Graph States Under the Action of Local Clifford Group in Non-Binary Case

TL;DR

This paper investigates the properties and equivalence classes of non-binary graph states under local Clifford group actions, providing bounds, transformations, and an efficient algorithm for state comparison.

Contribution

It extends the analysis of graph states to the non-binary case, establishing bounds and developing an algorithm for local equivalence verification.

Findings

Bound on the number of non-isomorphic, non-equivalent graph states

Transformation rules for non-binary graph states under local Clifford group

Efficient algorithm for verifying local equivalence of non-binary graph states

Abstract

Graph states are well-entangled quantum states that are defined based on a graph. Of course, if two graphs are isomorphic their associated states are the same. Also, we know local operations do not change the entanglement of quantum states. Therefore, graph states that are either isomorphic or equivalent under the local Clifford group have the same properties. In this paper, we first establish a bound on the number of graph states which are neither isomorphic nor equivalent under the action of local Clifford group. Also, we study graph states in non-binary case. We translate the action of local Clifford group, as well as measurement of Pauli operators, into transformations on their associated graphs. Finally, we present an efficient algorithm to verify whether two graph states, in non-binary case, are locally equivalent or not.