Discrete mathematics

Physics

Graph

Qubit

Mathematics

Combinatorial optimization

Hamiltonian path

Hamiltonian path problem

Algorithm

Mathematical optimization

Hamiltonian (control theory)

Optimization problem

Political science

Travelling salesman problem

Computer science

Quantum mechanics

Unitary state

Quantum

Computers, materials & continua/Computers, materials & continua (Print)

The Quantum Approximate Optimization Algorithm (QAOA) is an algorithmic framework for finding approximate solutions to combinatorial optimization problems. It consists of interleaved unitary transformations induced by two operators labelled the mixing and problem Hamiltonians. To fit this framework, one needs to transform the original problem into a suitable form and embed it into these two Hamiltonians. In this paper, for the well-known NP-hard Traveling Salesman Problem (TSP), we encode its constraints into the mixing Hamiltonian rather than the conventional approach of adding penalty terms to the problem Hamiltonian. Moreover, we map edges (routes) connecting each pair of cities to qubits, which decreases the search space significantly in comparison to other approaches. As a result, our method can achieve a higher probability for the shortest round-trip route with only half the number of qubits consumed compared to IBM Q's approach. We argue the formalization approach presented in this paper would lead to a generalized framework for finding, in the context of QAOA, high-quality approximate solutions to NP optimization problems.

The Quantum Approximate Algorithm for Solving Traveling Salesman Problem