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

Figures (2)

• Figure 1: Implementation of a general duality with symmetry-twisted closed boundary conditions as a linear depth circuit. Given a state $\ket{\psi}$ with a symmetry twisted boundary condition, we add ancillary states $\ket{\mathbb 1}$ and $\ket{+}$ as defined in the text, before sequentially acting with the duality gate on the physical degrees of freedom $\{Y_{\mathsf i-1/2}\}_{\mathsf i=1}^L$. We then act with a composition gate on the symmetry defect strand and the duality strand to its right, and compose the resulting duality strand with an opposite duality strand to obtain a symmetry twist.
• Figure 2: Implementation of a nilpotent duality as a constant depth circuit. Given an initial state with some symmetry twisted boundary condition, we begin by adding ancillary states $\ket{\mathbb 1}$ and $\ket{+}$ in between every physical site before acting with inverse composition gates, followed by the action of duality gates. Then, the boundary condition is composed with its neighboring duality strand while the corresponding multiplicity degree of freedom $i_1$ is measured. The first stage is concluded by composing all adjacent duality strands and measuring the resulting multiplicity degrees of freedom resulting in symmetry twists in $\mathcal{D}^\star_\mathcal{N} =: \mathcal{C}^{(0)}$. The second stage amounts to inserting new ancillary states to the left of the symmetry twists via \ref{['eq:cupC']}, before acting with symmetry gates. The second stage is concluded by composing all adjacent symmetry strands and measuring the resulting multiplicity degrees of freedom, resulting almost exclusively in symmetry twists in $\mathcal{C}^{(1)}$. Repeating this process $n-1$ more times results in trivial symmetry twists---apart from the leftmost one---labeled by the single simple object in $\mathcal{C}^{(n)} = \mathsf{Vec}$.