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arXiv:2606.07322

Towards Implementable Quantum Divide and Conquer: A TSP Solver with Improved Exponential Base over Held-Karp

Xujun Bai, Yun Shang, Honghong Lin

quant-phcs.CCSubmitted Jun 5, 2026• Announced Jun 8, 2026• 0 citations
v1 · Jun 5, 2026v2 · Jul 5, 2026
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Abstract

The traveling salesman problem (TSP) is a significant classical NP-hard combinatorial optimization problem. In this work, we demonstrate that combining classical dynamic programming with quantum search can yield an achievable quantum advantage for TSP on the basis of excellent work by the authors of~\cite{ambainis2019quantum}. We design the quantum divide and conquer strategy to provide a parameterized spectrum for this combination. The hybrid algorithm proposed in~\cite{ambainis2019quantum} corresponds to a specific case in this spectrum, while the two extremes of the spectrum represent the purely classical Held-Karp and the purely quantum search algorithm, respectively. Within our parameterized spectrum, we prove that the optimal query complexity is $O^*(1.865666\ldots^n)$, achieved with the 4-subset scheme, while the counting in~\cite{ambainis2019quantum} overlooked half of the recursive branches. The correct query complexity of their algorithm is $O^*(2.225880\ldots^n)$ at their chosen parameter ($α\approx0.055362$), and cannot fall below $O^*(2^n)$ for any $α$ - meaning their $8$-subset scheme, correctly analyzed, never surpasses the classical Held-Karp bound. Furthermore, in previous studies on quantum advantages for NP-hard combinatorial optimization problems, researchers focused only on improvements in query complexity. Our work, however, points out that the quantum advantage stems not only from the quadratic speedup of quantum search but also from the structured quantum state preparation. We argue that structured state preparation is indispensable for realizing the oracle operator while maintaining the total time complexity of $O^*(1.865666\ldots^n)$. Therefore, we design an elegant method for preparing the set partition state, which makes our TSP solver practically executable.

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The traveling salesman problem (TSP) is a significant classical NP-hard combinatorial optimization problem. In this work, we demonstrate that combining classical dynamic programming with quantum search can yield an achievable quantum advantage for TSP on the basis of excellent work by the authors of~\cite{ambainis2019quantum}. We design the quantum divide and conquer strategy to provide a parameterized spectrum for this combination. The hybrid algorithm proposed in~\cite{ambainis2019quantum} corresponds to a specific case in this spectrum, while the two extremes of the spectrum represent the purely classical Held-Karp and the purely quantum search algorithm, respectively. Within our parameterized spectrum, we prove that the optimal query complexity is O∗(1.865666…n)O^*(1.865666\ldots^n)O∗(1.865666…n), achieved with the 4-subset scheme, while the counting in~\cite{ambainis2019quantum} overlooked half of the recursive branches. The correct query complexity of their algorithm is O∗(2.225880…n)O^*(2.225880\ldots^n)O∗(2.225880…n) at their chosen parameter (α≈0.055362α\approx0.055362α≈0.055362), and cannot fall below O∗(2n)O^*(2^n)O∗(2n) for any ααα - meaning their 888-subset scheme, correctly analyzed, never surpasses the classical Held-Karp bound.
Furthermore, in previous studies on quantum advantages for NP-hard combinatorial optimization problems, researchers focused only on improvements in query complexity. Our work, however, points out that the quantum advantage stems not only from the quadratic speedup of quantum search but also from the structured quantum state preparation. We argue that structured state preparation is indispensable for realizing the oracle operator while maintaining the total time complexity of O∗(1.865666…n)O^*(1.865666\ldots^n)O∗(1.865666…n). Therefore, we design an elegant method for preparing the set partition state, which makes our TSP solver practically executable.
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 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) asks for the shortest tour that visits each city once and returns to the start. It is a classic NP-hard problem and a benchmark for optimization, logistics, and network design. This paper surveys classical, quantum, and hybrid algorithms for the TSP using clear, step-by-step explanations. We summarize how each method works, what it guarantees, and how it scales with the number of cities. We also discuss practical limits of current quantum hardware and outline open problems and promising research directions. While quantum techniques can speed up certain subroutines or offer strong heuristics, no known method solves general TSP in polynomial time. Our goal is to offer a readable yet rigorous map of the landscape for students and practitioners.
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.
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.
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.