Quantum solutions for densest k -subgraph problems

In this paper, we present, for the first time, quantum annealing solutions for densest k -subgraph problems which have many applications in computational biology. Our solutions are formulated as solutions for quadratic unconstrained binary optimization (QUBO) and integer programming problems, proved to be equivalent with the densest k -subgraph problems and then tested on an D-Wave 2X machine. The QUBO formulations are optimal in terms of the number of logical qubits, but require the highest number of physical qubits after embedding. Experimental work reported here suggests that the D-Wave 2X model cannot handle problems of this difficulty. The next generation of D-wave hardware architecture—the Pegasus architecture—is much denser than the current one, so dense QUBOs will be easier to embed. The experimental work also suggests that the current built-in post-processing optimization method does not work very well for some problems and the default setting (post-processing optimization on) should be avoided (or at least tested before being turned on).