Abstract The traveling salesman problem (TSP) forms the foundation for many logistics and optimization tasks. At the same time, this problem is NP-hard and requires significant computational resources. Quantum computing promises to substantially accelerate computations or increase solution accuracy. The TSP can be conveniently mapped onto the quadratic unconstrained binary optimization (QUBO) formulation, which in turn can be solved using variational quantum eigensolver (VQE) and quantum approximate optimization algorithm (QAOA) methods, but this mapping requires a quadratic number of qubits. The higher-order binary optimization (HOBO) formulation requires significantly fewer qubits, but it contains higher-order terms. In our work, we thoroughly investigate both formulations and provide a practical recipe for solving the TSP. We give a comprehensive study and compare in detail QUBO and HOBO formulations.
Abstract The traveling salesman problem (TSP) forms the foundation for many logistics and optimization tasks. At the same time, this problem is NP-hard and requires significant computational resources. Quantum computing promises to substantially accelerate computations or increase solution accuracy. The TSP can be conveniently mapped onto the quadratic unconstrained binary optimization (QUBO) formulation, which in turn can be solved using variational quantum eigensolver (VQE) and quantum approximate optimization algorithm (QAOA) methods, but this mapping requires a quadratic number of qubits. The higher-order binary optimization (HOBO) formulation requires significantly fewer qubits, but it contains higher-order terms. In our work, we thoroughly investigate both formulations and provide a practical recipe for solving the TSP. We give a comprehensive study and compare in detail QUBO and HOBO formulations.
The Travelling Salesman Problem (TSP) is an important combinatorial optimisation problem, and is usually solved on a quantum computer using a Quadratic Unconstrained Binary Optimisation (QUBO) formulation or a Higher Order Binary Optimisation(HOBO) formulation. In these formulations, penalty terms are added to the objective function for outputs that don't map to valid routes. We present a novel formulation which needs fewer binary variables, and where, by design, there are no penalty terms because all outputs from the quantum device are mapped to valid routes. Simulations of a quantum boson sampler were carried out which demonstrate that larger networks can be solved with this penalty-free formulation than with formulations with penalties. Simulations were successfully translated to hardware by running a non-QUBO formulation with penalties on an early experimental prototype ORCA PT-1 boson sampler. Although we worked with a boson sampler, we believe that this novel formulation is relevant to other quantum devices. This work shows that a good embedding for combinatorial optimisation problems can solve larger problems with the same quantum computing resource. The flexibility of boson sampling quantum devices is a powerful asset in solving combinatorial optimisation problem, because it enables formulations where the output string is always mapped to a valid solution, avoiding the need for penalties.
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 being used heavily as a vehicle to study the feasibility of the quantum computing paradigm for this class of problems. In this work, we formulate the symmetric TSP as an optimization problem, which we solve using the quantum approximate optimization algorithm (QAOA) approach. By adopting an improved qubit encoding strategy and a layerwise learning optimization protocol, we obtain numerical results on the gate-based digital quantum simulator for the 3-, 4-, and 5-city TSPs. Specifically, we focus on three QAOA mixer designs to evaluate their performances in terms of numerical accuracy and optimization cost. Based on our results, we propose that a well-balanced QAOA mixer design is more prominent on gate-based simulators or realistic quantum devices in the near future. In addition, we study the sensitivity of TSP graph properties such as graph skewness and penalty weight in the TSP-QAOA simulation. Overall, our results prove digital quantum simulation is a powerful candidate for obtaining the optimal solution to the TSP.
In this paper, we study the application of emerging photonic and quantum computing architectures to solving the Traveling Salesman Problem (TSP), a well-known NP-hard optimization problem. We investigate several approaches: Simulated Annealing (SA), Quadratic Unconstrained Binary Optimization (QUBO-Ising) methods implemented on quantum annealers and Optical Coherent Ising Machines, as well as the Quantum Approximate Optimization Algorithm (QAOA) and the Quantum Phase Estimation (QPE) algorithm on gate-based quantum computers. QAOA and QPE were tested on the IBM Quantum platform. The QUBO-Ising method was explored using the D-wave quantum annealer, which operates on superconducting Josephson junctions, and the Quantum Computing Inc (QCi) Dirac-1 entropy quantum optimization machine. Gate-based quantum computers demonstrated accurate results for small TSP instances in simulation. However, real quantum devices are hindered by noise and limited scalability. Circuit complexity grows with problem size, restricting performance to TSP instances with a maximum of six nodes. In contrast, Ising-based architectures show improved scalability for larger problem sizes. SQUID-based Ising machines can handle TSP instances with up to 12 nodes, while entropy computing implemented in hybrid optoelectronic components extend this capability to 18 nodes. Nevertheless, the solutions tend to be suboptimal due to hardware limitations and challenges in achieving ground state convergence as the problem size increases. Despite these limitations, Ising machines demonstrate significant time advantages over classical methods, making them a promising candidate for solving larger-scale TSPs efficiently.
Ising formulation is important for many NP problems (Lucas, 2014). This formulation enables implementing novel quantum computing methods including Quantum Approximate Optimization Algorithm and Variational Quantum Eigensolver (VQE). Here, we investigate closely the traveling salesman problem (TSP).
First, we present some non-trivial issues related to Ising model view versus a realistic salesman.
Then, focusing on VQE we discuss and clarify the use of: a.-- Conventional VQE and how it is relevant as a novel SAT-solver; b.-- Qubit efficiency and its importance in the Noisy Intermediate Scale Quantum-era; and c.-- the relevance and importance of a novel approach named Discrete Quantum Exhaustive Search (Alfassi, Meirom, and Mor, 2024), for enhancing VQE and other methods using mutually unbiased bases.
The approach we present here in details can potentially be extended for analyzing approximating and solving various other NP complete problems. Our approach can also be extended beyond the Ising model and beyond the class NP, for example to the class Quantum Merlin Arthur (QMA) of problems, relevant for quantum chemistry and for general spin problems.
Abstract Quantum computing is offering a novel perspective for solving combinatorial optimization problems. To fully explore the possibilities offered by quantum computers, the problems need to be formulated as unconstrained binary models, taking into account limitation and advantages of quantum devices. In this work, we provide a detailed analysis of the travelling salesman problem with time windows (TSPTW) in the context of solving it on a quantum computer. We introduce quadratic unconstrained binary optimization and higher-order binary optimization formulations of this problem. We demonstrate the advantages of edge-based and node-based formulations of the TSPTW problem. Additionally, we investigate the experimental realization of the presented methods on a quantum annealing device. The provided results pave the path for utilizing quantum computer for a variety of real-world tasks which can be cast in the form of travelling salesman problem with time windows.
The Traveling Salesman Problem (TSP) is a fundamental challenge in combinatorial optimization, widely applied in logistics and transportation. As the size of TSP instances grows, traditional algorithms often struggle to produce high-quality solutions within reasonable timeframes. This study investigates the potential of the Quantum Approximate Optimization Algorithm (QAOA), a hybrid quantum-classical method, to solve TSP under realistic constraints. We adopt a QUBO-based formulation of TSP that integrates real-world logistical constraints reflecting operational conditions, such as vehicle capacity, road accessibility, and time windows, while ensuring compatibility with the limitations of current quantum hardware. Our experiments are conducted in a simulated environment using high-performance computing (HPC) resources to assess QAOA's performance across different problem sizes and quantum circuit depths. In order to improve scalability, we propose clustering QAOA (Cl-QAOA), a hybrid approach combining classical machine learning with QAOA. This method decomposes large TSP instances into smaller sub-problems, making quantum optimization feasible even on devices with a limited number of qubits. The results offer a comprehensive evaluation of QAOA's strengths and limitations in solving constrained TSP scenarios. This study advances quantum optimization and lays groundwork for future large-scale applications.