Quantum Graphical Models

• Quantum Graphical Models (QGMs) are a framework that quantizes graph relations by encoding classical graphs as quantum states in a Hilbert space.
• They utilize an occupation graph basis with ladder operators and symmetrization to construct both labeled and unlabeled quantum graphs.
• QGMs reveal thermodynamic phase transitions and critical phenomena, highlighting nontrivial quantum statistical mechanics in relational systems.

A dynamical quantum multigraph is a quantum mechanical generalization of a finite multigraph in which the edges (and potentially vertices) are elevated from static classical relations to quantum degrees of freedom, represented as basis states in a Hilbert space. This framework enables the treatment of the graphs themselves—including both labeled and unlabeled cases—as dynamical variables rather than as mere descriptors of interactions among underlying quantum systems. The quantization of graphs provides a microscopic foundation for treating relational structures in quantum theory, and supports the paper of their thermodynamics and collective phenomena, such as phase transitions. These constructions are directly relevant to the development and understanding of Quantum Graphical Models (QGMs), embedding the relations themselves as quantum observables.

1. Quantization of Labeled and Unlabeled Multigraphs

In the quantization scheme, a finite (possibly simple) undirected multigraph with maximum edge count is encoded into a Hilbert space via the following prescription. For N vertices, every unordered pair (i, j) is associated with a “single-particle” Hilbert space H_{ij} of dimension D (for simple graphs, D=2). The total Hilbert space for labeled graphs is: $H_{\mathrm{MG}} = \bigotimes_{j>i} H_{ij}$ with total dimension $2^{N(N-1)/2}$ in the undirected, simple (D=2) case. Each basis vector corresponds directly to a choice of occupation numbers (edge multiplicities) for each potential edge, i.e., the adjacency matrix entries are quantized.

Moving from labeled to unlabeled graphs requires projecting onto S_N-invariant subspaces, where S_N is the symmetric group acting on the vertex labels. The projection identifies states related by vertex relabeling, such that

$$\pi |G\rangle = |G\rangle \;\; \text{(symmetric case)}$$

for any permutation $\pi \in S_N$, or with a sign for the antisymmetric case. Thus, the quantum state space for unlabeled graphs consists of equivalence classes of labeled graphs under relabeling.

This framework allows both labeled and unlabeled quantum graphs to be rigorously described using quantum mechanical principles, generalizing classical combinatorial notions to Hilbert spaces of relations.

2. Hilbert Space Structure and Operators

A key structural element is the construction of an “occupation graph basis.” Here, the basis states are labeled by the occupation numbers of each edge, grouped by one-particle states (e.g., present or absent, for D=2). Ladder (raising and lowering) operators $L^+, L^-$ are defined for each edge as

$$L^+ |n\rangle = |n+1\rangle,\qquad L^- |n\rangle = |n-1\rangle$$

These operators enable any basis graph state to be generated from a “ground state” (all edges absent), building up more complex graphs via successive application.

For unlabeled cases, a projection operator (symmetrizer or antisymmetrizer) manifests as a sum over all label permutations, ensuring that the resulting quantum graphs are defined only up to isomorphism. Explicitly, a state in the unlabeled case is constructed as

$$|G_\mathrm{unlab}\rangle = \mathcal{S} |G_\mathrm{lab}\rangle$$

where $\mathcal{S}$ is the (anti)symmetrization operator.

This explicit Hilbert space structure encapsulates both topological and quantum features, making it suitable for describing quantum relational systems and forming the foundation for relational QGMs.

3. Thermodynamic Behavior and Hamiltonian Dynamics

The thermodynamics of quantum simple graphs is studied under two kinds of Hamiltonians:

a) Free Hamiltonian: Each edge contributes an energy determined by its occupation state: $$H_0 |G_0, G_1\rangle = (E_0 n_0 + E_1 n_1)|G_0, G_1\rangle$$ The partition function factorizes across edges, leading to a closed-form expression for labeled graphs: $$Z_N^l = \exp\left(-\beta J E_1 \frac{N(N-1)}{2}\right) \left(1 + \exp[\beta J (E_1 - E_0)]\right)^{N(N-1)/2}$$ This model is thermodynamically trivial for labeled graphs, matching the analytic free energy and absence of phase transitions observed in the Erdős–Rényi–Gilbert $G(N,p)$ ensemble.

b) Ising-type Hamiltonian: Inter-edge interactions are introduced via a Hamiltonian that counts the number of “angle subgraphs” (subgraphs of two edges sharing a vertex), or alternately sums over vertex degrees: $$H_\mathrm{Ising} = \sum_k E_k \left( \sum_{i=1}^N \binom{d_i^k}{2} \right )$$ This Hamiltonian can be mapped to an infinite-range Ising model on the line graph of the complete graph, introducing nontrivial correlations between edges.

These constructions make explicit use of quantum control over the relational structure, with the possibility of superposing and interacting different graph topologies.

4. Thermodynamic Phase Transitions in Unlabeled Quantum Graphs

A primary result is the distinction between the thermodynamic behavior of labeled and unlabeled quantum multigraphs. For labeled graphs, the free theory is equivalent to the classical Erdős–Rényi random graph model, with the edge occupation probability

$p = \frac{1}{1 + \exp( -\beta J \Delta E )}$

and a corresponding analytic (smooth) free energy curve, i.e., no true phase transition.

In contrast, for unlabeled quantum graphs, both in the free and ferromagnetic Ising models, true thermodynamic phase transitions are observed. Monte Carlo simulations evidence divergent specific heat and critical slowing down (diverging autocorrelation times) near a critical temperature, governed by the interplay between graph automorphism classes and energy statistics. The energy levels and statistical weights are sensitive to the underlying isomorphism class, leading to nonanalyticities in the free energy for the unlabeled ensemble.

As temperature is varied, the “typical” excited-state graph in the ensemble can undergo a transition from one with significant symmetry (large automorphism group) to an almost rigid structure (trivial automorphism group), accompanied by the emergence of a giant component.

5. Order Parameters and Critical Phenomena

The order parameter for these quantum graph phase transitions is the fraction of vertices lying in the largest connected component ($s_1$) of the excited-state graph $G_1$: $s_1 = \frac{|S_1|}{N}$ Here, $S_1$ denotes the largest connected subgraph in $G_1$. This parameter captures the emergence of a “giant component,” paralleling the percolation transition in random graphs, but—crucially—here it indicates a real thermodynamic phase transition, evidenced by singularities in thermodynamic quantities such as specific heat and variance of $s_1$.

Unlike the $G(N,p)$ model, where the percolation threshold is a structural transition (without a free energy singularity), the quantum unlabeled multigraph ensemble shows true thermodynamic criticality.

6. Relevance and Implications for Quantum Graphical Models

The quantization of multigraphs and the associated Hilbert space formalism provide a rigorous foundation for “relational” quantum theory, in which relations (edges) are dynamical quantum degrees of freedom on par with the states of particles (vertices). This aligns with the goals of Quantum Graphical Models (QGMs), where the emphasis extends beyond statistical correlations of fixed classical graphs to situations where the graph structure itself is quantum and dynamical.

The explicit use of occupation number bases, ladder operators, and symmetrization equips QGM constructions with the tools to:

• Encode superpositions and entangled superpositions of relational structures.
• Analyze quantum dynamics and collective behavior (e.g., phase transitions) arising from the quantum nature of relations.
• Bridge to statistical mechanical models, such as generalized quantum Ising models on line graphs, connecting QGM research to established statistical and condensed matter frameworks.

The phase phenomena found in unlabeled quantum graph ensembles suggest that relational degrees of freedom can drive nontrivial critical phenomena, potentially relevant not only in quantum information processing but also in models of emergent geometry (e.g., in background-independent quantum gravity).

In summary, the dynamical quantum multigraph approach advances both the mathematical foundations and applicative scope of quantum graphical models by promoting relational data to first-class quantum variables, enabling new types of collective phenomena—including genuine thermodynamic phase transitions unavailable in conventional (labeled) random graph models—and offering a route for the systematic paper of quantum correlations, information flow, and emergent structures in general QGMs.