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Журнал Экспериментальной и Теоретической Физики

A Quantum Algorithm for Solving the Travelling Salesman Problem by Quantum Phase Estimation and Quantum Search

Ch. Tszyun'si, Ilya I. Beterov

Published Dec 15, 2023• 0 citations
Publisher

Abstract

A quantum algorithm for solving the traveling salesman problem by the quantum phase estimation and quantum search method is considered. An approach is developed that was previously proposed for solving this problem. A quantum register is used to encode the eigenstates of a unitary operator whose phase determines the length of each possible route. The quantum phase estimation algorithm is used to estimate the length of a route. Then, to find the minimum route length, the measured values of length are encoded into the states of the second quantum register, and the search for the optimal route is carried out using a modified Grover algorithm. Numerical simulation of the proposed quantum algorithm is carried out using the Qiskit library for one and two iterations of the modified Grover algorithm.

DOI
10.31857/s0044451023080102

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A quantum algorithm for solving the traveling salesman problem by the quantum phase estimation and quantum search method is considered. An approach is developed that was previously proposed for solving this problem. A quantum register is used to encode the eigenstates of a unitary operator whose phase determines the length of each possible route. The quantum phase estimation algorithm is used to estimate the length of a route. Then, to find the minimum route length, the measured values of length are encoded into the states of the second quantum register, and the search for the optimal route is carried out using a modified Grover algorithm. Numerical simulation of the proposed quantum algorithm is carried out using the Qiskit library for one and two iterations of the modified Grover algorithm.
Abstract A quantum algorithm for solving the traveling salesman problem by the quantum phase estimation and quantum search method is considered. An approach is developed that was previously proposed for solving this problem. A quantum register is used to encode the eigenstates of a unitary operator whose phase determines the length of each possible route. The quantum phase estimation algorithm is used to estimate the length of a route. Then, to find the minimum route length, the measured values of length are encoded into the states of the second quantum register, and the search for the optimal route is carried out using a modified Grover algorithm. Numerical simulation of the proposed quantum algorithm is carried out using the Qiskit library for one and two iterations of the modified Grover algorithm.
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.
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.
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 paper proposes a quantum algorithm for the traveling salesman problem (TSP) based on the Grover Adaptive Search (GAS), which can be successfully executed on IBM's Qiskit library. Under the GAS framework, there are at least two fundamental difficulties that limit the application of quantum algorithms for combinatorial optimization problems. One difficulty is that the solutions given by the quantum algorithms may not be feasible. The other difficulty is that the number of qubits of current quantum computers is still very limited, and it cannot meet the minimum requirements for the number of qubits required by the algorithm. In response to the above difficulties, we designed and improved the Hamiltonian Cycle Detection (HCD) oracle based on mathematical theorems. It can automatically eliminate infeasible solutions during the execution of the algorithm. On the other hand, we design an anchor register strategy to save the usage of qubits. The strategy fully considers the reversibility requirement of quantum computing, overcoming the difficulty that the used qubits cannot be simply overwritten or released. As a result, we successfully implemented the numerical solution to TSP on IBM's Qiskit. For the seven-node TSP, we only need 31 qubits, and the success rate in obtaining the optimal solution is 86.71%.