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Academia quantum.

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

B. Collignon

Published May 27, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.20935/acadquant8290

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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.
This paper explores the potential of quantum-inspired algorithms for graph search problems. Traditional random walk approaches often suffer from inefficiencies, particularly in complex graphs. We propose a novel framework that leverages concepts from quantum walks – specifically interference and tunneling – within a classical computational environment. Our algorithms dynamically adjust the probability distribution of node visits based on graph connectivity and the history of previous steps, mimicking the probabilistic nature of quantum mechanics. We demonstrate that this approach can, in certain scenarios, outperform standard random walks, offering a more adaptable and potentially faster search strategy. The core of the method lies in approximating the quantum effects through carefully designed classical simulations, without requiring actual quantum hardware. We present a mathematical formulation of the core algorithm and discuss its key parameters, focusing on the balance between classical simulation fidelity and computational cost. The results suggest a promising direction for graph search optimization, drawing inspiration from the principles of quantum mechanics.
We present a discrete graph-based model in which quantum interference emerges from bidirectional update dynamics over a finite history space. An action functional is defined over graph evolution histories, and complex amplitudes are assigned via phase weighting. Summation over histories produces constructive and destructive interference consistent with the Born probability rule. Numerical simulations recover classical statistics when backward updates are disallowed and yield Bell-type correlations violating classical bounds up to the Tsirelson limit. These results indicate that quantum behavior can arise from minimal relational structure without assuming spacetime geometry, wavefunctions, or continuous fields.
Classical simulation of quantum circuits is expensive for two distinct reasons. The obvious one is state-space size: an n-qubit system requires exponentially many amplitudes. The less obvious one is interference: useful output distributions emerge only after many computational histories have been coherently combined at common endpoints, and this aggregation step is itself a substantial source of cost. We introduce statistical interference sampling, a framework that makes this second bottleneck explicit by treating endpoint interference as a separately schedulable computation. Using the Chemical Abstract Machine (ChAM) as our model, weighted path contributions evolve as concurrent molecular species, and interference reactions combine contributions that share a common output state. A threshold rule terminates the process once an endpoint accumulates sufficient amplitude, discarding the remaining reactions. The method does not improve worst-case complexity and is not intended as a general-purpose simulator. Its purpose is to ask a more targeted question: how much of the interference calculation can be skipped while still recovering a useful output distribution? On benchmark circuits for Deutsch-Jozsa, Grover search, Simon's problem, and small Shor period-finding instances, we find that nearly 50% of endpoint interference reactions can be omitted while maintaining over 90% output accuracy for most algorithms tested. These results suggest that interference arithmetic is a structured resource that admits meaningful approximation, and that exposing it explicitly opens new opportunities for pruning strategies across path-sum, Pauli-path, and tensor-network simulation methods.
We provide a rigorous theoretical analysis of quantum algorithms for shortest path problems in weighted graphs, establishing complexity-theoretic foundations for quantum acceleration in this fundamental computational domain. Building upon recent quantum adaptations of classical graph algorithms, we develop a unified framework that characterizes when and why quantum approaches offer computational advantages for single-source and all-pairs shortest path computations. Our analysis encompasses both positive-weight and negative-weight scenarios, deriving conditional speedup theorems that depend on graph structure, weight distribution, and oracle access models. We prove that quantum amplitude amplification and quantum walks can provide polynomial to exponential speedups under specific structural assumptions, while establishing hardness results that identify problem instances where quantum advantage is provably absent. The theoretical framework introduces novel complexity measures that capture graph-theoretic properties relevant to quantum algorithm performance, including quantum diameter, entanglement capacity, and interference efficiency. We extend our analysis to dynamic graph scenarios where edge weights evolve over time, establishing bounds on quantum advantage preservation under graph modifications. This work contributes fundamental insights into quantum algorithmic graph theory, clarifying the boundaries of quantum computational advantage and providing theoretical guidance for algorithm design in network optimization, routing, and graph analytics applications.
We propose a novel and interdisciplinary conceptual framework that bridges Feynman’s path integral formulation of quantum mechanics with reinforcement learning (RL) in artificial intelligence. In the path integral approach, a quantum system does not follow a single predetermined trajectory but instead explores all possible paths simultaneously, assigning each a complex amplitude weighted by eiS/h, where S represents the classical action. Constructive and destructive interference across these paths acts as a natural filter, amplifying trajectories of stationary action and suppressing suboptimal ones, thereby leaving paths of least action as the observable outcomes. We argue this process is strongly analogous to a quantum agent evaluating an entire policy space in superposition, where interference effectively encodes a reward mechanism that eliminates non-optimal policies. This perspective not only deepens our understanding of light’s propagation in complex refractive media but also inspires the design of quantum-inspired Reinforcement learning architectures capable of leveraging the intrinsic parallelism of quantum mechanics. Furthermore, the advent of quantum computing, with its inherent properties of superposition, entanglement, and quantum interference, provides a tangible pathway for implementing such algorithms in practice. To illustrate this paradigm, we propose a toy model wherein policies are encoded as quantum states, rewards are mapped to phase shifts, and measurement collapses the superposed state into an optimal policy. The implications of this framework extend beyond algorithmic innovation, offering insights into the possibility that nature itself operates as a quantum learning system, with physical laws emerging from a process akin to reinforcement learning.
This paper explores the application of quantum walk algorithms to graph traversal and search problems. Classical algorithms often struggle with the computational complexity of traversing and searching large graphs, leading to significant time and resource limitations. Quantum walks, utilizing superposition and interference, offer a potential pathway to achieve exponential speedups. This work details the theoretical framework for applying quantum walks to graph problems, specifically focusing on traversal and search tasks. We present a generalized approach for mapping graph structures onto quantum walk systems and discuss the key mathematical formulations involved. The core of our investigation revolves around the construction of quantum walk operators that mimic the movement and interaction rules inherent in graph traversal. We analyze the evolution of the quantum walk state and derive expressions for the probability of reaching a specific node in the graph. Furthermore, we examine the impact of graph topology on the performance of the quantum walk, highlighting the potential for tailored algorithms to exploit specific graph structures. The results demonstrate the theoretical feasibility of quantum walk algorithms for significantly accelerating graph traversal and search operations, contributing to advancements in areas such as network analysis, database management, and artificial intelligence.