Optics Express

An Optical Solution For The Traveling Salesman Problem

Tobias J. Haist, Wolfgang Osten

Published Jan 1, 2007• 58 citations

Abstract

We introduce an optical method based on white light interferometry in order to solve the well-known NP-complete traveling salesman problem. To our knowledge it is the first time that a method for the reduction of non-polynomial time to quadratic time has been proposed. We will show that this achievement is limited by the number of available photons for solving the problem. It will turn out that this number of photons is proportional to N(N) for a traveling salesman problem with N cities and that for large numbers of cities the method in practice therefore is limited by the signal-to-noise ratio. The proposed method is meant purely as a gedankenexperiment.

DOI: 10.1364/oe.15.010473

Related papers

An Efficient Quantum Algorithm for the Traveling Salesman Problem

Anant Sharma, Nupur Deshpande, Sanchita Purohit Ghosh +2 more

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 Information and ComputationFeb 22, 20251 citation

Efficient quantum algorithm for solving travelling salesman problem: An IBM quantum experience

Karthik Srinivasan, Saipriya Satyajit, Bikash K. Behera +1 more

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.

arXiv:1805.10928May 28, 201862 citations

Ultra-fast wave-optical computing

Tobias J. Haist

I present a method for optical computing based on white-light interferometry. The problem to be solved is coded by optical path lengths and the superposition of all possible paths that a photon can travel is used for computing the solution. The solution itself is chosen by interference with the reference light. Several gedankenexperiments demonstrate how this method can be used for solving computational hard problems. Especially, I will introduce the basic principle with two maze-type puzzles and then concentrate on how the method can be employed for performing ultra-fast (sub-ps) digital-optical arithmetic with arbitrary precision.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIEAug 28, 20082 citations

Quantum speedup of the traveling-salesman problem for bounded-degree graphs

Dominic J. Moylett, Noah Linden, Ashley Montanaro

The traveling-salesman problem is an iconic route-finding task, with applications from chip design to planning and logistics. Here the authors show that if the graph of cities to be visited is of low degree, a traveling salesman armed with a quantum satnav can find the best route quadratically faster than using any known classical method.

arXiv:1612.06203Physical Review AMar 22, 201743 citations

The Computational Complexity of the Traveling Salesman Problem

Craig Alan Feinstein

In this note, we show that the Traveling Salesman Problem cannot be solved in polynomial-time on a classical computer.

arXiv:cs/0611082Nov 17, 20064 citations

Sensing the insensible using optical schemes: converting the maze problem into a quantum search problem

Debabrata Goswami

Many large image processing and data processing scenarios can soon develop into maze problems requiring, say, finding the longest possible path, which are unresponsive, intractable, and challenging to analyze! Such maze problems, of which the traveling salesman decision problem is a special case, are of the Non-determinate Polynomial (NPcomplete) class of problems that are often impossible to solve with finite time and storage. We propose a novel methodology to approach this class of NP-complete problems. We convert a suitably formulated maze problem into a Quantum Search Problem (QSP), and the desired solutions are then sought using the iterative Grover’s Search Algorithm. Thus, we reformulate the entire class of such NP-problems into QSPs. Our current solution deals with two-dimensional perfect mazes with no closed loops. We encode all possible individual paths from the starting point of the maze into a quantum register. A quantum fitness operator applied to the register encodes each qubit with its fitness value. We propose the design of an optical oracle that marks all entities above a certain fitness value and uses the Grover search algorithm to find the optimal marked state in an iterative manner.

Apr 23, 20200 citations