Graph states and local unitary transformations beyond local Clifford operations

TL;DR

This paper explores the effects of non-Clifford local unitary operations on graph and hypergraph states, revealing new equivalences and counterexamples to the LU-LC conjecture with a systematic approach.

Contribution

It introduces a graphical hypergraph description for certain non-Clifford transformations and systematically constructs local unitary equivalent states that are not locally Clifford equivalent.

Findings

Identified hypergraph states equivalent to graph states under local unitaries.

Constructed pairs of graph states LU-equivalent but not LC-equivalent.

Provided a systematic method reproducing known counterexamples to the LU-LC conjecture.

Abstract

Graph states are quantum states that can be described by a stabilizer formalism and play an important role in quantum information processing. We consider the action of local unitary operations on graph states and hypergraph states. We focus on non-Clifford operations and find for certain transformations a graphical description in terms of weighted hypergraphs. This leads to the indentification of hypergraph states that are locally equivalent to graph states. Moreover, we present a systematic way to construct pairs of graph states which are equivalent under local unitary operations, but not equivalent under local Clifford operations. This generates counterexamples to a conjecture known as LU-LC conjecture. So far, the only counterexamples to this conjecture were found by random search. Our method reproduces the smallest known counterexample as a special case and provides a physical interpretation.