Stabilizer R\'enyi entropy of 3-uniform hypergraph states

TL;DR

This paper studies the nonstabilizerness of 3-uniform hypergraph states using stabilizer Rényi entropy, providing a computationally efficient method and evaluating SREs for various hypergraph states.

Contribution

It introduces a matrix rank-based expression for SRE of 3-uniform hypergraph states, significantly reducing computational complexity and enabling exact and numerical evaluations.

Findings

SRE of 3-uniform hypergraph states can be expressed via matrix rank.

The method reduces computational cost from exponential to polynomial in N.

Exact and numerical SRE values are obtained for various hypergraph states.

Abstract

Nonstabilizerness, also known as magic, plays a central role in universal quantum computation. Hypergraph states are nonstabilizer generalizations of graph states and constitute a key class of quantum states in various areas of quantum physics, such as the demonstration of quantum advantage, measurement-based quantum computation, and the study of topological phases. In this work, we investigate nonstabilizerness of 3-uniform hypergraph states, which are solely generated by controlled-controlled-Z gates, in terms of the stabilizer R\'{e}nyi entropy (SRE). We find that the SRE of 3-uniform hypergraph states can be expressed using the matrix rank, which reduces computational cost from $\mathcal{O}(2^{3N})$ to $\mathcal{O}(N^3 2^{N})$ for $N$-qubit states. Based on this result, we exactly evaluate SREs of one-dimensional hypergraph states. We also present numerical results of SREs of several large-scale 3-uniform hypergraph states. Our results would contribute to an understanding of the role of nonstabilizerness in a wide range of physical settings where hypergraph states are employed.