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arXiv:2405.07129 · IEEE Transactions on Quantum Engineering

Two-Step Quantum Search Algorithm for Solving Traveling Salesman Problems

Rei Sato, Cui Gordon, Kazuhiro Saito, Hideyuki Kawashima, Tetsuro Nikuni, Shohei Watabe

Published Jan 1, 2025• 6 citations
arXivHTMLPDF

Abstract

Quantum search algorithms, such as Grover's algorithm, are anticipated to efficiently solve constrained combinatorial optimization problems. However, applying these algorithms to the traveling salesman problem (TSP) on a quantum circuit presents a significant challenge. Existing quantum search algorithms for the TSP typically assume that an initial state—an equal superposition of all feasible solutions satisfying the problem's constraints—is pre-prepared. The query complexity of preparing this state using brute-force methods scales exponentially with the factorial growth of feasible solutions, creating a significant hurdle in designing quantum circuits for large-scale TSPs. To address this issue, we propose a two-step quantum search (TSQS) algorithm that employs two sets of operators. In the first step, all the feasible solutions are amplified into their equal superposition state. In the second step, the optimal solution state is amplified from this superposition state. The TSQS algorithm demonstrates greater efficiency compared to conventional search algorithms that employ a single oracle operator for finding a solution within the encoded space. Encoded in the higher order unconstrained binary optimization representation, our approach significantly reduces the qubit requirements. This enables efficient initial state preparation through a unified circuit design, offering a quadratic speedup in solving the TSP without prior knowledge of feasible solutions.

DOI
10.1109/tqe.2025.3548706

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Quantum search algorithms, such as Grover's algorithm, are anticipated to efficiently solve constrained combinatorial optimization problems. However, applying these algorithms to the traveling salesman problem (TSP) on a quantum circuit presents a significant challenge. Existing quantum search algorithms for the TSP typically assume that an initial state—an equal superposition of all feasible solutions satisfying the problem's constraints—is pre-prepared. The query complexity of preparing this state using brute-force methods scales exponentially with the factorial growth of feasible solutions, creating a significant hurdle in designing quantum circuits for large-scale TSPs. To address this issue, we propose a two-step quantum search (TSQS) algorithm that employs two sets of operators. In the first step, all the feasible solutions are amplified into their equal superposition state. In the second step, the optimal solution state is amplified from this superposition state. The TSQS algorithm demonstrates greater efficiency compared to conventional search algorithms that employ a single oracle operator for finding a solution within the encoded space. Encoded in the higher order unconstrained binary optimization representation, our approach significantly reduces the qubit requirements. This enables efficient initial state preparation through a unified circuit design, offering a quadratic speedup in solving the TSP without prior knowledge of feasible solutions.
The Traveling Salesman Problem (TSP) is a classical NP-hard problem that plays a crucial role in combinatorial optimization. In this paper, we are interested in the quantum search framework for the TSP because it has robust theoretical guarantees. However, we need to first search for all Hamiltonian cycles from a very large solution space, which greatly weakens the advantage of quantum search algorithms. To address this issue, one can first prepare a superposition state of all feasible solutions, and then amplify the amplitude of the optimal solution from it. We propose a quantum algorithm to generate the uniform superposition state of all N-length Hamiltonian cycles as an initial state within polynomial gate complexity based on pure quantum dynamic programming with very few ancillary qubits, which achieves exponential acceleration compared to the previous initial state preparation algorithm. As a result, we realized the theoretical minimum query complexity of quantum search algorithms for a general TSP. Compared to some algorithms that theoretically have lower query complexities but lack practical implementation solutions, our algorithm has feasible circuit implementation. Our work provides a meaningful research case on how to fully utilize the structures of specific problems to unleash the acceleration capability of the quantum search algorithms.
The Travelling Salesman Problem (TSP) is a classical NP-hard problem that plays a crucial role in combinatorial optimization. In this paper, we are interested in the quantum search framework for the TSP because it has robust theoretical guarantees. However, we need to first search for all Hamiltonian cycles from a very large solution space, which greatly weakens the advantage of quantum search algorithms. To address this issue, one can first prepare a superposition state of all feasible solutions, and then amplify the amplitude of the optimal solution from it. We propose a quantum algorithm to generate the uniform superposition state of all N-length Hamiltonian cycles as an initial state within polynomial gate complexity based on pure quantum dynamic programming with very few ancillary qubits, which achieves exponential acceleration compared to the previous initial state preparation algorithm. As a result, we realized the theoretical minimum query complexity of quantum search algorithms for a general TSP. Compared to some algorithms that theoretically have lower query complexities but lack practical implementation solutions, our algorithm has feasible circuit implementation.
Quantum search algorithms, such as the Grover algorithm, are investigated as one of the ways to solve a traveling salesman problem (TSP). In TSP, the quantum search starts from an equal superposition of all feasible solutions. The embedding cost of all feasible solutions is up to exponential, i.e.,O(n!)\mathcal{O}(n!)O(n!)withnnncities, and makes it difficult to solve large-scale TSP. Hence, preparing the initial state of TSP at a lower embedding cost is an important issue. Here, we propose a quantum search algorithm based on the divide-and-conquer approach, which divides each constraint term of the TSP into subproblems and applies the quantum search to each subproblem. We investigate the embedding quality of all feasible solutions by comparing our proposed method with the Grover algorithm regarding optimal iterations and success probability. The results show that our quantum divide-and-conquer approach can significantly reduce the embedding cost while the success probability decreases.
The Traveling Salesman Problem (TSP) is a classical NP-hard problem in combinatorial optimization, where determining the shortest route among a set of cities becomes computationally prohibitive as the problem size increases. This work explores quantum computing as an alternative approach to address this complexity. Unlike existing methods that primarily rely on quantum annealing, we propose a quantum iterative framework integrating Quantum Phase Estimation (QPE) and Grover's search algorithm. Route costs are encoded as quantum phases, enabling QPE to efficiently evaluate them, while Amplitude Amplification, implemented via the Grover-Long algorithm, iteratively refines the solution space toward the optimal route. A proof-of-concept case study on a small-scale TSP instance demonstrates the feasibility of this approach and its potential for scaling to larger optimization problems. Furthermore, under an expectation-based analysis, the algorithm exhibits an expected computational complexity of O(m2log⁡2(m)log⁡2(1/ε)ε)O(\frac{m^2\log_2(m)\log_2(1/ε)}{\sqrtε})O(ε​m2log2​(m)log2​(1/ε)​) which depends on the error tolerance parameter εεε. This estimation omits the initialization term, which we expect future refinements to render subdominant to Phase Estimation.
The famous Travelling Salesman Problem (TSP) is an important category of optimization problems that is mostly encountered in various areas of science and engineering. Studying optimization problems motivates to develop advanced techniques more suited to contemporary practical problems. Among those, especially the NP hard problems provide an apt platform to demonstrate supremacy of quantum over classical technologies in terms of resources and time. TSP is one such NP hard problem in combinatorial optimization which takes exponential time order for solving by brute force method. Here we propose a quantum algorithm to solve the travelling salesman problem using phase estimation technique. We approach the problem by encoding the given distances between the cities as phases. We construct unitary operators whose eigenvectors are the computational basis states and eigenvalues are various combinations of these phases. Then we apply phase estimation algorithm to certain eigenstates which give us all the total distances possible for all the routes. After obtaining the distances we can search through this information using the quantum search algorithm for finding the minimum to find the least possible distance as well the route taken. This provides us a quadratic speedup over the classical brute force method for a large number of cities. In this paper, we illustrate an example of the travelling salesman problem by taking four cities and present the results by simulating the codes in the IBM's quantum simulator.
Grover's search algorithm is one of the basic building block in the world of quantum algorithms. Successfully applying it to combinatorial optimization problems is a subtle challenge. As a quadratic speedup is not enough to naively search an exponentially large space, the search has to be complemented with a state preparation routine which increases the amplitudes of promising states by exploiting the problem structure. In this paper, we build upon previous work by Baertschi and Eidenbenz to construct heuristic state preparation routines for the Traveling Salesperson Problem (TSP), mimicking the well-known classical Lin-Kernighan heuristic. With our heuristic, we aim to achieve a reasonable approximation ratio with only a polynomial number of Grover iterations. Further, we compare several algorithmic settings relating to termination criteria and the choice of Grover iterations when the number of marked solutions is unknown.