# Enhancing quantum annealing performance for the molecular similarity problem

- 82 Downloads

## Abstract

Quantum annealing is a promising technique which leverages quantum mechanics to solve hard optimization problems. Considerable progress has been made in the development of a physical quantum annealer, motivating the study of methods to enhance the efficiency of such a solver. In this work, we present a quantum annealing approach to measure similarity among molecular structures. Implementing real-world problems on a quantum annealer is challenging due to hardware limitations such as sparse connectivity, intrinsic control error, and limited precision. In order to overcome the limited connectivity, a problem must be reformulated using minor-embedding techniques. Using a real data set, we investigate the performance of a quantum annealer in solving the molecular similarity problem. We provide experimental evidence that common practices for embedding can be replaced by new alternatives which mitigate some of the hardware limitations and enhance its performance. Common practices for embedding include minimizing either the number of qubits or the chain length and determining the strength of ferromagnetic couplers empirically. We show that current criteria for selecting an embedding do not improve the hardware’s performance for the molecular similarity problem. Furthermore, we use a theoretical approach to determine the strength of ferromagnetic couplers. Such an approach removes the computational burden of the current empirical approaches and also results in hardware solutions that can benefit from simple local classical improvement. Although our results are limited to the problems considered here, they can be generalized to guide future benchmarking studies.

### Keywords

Quantum annealing Quantum optimization Molecular similarity Minor embedding Parameter setting Quadratic unconstrained binary optimization## Notes

### Acknowledgements

This research was supported by 1QBit. The authors would like to thank Clemens Adolphs, Hamed Karimi, Anna Levit, Dominic Marchand, and Arman Zaribafiyan for useful discussions, Robyn Foerster for valuable support, and Marko Bucyk for editorial help. We thank Helmut Katzgraber for reviewing the manuscript. We also acknowledge the support of the Universities Space Research Association (USRA) Quantum Artificial Intelligence Laboratory Research Opportunity program.

### References

- 1.Abu-Khzam, F.N., Samatova, N.F., Rizk, M.A., Langston, M.A.: The maximum common subgraph problem: faster solutions via vertex cover. In: IEEE/ACS International Conference on Computer Systems and Applications, pp. 367–373 (2007)Google Scholar
- 2.Amin, M.H.: Searching for quantum speedup in quasistatic quantum annealers. Phys. Rev. A
**92**, 052,323 (2015)CrossRefGoogle Scholar - 3.Balasundaram, B., Butenko, S., Hicks, I.V.: Clique relaxations in social network analysis: the maximum \(k\)-plex problem. Oper. Res.
**59**, 133–142 (2011)MathSciNetCrossRefMATHGoogle Scholar - 4.Balasundaram, B., Mahdavi Pajouh, F.: Graph theoretic clique relaxations and applications. In: Pardalos, P.M., Du, D.Z., Graham, R.L. (eds.) Handbook of Combinatorial Optimization, pp. 1559–1598. Springer, New York (2013)CrossRefGoogle Scholar
- 5.Baum, D.: A point-based algorithm for multiple 3D surface alignment of drug-sized molecules. Ph.D. thesis, Free University of Berlin (2007)Google Scholar
- 6.Bian, Z., Chudak, F., Israel, R., Lackey, B., Macready, W.G., Roy, A.: Mapping constrained optimization problems to quantum annealing with application to fault diagnosis. arXiv preprint arXiv:1603.03111 (2016)
- 7.Boixo, S., Ronnow, T.F., Isakov, S.V., Wang, Z., Wecker, D., Lidar, D.A., Martinis, J.M., Troyer, M.: Evidence for quantum annealing with more than one hundred qubits. Nat. Phys.
**10**, 218–224 (2014)CrossRefGoogle Scholar - 8.Boixo, S., Smelyanskiy, V.N., Shabani, A., Isakov, S.V., Dykman, M., Denchev, V.S., Amin, M.H., Smirnov, A.Y., Mohseni, M., Neven, H.: Computational multiqubit tunnelling in programmable quantum annealers. Nat. Commun. (2016). doi: 10.1038/ncomms10327 Google Scholar
- 9.Boothby, T., King, A.D., Roy, A.: Fast clique minor generation in chimera qubit connectivity graphs. Quantum Inf. Process
**15**, 495–508 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar - 10.Boros, E., Gruber, A.: On quadratization of pseudo-Boolean functions. In: International Symposium on Artificial Intelligence and Mathematics (2012)Google Scholar
- 11.Cai, J., Macready, W.G., Roy, A.: A practical heuristic for finding graph minors. arXiv preprint arXiv:1406.2741 (2014)
- 12.Callaway, D.S., Newman, M.E.J., Strogatz, S.H., Watts, D.J.: Network robustness and fragility: percolation on random graphs. Phys. Rev. Lett.
**85**, 5468–5471 (2000)ADSCrossRefGoogle Scholar - 13.Choi, V.: Minor-embedding in adiabatic quantum computation: I. The parameter setting problem. Quantum Inf. Process.
**7**, 193–209 (2008)MathSciNetCrossRefMATHGoogle Scholar - 14.Choi, V.: Minor-embedding in adiabatic quantum computation: II. Minor-universal graph design. Quantum Inf. Process.
**10**, 343–353 (2011)MathSciNetCrossRefMATHGoogle Scholar - 15.Clarke, B.S., Barron, A.R.: Jeffreys’ prior is asymptotically least favorable under entropy risk. J. Stat. Plan. Inference
**41**, 37–60 (1994)MathSciNetCrossRefMATHGoogle Scholar - 16.Das, A., Chakrabarti, B.K.: Colloquium: quantum annealing and analog quantum computation. Rev. Mod. Phys.
**80**, 1061–1081 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar - 17.Denchev, V.S., Boixo, S., Isakov, S.V., Ding, N., Babbush, R., Smelyanskiy, V., Martinis, J., Neven, H.: What is the computational value of finite-range tunneling? Phys. Rev. X
**6**, 031,015 (2016)Google Scholar - 18.Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness II: On completeness for \(w[1]\). Theor. Comput. Sci
**141**, 109–131 (1995)MathSciNetCrossRefMATHGoogle Scholar - 19.Farhi, E., Goldstone, J., Gutmann, S., Lapan, J., Lundgren, A., Preda, D.: A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem. Science
**292**, 472–475 (2001)ADSMathSciNetCrossRefMATHGoogle Scholar - 20.Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, New York (1979)MATHGoogle Scholar
- 21.Giuseppe, E.S., Erio, T.: Optimization using quantum mechanics: quantum annealing through adiabatic evolution. J. Phys. A Math. Gen.
**39**, R393 (2006)MathSciNetCrossRefMATHGoogle Scholar - 22.Hamze, F., Freitas, N.d.: From fields to trees. In: Proceedings of the 20th Conference on Uncertainty in Artificial Intelligence, pp. 243–250 (2004)Google Scholar
- 23.Hen, I., Job, J., Albash, T., Rønnow, T.F., Troyer, M., Lidar, D.A.: Probing for quantum speedup in spin-glass problems with planted solutions. Phys. Rev. A
**92**, 042,325 (2015)CrossRefGoogle Scholar - 24.Hernandez, M., Zaribafiyan, A., Aramon, M., Naghibi, M.: A novel graph-based approach for determining molecular similarity. arXiv preprint arXiv:1601.06693 (2016)
- 25.Johnson, M.W., Amin, M.H.S., Gildert, S., Lanting, T., Hamze, F., Dickson, N., Harris, R., Berkley, A.J., Johansson, J., Bunyk, P., Chapple, E.M., Enderud, C., Hilton, J.P., Karimi, K., Ladizinsky, E., Ladizinsky, N., Oh, T., Perminov, I., Rich, C., Thom, M.C., Tolkacheva, E., Truncik, C.J.S., Uchaikin, S., Wang, J., Wilson, B., Rose, G.: Quantum annealing with manufactured spins. Nature
**473**, 194–198 (2011)ADSCrossRefGoogle Scholar - 26.Kadowaki, T., Nishimori, H.: Quantum annealing in the transverse Ising model. Phys. Rev. E
**58**, 5355–5363 (1998)ADSCrossRefGoogle Scholar - 27.Katzgraber, H.G., Hamze, F., Zhu, Z., Ochoa, A.J., Munoz-Bauza, H.: Seeking quantum speedup through spin glasses: the good, the bad, and the ugly. Phys. Rev. X
**5**, 031,026 (2015)Google Scholar - 28.King, A.D., McGeoch, C.C.: Algorithm engineering for a quantum annealing platform. arXiv preprint arXiv:1410.2628 (2014)
- 29.Lanting, T., Przybysz, A.J., Smirnov, A.Y., Spedalieri, F.M., Amin, M.H., Berkley, A.J., Harris, R., Altomare, F., Boixo, S., Bunyk, P., Dickson, N., Enderud, C., Hilton, J.P., Hoskinson, E., Johnson, M.W., Ladizinsky, E., Ladizinsky, N., Neufeld, R., Oh, T., Perminov, I., Rich, C., Thom, M.C., Tolkacheva, E., Uchaikin, S., Wilson, A.B., Rose, G.: Entanglement in a quantum annealing processor. Phys. Rev. X
**4**, 021,041 (2014)Google Scholar - 30.Mandrà, S., Zhu, Z., Wang, W., Perdomo-Ortiz, A., Katzgraber, H.G.: Strengths and weaknesses of weak-strong cluster problems: a detailed overview of state-of-the-art classical heuristics vs quantum approaches. arXiv preprint arXiv:1604.01746 (2016)
- 31.Newman, M.E.J., Strogatz, S.H., Watts, D.J.: Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E.
**64**, 026,118 (2001)CrossRefGoogle Scholar - 32.Perdomo-Ortiz, A., Fluegemann, J., Biswas, R., Smelyanskiy, V.N.: A performance estimator for quantum annealers: gauge selection and parameter setting. arXiv preprint arXiv:1503.01083 (2015)
- 33.Perdomo-Ortiz, A., O’Gorman, B., Fluegemann, J., Biswas, R., Smelyanskiy, V.N.: Determination and correction of persistent biases in quantum annealers. Sci. Rep. (2016). doi: 10.1038/srep18628 Google Scholar
- 34.Popelier, P.L.A.: Quantum molecular similarity. 1. BCP space. J. Phys. Chem. A
**103**(15), 2883–2890 (1999)CrossRefGoogle Scholar - 35.Pudenz, K.L., Albash, T., Lidar, D.A.: Error-corrected quantum annealing with hundreds of qubits. Nat. Commun. (2014). doi: 10.1038/ncomms4243 Google Scholar
- 36.Rarey, M., Dixon, J.S.: Feature trees: a new molecular similarity measure based on tree matching. J. Comput. Aided Mol. Des.
**12**(5), 471–490 (1998)ADSCrossRefGoogle Scholar - 37.Rieffel, E.G., Venturelli, D., O’Gorman, B., Do, M.B., Prystay, E.M., Smelyanskiy, V.N.: A case study in programming a quantum annealer for hard operational planning problems. Quantum Inf. Process.
**14**, 1–36 (2015)ADSCrossRefMATHGoogle Scholar - 38.Rønnow, T.F., Wang, Z., Job, J., Boixo, S., Isakov, S.V., Wecker, D., Martinis, J.M., Lidar, D.A., Troyer, M.: Defining and detecting quantum speedup. Science
**345**, 420–424 (2014)ADSCrossRefGoogle Scholar - 39.Rosenberg, G., Haghnegahdar, P., Goddard, P., Carr, P., Wu, J., de Prado, M.: Solving the optimal trading trajectory problem using a quantum annealer. IEEE J. Sel. Top. Signal Process.
**10**, 1053–1060 (2016)ADSCrossRefGoogle Scholar - 40.Selby, A.: Efficient subgraph-based sampling of Ising-type models with frustration. arXiv preprint arXiv:1409.3934 (2014)
- 41.Venturelli, D., Mandrà, S., Knysh, S., O’Gorman, B., Biswas, R., Smelyanskiy, V.: Quantum optimization of fully connected spin glasses. Phys. Rev. X
**5**, 031,040 (2015)Google Scholar - 42.Venturelli, D., Marchand, D.J.J., Rojo, G.: Quantum annealing implementation of job-shop scheduling. arXiv preprint arXiv:1506.08479 (2015)
- 43.Vinci, W., Albash, T., Paz-Silva, G., Hen, I., Lidar, D.A.: Quantum annealing correction with minor embedding. Phys. Rev. A
**92**, 042,310 (2015)CrossRefGoogle Scholar - 44.Xu, C., Cheng, F., Chen, L., Du, Z., Li, W., Liu, G., Lee, P.W., Tang, Y.: In silico prediction of chemical Ames mutagenicity. J. Chem. Inf. Model.
**52**, 2840–2847 (2012)CrossRefGoogle Scholar - 45.Zhu, Z., Ochoa, A.J., Schnabel, S., Hamze, F., Katzgraber, H.G.: Best-case performance of quantum annealers on native spin-glass benchmarks: how chaos can affect success probabilities. Phys. Rev. A
**93**, 012,317 (2016)CrossRefGoogle Scholar - 46.Zick, K.M., Shehab, O., French, M.: Experimental quantum annealing: case study involving the graph isomorphism problem. Sci. Rep. (2015). doi: 10.1038/srep11168 Google Scholar