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arXiv:2311.01439 · Physical Review Letters

Low-Depth Unitary Quantum Circuits for Dualities in One-Dimensional Quantum Lattice Models

Laurens Lootens, Clement Delcamp, Dominic J. Williamson, Frank Verstraete

Published Apr 4, 2025• 16 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.134.130403

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It has been a long-standing open problem to construct a general framework for relating the spectra of dual theories to each other. Here, we solve this problem for the case of one-dimensional quantum lattice models with symmetry-twisted boundary conditions. In Ref. [PRX Quantum 4, 020357], dualities are defined between (categorically) symmetric models that only differ in a choice of module category. Using matrix product operators, we construct from the data of module functors explicit symmetry operators preserving boundary conditions as well as intertwiners mapping topological sectors of dual models onto one another. We illustrate our construction with a family of examples that are in the duality class of the spin- 12 Heisenberg XXZ model. One model has symmetry operators forming the fusion category Rep(S3) of representations of the group S3 . We find that the mapping between its topological sectors and those of the XXZ model is associated with the nontrivial braided autoequivalence of the Drinfel’d center of Rep(S3) . Published by the American Physical Society 2024
Finite-depth quantum circuits preserve the long-range entanglement structure in quantum states and map between states within a gapped phase. To map between states of different gapped phases, we can use sequential quantum circuits, which apply unitary transformations to local patches, strips, or other subregions of a system in a sequential way. The sequential structure of the circuit, on the one hand, preserves entanglement area law and hence the gappedness of the quantum states. On the other hand, the circuit has generically a linear depth, hence, it is capable of changing the long-range correlation and entanglement of quantum states and the phase they belong to. In this paper, we systematically discuss the definition, basic properties, and prototypical examples of sequential quantum circuits that map product states to Greenberger-Horne-Zeilinger states, symmetry-protected topological states, intrinsic topological states, and fracton states. We discuss the physical interpretation of the power of the circuits through connection to condensation, Kramers-Wannier duality, and the notion of foliation for fracton phases.
We provide explicit constant-depth local unitary circuits that realize general anyon permutations in Kitaev's quantum double models. This construction can be naturally understood through a correspondence between anyon permutation symmetries of two-dimensional topological orders and self-dualities in one-dimensional systems, where local gates implement self-duality transformations on the boundaries of microscopic regions. From this holographic perspective, general anyon permutations in the D(G)D(G)D(G) quantum double correspond to compositions of three classes of one-dimensional self-dualities, including gauging of certain subgroups of GGG, stacking with GGG symmetry-protected topological phases, and outer automorphisms of the group GGG. We construct circuits realizing the first class by employing self-dual unitary gauging maps, and present transversal circuits for the latter two classes.
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.