Entanglement of random hypergraph states

TL;DR

This paper analyzes the entanglement properties of random hypergraph states generated by CZ and CCZ gates, revealing differences in entanglement fluctuation and implications for quantum chaos.

Contribution

It provides an analytical comparison of entanglement features in hypergraph states generated by different controlled-phase gates, highlighting their typicality and complexity.

Findings

Both ensembles exhibit volume law for entanglement entropy.

CCZ ensemble has small, universal purity fluctuation.

CZ ensemble shows large purity fluctuation.

Abstract

Random quantum states and operations are of fundamental and practical interests. In this work, we investigate the entanglement properties of random hypergraph states, which generalize the notion of graph states by applying generalized controlled-phase gates on an initial reference product state. In particular, we study the two ensembles generated by random Controlled-Z(CZ) and Controlled-Controlled-Z(CCZ) gates, respectively. By applying tensor network representation and combinational counting, we analytically show that the average subsystem purity and entanglement entropy for the two ensembles feature the same volume law, but greatly differ in typicality, namely the purity fluctuation is small and universal for the CCZ ensemble while it is large for the CZ ensemble. We discuss the implications of these results for the onset of entanglement complexity and quantum chaos.