The Structure of Circle Graph States

TL;DR

This paper investigates the local equivalence and classical simulability of circle graph states, revealing their structural properties and connections to planar code states, and proves the computational hardness of counting LU-equivalent states.

Contribution

It proves that circle graph states are closed under local complementations, establishes a correspondence between bipartite circle graph states and planar code states, and shows counting LU-equivalent states is -hard.

Findings

Circle graph states are closed under r-local complementations.

Bipartite circle graph states correspond to planar code states.

Counting LU-equivalent graph states is -hard.

Abstract

Circle graph states are a structurally important family of graph states. The family's entanglement is a priori high enough to allow for universal measurement-based quantum computation (MBQC); however, MBQC on circle graph states is actually efficiently classically simulable. In this work, we paint a detailed picture of the local equivalence of circle graph states. First, we consider the class of all graph states that are local unitary (LU)-equivalent to circle graph states. In graph-theoretical terms, this LU-equivalence class is the set of all graphs reachable from the family of circle graphs by applying $r$-local complementations. We prove that the only graph states that are LU-equivalent to circle graph states are circle graph states themselves: circle graphs are closed under $r$-local complementation. Second, we show that bipartite circle graph states, i.e., 2-colorable circle graph states, are in one-to-one correspondence with planar code states, on which MBQC is known to be efficiently classically simulable. Leveraging this correspondence, we present alternative, simple proofs that (1) if a planar code state is LU-equivalent to a stabilizer state, they are in fact local Clifford (LC)-equivalent to it and that (2) MBQC on all circle graph states is efficiently classically simulable. Third and finally, we demonstrate that the problem of counting the number of graph states LU-equivalent to a given graph state is $\#\mathsf{P}$-hard.