Two-Step Quantum Search Algorithm for Solving Traveling Salesman Problems

TL;DR

This paper introduces a two-step quantum search algorithm that efficiently solves the traveling salesman problem by reducing qubit requirements and achieving quadratic speedup, overcoming previous limitations in initial state preparation.

Contribution

The proposed TSQS algorithm innovatively amplifies feasible solutions and the optimal solution in two steps, improving efficiency and reducing qubit needs for large-scale TSPs.

Findings

Quadratic speedup over traditional quantum search methods

Reduced qubit requirements via HOBO representation

Efficient initial state preparation with a unified circuit

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 (HOBO) 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.