This paper discusses techniques for solving discrete optimization problems using quantum annealing. Practical issues likely to affect the computation include precision limitations, finite temperature, bounded energy range, sparse connectivity, and small numbers of qubits. To address these concerns they propose a way of finding energy representations with large classical gaps between ground and first excited states, efficient algorithms for mapping non-compatible Ising models into the hardware, and the use of decomposition methods for problems that are too large to fit in hardware. They validate the approach by describing experiments with D-Wave quantum hardware for low density parity check decoding with up to 1000 variables.
They have outlined a general approach for coping with intrinsic issues related to the practical use of quantum annealing. To address these issues we proposed methods for finding Ising problem representations that have a large classical gap between ground states and first excited states, practical methods for embedding Ising models that are not compatible with the hardware graph, and decomposition methods to solve problems that are larger than the hardware. As an application of our techniques, we described how we implemented LDPC decoding problems in D-Wave hardware. Our approach has enabled us to solve LDPC decoding problems of up to 1000 variables. The current hardware implementation of QA tested here is roughly as fast as an efficient implementation of simulated annealing, but these results offer the promise of hybrid quantum/classical algorithms that surpass purely classical solution as QA hardware matures.
As future work, they would like to improve upon the scalability of the current method for constructing penalty functions with large gaps. This would allow larger component subproblems and reduce the need for minor embedding between subproblems. Further, the methods they ave described here for finding penalty functions assume an assignment of decision variables to qubits. Different assignment choices lead to different results and different hardware performance. They do not currently have an effective method for this assignment.