A graph-based quantum-inspired approximation of the Feynman path integral

Introduction: We introduce a novel quantum-inspired simulator that approximates the Feynman path integral within a discrete, graph-theoretic framework, mapping logical qubits to graph vertices. Materials and methods: The entire pathfinding process is driven by gate operations, forming the core of our unique, three-tier search: (1) classical path probability boosting to identify primary routes, (2) Feynman-inspired vertex amplification to weight critical nodes, and (3) complex amplitude calculations, based on logical qubit interactions (edge-weighted), which are crucial for reproducing quantum interference. Results: While methods (1) and (2) identify the shortest classical paths, only the third tier successfully approximates superposition and interference effects through complex amplitude summation, where distinct phase contributions lead to either amplification or suppression. We show that interference patterns align with pairwise Pauli-Z correlations that effectively emulate entanglement phenomena. High gate dynamism, especially 4-qubit gate operations, are essential for the broad search of optimal routes, while boundary conditions may limit continuous exploration and refinement of pathways, emphasizing the need for fine parameter tuning. Simulations achieve convergence, with entropy plateaus confirming stability given a sufficient tolerance threshold for systems up to 8 vertices. The temporal evolution of vertex occurrence counts exhibits Gamma–Poisson (Cox process) statistics, with variance-to-mean ratios increasing over time, indicating non-Poissonian dynamics. A key finding is that quantum pathfinding, driven by destructive interference, effectively reveals dominant paths that may not be classically apparent. The observed destructive interference directly results from the summation of complex amplitudes where the phase differences equal approximately π radians (180), revealing complementary path and “anti-path” structures. Conclusions: The novelty of our simulator lies in its dual vertex-wave nature that provides new insights for quantum algorithm design and the study of quantum dynamics. Beyond the core framework, structural and mathematical analogies emerging from Simulation H—including a correlation structure consistent with a light-cone pattern—are reported as posthoc observations of the simulation output. These are not physical derivations, but suggest directions for further investigation at the intersection of graph-theoretic quantum simulation and relativistic quantum mechanics.