An invariant of topologically ordered states under local unitary transformations

TL;DR

This paper introduces an invariant related to topologically ordered states that can be derived from ground state wave functions, providing insights into their braiding properties and entanglement structure, with implications for quantum circuit complexity.

Contribution

It defines a Hamiltonian-independent invariant equivalent to the topological S-matrix, applicable to ground states without degeneracy, and uses it to establish lower bounds on quantum circuit depth.

Findings

Invariant can be computed from any Hamiltonian with the given ground state.

Any local quantum circuit connecting certain ground states must have linear depth.

Introduces locally invisible operators as a tool for entanglement detection.

Abstract

For an anyon model in two spatial dimensions described by a modular tensor category, the topological S-matrix encodes the mutual braiding statistics, the quantum dimensions, and the fusion rules of anyons. It is nontrivial whether one can compute the S-matrix from a single ground state wave function. Here, we define a class of Hamiltonians consisting of local commuting projectors and an associated matrix that is invariant under local unitary transformations. We argue that the invariant is equivalent to the topological S-matrix. The definition does not require degeneracy of the ground state. We prove that the invariant depends on the state only, in the sense that it can be computed by any Hamiltonian in the class of which the state is a ground state. As a corollary, we prove that any local quantum circuit that connects two ground states of quantum double models (discrete gauge theories) with non-isomorphic abelian groups, must have depth that is at least linear in the system's diameter. As a tool for the proof, a manifestly Hamiltonian-independent notion of locally invisible operators is introduced. This gives a sufficient condition for a many-body state not to be generated from a product state by any small depth quantum circuit; this is a many-body entanglement witness.