Magic of quantum hypergraph states

Abstract: Magic, or nonstabilizerness, characterizes the deviation of a quantum state from the set of stabilizer states and plays a fundamental role from quantum state complexity to universal fault-tolerant quantum computing. However, analytical or even numerical characterizations of magic are very challenging, especially in the multi-qubit system, even with a moderate qubit number. Here we systemically and analytically investigate the magic resource of archetypal multipartite quantum states -- quantum hypergraph states, which can be generated by multi-qubit Controlled-phase gates encoded by hypergraphs. We first give the magic formula in terms of the stabilizer R$\mathrm{\acute{e}}$nyi-$\alpha$ entropies for general quantum hypergraph states and prove the magic can not reach the maximal value, if the average degree of the corresponding hypergraph is constant. Then we investigate the statistical behaviors of random hypergraph states and prove the concentration result that typically random hypergraph states can reach the maximal magic. This also suggests an efficient way to generate maximal magic states with random diagonal circuits. Finally, we study some highly symmetric hypergraph states with permutation-symmetry, such as the one whose associated hypergraph is $3$-complete, i.e., any three vertices are connected by a hyperedge. Counterintuitively, such states can only possess constant or even exponentially small magic for $\alpha\geq 2$. Our study advances the understanding of multipartite quantum magic and could lead to applications in quantum computing and quantum many-body physics.