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arXiv:2102.13298 · Journal of the Physical Society of Japan

Many-Qudit Representation for the Travelling Salesman Problem Optimisation

Vladimir Vargas-Calderón, Nicolas Parra-A., Herbert Vinck-Posada, Fabio A. González

Published Oct 6, 2021• 10 citations
arXivHTMLPDF

Abstract

We present a map from the travelling salesman problem (TSP), a prototypical NP-complete combinatorial optimisation task, to the ground state associated with a system of many-qudits. Conventionally, the TSP is cast into a quadratic unconstrained binary optimisation (QUBO) problem, that can be solved on an Ising machine. The size of the corresponding physical system's Hilbert space is $2^{N^2}$, where $N$ is the number of cities considered in the TSP. Our proposal provides a many-qudit system with a Hilbert space of dimension $2^{N\log_2N}$, which is considerably smaller than the dimension of the Hilbert space of the system resulting from the usual QUBO map. This reduction can yield a significant speedup in quantum and classical computers. We simulate and validate our proposal using variational Monte Carlo with a neural quantum state, solving the TSP in a linear layout for up to almost 100 cities.

DOI
10.7566/jpsj.90.114002

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We present a map from the travelling salesman problem (TSP), a prototypical NP-complete combinatorial optimisation task, to the ground state associated with a system of many-qudits. Conventionally, the TSP is cast into a quadratic unconstrained binary optimisation (QUBO) problem, that can be solved on an Ising machine. The size of the corresponding physical system's Hilbert space is 2N22^{N^2}2N2, where NNN is the number of cities considered in the TSP. Our proposal provides a many-qudit system with a Hilbert space of dimension 2Nlog⁡2N2^{N\log_2N}2Nlog2​N, which is considerably smaller than the dimension of the Hilbert space of the system resulting from the usual QUBO map. This reduction can yield a significant speedup in quantum and classical computers. We simulate and validate our proposal using variational Monte Carlo with a neural quantum state, solving the TSP in a linear layout for up to almost 100 cities.
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.
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.
Ising machines have recently gained a significant attention as a promising approach to combinatorial optimization problems, which are notoriously challenging for conventional computers to solve efficiently. In contrast to conventional computers, Ising machines operate based on the principles of the Ising model derived from statistical mechanics. Ising machines tackle combinatorial optimization problems by transforming them into Quadratic Unconstrained Binary Optimization (QUBO) formulations. The Traveling Salesperson Problem (TSP) is a prominent example of such optimization challenges and is widely recognized for its computational difficulty on conventional computers. Although Ising machines offer the potential for efficiently solving TSPs, the process of a QUBO formulation, namely the generation of a QUBO matrix that represents a TSP, demands substantial memory resources and considerable computation overheads. This study proposes a compressed QUBO format for TSP. This format reduces the dimensionality of the QUBO matrix for TSP by leveraging repetitive values, structural patterns, and sparsity within the matrix. The proposed format enhances the efficiency of data preprocessing, significantly reducing both the computational time required for QUBO matrix generation and the associated memory usage. These improvements contribute to the practical applicability of digital hardware-based Ising machines for addressing large-scale combinatorial optimization problems, thereby expanding their utility in solving complex real-world challenges.
Abstract The travelling salesman problem (TSP) is a popular NP-hard combinatorial optimization problem that requires finding the optimal way for a salesman to travel through different cities once and return to the initial city. The existing methods of solving TSPs on quantum systems are either gate-based or binary variable-based encoding. Both approaches are resource-expensive in terms of the number of qubits, while performing worse compared to existing classical algorithms, even for small-sized problems. A novel encoding scheme is needed to map the TSP problem onto a quantum system, which is addressed in this work. We introduce a distinct geometric approach to encode the TSP on a single qubit and present a quantum-inspired algorithm to solve the problem by invoking the principle of quantum superposition . The cities are represented as quantum states on the Bloch sphere, while the preparation of superposition states allows us to traverse multiple paths at once. The underlying framework of our algorithm is a quantum-inspired version of the classical Brachistochrone approach. Optimal control methods are employed to create a selective superposition of the quantum states to find the shortest route of a given TSP. The numerical simulations solve a sample of four to nine cities for which exact solutions are obtained. The algorithm can be implemented on any quantum platform capable of efficiently rotating a qubit and allowing state tomography measurements. For the TSP problem sizes considered in this work, our algorithm is more resource-efficient and accurate than existing quantum algorithms, with the potential for scalability.
The Multiple Traveling Salesman Problem (m-TSP) is a well-known combinatorial optimisation problem with diverse applications in various fields, including transportation, logistics, and network design. The m-TSP involves finding the optimal routes for multiple salesmen to visit a set of cities, minimising the total travel distance while ensuring that each city is visited exactly once. Traditional computational approaches struggle to efficiently solve large-scale instances of the m-TSP due to its inherent complexity. In recent years, quantum computing has emerged as a promising field for tackling combinatorial optimisation problems. In this study, we investigate the application of the Quantum Approximate Optimisation Algorithm (QAOA) to solve the m-TSP, leveraging the inherent parallelism and quantum computational power to potentially obtain high-quality solutions.
In this paper, we consider a method for solving difficult combinatorial optimization problems on real quantum computers. We focus on the traveling salesman problem as a representative problem for a group of problems where the solution is represented by a permutation. Typically, existing algorithmic solutions use binary matrices to store this permutation – the QUBO (quadratic unconstrained binary optimization) model. We propose a new way of encoding permutations on quantum computers, using a significantly smaller number of qubits than binary matrix encodings. Our method allows for significant performance improvements for any problem whose input or solution is a permutation. We demonstrate an example implementation of the traveling salesman problem on the IQM quantum computers: IQM Spark 5-qubit ‘ODRA-5’ computer and IQM Radiance ‘Garnet’ 20-qubit computer.