Low-depth unitary quantum circuits for dualities in one-dimensional quantum lattice models

TL;DR

This paper develops low-depth quantum circuits implementing dualities in 1D quantum lattice models, enabling efficient state preparation and boundary mapping, with potential applications in topological quantum computation.

Contribution

It introduces a method to convert duality operators into low-depth quantum circuits using ancillary degrees of freedom, including constant depth circuits for certain symmetry dualities.

Findings

Duality operators can be implemented as linear depth quantum circuits.

With measurements, dualities related to nilpotent fusion categories can be realized in constant depth.

Circuits can prepare entangled states and map between gapped boundaries in topological models.

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 non-trivial way in which dualities intertwine closed boundary conditions and charge sectors, these can be implemented by unitary matrix product operators. In this manuscript, 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 realised 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.