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Genetic and Evolutionary Computation Conference Companion (GECCO Companion)

Quantum Optimized Crossover for the Travelling Salesman and Minimization of Makespan with Setup Times

Anton V. Eremeev, Yulia Kovalenko

Published Jul 13, 2026• 0 citations
Publisher

Abstract

This paper proposes quantum-accelerated optimal recombination operators (optimized crossovers) to be used in genetic algorithms for the traveling salesman problem (TSP) and makespan minimization on a single machine with setup times. For the symmetric TSP with adjacency-based encoding, the Optimal Recombination Problem (ORP) reduces to a TSP on a graph with maximum vertex degree 4 and a set of forced edges. Applying the quantum speedup result of Moylett, Linden, and Montanaro (2017), we obtain a bounded-error quantum crossover algorithm. In the case of asymmetric TSP (ATSP) with adjacency-based encoding, the ORP reduces to the ATSP on qubic graphs, so a modification of the quantum algorithm from Moylett, Linden, and Montanaro (2017) yields a bounded-error optimized crossover. Both quantum crossovers for the TSP and ATSP have time complexity bounds, smaller by exponential factors (in the number of vertices), compared to previously known optimized crossovers for these problems. For makespan minimization on a single machine with position-based encoding, the ORP reduces to a quadratic unconstrained binary optimization (QUBO) formulation, enabling solution by quantum annealers. The QUBO has at most ⌊k/2⌋ binary variables and, for almost all problem instances, only O(log k) binary variables, where k is the number of jobs.

DOI
10.1145/3795101.3814656

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This paper proposes quantum-accelerated optimal recombination operators (optimized crossovers) to be used in genetic algorithms for the traveling salesman problem (TSP) and makespan minimization on a single machine with setup times. For the symmetric TSP with adjacency-based encoding, the Optimal Recombination Problem (ORP) reduces to a TSP on a graph with maximum vertex degree 4 and a set of forced edges. Applying the quantum speedup result of Moylett, Linden, and Montanaro (2017), we obtain a bounded-error quantum crossover algorithm. In the case of asymmetric TSP (ATSP) with adjacency-based encoding, the ORP reduces to the ATSP on qubic graphs, so a modification of the quantum algorithm from Moylett, Linden, and Montanaro (2017) yields a bounded-error optimized crossover. Both quantum crossovers for the TSP and ATSP have time complexity bounds, smaller by exponential factors (in the number of vertices), compared to previously known optimized crossovers for these problems. For makespan minimization on a single machine with position-based encoding, the ORP reduces to a quadratic unconstrained binary optimization (QUBO) formulation, enabling solution by quantum annealers. The QUBO has at most ⌊k/2⌋ binary variables and, for almost all problem instances, only O(log k) binary variables, where k is the number of jobs.
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.
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.
Quantum computing has paved a new way for faster and more efficient solutions to large-scale, real-world optimization problems that are challenging for classical computing systems. For instance, selective traveling salesman problem (sTSP) that is famous in such fields as logistic optimization and has attracted increasing attention from the research community, however, is known as an NP-Hard problem. Solving the sTSP is, therefore, extremely complex because the optimization function potentially comes with an exponential number of variables that cannot be solved in polynomial time in general. To this end, we propose a quantum annealing framework for time-bounded and near-optimal solutions for the sTSP, overcoming hardware limits of near-term quantum devices. In particular, we put forth an efficient Hamiltonian (QUBO) to encode the complex decision-making for the sTSP on noisy intermediate-scale quantum (NISQ) annealer. Furthermore, experimental results we obtained on the D-Wave 2000QQQquantum hardware demonstrate that the optimal solutions for several instances can be attained.
This study explores quantum computing for combinatorial optimization, focusing on the Traveling Salesman Problem (TSP). Using Qiskit, the Quantum Approximate Optimization Algorithm (QAOA) is implemented on simulators and quantum hardware, and compared with classical optimization using Gurobi. TSP instances are encoded as Quadratic Unconstrained Binary Optimization (QUBO) problems derived from the Miller–Tucker–Zemlin formulation, with quadratic penalties enforcing degree and subtour constraints. Results indicate that QAOA consistently generates feasible tours, but classical branch-and-bound methods achieve better solution quality and runtime. On Noisy Intermediate-Scale Quantum (NISQ) devices, performance degrades with problem size due to sensitivity to parameter tuning, penalty scaling, and hardware noise. Among the tested classical optimizers, COBYLA provides the best balance of efficiency and noise tolerance. Overall, this work establishes a reproducible quantum-classical workflow, provides performance baselines, and identifies critical factors such as encoding design, optimizer choice, and error mitigation for improving near-term quantum heuristic performance.
Abstract The traveling salesman problem is the problem of finding out the shortest route in a network of cities, that a salesman needs to travel to cover all the cities, without visiting the same city more than once. This problem is known to be N P -hard with a brute-force complexity of O ( N N ) or O ( N 2 N ) for N number of cities. This problem is equivalent to finding out the shortest Hamiltonian cycle in a given graph, if at least one Hamiltonian cycle exists in it. Quantum algorithms for this problem typically provide with a quadratic speedup only, using Grover’s search, thereby having a complexity of O ( N N /2 ) or O ( N N ). We present a bounded-error quantum polynomial-time (BQP) algorithm for solving the problem, providing with an exponential speedup. The overall complexity of our algorithm is O ( N 3 log( N ) κ / ϵ + 1/ ϵ 3 ), where the errors ϵ are O (1/poly( N )), and κ is the not-too-large condition number of the matrix encoding all Hamiltonian cycles.
This thesis explores quantum computing for combinatorial optimization through the Traveling Salesman Problem (TSP), which aims to find a minimum-cost Hamiltonian cycle visiting each city exactly once. Using Qiskit, we implement the Quantum Approximate Optimization Algorithm (QAOA) on both simulators and quantum hardware, and compare its performance with that of classical exact optimization via the Gurobi solver. TSP instances are encoded as Quadratic Unconstrained Binary Optimization (QUBO) problems derived from the Miller–Tucker–Zemlin formulation, with degree and subtour constraints embedded using quadratic penalties. Results indicate that QAOA consistently generates feasible tours, but classical branch-and-bound solvers outperform it in both solution quality and runtime. On Noisy Intermediate-Scale Quantum (NISQ) devices, the optimality gap grows with problem size, reflecting sensitivity to variational parameter tuning, penalty scaling, and hardware noise. Among classical optimizers for QAOA, COBYLA offers the most effective tradeoff between efficiency and noise tolerance. Overall, this work establishes a reproducible quantum-classical workflow, provides performance baselines, and identifies critical factors such as encoding design, optimizer choice, and error mitigation for improving near-term quantum heuristic performance.