This paper investigates quantum heuristics based on Mixer Hamiltonians, which allow the search to be restricted to a specific subspace and enable warm-start strategies for solving the Traveling Salesman Problem (TSP). Approaches involving Mixer Hamiltonians can be integrated into the Quantum Approximate Optimization Algorithm (QAOA), where the Mixer acts as a mapping function that transforms qubit strings into feasible solution sets. We first introduce a swap-based mixer tailored to the TSP, which ensures that only qubit strings representing valid TSP solutions are explored during the QAOA process. Second, we propose a warm-start technique that initializes QAOA with a solution generated by any classical heuristic, thereby promoting faster convergence. These two contributions are combined into a Warm-Start QAOA framework with a Swap-Based Mixer, leveraging both structural and initialization advantages. Experimental results on a custom TSP instance involving five customers demonstrate the effectiveness of this approach, providing, for the first time, a viable integration of warm-start and swap-based mixers for the TSP within a quantum optimization framework.
This paper investigates quantum heuristics based on Mixer Hamiltonians, which allow the search to be restricted to a specific subspace and enable warm-start strategies for solving the Traveling Salesman Problem (TSP). Approaches involving Mixer Hamiltonians can be integrated into the Quantum Approximate Optimization Algorithm (QAOA), where the Mixer acts as a mapping function that transforms qubit strings into feasible solution sets. We first introduce a swap-based mixer tailored to the TSP, which ensures that only qubit strings representing valid TSP solutions are explored during the QAOA process. Second, we propose a warm-start technique that initializes QAOA with a solution generated by any classical heuristic, thereby promoting faster convergence. These two contributions are combined into a Warm-Start QAOA framework with a Swap-Based Mixer, leveraging both structural and initialization advantages. Experimental results on a custom TSP instance involving five customers demonstrate the effectiveness of this approach, providing, for the first time, a viable integration of warm-start and swap-based mixers for the TSP within a quantum optimization framework.
Quantum optimization algorithms, such as the Quantum Approximate Optimization Algorithm, are emerging as promising heuristics for solving complex combinatorial problems. To improve performance, several extensions to the standard QAOA framework have been proposed in recent years. Two notable directions include: warm-starting techniques, which incorporate classical approximate solutions to guide the quantum evolution, and custom mixer Hamiltonians, such as XY mixers, which constrain the search to feasible subspaces aligned with the structure of the problem. In this work, we propose an approach that integrates these two strategies: a warm-start initialization with an XY mixer ansatz, enabling constraint-preserving quantum evolution biased toward high-quality classical solutions. The method begins by reformulating the combinatorial problem as a MaxCut instance, solved approximately using the Goemans-Williamson algorithm. The resulting binary solution is relaxed and used to construct a biased superposition over valid one-hot quantum states, maintaining compatibility with the XY mixer's constraints. We evaluate the approach on 5-city instances of the Traveling Salesperson Problem, a canonical optimization problem frequently encountered as a subroutine in real-world Vehicle Routing Problems. Our method is benchmarked against both the standard XY-mixer QAOA and a warm-start-only variant based on MaxCut relaxation. Results show that the proposed combination consistently outperforms both baselines in terms of the percentage and rank of optimal solutions, demonstrating the effectiveness of combining structured initializations with constraint-aware quantum evolution for optimization problems.
The Quantum Approximate Optimization Algorithm (QAOA) is a leading hybrid heuristic for combinatorial optimization, but efficiently handling hard constraints remains a significant challenge. XY-mixers successfully confine quantum state evolution to a feasible subspace, such as the Hamming-weight-1 sector for one-hot constraints. On the contrary, warm-starting biases the search toward promising regions based on preliminary solutions. Combining these two techniques requires maintaining the essential alignment between the initial state and the mixer Hamiltonian to preserve convergence guarantees. Previous work demonstrated warm-starting with XY-mixers via a biased initial state, but relying only on standard mixer Hamiltonians. Consequently, the initial state is no longer a ground state of the mixer. In this work, we overcome these limitations by formulating a warm-started XY-mixer Hamiltonian for one-hot constraints and proving its ground-state properties. Furthermore, we provide a shallow circuit implementation suitable for NISQ implementations. We embed the warm-starting into a classical heuristic that iteratively updates the bias based on previous samples, called Iterative Warm-Starting (IWS). Extensive numerical simulations on Max-k-Cut and Traveling Salesperson Problem instances demonstrate that IWS-QAOA significantly accelerates the solution-finding process, increasing the probability of sampling optimal solutions by orders of magnitude compared to standard XY-QAOA. Finally, we validate our approach on the ibm_boston QPU using hardware-tailored 144-qubit problem instances. By coupling IWS-QAOA with a greedy steepest-descent post-processing strategy to repair infeasible measurements caused by hardware noise, we successfully identify optimal solutions on actual quantum devices.
This paper concerns quantum heuristics based on Mixer Hamiltonians that allow to restrict investigation on a specific subspace. Mixer Hamiltonian based approaches can be included in QAOA algorithm and we can state that Mixer Hamiltonians are mapping functions from the set of qubit-strings to the set of solutions. Mixer Hamiltonian offers an approach very similar to indirect representations commonly used in routing or in scheduling community for decades. After the initial publication of Cheng et al. in 1996 (Cheng et al., 1996), numerous propositions in OR lies on 1-to-n mapping functions, including the split algorithm that transform one TSP solution into a VRP solution. The objective is at first to give a compact and readable presentation of these Mixer Hamiltonians considering the functional analogies that exist between the OR community practices and the quantum field. Our experiments encompass numerical evaluations of circuit using the Qiskit library of IBM meeting the theoretical considerations.
The Quantum Approximate Optimization Algorithm (QAOA) was developed to tackle combinatorial optimization problems, such as the Travelling Salesman Problem (TSP), by approximating solutions through quantum mechanics. QAOA utilizes a unitary transformation to prepare a quantum state that encodes a potential solution, with the parameters of the unitary optimized by a classical optimizer. This study evaluates QAOA's performance on the NP-hard TSP, comparing the standard version of the algorithm from Farhi \cite{qaoa} and an enhancement, Warm-Start QAOA \cite{warmstart}, to a classical solver. The TSP seeks the shortest possible route that visits a set of cities, a problem that is amenable to quantum formulation using platforms like Qiskit. Warm-Start QAOA initializes the algorithm's parameters based on a solution to a relaxed version of the problem, offering potential improvements in optimization efficiency. By comparing these quantum algorithms to a classical solver, we provide a comprehensive assessment of their performance across key metrics, including route quality, execution time, and scalability. Through this analysis, we highlight the relative strengths and weaknesses of QAOA and its warm-start variant for solving the TSP, alongside the classical approach, offering insights into their potential for route optimization and practical applications in realworld settings. We conclude that, while the classical solver significantly outperforms QAOA and warm-start QAOA at smaller-scale problem cases, warm-Start QAOA has the potential to yield advantages at a larger scale when run on NISQ-era noisy hardware.
Quantum optimization algorithms, particularly the Quantum Approximate Optimization Algorithm (QAOA), have significantly addressed combinatorial optimization problems. While QAOA has been applied to various NP-hard problems, such as Max-Cut and the Traveling Salesman Problem (TSP), its application to the Degree-Constrained Minimum Spanning Tree (DCMST) problem remains unexplored. Inspired by Fowler’s formulation, this work presents the first implementation of the DCMST Hamiltonian within QAOA and insights from its benefits to graph-based machine learning algorithms. We investigated two approaches: (i) the standard QAOA ansatz with an X mixer and (ii) a warm-started QAOA utilizing classical preprocessing to enhance convergence. We used these strategies to provide numerical results for instances with 3 and 4 nodes, employing the COBYLA optimizer and metaheuristic optimization techniques. Our findings serve as a proof of concept, demonstrating the feasibility of applying QAOA to this problem; however, the number of qubits scales as O(N2) with the number of nodes, limiting scalability. Current research focuses on developing more efficient implementations that encode the same number of binary variables using fewer qubits. This study contributes to the expanding field of quantum optimization, highlighting the potential of hybrid quantum-classical algorithms, especially with resource-efficient mixers and warm-starting techniques, to solve complex combinatorial problems and advance the development of scalable quantum algorithms.
The traveling salesman problem (TSP) is one of the most often-used NP-hard problems in computer science to study the effectiveness of computing models and hardware platforms. In this regard, it is also heavily used as a vehicle to study the feasibility of the quantum computing paradigm for this class of problems. In this paper, we tackle the TSP using the quantum approximate optimization algorithm (QAOA) approach by formulating it as an optimization problem. By adopting an improved qubit encoding strategy and a layer-wise learning optimization protocol, we present numerical results obtained from the gate-based digital quantum simulator, specifically targeting TSP instances with 3, 4, and 5 cities. We focus on the evaluations of three distinctive QAOA mixer designs, considering their performances in terms of numerical accuracy and optimization cost. Notably, we find that a well-balanced QAOA mixer design exhibits more promising potential for gate-based simulators and realistic quantum devices in the long run, an observation further supported by our noise model simulations. Furthermore, we investigate the sensitivity of the simulations to the TSP graph. Overall, our simulation results show that the digital quantum simulation of problem-inspired ansatz is a successful candidate for finding optimal TSP solutions.