Figure 1

Original formulation of weighted graphs for ideal CV cluster states. Ideal CV cluster states are represented by undirected graphs with real-weighted edges [4, 5, 6, 14, 21, 22, 23, 24]. (Unweighted graphs are a special case with all weights equal to 1.) Each graph uniquely defines a recipe (i.e., a quantum circuit) for creating a CV cluster state, as illustrated: (1) each node represents a state that is infinitely squeezed in the $\mathrm{p̂}$ quadrature $|0\rangle {}_p$; and (2) ${\mathrm{C}}_Z$ gates are applied between modes in accordance with the graph, with the weight $g$ of an edge corresponding to the strength of the interaction ${\stackrel{\wedge }{\mathrm{C}}}_Z(g)=e^{\mathit{ig}\mathrm{q̂}\otimes \mathrm{q̂}}$ between the two nodes connected. These states are unphysical because they cannot be normalized. Instead they are approximated in physical applications by very highly squeezed states.Reuse & Permissions

Figure 2

Complex-weighted graph representation of approximate CV cluster states. Nodes no longer specify any particular input state on their own and are henceforth colored black to emphasize this distinction. Instead, the state represented by the graph is entirely specified by the edge weights, which can now be complex. The real part of the graph, $\mathbf{V},$ still has the same interpretation as in the original formalism, i.e., a collection of weights for the respective $C_Z$ gates applied between the linked nodes, while the imaginary part, $\mathbf{U},$ corresponds to initial multimode squeezing that only mixes $\mathrm{q̂}$s and $\mathrm{p̂}$s separately (see text). (a) The graph from Fig. 1 is reinterpreted in this formalism. Because the graph has only real weights, it does not represent a physical state (since $\mathbf{U}>/0$)—and rightly so since ideal CV cluster states are infinitely squeezed. (b) Graph for an approximate CV cluster state made by the canonical method [5, 6]. The imaginary-weighted self-loops indicate the amount of initial single-mode squeezing applied. In the limit of large squeezing ($r\to \infty$), these physical Gaussian pure states limit to the ideal graph in (a). Note that in this context, states with large squeezing in $\mathrm{p̂}$ are represented by black nodes with a vanishing imaginary self-loop weight, rather than by the white nodes in Fig. 1. (c) Another approximate CV cluster state, distinct from that in (b), that also limits to the ideal case (a) when $\alpha \to \infty$. States represented in (b) and (c) are physically distinct, but because they have the same large-squeezing behavior, they cannot be distinguished in the original graphical formalism (see Fig. 1). This new formalism allows them to be distinguished at the graph level.Reuse & Permissions