Optimizing Tourism Routes: A Quantum Approach to the Profitable Tour Problem

The Profitable Tour Problem is a well-known NP-hard optimization challenge central to tourism planning, aiming to maximize collected profit while minimizing travel costs. While classical heuristics provide approximate solutions, they often struggle with finding globally optimal routes. This paper explores the application of near-term quantum computing to this problem. We propose a framework based on the Variational Quantum Eigensolver to find high-quality solutions for the Profitable Tour Problem. The core of our contribution is a novel methodology for constructing a constraint-aware variational ansatz that directly encodes the problem's hard constraints. This approach circumvents the need for large penalty terms in the Hamiltonian problem, which are often a source of optimization challenges. We validate our method through numerical simulations on a representative tourism scenario of up to 25 qubits. The results demonstrate the viability of the approach, achieving high solution accuracy consistent with brute-force enumeration for smaller instances. This work serves as a proof-of-concept for applying Variational Quantum Eigensolver to complex tourism optimization problems and provides a basis for future exploration on real quantum hardware.