Sequential Circuit as Generalized Symmetry on Lattice

TL;DR

This paper explores how generalized symmetries on lattices can be represented and constrained by sequential quantum circuits, revealing distinctions between annihilable and unannihilable symmetries and their fusion properties.

Contribution

It establishes a framework connecting sequential circuits to generalized lattice symmetries, including non-invertible cases, and identifies constraints on their fusion and representation.

Findings

Sequential circuits fully determine annihilable symmetry actions.

Unannihilable symmetries require additional circuits for full description.

Matrix product and tensor network representations are crucial tools.

Abstract

Generalized symmetry extends the usual notion of symmetry to ones that are of higher-form, acting on subsystems, non-invertible, etc. The concept was originally defined in the field theory context using the idea of topological defects. On the lattice, an immediate consequence is that a symmetry twist is moved across the system by a sequential quantum circuit. In this paper, we ask how to obtain the full, potentially non-invertible symmetry action from the unitary sequential circuit and how the connection to sequential circuit constrains the properties of the generalized symmetries. We find that for symmetries that contain the trivial symmetry operator as a fusion outcome, which we call annihilable symmetries, the sequential circuit fully determines the symmetry action and puts various constraints on their fusion. In contrast, for unannihilable symmetries, like that whose corresponding twist is the Cheshire string, a further 1D sequential circuit is needed for the full description. Matrix product operator and tensor network operator representations play an important role in our discussion.