  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.
