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arXiv:2112.12757 · Physical review. B./Physical review. B

Mapping between Morita-equivalent string-net states with a constant depth quantum circuit

Laurens Lootens, Bram Vancraeynest-De Cuiper, Norbert Schuch, Frank Verstraete

Published Feb 17, 2022• 24 citations
arXivHTMLPDF

Abstract

We construct a constant depth quantum circuit that maps between Morita-equivalent string-net models. Due to its constant depth and unitarity, the circuit cannot alter the topological order, which demonstrates that Morita-equivalent string nets are in the same phase. The circuit is constructed from an invertible bimodule category connecting the two input fusion categories of the relevant string-net models, acting as a generalized Fourier transform for fusion categories. The circuit not only acts on the ground state subspace but also acts unitarily on the full Hilbert space when supplemented with ancillas.

DOI
10.1103/physrevb.105.085130

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We construct a constant depth quantum circuit that maps between Morita-equivalent string-net models. Due to its constant depth and unitarity, the circuit cannot alter the topological order, which demonstrates that Morita-equivalent string nets are in the same phase. The circuit is constructed from an invertible bimodule category connecting the two input fusion categories of the relevant string-net models, acting as a generalized Fourier transform for fusion categories. The circuit not only acts on the ground state subspace but also acts unitarily on the full Hilbert space when supplemented with ancillas.
We prove the conjectured classification of topological phases in two spatial dimensions with gappable boundary, in a simplified setting. Two gapped ground states of lattice Hamiltonians are in the same quantum phase of matter, or topological phase, if they can be connected by a constant-depth quantum circuit. It is conjectured that the Levin-Wen string-net models exhaust all possible gapped phases with gappable boundary, and these phases are labeled by unitary modular tensor categories. We prove this under the assumption that every phase has a representative state with zero correlation length satisfying the entanglement bootstrap axioms, or a strict form of area law. Our main technical development is to transform these states into string-net states using constant-depth quantum circuits.
A systematic approach to dualities in symmetric (1+1)d quantum lattice models has recently been proposed in terms of module categories over the symmetry fusion categories. By characterizing the nontrivial way in which dualities intertwine closed boundary conditions and charge sectors, these can be implemented by unitary matrix product operators. In this Letter, we explain how to turn such duality operators into unitary linear depth quantum circuits via the introduction of ancillary degrees of freedom that keep track of the various sectors. The linear depth is consistent with the fact that these dualities change the phase of the states on which they act. When supplemented with measurements, we show that dualities with respect to symmetries encoded into nilpotent fusion categories can be realized in constant depth. The resulting circuits can for instance be used to efficiently prepare short- and long-range entangled states or map between different gapped boundaries of (2+1)d topological models.
We exhibit a mapping identifying Kitaev's quantum double lattice models explicitly as a subclass of Levin and Wen's string-net models via a completion of the local Hilbert spaces with auxiliary degrees of freedom. This identification allows one to carry over to these string-net models the representation-theoretic classification of the excitations in quantum double models as well as define them in arbitrary lattices and provides an illustration of the abstract notion of Morita equivalence. The possibility of generalizing the map to broader classes of string nets is considered.
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The finding of physical realizations of topologically ordered states in experimental settings, from condensed matter to artificial quantum systems, has been the main challenge en route to utilizing their unconventional properties.We show how to realize a large class of topologically ordered states and simulate their quasiparticle excitations on a digital quantum computer.To achieve this, we design a set of linear-depth quantum circuits to generate ground states of general string-net models together with unitary open-string operators to simulate the creation and braiding of Abelian and non-Abelian anyons.We show that the Abelian (non-Abelian) unitary string operators can be implemented with a constant-(linear-) depth quantum circuit.Our scheme allows us to directly probe characteristic topological properties, including topological entanglement entropy, braiding statistics, and fusion channels of anyons.Moreover, this set of efficiently prepared topologically ordered states has potential applications in the development of fault-tolerant quantum computers.
We study the condensation of Abelian bosons in string-net models by constructing a family of Hamiltonians that can be tuned through any such transition. We show that these Hamiltonians admit two exactly solvable, string-net limits: one deep in the uncondensed phase, described by an initial, uncondensed string-net Hamiltonian, and one deep in the condensed phase, described by a final, condensed string-net model. We give a systematic description of the condensed string-net model in terms of the uncondensed string-net and the data associated with the condensing Abelian bosons. Specifically, if the uncondensed string-net is described by a fusion category C\mathcal{C}C, we show how the string labels and fusion data of the fusion category $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{C}}$ describing the condensed string-net can be obtained from that of C\mathcal{C}C and the data describing the string operators that create the condensing boson. This construction generalizes previous approaches to anyon condensation in string-nets by allowing the condensation of arbitrary Abelian bosons, including chiral bosons in string-nets constructed from, for example, Chern-Simons theories, which describe time-reversal invariant bilayer states. This gives a method for obtaining the full data for string-nets without explicit time-reversal symmetry from such bilayer models. We illustrate our approach with several examples.