# Journal of Advances in Applied Mathematics

### Mathematical Methods for a Quantum Annealing Computer

Download PDF (465.7 KB) PP. 82 - 90 Pub. Date: July 1, 2018

### Author(s)

**Richard H. Warren**^{*}

Lockheed Martin Corporation-Retired

### Abstract

### Keywords

### References

[1] S. H. Adachi and M. P. Henderson. 2015. Application of quantum annealing to training of deep neural networks. arXiv:1510.06356, 18 pages.

[2] M. Benedetti, J. Realpe-Gomez, R. Biswas, and A. Perdomo-Ortiz. 2016. Estimation of effective temperatures in quantum annealers for sampling applications: A case study with possible applications in deep learning. Phys. Rev. A 94, 022308.

[3] Z. Bian, F. Chudak, W. G. Macready, G. Rose. 2010. The Ising model: teaching an old problem new tricks. DWave Systems. http://www.dwavesys.com/sites/default/files/weightedmaxsat_v2.pdf

[4] Z. Bian, F. Chudak, W. G. Macready, L. Clark, F. Gaitan. 2013. Experimental determination of Ramsey numbers. Phys. Rev. Lett. 111, 130505.

[5] Z. Bian, F. Chudak, R. Israel, B. Lackey, W. G. Macready, A. Roy. 2014. Discrete optimization using quantum annealing on sparse Ising models. Front. Phys. 2 Article 56.

[6] E. Boros and P. L. Hammer. 2001. Pseudo-Boolean Optimization. http://rutcor.rutgers.edu/~boros/Papers/2002-DAM-BH.pdf

[7] S. Boixo, T. F. R?nnow, S. V. Isakov, Z. Wang, D. Wecker, D. A. Lidar, J. M. Martinis, M. Troyer. 2014. Evidence for quantum annealing with more than one hundred qubits. Nature Physics 10, 218-224. DOI: 10.1038/nphys2900

[8] S. Boixo, G. Ortiz, R. Somma. 2015. Fast quantum methods for optimization. Eur. Phys. J. Special Topics 224, 35-49.

[9] M. Henderson, J. Novak, T. Cook. 2018. Leveraging adiabatic quantum computation for election forecasting. arXiv:1802.00069

[10] V. Horan, S. Adachi, S. Bak. 2016. A comparison of approaches for finding minimum identifying codes on graphs. Quantum Inf. Process. 15, 1827-1848.

[11] Z. Jiang and E. G. Rieffel. 2017. Non-commuting two-local Hamiltonians for quantum error suppression. Quantum Inf. Process. 16, Article 89. https://doi.org/10.1007/s11128-017-1527-9

[12] M. W. Johnson et al. 2011. Quantum annealing with manufactured spins. Nature 473, 194-198. DOI: 10.1038/nature10012

[13] K. Karimi, N. G. Dickson, F. Hamze, M. H. S. Amin, M. Drew-Brook, F. A. Chudak, P. I. Bunyk, W. G. Macready, G. Rose. 2012. Investigating the performance of an adiabatic quantum optimization processor. Quantum Inf. Process. 11, 77-88.

[14] H. G. Katzgraber, F. Hamze, R. S. Andrist. 2014. Glassy chimeras could be blind to quantum speedup: Designing better benchmarks for quantum annealing machines. Phys. Review X 4, 021008.

[15] J. King, S. Yarkoni, M. M. Nevisi, J. P. Hilton, C. C. McGeoch. 2015. Benchmarking a quantum annealing processor with the time-to-target metric. arXiv:1508.05087

[16] D. Korenkevych, Y. Xue, Z. Bian, F. Chudak, W. G. Macready, J. Rolfe, E. Andriyash. 2016. Benchmarking quantum hardware for training of fully visible Boltzmann machines. arXiv:1611.04528

[17] T. Lanting et al. 2014. Entanglement in a quantum annealing processor. Phys. Rev. X 4, 021041. DOI: 10.1103/PhysRevX.4.021041

[18] A. Lucas. 2014. Ising formulations of many NP problems. Front. Phys. 2 Article 5, 15 pages. DOI:10.3389/fphy.2014.00005

[19] V. Martin-Mayor and I. Hen. 2015. Unraveling quantum annealers using classical hardness. Scientific Reports 5, 15324.

[20] C. C. McGeoch and C. Wang. 2013. Experimental evaluation of an adiabatic quantum system for combinatorial optimization, Proceedings of the ACM International Conference on Computing Frontiers, Article No. 23, ACM Press. DOI:10.1145/2482767.2482797

[21] C. C. McGeoch. 2014. Adiabatic Quantum Computation and Quantum Annealing: Theory and Practice, Morgan & Claypool.

[22] F. Neukart, D. Von Dollen, C. Seidel, G. Compostella. 2018. Quantum-enhanced reinforcement learning for finite-episode games with discrete state spaces. Front. Phys. 5, Article 71. DOI:10.3389/fphy.2017.00071

[23] Programming with QUBOs. 2016. Release 2.4, D-Wave Systems. Available upon request at inquiry@dwavesys.com.

[24] K. L. Pudenz, T. Albash, D. A. Lidar. 2014. Error-corrected quantum annealing with hundreds of qubits. Nature Communications 5, Article No. 3243.

[25] E. G. Rieffel, D. Venturelli, B. O’Gorman, M. B. Do, E. M. Prystay, V. N. Smelyanskiy. 2015. A case study in programming a quantum annealer for hard operational planning problems. Quantum Inf. Process. 14, 1-36.

[26] T. F. R?nnow, Z. Wang, J. Job, S. Boixo, S. V. Isakov, D. Wecker, J. M. Martinis, D. A. Lidar, M. Troyer. 2014. Defining and detecting quantum speedup. Nature 345, 420-424.

[27] G. E. Santoro and E. Tosatti. 2006. Optimization using quantum mechanics: quantum annealing through adiabatic evolution. J. Phys. A 39, R393-R431. DOI:10.1088/0305-4470/39/36/R01

[28] P. Shor. 1997. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM J. Comput. 26, 1484-1509.

[29] S. Suzuki, J. Inoue, B. K. Chakrabarti. 2013. Quantum Ising Phases and Transitions in Transverse Ising Models, 2nd edition, Springer-Verlag.

[30] R. H. Warren. 2013. Adapting the traveling salesman problem to an adiabatic quantum computer, Quantum Inf. Process. 12, 1781-1785. DOI:10.1007/s11128-012-0490-8

[31] J. D. Whitfield, M. Faccin, J. D. Biamonte. 2012. Ground-state spin logic. Europhysics Letters 99, 57004. DOI:10.1209/0295-5075/99/57004

[32] T. Albash and D. A. Lidar. 2018. Adiabatic quantum computing, Rev. Mod. Phys. 90, 015002. DOI:10.1103/RevModPhys.90.015002