Encoding hypergraphs into quantum states

Ionicioiu and Spiller [Phys. Rev. A 85, 062313 (2012)] have recently presented an axiomatic framework for mapping graphs to quantum states of a suitable physical system. Based on their study, we first extend the axiomatic framework to hypergraphs by means of modifying its axioms and consistency conditions. Then we use the axiomatic approach to encode hypergraphs into a different family of quantum states, called the hypergraph states. Moreover, we also try to do the following: (i) show that real equally weighted states, which occur in Grover and Deutsch-Jozsa algorithms, are equivalent to hypergraph states; (ii) describe the relations among hypergraph states, graph states, and stabilizer states; (iii) provide some transformation rules, stated in purely hypergraph theoretical terms, which completely characterize the evolution of hypergraph states under some local operations, including operators in the Pauli group and some special local Pauli measurements; and (iv) investigate some properties of multipartite entanglement of hypergraph states by hypergraph theory.

Figure 1

Examples of hypergraphs. The hypergraphs (a)–(d) have the same vertex set $\{1,2,3,4\}$. The hypergraph $g_a$ in (a) has 4 hyperedges: $\left\{1\right\},\{2,3\},\{3,4\}$ and $\{1,2,3\}$. In (b), the hypergraph $g_b$ also has 4 hyperedges: $\left\{\Phi \right\},\left\{1\right\},\{3,4\}$ and $\{1,2,3\}$. Two hyperedges, i.e., $\{2,3\}$ and $\{3,4\}$, constitute the hyperedge set of $g_c$ in (c). The hypergraph $g_d$ in (d) has 2 hyperedges: $\left\{\Phi \right\}$ and $\{3,4\}$. The hypergraphs $g_c$ and $g_d$ are respectively corresponding to $g_a{-}_{+}\left\{1\right\}$ and $g_b{-}_{-}\left\{1\right\}$ while $g_b=g_a+\{\left\{\Phi \right\},\{2,3\}\}$.

Figure 2

The relations among hypergraph states, graph states and stabilizer states. Graph states constitute a subclass of hypergraph states or stabilizer states. The intersection of hypergraph states and stabilizer states of $n$ qubits is the set $\left\{\right|g\rangle |g\in {\Theta }_n\wedge \mathit{ran}\left(g\right)\le 2\}$.