Abstract
We illustrate how the D-Wave Two quantum computer is programmed and works by solving the Broadcast Time Problem. We start from a concise integer program formulation of the problem and apply some simple transformations to arrive at the QUBO form which can be run on the D-Wave quantum computer. Finally, we explore the feasibility of this method on several well-known graphs.
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Notes
- 1.
D-Wave Two is capable of using it.
- 2.
Solving the problem for other originators can be easily done by relabelling the vertices of the graph or doing obvious modifications in the formulation below.
References
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Acknowledgments
This work was supported in part by the Quantum Computing Research Initiatives at Lockheed Martin. We thank A. Fowler for comments which improved the presentation.
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Appendix: Quadratic IP Formulation for Broadcasting in \(Q_3\)
Appendix: Quadratic IP Formulation for Broadcasting in \(Q_3\)
The output of our integer programming formulation from Sect. 17.5 with the hypercube \(Q_3\) as input is given below.
\(x_0 \le 7\) (1) time \(t\)
\(x_1 \le x_0\) (2) vertices \(v_0 \ldots v_7\) informed times \(\le t\)
\(x_2 \le x_0\)
\(x_3 \le x_0\)
\(x_4 \le x_0\)
\(x_5 \le x_0\)
\(x_6 \le x_0\)
\(x_7 \le x_0\)
\(x_8 \le x_0\)
\(x_9 + x_{10} + x_{11} \le 0\) (3) originator has no parent
\(x_{12} + x_{13} + x_{14} \le 1\) (4) other vertices have one parent
\(x_{15} + x_{16} + x_{17} \le 1\)
\(x_{18} + x_{19} + x_{20} \le 1\)
\(x_{21} + x_{22} + x_{23} \le 1\)
\(x_{24} + x_{25} + x_{26} \le 1\)
\(x_{27} + x_{28} + x_{29} \le 1\)
\(x_{30} + x_{31} + x_{32} \le 1\)
\(x_{12} + x_{12} * x_1 - x_{12} * x_2 \le 0\) (5) parent time less than child time
\(x_{15} + x_{15} * x_1 - x_{15} * x_3 \le 0\)
\(x_{21} + x_{21} * x_1 - x_{21} * x_5 \le 0\)
\(x_9 + x_9 * x_2 - x_9 * x_1 \le 0\)
\(x_{18} + x_{18} * x_2 - x_{18} * x_4 \le 0\)
\(x_{24} + x_{24} * x_2 - x_{24} * x_6 \le 0\)
\(x_{10} + x_{10} * x_3 - x_{10} * x_1 \le 0\)
\(x_{19} + x_{19} * x_3 - x_{19} * x_4 \le 0\)
\(x_{27} + x_{27} * x_3 - x_{27} * x_7 \le 0\)
\(x_{13} + x_{13} * x_4 - x_{13} * x_2 \le 0\)
\(x_{16} + x_{16} * x_4 - x_{16} * x_3 \le 0\)
\(x_{30} + x_{30} * x_4 - x_{30} * x_8 \le 0\)
\(x_{11} + x_{11} * x_5 - x_{11} * x_1 \le 0\)
\(x_{25} + x_{25} * x_5 - x_{25} * x_6 \le 0\)
\(x_{28} + x_{28} * x_5 - x_{28} * x_7 \le 0\)
\(x_{14} + x_{14} * x_6 - x_{14} * x_2 \le 0\)
\(x_{22} + x_{22} * x_6 - x_{22} * x_5 \le 0\)
\(x_{31} + x_{31} * x_6 - x_{31} * x_8 \le 0\)
\(x_{17} + x_{17} * x_7 - x_{17} * x_3 \le 0\)
\(x_{23} + x_{23} * x_7 - x_{23} * x_5 \le 0\)
\(x_{32} + x_{32} * x_7 - x_{32} * x_8 \le 0\)
\(x_{20} + x_{20} * x_8 - x_{20} * x_4 \le 0\)
\(x_{26} + x_{26} * x_8 - x_{26} * x_6 \le 0\)
\(x_{29} + x_{29} * x_8 - x_{29} * x_7 \le 0\)
\(x_{12} + x_{15} - sqr( x_2 - x_3 ) \le 1\) (6) each child with different times
\(x_{12} + x_{21} - sqr( x_2 - x_5 ) \le 1\)
\(x_{15} + x_{21} - sqr( x_3 - x_5 ) \le 1\)
\(x_9 + x_{18} - sqr( x_1 - x_4 ) \le 1\)
\(x_9 + x_{24} - sqr( x_1 - x_6 ) \le 1\)
\(x_{18} + x_{24} - sqr( x_4 - x_6 ) \le 1\)
\(x_{10} + x_{19} - sqr( x_1 - x_4 ) \le 1\)
\(x_{10} + x_{27} - sqr( x_1 - x_7 ) \le 1\)
\(x_{19} + x_{27} - sqr( x_4 - x_7 ) \le 1\)
\(x_{13} + x_{16} - sqr( x_2 - x_3 ) \le 1\)
\(x_{13} + x_{30} - sqr( x_2 - x_8 ) \le 1\)
\(x_{16} + x_{30} - sqr( x_3 - x_8 ) \le 1\)
\(x_{11} + x_{25} - sqr( x_1 - x_6 ) \le 1\)
\(x_{11} + x_{28} - sqr( x_1 - x_7 ) \le 1\)
\(x_{25} + x_{28} - sqr( x_6 - x_7 ) \le 1\)
\(x_{14} + x_{22} - sqr( x_2 - x_5 ) \le 1\)
\(x_{14} + x_{31} - sqr( x_2 - x_8 ) \le 1\)
\(x_{22} + x_{31} - sqr( x_5 - x_8 ) \le 1\)
\(x_{17} + x_{23} - sqr( x_3 - x_5 ) \le 1\)
\(x_{17} + x_{32} - sqr( x_3 - x_8 ) \le 1\)
\(x_{23} + x_{32} - sqr( x_5 - x_8 ) \le 1\)
\(x_{20} + x_{26} - sqr( x_4 - x_6 ) \le 1\)
\(x_{20} + x_{29} - sqr( x_4 - x_7 ) \le 1\)
\(x_{26} + x_{29} - sqr( x_6 - x_7 ) \le 1\)
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Calude, C.S., Dinneen, M.J. (2017). Solving the Broadcast Time Problem Using a D-wave Quantum Computer. In: Adamatzky, A. (eds) Advances in Unconventional Computing. Emergence, Complexity and Computation, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-319-33924-5_17
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