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Mapping graph coloring to quantum annealing

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Abstract

Quantum annealing provides a method to solve combinatorial optimization problems in complex energy landscapes by exploiting thermal fluctuations that exist in a physical system. This work introduces the mapping of a graph coloring problem based on pseudo-Boolean constraints to a working graph of the D-Wave Systems Inc. We start from the problem formulated as a set of constraints represented in propositional logic. We use the SATyrus approach to transform this set of constraints to an energy minimization problem. We convert the formulation to a quadratic unconstrained binary optimization problem (QUBO), applying polynomial reduction when needed, and solve the problem using different approaches: (a) classical QUBO using simulated annealing in a von Neumann machine; (b) QUBO in a simulated quantum environment; (c) actual quantum 1, QUBO using the D-Wave quantum machine and reducing polynomial degree using a D-Wave library; and (d) actual quantum 2, QUBO using the D-Wave quantum machine and reducing polynomial degree using our own implementation. We study how the implementations using these approaches vary in terms of the impact on the number of solutions found (a) when varying the penalties associated with the constraints and (b) when varying the annealing approach, simulated (SA) versus quantum (QA). Results show that both SA and QA produce good heuristics for this specific problem, although we found more solutions through the QA approach.

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Notes

  1. The problem formulated this way assumes that there is a solution, suboptimal, that assigns one distinct color to each region in the map.

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Authors and Affiliations

  1. Department of Computer Science, Faculty of Sciences, University of Porto, Porto, Portugal

    Carla Silva

  2. Faculty of Engineering, Instituto de Telecomunicações and Department of Electrical and Computer Engineering, Porto, Portugal

    Ana Aguiar

  3. PESC/COPPE/CT and Tercio Pacitti Institute/CCMN and Federal University of Rio de Janeiro, Rio de Janeiro, Brazil

    Priscila M. V. Lima

  4. CINTESIS and Department of Computer Science, Faculty of Sciences, University of Porto, Porto, Portugal

    Inês Dutra

Authors
  1. Carla Silva
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  2. Ana Aguiar
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  3. Priscila M. V. Lima
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  4. Inês Dutra
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Correspondence to Carla Silva.

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Appendices

Appendix 1: Varying α and β

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figure 8

Solutions for pure classical C and classical quantum simulator C_Q_sim and both methods of polynomial reduction Q_mq and Q_ms (varying α and β)

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Fig. 9
figure 9

Energy values of classical C and both methods of polynomial reduction (Q_mq and Q_ms) (varying α and β)

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Fig. 10
figure 10

Solutions for pure classical C and classical quantum simulator C_Q_sim and both methods of polynomial reduction Q_mq and Q_ms (varying α)

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figure 11

Energy values of classical C and both methods of polynomial reduction (Q_mq and Q_ms) (varying α)

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Fig. 12
figure 12

Solutions for pure classical C and classical quantum simulator C_Q_sim and both methods of polynomial reduction Q_mq and Q_ms (varying β)

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figure 13

Energy values of classical C and both methods of polynomial reduction (Q_mq and Q_ms) (varying β)

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Appendix 2: Problemmapped onto the QPU

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Mapping variables to assigned qubits on both Q_mq and Q_ms methods for α = 5 and β = 755 for test in Fig. 15

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Fig. 15
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Binary optimal results: (α= 1, β= 151), (α= 2, β= 302), (α= 3, β= 453), (α= 4, β= 604), (α= 5, β= 755) on both Q_mq and Q_ms methods for test (varying α and β)

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Silva, C., Aguiar, A., Lima, P.M.V. et al. Mapping graph coloring to quantum annealing. Quantum Mach. Intell. 2, 16 (2020). https://doi.org/10.1007/s42484-020-00028-4

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  • DOI: https://doi.org/10.1007/s42484-020-00028-4

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