Dual correspondence between classical spin models and quantum Calderbank-Shor-Steane states

The correspondence between classical spin models and quantum states has attracted much attention in recent years. However, it remains an open problem as to which specific spin model a given (well-known) quantum state maps. Here, we provide such an explicit correspondence for an important class of quantum states where a duality relation is proved between classical spin models and quantum Calderbank-Shor-Steane (CSS) states. In particular, we employ graph-theoretic methods to prove that the partition function of a classical spin model on a hypergraph $H$ is equal to the inner product of a product state with a quantum CSS state on a dual hypergraph $\tilde{H}$. We next use this dual correspondence to prove that the critical behavior of the classical system corresponds to a relative stability of the corresponding CSS state to bit-flip (or phase-flip) noise, thus called critical stability. We finally conjecture that such critical stability is related to the topological order in quantum CSS states, thus providing a possible practical characterization of such states.

Figure 1

(a) A simple hypergraph where we use purple (dark) for vertices and yellow (light) for hyperedges. We use a loop for an edge containing only one vertex, a link for edges containing two vertices, and closed curves for edges containing more than two vertices. (b) Dual hypergraph where we use yellow (light) for vertices and purple (dark) for hyperedges and (c) orthogonal hypergraph of (a).

Figure 2

(a) A hypergraph corresponding to a simple spin model. Spin variables correspond to vertices and each edge refers to an interaction term. (b) Corresponding to each edge of the original hypergraph, a new vertex is assigned with a new spin variable $S_{\left(m\right)}$. A constraint on these variables is denoted by a closed curve. This new hypergraph is denoted $H_C$. (c) The dual of the original hypergraph is orthogonal to $H_C$.

Figure 3

(a) Plaquette and vertex operators of the TC are shown on a simple square lattice. (b) Corresponding to each vertex of the original graph we consider an edge of a hypergraph denoted by yellow (light) where the TC state can be redefined as a quantum CSS state on that hypergraph. (c) Since neighboring edges of the hypergraph have only one common vertex, the dual of the hypergraph is the same original graph.

Figure 4

Left: TC on a 3D lattice; each node of the lattice corresponds to a hyperedge of a hypergraph which is denoted by yellow (light). Right: In the dual space, a vertex of the $\tilde{H}$ denoted by yellow (light) circles corresponds to each edge of the $H$. Each edge of the $\tilde{H}$ denoted by a purple (dark) link involves two vertices.

Figure 5

(a) A hexagonal lattice is an example of 2-colexes where edges are three colorable. (b) Corresponding to each plaquette of the lattice we consider an edge of a hypergraph denoted by yellow (light) where the CC can be defined as a quantum CSS state on that hypergraph. (c) Dual of the hypergraph is a triangular lattice where three vertices of each triangle belong to an edge of the hypergraph.

Figure 6

A 3-colex where all vertices are tetravalent and all edges are colored by four different colors.

Figure 7

(a) Each vertex of the 3-colex is a tetravalent colored by four different colors. (b) Since each vertex is a member of four cells (or edges of the hypergraph) of the 3-colex, the dual of the 3-colex is a tetrahedron lattice.