ILP-Based QUBO Encodings of the Traveling Salesman Problem
Derivation and Benchmarking on Quantum Annealing Hardware
Abstract
The Traveling Salesman Problem (TSP) has been one of the most intensively studied problems in theoretical computer science for decades. Being NP-hard, it is unlikely to be solved in polynomial time; hence, despite numerous classical exact and approximate methods, quantum-inspired approaches are also explored. In this study, we analyze several formulations of the TSP suitable for quantum annealing, focusing not only on qubit requirements but also on practical performance, i.e., how frequently feasible and
optimal solutions are obtained. We consider both complete and non-complete graphs, reflecting common real-world scenarios.
One of the analyzed formulations corresponds to a previously established Quadratic Unconstrained Binary Optimization model, while the others originate from classical Integer Linear Programming formulations. We provide a unified, step-by-step translation
of selected compact ILP TSP formulations into QUBO models. We analytically show that for non-complete graphs these encodings are asymptotically smaller than the commonly used baseline formulation. We benchmark the formulations using simulated and quantum annealing to evaluate their feasibility and optimality rates. The results are discussed in the context of current quantum annealing hardware limitations, including QPU size and embedding overhead.
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