Topological phases with generalized global symmetries

TL;DR

This paper introduces lattice models for symmetry-protected topological phases with generalized global symmetries, demonstrating their properties and potential applications in quantum error correction.

Contribution

It provides explicit lattice constructions of SPT phases with $q$-form symmetries using multi-qubit phase gates, expanding understanding of topological phases with higher-dimensional charged excitations.

Findings

Models support coexisting excitations of different dimensions.

Protected boundary modes confirm non-triviality of models.

Applications to quantum error-correcting codes and fault-tolerant gates.

Abstract

We present simple lattice realizations of symmetry-protected topological (SPT) phases with $q$-form global symmetries where charged excitations have $q$ spatial dimensions. Specifically, we construct $d$ space-dimensional models supported on a $(d+1)$-colorable graph by using a family of unitary phase gates, known as multi-qubit control-$Z$ gates in quantum information community. In our construction, charged excitations of different dimensionality may coexist and form a short-range entangled state which is protected by symmetry operators of different dimensionality. Non-triviality of proposed models, in a sense of quantum circuit complexity, is confirmed by studying protected boundary modes, gauged models and corresponding gapped domain walls. We also comment on applications of our construction to quantum error-correcting codes, and discuss corresponding fault-tolerant logical gates.