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Quantum solutions for densest k-subgraph problems

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Abstract

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).

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Notes

  1. This particular formulation is actually a \(0-1\) IP formulation as well.

  2. These publications refer to the map size as ‘chain length’ which could be somewhat misleading since the set of physical qubits do not necessarily form a path.

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Acknowledgements

This work was supported in part by the Quantum Computing Research Initiatives at Lockheed Martin. We thank the anonymous referees for useful comments which improved the presentation.

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Authors and Affiliations

  1. School of Computer Science, University of Auckland, Auckland, New Zealand

    Cristian S. Calude, Michael J. Dinneen & Richard Hua

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  1. Cristian S. Calude
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  2. Michael J. Dinneen
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  3. Richard Hua
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Correspondence to Cristian S. Calude.

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Appendices

Appendix 1: Python script to generate random bipartite graphs

figure d

Appendix 2: Python script to generate QUBOs

figure e

Appendix 3: Sage script to solve the edge-weighted densest k-subgraph problem

figure f

Appendix 4: Randomized algorithm to solve the edge-weighted densest k-subgraph problem

figure g

Appendix 5: Drawing of a sample test graph

figure h

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Calude, C.S., Dinneen, M.J. & Hua, R. Quantum solutions for densest k-subgraph problems. J Membr Comput 2, 26–41 (2020). https://doi.org/10.1007/s41965-019-00030-1

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