Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stabilizer Rényi entropy of 3-uniform hypergraph states

Daichi Kagamihara, Shunji Tsuchiya

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 Rényi 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.

Stabilizer Rényi entropy of 3-uniform 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 Rényi entropy (SRE). We find that the SRE of 3-uniform hypergraph states can be expressed using the matrix rank, which reduces computational cost from to for -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.
Paper Structure (11 sections, 3 theorems, 57 equations, 3 figures, 1 table)

This paper contains 11 sections, 3 theorems, 57 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Results
  4. Physical meaning
  5. Upper bound of SRE of 3-uniform hypergraph states
  6. Exact evaluation of SREs of one-dimensional hypergraph states
  7. Numerical evaluation of SRE of triangulated lattice hypergraph states
  8. Summary
  9. Review of the theory of quadratic forms over GF(2)
  10. Relation between the rank of $C+C^T$ and the non-degenerate subspace dimension of quadratic form $Q$
  11. Detailed calculation of $h_k$ and $\Delta_k$

Key Result

Theorem 1

The PL moment of a 3-uniform hypergraph state is given by where $2h(\bm{x})$ (always even) is the rank of the GF(2) symmetric matrix $C(\bm{x}) + C(\bm{x})^T$.

Figures (3)

  • Figure 1: (a) Union Jack and (b) triangular lattice hypergraph states for the linear length $L=3$. Both filled and unfilled dots denote qubits. CCZ operations are applied to each of the smallest triangles.
  • Figure 2: An example of the 3-uniform hypergraph state whose PL moment can be exactly calculated. We show the $N = 8$ case. Dots denote qubits and triangles represent CCZ operations.
  • Figure 3: Evaluated $\alpha=2$ SREs of Union Jack and triangular lattice hypergraph states. Dots and crosses with error bars represent SREs of Union Jack and triangular lattice hypergraph states, respectively. Solid and dashed lines show linear fits. The dotted line is the upper bounds of $\alpha=2$ SRE of these states, Eq. \ref{['eq:upper_bound_present_work']} with $\bar{h} = 2$.

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Lemma 2: 6.29 in Ref. Lidl_Niederreiter_1996
  • proof
  • Theorem 3: 6.30 in Ref. Lidl_Niederreiter_1996
  • proof