Quantum Weighted Hypergraph Operator

Many learning problems depend on higher-order interactions among groups of variables. Hypergraphs make these interactions explicit, motivating quantum representations that preserve their structure and support recovery of their weights. We introduce the Quantum Weighted Hypergraph Operator (QWHO), a unified framework for encoding weighted hypergraphs into quantum circuits and decoding their interaction weights. Each hyperedge defines a gate built from a shared non-scalar local Hermitian matrix. This choice gives the circuit a common tensor spectral structure while preserving its hypergraph organization. From this structure, we derive an explicit inverse and exact weight recovery conditions for real, continuous-phase, and discrete weights. The operator acts on arbitrary input states; decoding uses exact classical descriptions of a known input-output state-vector pair with nonzero input amplitudes throughout the product eigenbasis. The same tensor factorization supports in-place state-vector simulation and decoding without constructing a dense unitary matrix. Reversible coordinate transformations further show how operator choice can express the same circuit action up to global phase with fewer nonzero hyperedge weights. Graph states, weighted hypergraph states, and commuting Pauli-Z gates follow from particular choices of the local matrix, weights, and input. Together, these results provide a common framework for constructing, recovering, and adapting quantum representations of higher-order interactions.