Enhancing quantum annealing performance for the molecular similarity problem

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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Notes

  1. 1.

    H. Katzgraber, personal communication, 2016.

References

  1. 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)

  2. 2.

    Amin, M.H.: Searching for quantum speedup in quasistatic quantum annealers. Phys. Rev. A 92, 052,323 (2015)

    Article  Google Scholar 

  3. 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)

    MathSciNet  Article  MATH  Google Scholar 

  4. 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)

    Chapter  Google Scholar 

  5. 5.

    Baum, D.: A point-based algorithm for multiple 3D surface alignment of drug-sized molecules. Ph.D. thesis, Free University of Berlin (2007)

  6. 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. 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)

    Article  Google Scholar 

  8. 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. 9.

    Boothby, T., King, A.D., Roy, A.: Fast clique minor generation in chimera qubit connectivity graphs. Quantum Inf. Process 15, 495–508 (2016)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Boros, E., Gruber, A.: On quadratization of pseudo-Boolean functions. In: International Symposium on Artificial Intelligence and Mathematics (2012)

  11. 11.

    Cai, J., Macready, W.G., Roy, A.: A practical heuristic for finding graph minors. arXiv preprint arXiv:1406.2741 (2014)

  12. 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)

    ADS  Article  Google Scholar 

  13. 13.

    Choi, V.: Minor-embedding in adiabatic quantum computation: I. The parameter setting problem. Quantum Inf. Process. 7, 193–209 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Choi, V.: Minor-embedding in adiabatic quantum computation: II. Minor-universal graph design. Quantum Inf. Process. 10, 343–353 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Clarke, B.S., Barron, A.R.: Jeffreys’ prior is asymptotically least favorable under entropy risk. J. Stat. Plan. Inference 41, 37–60 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Das, A., Chakrabarti, B.K.: Colloquium: quantum annealing and analog quantum computation. Rev. Mod. Phys. 80, 1061–1081 (2008)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  17. 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. 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)

    MathSciNet  Article  MATH  Google Scholar 

  19. 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)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  20. 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)

    MATH  Google Scholar 

  21. 21.

    Giuseppe, E.S., Erio, T.: Optimization using quantum mechanics: quantum annealing through adiabatic evolution. J. Phys. A Math. Gen. 39, R393 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  22. 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)

  23. 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)

    Article  Google Scholar 

  24. 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. 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)

    ADS  Article  Google Scholar 

  26. 26.

    Kadowaki, T., Nishimori, H.: Quantum annealing in the transverse Ising model. Phys. Rev. E 58, 5355–5363 (1998)

    ADS  Article  Google Scholar 

  27. 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. 28.

    King, A.D., McGeoch, C.C.: Algorithm engineering for a quantum annealing platform. arXiv preprint arXiv:1410.2628 (2014)

  29. 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. 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. 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)

    Article  Google Scholar 

  32. 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. 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. 34.

    Popelier, P.L.A.: Quantum molecular similarity. 1. BCP space. J. Phys. Chem. A 103(15), 2883–2890 (1999)

    Article  Google Scholar 

  35. 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. 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)

    ADS  Article  Google Scholar 

  37. 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)

    ADS  Article  MATH  Google Scholar 

  38. 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)

    ADS  Article  Google Scholar 

  39. 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)

    ADS  Article  Google Scholar 

  40. 40.

    Selby, A.: Efficient subgraph-based sampling of Ising-type models with frustration. arXiv preprint arXiv:1409.3934 (2014)

  41. 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. 42.

    Venturelli, D., Marchand, D.J.J., Rojo, G.: Quantum annealing implementation of job-shop scheduling. arXiv preprint arXiv:1506.08479 (2015)

  43. 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)

    Article  Google Scholar 

  44. 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)

    Article  Google Scholar 

  45. 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)

    Article  Google Scholar 

  46. 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 

Download references

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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Maritza Hernandez.

Appendix

Appendix

Theoretical parameter setting

As mentioned in Sect. 3.2.2, we generalize the approach presented by Choi [13] to theoretically set the parameters of the embedded graph, in which the connected subgraphs representing the logical qubits are not necessarily subtrees of the hardware graph. The procedure used in this paper is explained in Algorithm 1. It is worth mentioning that we do not discuss the proof for the validity of the theoretical approach. The key idea as presented by Choi [13] is to ensure that the ground state of the input graph before and after embedding matches.

Let us consider a conflict graph \(G_{\hbox {c}} =(V_{\hbox {c}}, E_{\hbox {c}})\) with logical local fields and couplings denoted by h and J, respectively. Assume that logical qubit \(i \in V_{\hbox {c}}\) is represented by \(n_i\) physical qubits forming the physical subgraph \(S_i = (V_{S_i}, E_{S_i})\). Further assume that \(\hbox {lnb}(i)\) is the set of neighbouring vertices of the logical qubit i in the logical graph \(G_{\hbox {c}}\) and \(\hbox {pnb}(l)\) is the set of neighbouring vertices of the physical qubit \(l \in V(S_i)\), \(\forall i \in V_{\hbox {c}}\), excluding the vertices representing the same logical qubit.

The theoretical approach detailed below has an iterative pre-processing step in which several logical qubits might be removed from the logical graph, since their optimal values can be inferred in advance. The parameters are then set on the reduced logical graph.

figurea

Time-to-solution estimation

Since the quantum annealer is a stochastic solver, we consider the successive annealing runs as a sequence of binary experiments that might succeed in returning the ground state with some probability. Let us formally define \(X_1, X_2, \ldots , X_n\) as a sequence of random independent outcomes of n annealing runs, where \({\mathbbm {P}}(X_i=1)=\theta \) denotes the probability of observing the ground state at the i-th anneal. Defining Y as the number of successes observed in n anneals (\(Y=\sum _{i=1}^{n} X_i\)), we have \({\mathbbm {P}}(Y = y|\theta ) = {n \atopwithdelims ()y} (1-\theta )^{n-y}\theta ^y\) (\(Y|\theta \sim \hbox {Bin}(n, \theta )\)). That is, \(Y|\theta \) has a binomial distribution with parameters n and \(\theta \). The \(\hbox {R}_{99}\) then equals n such that \({\mathbbm {P}}(Y \ge 1|\theta ) = 0.99\). It is easy to verify that \(\hbox {R}_{99} = \log (1-0.99) / \log (1-\theta )\). Since the probability of success \(\theta \) is unknown, the challenge is to estimate \(\theta \).

We follow the Bayesian inference technique to estimate the probability of success for each instance i [23]. In the Bayesian inference framework, we start with a guess on the distribution of \(\theta \) known as prior and update it based on the observations from the device in order to get the posterior. Since the observations from the device have a binomial distribution, the proper choice of prior is a beta distribution which is the conjugate prior of the binomial distribution. This choice guarantees that the posterior also has a beta distribution. The beta distribution with parameters \(\alpha =0.5\) and \(\beta =0.5\) (the Jeffery prior) is chosen as prior since it is invariant under reparameterization of the space and it learns the most from the data [15].

Updating the Jeffery prior based on the data from the device, the posterior distribution denoted by \(\pi _i(\theta )\) is then

$$\begin{aligned} \pi _i(\theta ) \sim {\hbox {Beta}} \bigg (0.5+\sum _{c=1}^{C} y_{ci}, 0.5+NC-\sum _{c=1}^{C} y_{ci} \bigg ), \end{aligned}$$
(4)

where C is the number of calls to the quantum annealer, N is the number of anneals in each call, and \(y_{ci}\) is the number of times that the ground state of instance i is observed at the c-th call.

To estimate the TTS (or \(\hbox {R}_{99}\)) for the entire population of instances with similar parameters, let us assume that there are I instances with similar properties, for example, with the same number of variables. We are interested in using the data on these I instances to estimate the TTS for the entire population of instances with the same number of variables. After finding the posterior distribution \(\pi _i(\theta )\) for all instances in set \(\{I\}\), we use the bootstrap methodology to estimate the distribution of the q-th percentile of the TTS. The procedure is described in Algorithm 2.

figureb

Percolation threshold of the molecular similarity problem instances

The QUBO graphs of molecular similarity problem instances can be considered random graphs. A random graph is a collection of vertices with edges connecting pairs of them randomly. Newman et al. [31] and Callaway et al. [12] developed an approach based on generating functions to determine the statistical properties of random graphs with arbitrary degree distribution. Here we review their approach to determine the percolation threshold.

Let us denote the probability that a randomly chosen vertex has degree k by \(p_k\). Let us further define

$$\begin{aligned} G_0(x) = \sum _{k=0}^{\infty } p_k x^k, ~~~~G_1(x) = \frac{1}{z} G^{\prime }_0(x),~~~~{\hbox {and}} ~~~~z = G^{\prime }_0(1), \end{aligned}$$

where \(G_0(x)\) and \(G_1(x)\) are the generating functions for the probability distribution of vertex degrees and outgoing edges, respectively, and z is the average vertex degree. Callaway et al. [12] have shown that the percolation threshold, or critical probability, can be calculated as follows:

$$\begin{aligned} p_{\hbox {c}} = \frac{1}{G^{\prime }_1(1)} = \frac{G^{\prime }_0(1)}{G^{\prime \prime }_0(1)}. \end{aligned}$$

The details of the derivation can be found in [12, 31]. The key idea is that the critical probability is the point at which the mean cluster size goes to infinity.

The context in which we apply this criterion is on random graphs whose degree distribution is known. It is known because the degree distribution can be measured directly [31]. Below, we provide an example with a known degree distribution for which we calculate the critical probability.

Example  Consider a graph similarity problem instance with 18 vertices and \(d \in [65, 75]\). The number of vertices with degree 11, 12, and 13 are, in respective terms, 8, 8, and 2. The distribution of vertex degrees can be generated by

$$\begin{aligned} G_0(x) = \frac{8x^{11} + 8x^{12} + 2x^{13}}{18}. \end{aligned}$$

Applying the formula above, the critical probability is \(p_{\hbox {c}} = 0.0934\).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Hernandez, M., Aramon, M. Enhancing quantum annealing performance for the molecular similarity problem. Quantum Inf Process 16, 133 (2017). https://doi.org/10.1007/s11128-017-1586-y

Download citation

Keywords

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