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.
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.
Routing problems are a common optimization problem in industrial applications, which occur on a large scale in supply chain planning. Due to classical limitations for solving NP-hard problems, quantum computing hopes to improve upon speed or solution quality. Several suggestions have been made for encodings of routing problems to solve them with variational quantum algorithms. However, for an end user it is hard to decide a priori which encoding will give the best solutions according to their needs. In this work, we investigate different encodings for the Travelling Salesperson Problem. We compare their scaling and performance when using the Quantum Approximate Optimization Algorithm and the Variational Quantum Eigensolver and provide a clear guide for users when to choose which encoding. For small instances, we find evidence that the permutation encoding can yield good results since it does not suffer from feasibility issues.
This thesis explores quantum computing for combinatorial optimization through the Traveling Salesman Problem (TSP), which aims to find a minimum-cost Hamiltonian cycle visiting each city exactly once. Using Qiskit, we implement the Quantum Approximate Optimization Algorithm (QAOA) on both simulators and quantum hardware, and compare its performance with that of classical exact optimization via the Gurobi solver. TSP instances are encoded as Quadratic Unconstrained Binary Optimization (QUBO) problems derived from the Miller–Tucker–Zemlin formulation, with degree and subtour constraints embedded using quadratic penalties. Results indicate that QAOA consistently generates feasible tours, but classical branch-and-bound solvers outperform it in both solution quality and runtime. On Noisy Intermediate-Scale Quantum (NISQ) devices, the optimality gap grows with problem size, reflecting sensitivity to variational parameter tuning, penalty scaling, and hardware noise. Among classical optimizers for QAOA, COBYLA offers the most effective tradeoff between efficiency and noise tolerance. Overall, this work establishes a reproducible quantum-classical workflow, provides performance baselines, and identifies critical factors such as encoding design, optimizer choice, and error mitigation for improving near-term quantum heuristic performance.
The Quantum Approximate Optimization Algorithm (QAOA) is an algorithmic framework for finding approximate solutions to combinatorial optimization problems. It consists of interleaved unitary transformations induced by two operators labelled the mixing and problem Hamiltonians. To fit this framework, one needs to transform the original problem into a suitable form and embed it into these two Hamiltonians. In this paper, for the well-known NP-hard Traveling Salesman Problem (TSP), we encode its constraints into the mixing Hamiltonian rather than the conventional approach of adding penalty terms to the problem Hamiltonian. Moreover, we map edges (routes) connecting each pair of cities to qubits, which decreases the search space significantly in comparison to other approaches. As a result, our method can achieve a higher probability for the shortest round-trip route with only half the number of qubits consumed compared to IBM Q’s approach. We argue the formalization approach presented in this paper would lead to a generalized framework for finding, in the context of QAOA, high-quality approximate solutions to NP optimization problems.
Many relevant problems in industrial settings result in NP-hard optimization problems, such as the Capacitated Vehicle Routing Problem (CVRP) or its reduced variant, the Travelling Salesperson Problem (TSP). Even with today's most powerful classical algorithms, the CVRP is challenging to solve classically. Quantum computing may offer a way to improve the time to solution, although the question remains open as to whether Noisy Intermediate-Scale Quantum (NISQ) devices can achieve a practical advantage compared to classical heuristics. The most prominent algorithms proposed to solve combinatorial optimization problems in the NISQ era are the Quantum Approximate Optimization Algorithm (QAOA) and the more general Variational Quantum Eigensolver (VQE). However, implementing them in a way that reliably provides high-quality solutions is challenging, even for toy examples. In this work, we discuss decomposition and formulation aspects of the CVRP and propose an application-driven way to measure solution quality. Considering current hardware constraints, we reduce the CVRP to a clustering phase and a set of TSPs. For the TSP, we extensively test both QAOA and VQE and investigate the influence of various hyperparameters, such as the classical optimizer choice and strength of constraint penalization. Results of QAOA are generally of limited quality because the algorithm does not reach the energy threshold for feasible TSP solutions, even when considering various extensions such as recursive and constraint-preserving mixer QAOA. On the other hand, the VQE reaches the energy threshold and shows a better performance. Our work outlines the obstacles to quantum-assisted solutions for real-world optimization problems and proposes perspectives on how to overcome them.
The Travelling Salesman Problem (TSP) is a well-known NP-hard combinatorial optimisation problem with important industrial applications. Although extensively studied, quantum solutions for instances beyond a dozen locations remain uncommon. We present a hybrid penalty-free Variational Quantum Algorithm (VQA) that significantly reduces both the qubit count and the runtime through a compact penalty-free formulation scaling as O(nlog2n) qubits compared to conventional formulations with O(n2) scaling. The VQA achieves solution qualities exceeding 90\% on Rigetti quantum hardware for instances of up to 17 locations. To the best of our knowledge, this represents the largest TSP instance solved on gate-based hardware. We systematically evaluate multiple encoding and optimisation strategies, and show that warm-start initialisation improves solution quality, while runtime is reduced by nearly two orders of magnitude through the use of gradient-free optimisation methods and cost-function caching. We additionally introduce a novel VQA inspired classical machine learning model (VQAi-CML) and benchmark the VQA against VQAi-CML, a greedy heuristic, a Monte Carlo baseline, and a strong classical optimisation baseline. Across the networks evaluated, the VQA consistently achieves superior solution quality relative to VQAi-CML, greedy heuristic, and Monte Carlo baselines, but not the classical solver
NP-hard problems are one of the most computationally intensive challenges in computer science, appearing in many domains such as logistics, cybersecurity, drug design, and financial modelling. Classical algorithms become impractical as problem size increases due to combinatorial explosion. This paper presents a structured framework for analysing quantum algorithms targeting NP-hard optimization problems, like the Traveling Salesman Problem (TSP), Max-Cut, and Boolean Satisfiability (SAT). We have developed a taxonomy which categorizes quantum approaches—quantum annealing, variational algorithms (QAOA/VQE), and Grover-based search—based on problem encoding strategies, resource requirements, and theoretical guarantees. From the comparative analysis of existing literature, we: 1) Map algorithm classes to problem domains (e.g., QAOA for Max-Cut, annealing for TSP) 2) Identify performance trade-offs between solution quality, qubit counts, and circuit depth 3) Analyse hardware compatibility constraints across NISQ platforms 4) Reveal fundamental scalability barriers including noise susceptibility and embedding overhead The framework establishes implementation guidelines for near- term quantum hardware and proposes hybrid quantum-classical pathways to bridge theoretical potential and practical deployment. This systematic analysis prioritizes research directions for achieving quantum advantage in combinatorial optimization.