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Quantum vs. Classical Routing: A Hardware-Aware Study of QAOA and Dijkstra with Constrained Shortest Path and Constraint-Preserving Mixers

Quang Bui

Published Jan 1, 2026• 0 citations
Publisher
DOI
10.2139/ssrn.6386198

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•Mar 1, 2009•12 citations
Transportation routing problems such as the Traveling Salesperson Problem (TSP) and the Vehicle Routing Problem (VRP) are characterized by strict feasibility requirements involving customer assignment and visit rules, route sequencing, and depot-return logic alongside cost minimization. Most quantum routing formulations adopt Quadratic Unconstrained Binary Optimization (QUBO) encodings, where feasibility is incorporated indirectly via penalty terms in the cost Hamiltonian. While convenient for standard implementations of the Quantum Approximate Optimization Algorithm (QAOA), QUBO encodings allow the quantum search dynamics to allocate substantial probability to infeasible route configurations. This study develops a transportation-grounded constraint-aware Quantum Alternating Operator Ansatz (QAOA+) framework that embeds feasibility-preserving logic directly into the search operator. We introduce a custom mixer that functions as a quantum analogue of feasibility-preserving routing neighborhoods, using column-wise swap moves, it restricts evolution to feasible configurations while enabling structured exploration of valid routes. We compare three constraint-handling architectures: penalty-based QUBO QAOA, penalty free QAOA+ with the feasibility-preserving mixer, and a Hybrid QAOA+ combining mixer based feasibility with and penalty guidance. Results on small TSP and VRP instances show that constraint-handling architecture strongly influences feasible-route sampling, convergence behavior, and probability concentration over low-cost feasible routes. These findings position constraint-aware quantum search as a methodological extension of transportation routing search approaches, where feasibility is enforced through admissible quantum transitions rather than post-hoc penalties.
Routing in wireless communication networks is shaped by mobility, interference, congestion, and competing service requirements, making route selection a high-dimensional constrained optimization problem rather than a simple shortest-path task. This paper investigates the use of hybrid classical--quantum methods for wireless routing, focusing on the Quantum Approximate Optimization Algorithm (QAOA) and quantum walks as candidate mechanisms for exploring complex routing spaces. The paper examines how wireless routing can be expressed as a constrained graph optimization problem in which routing objectives, flow constraints, connectivity requirements, and interference effects are mapped into quantum-compatible Hamiltonian representations. It then discusses how these approaches can be integrated into a hybrid architecture in which classical systems perform network monitoring, graph construction, pre-processing, and deployment, while quantum subroutines are used for selected optimization components. The analysis shows that the potential value of quantum routing lies primarily in the treatment of difficult combinatorial subproblems rather than end-to-end replacement of classical routing frameworks. The paper also highlights practical limitations arising from state preparation, constraint encoding, oracle construction, hardware noise, limited qubit resources, and hybrid execution overhead. It is argued that any meaningful near-term advantage will depend on careful problem decomposition, compact encoding, and tight classical--quantum integration.
Classical algorithms have been used to search over some space for finding the shortest paths problem between two points in a network and a minimal weight spanning tree for routing. Any classical algorithm deterministic or probabilistic will clearly used O(N) steps since on the average it will measure a large fraction of N records. Quantum algorithm is the fastest possible algorithm that can do several operations simultaneously due to their wave like properties. This wave gives an O( N ) steps quantum algorithm for identifying that record, where was used classical Dijkstra’s algorithm for finding shortest path problem in the graph of network and implement quantum search. Also we proposed the structure for non-classical algorithms and design the various phases of the probabilistic quantum-classical algorithm for classical and quantum parts. Finally, we represent the result of implementing and simulating Dijkstra's algorithm as the probabilistic quantum-classical algorithm.