kato, Kohtaro; Liu, Ziwen; Hsieh, Timothy H.

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

Cited by 19 publications

(7 citation statements)

References 94 publications

(210 reference statements)

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“…There is some equivocation on terminology here: in particular, the definition of "long-range magic" in[55] differs from the definition in this paper.…”

mentioning

confidence: 83%

“…It is an open question if there is a suitable linear combination of magic monotones to unambiguously detect long-range magic (akin to the Kitaev-Preskill/Levin-Wen measure of long-range entanglement [53,54]), in more physically relevant systems, however very recent work indicates that "long-range magic" 17 is a useful diagnostic of symmetry protected and many body phases [55,56]. Developing continuum effective field theory descriptions of these investigations is an interesting avenue for future research.…”

Section: Diagnosing Topologically Insulating Phasesmentioning

confidence: 99%

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Knots, links, and long-range magic

Fliss

2020

Preprint

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We study the extent to which knot and link states (that is, states in 3d Chern-Simons theory prepared by path integration on knot and link complements) can or cannot be described by stabilizer states. States which are not classical mixtures of stabilizer states are known as "magic states" and play a key role in quantum resource theory. By implementing a particular magic monotone known as the "mana" we quantify the magic of knot and link states. In particular, for SU (2) k Chern-Simons theory we show that knot and link states are generically magical. For link states, we further investigate the mana associated to correlations between separate boundaries which characterizes the state's long-range magic. Our numerical results suggest that the magic of a majority of link states is entirely long-range. We make these statements sharper for torus links.

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“…There is some equivocation on terminology here: in particular, the definition of "long-range magic" in[55] differs from the definition in this paper.…”

mentioning

confidence: 83%

Section: Diagnosing Topologically Insulating Phasesmentioning

confidence: 99%

Knots, links, and long-range magic

Fliss

2020

Preprint

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“…As a, b can be arbitrarily far apart, the MIE does not decay. This is the "quantum wire" property of 1d SPTs [13], a manifestation of a symmetry-protected sign problem in the symmetry charge (Z) basis [10].…”

Section: A Bound On Mie For Sign-free Stabilizer Statesmentioning

confidence: 99%

“…In Ref. [10], it was shown that many symmetry-protected topological (SPT) phases have a symmetry-protected sign problem, which no symmetric finite depth circuit can remove. Moreover, the sign structure of highly excited and random states is intimately connected to their volume law entanglement scaling [11].…”

Section: Introductionmentioning

confidence: 99%

Probing sign structure using measurement-induced entanglement

Lin¹,

Ye²,

Zou³

et al. 2022

Preprint

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The sign structure of quantum states is closely connected to quantum phases of matter, yet detecting such fine-grained properties of amplitudes is subtle. Here we employ as a diagnostic measurement-induced entanglement (MIE)-the average entanglement generated between two parties after measuring the rest of the system. We propose that for a sign-free state, the MIE upon measuring in the sign-free basis decays no slower than correlations in the state before measurement. Concretely, we prove that MIE is upper bounded by mutual information for sign-free stabilizer states (essentially CSS codes), which establishes a bound between scaling dimensions of conformal field theories describing measurement-induced critical points in stabilizer systems. We also show that for sign-free qubit wavefunctions, MIE between two qubits is upper bounded by a simple two-point correlation function, and we verify our proposal in several critical ground states of one-dimensional systems, including the transverse field and tri-critical Ising models. In contrast, for states with sign structure, such bounds can be violated, as we illustrate in critical hybrid circuits involving both Haar or Clifford random unitaries and measurements, and gapless symmetry-protected topological states.

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“…Firstly, numerical approaches that can handle the large system sizes needed for exploring quantum criticality require a sign-problem-free realization. Indeed, certain phases are known to have an intrinsic sign problem [55][56][57][58]. Secondly, in the absence of large continuous symmetry groups, direct continuous transitions are often interrupted by direct first order transitions or intervened by intermediate phases [59][60][61] (which can nevertheless be exotic such as the gapless stripe phase reported in Ref.…”

Section: Introductionmentioning

confidence: 99%

Building models of topological quantum criticality from pivot Hamiltonians

et al. 2023

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Progress in understanding symmetry-protected topological (SPT) phases has been greatly aided by our ability to construct lattice models realizing these states. In contrast, a systematic approach to constructing models that realize quantum critical points between SPT phases is lacking, particularly in dimension d>1d>1. Here, we show how the recently introduced notion of the pivot Hamiltonian—generating rotations between SPT phases—facilitates such a construction. We demonstrate this approach by constructing a spin model on the triangular lattice, which is midway between a trivial and SPT phase. The pivot Hamiltonian generates a U(1)U(1) pivot symmetry which helps to stabilize a direct SPT transition. The sign-problem free nature of the model—with an additional Ising interaction preserving the pivot symmetry—allows us to obtain the phase diagram using quantum Monte Carlo simulations. We find evidence for a direct transition between trivial and SPT phases that is consistent with a deconfined quantum critical point with emergent SO(5)SO(5) symmetry. The known anomaly of the latter is made possible by the non-local nature of the U(1)U(1) pivot symmetry. Interestingly, the pivot Hamiltonian generating this symmetry is nothing other than the staggered Baxter-Wu three-spin interaction. This work illustrates the importance of U(1)U(1) pivot symmetries and proposes how to generally construct sign-problem-free lattice models of SPT transitions with such anomalous symmetry groups for other lattices and dimensions.

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

Cited by 19 publications

References 94 publications

Knots, links, and long-range magic

Knots, links, and long-range magic

Probing sign structure using measurement-induced entanglement

Building models of topological quantum criticality from pivot Hamiltonians

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