Hypergraph States in SU(N)1, N odd prime, Chern-Simons Theory

TL;DR

This paper explores the construction of hypergraph and graph states within SU(N)1 Chern-Simons theory, revealing topological preparation methods and restrictions based on the value of N mod 4.

Contribution

It establishes a topological framework for constructing hypergraph states in SU(N)1 Chern-Simons theory and identifies limitations for N ≡ 5 mod 4.

Findings

Hypergraph states can be constructed topologically in SU(N)1 for N ≠ 5 mod 4.

Only stabilizer states are preparable on the n-torus Hilbert space when N ≡ 5 mod 4.

The level-rank duality relates state construction to topological properties of the theory.

Abstract

Graph states and hypergraph states can be constructed from products of basic operations that appear in SU(N)1. The level-rank dual of a theorem of Salton, Swingle, and Walter implies that these operations can be prepared topologically in the n-torus Hilbert space of Chern-Simons theory for N neq 5 mod 4. For SU(N)1, N = 5 mod 4, only stabilizer states can be prepared on the n-torus Hilbert space, which restricts the construction to graph states.