Abstract
This article addresses the formulation for implementing a single source, single-destination shortest path algorithm on a quantum annealing computer. Three distinct approaches are presented. In all the three cases, the shortest path problem is formulated as a quadratic unconstrained binary optimization problem amenable to quantum annealing. The first implementation builds on existing quantum annealing solutions to the traveling salesman problem, and requires the anticipated maximum number of vertices on the solution path $|P|$ to be provided as an input. For a graph with $|V|$ vertices, $|E|$ edges, and no self-loops, it encodes the problem instance using $|V||P|$ qubits. The second implementation adapts the linear programming formulation of the shortest path problem, and encodes the problem instance using $|E|$ qubits for directed graphs or $2|E|$ qubits for undirected graphs. The third implementation, designed exclusively for undirected graphs, encodes the problem in $|E| + |V|$ qubits. Scaling factors for penalty terms, complexity of coupling matrix construction, and numerical estimates of the annealing time required to find the shortest path are made explicit in the article.
Highlights
The shortest path problem is a well-studied primitive in graph theory
The energy function is generally represented as an Ising model, or a related quadratic unconstrained binary optimization (QUBO) formulation
We have chosen the four-vertex, five-edge undirected graph in Fig. 2, as it is amenable to all three quantum annealing formulations we have described and does not require too many qubits to be represented in any of them
Summary
The shortest path problem is a well-studied primitive in graph theory. The single source, single-destination variant of the problem is the simplest to describe: Given a graph with costs assigned to its edges and given a source and terminal vertex in the graph, find a continuous path from the source vertex to the terminal vertex whose constituent edges have minimum total cost. At the completion of the annealing, the qubits remain in the ground state of the desired problem Hamiltonian and, represent the global optimum [2]–[4]. The energy function is generally represented as an Ising model, or a related quadratic unconstrained binary optimization (QUBO) formulation. This article will detail a QUBO formulation of this shortest path (lowest cost) problem suitable for execution on a D-Wave quantum annealing machine or via a simulated annealing package. The following three sections develop three separate but related approaches to solving the shortest path problem. Krauss and McCollum: SOLVING THE NETWORK SHORTEST PATH PROBLEM ON A QUANTUM ANNEALER
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