# Boltzmann Sampling by Degenerate Optical Parametric Oscillator Network for Structure-Based Virtual Screening

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Coherent Ising Machine (CIM)

#### 2.1. Computation Principle of CIM

#### 2.2. Implementation of Zeeman Terms

## 3. The Problem Hamiltonians

#### 3.1. Lead Optimization Procedure

#### 3.2. Mapping to the Ising Hamiltonian

#### 3.3. Several Heuristic Modifications

**J**as the vector whose elements are non-zero elements of $\left\{{J}_{in;jm}\right\}$, the problem is finding $\mathbf{J}$ which satisfies

## 4. Numerical Simulation Results

## 5. Conclusions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Bohacek, R.S.; McMartin, C.; Guida, W.C. The art and practice of structure-based drug design: A molecular modeling perspective. Med. Res. Rev.
**1996**, 16, 3–50. [Google Scholar] [CrossRef] - Anderson, A.C. The process of structure-based drug design. Chem. Biol.
**2003**, 10, 787–797. [Google Scholar] [CrossRef] [PubMed] - Lounnas, V.; Ritschel, T.; Kelder, J.; McGuire, R.; Bywater, R.P.; Foloppe, N. Current progress in structure-based rational drug design marks a new mindset in drug discovery. Comput. Struct. Biotechnol. J.
**2013**, 5, 1–14. [Google Scholar] [CrossRef] [PubMed] - Yamazaki, K.; Kusunose, N.; Fujita, K.; Sato, H.; Asano, S.; Dan, A.; Kanaoka, M. Identification of phosphodiesterase-1 and 5 dual inhibitors by a ligand-based virtual screening optimized for lead evolution. Bioorg. Med. Chem. Lett.
**2006**, 16, 1371–1379. [Google Scholar] [CrossRef] [PubMed] - Yan, S.; Appleby, T.; Larson, G.; Wu, J.Z.; Hamatake, R.K.; Hong, Z.; Yao, N. Thiazolone-acylsulfonamides as novel HCV NS5B polymerase allosteric inhibitors: Convergence of structure-based drug design and X-ray crystallographic study. Bioorg. Med. Chem. Lett.
**2007**, 17, 1991–1995. [Google Scholar] [CrossRef] [PubMed] - Carosati, E.; Mannhold, R.; Wahl, P.; Hansen, J.B.; Fremming, T.; Zamora, I.; Cianchetta, G.; Baroni, M. Virtual screening for novel openers of pancreatic K(ATP) channels. J. Med. Chem.
**2007**, 50, 2117–2126. [Google Scholar] [CrossRef] [PubMed] - Ogata, K.; Isomura, T.; Yamashita, H.; Kubodera, H. A quantitative approach to the estimation of chemical space from a given geometry by the combination of atomic species. Mol. Inform.
**2007**, 26, 596–607. [Google Scholar] [CrossRef] - Ogata, K.; Isomura, T.; Kawata, S.; Yamashita, H.; Kubodera, H.; Wodak, S.J. Lead generation and optimization based on protein-ligand complementarity. Molecules
**2010**, 15, 4382–4400. [Google Scholar] [CrossRef] [PubMed] - Johnson, M.W.; Amin, M.H.S.; Gildert, S.; Lanting, T.; Hamze, F.; Dickson, N.; Harris, R.; Berkley, A.J.; Johansson, J.; Bunyk, P.; et al. Quantum annealing with manufactured spins. Nature
**2011**, 473, 194–198. [Google Scholar] [CrossRef] [PubMed] - Yamaoka, M.; Yoshimura, C.; Hayashi, M.; Okuyama, T.; Aoki, H.; Mizuno, H. 20k-spin Ising Chip for Combinational Optimization Problem with CMOS Annealing. In Proceedings of the IEEE International Solid-State Circuits Conference, San Francisco, CA, USA, 22–26 February 2015.
- Utsunomiya, S.; Takata, K.; Yamamoto, Y. Mapping of Ising models onto injection-locked laser systems. Opt. Express
**2011**, 19, 18091–18108. [Google Scholar] [CrossRef] [PubMed] - Lucas, A. Ising formulations of many NP problems. Front. Phys.
**2014**, 2. [Google Scholar] [CrossRef] - 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.
**2014**, 14, 1–36. [Google Scholar] [CrossRef] - Perdomo-Ortiz, A.; Dickson, N.; Drew-Brook, M.; Rose, G.; Aspuru-Guzik, A. Finding low-energy conformations of lattice protein models by quantum annealing. Sci. Rep.
**2012**, 2, 571. [Google Scholar] [CrossRef] [PubMed] - Denil, N.; Fretias, N. Toward the Implementation of a Quantum RBM. In Proceedings of the NIPS 2011 Deep Learning and Unsupervised Feature Learning Workshop, Cranada, Spain, 16 December 2011.
- Dumoulin, V.; Goodfellow, I.J.; Courville, A.; Bengio, Y. On the Challenges of Physical Implementations of RBMs. In Proceedings of the Twenty-Eighth AAAI Conference on Artificial Intelligence, Québec, QC, Canada, 27–31 July 2014; pp. 1199–1205.
- Adachi, S.H.; Henderson, M.P. Application of Quantum Annealing to Training of Deep Neural Networks. 2015; arXiv:1510.00635. [Google Scholar]
- Wang, Z.; Marandi, A.; Wen, K.; Byer, R.L.; Yamamoto, Y. Coherent Ising machine based on degenerate optical parametric oscillators. Phys. Rev. A
**2013**, 88, 063853. [Google Scholar] [CrossRef] - Marandi, A.; Wang, Z.; Takata, K.; Byer, R.L.; Yamamoto, Y. Network of Time-Multiplexed Optical Parametric Oscillators as a Coherent Ising Machine. Nat. Photonics
**2014**, 8, 937–942. [Google Scholar] [CrossRef] - Inagaki, T.; Inaba, K.; Hamerly, R.; Inoue, K.; Yamamoto, Y.; Takesue, H. Large-scale Ising spin network based on degenerate optical parametric oscillators. Nat. Photonics
**2016**, 10, 415–419. [Google Scholar] [CrossRef] - Haribara, Y.; Utsunomiya, S.; Yamamoto, Y. Computational principle and performance evaluation of coherent Ising machine based on degenerate optical parametric oscillator network. Entropy
**2016**, 18, 151. [Google Scholar] [CrossRef] - Takata, K.; Marandi, A.; Yamamoto, Y. Quantum correlation in degenerate optical parametric oscillators with mutual injections. Phys. Rev. A
**2015**, 92, 043821. [Google Scholar] [CrossRef] - Maruo, D.; Utsunomiya, S.; Yamamoto, Y. Truncated Wigner theory of coherent Ising machines based on degenerate optical parametric oscillator. Phys. Scr.
**2016**, 91, 083010. [Google Scholar] [CrossRef] - Penrose, R. A generalized inverse for matrices. Math. Proc. Camb. Philos. Soc.
**1955**, 51, 406–413. [Google Scholar] [CrossRef] - Molnár, B.; Ercsey-Ravasz, M. Asymmetric Continuous-Time Neural Networks without Local Traps for Solving Constraint Satisfaction Problems. PLoS ONE
**2013**, 8, e73400. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Coherent Ising machine (CIM) based on time-division-multiplexed (TDM) pulsed degenerate optical parametric oscillator (DOPO) with measurement and feedback control. Both local oscillator (LO) pulses and feedback (FB) pulses are taken from the pump laser. A parametric gain is provided by a periodically-poled ${\mathrm{LiNbO}}_{3}$ (PPLN) waveguide device and an optical ring cavity is formed by a fiber with ∼1 km length.

**Figure 2.**6-membered ring placing near Ala-Asp-Ala tripeptide. (

**a**) initial structure; (

**b**) benzene with the lowest energy and (

**c**) pyridine with the 2nd lowest energy.

**Figure 3.**The success probability of satisfying the constraints for 1000 identical trials. The parameter p is the final pump rate for gradual pumping. The parameters of the Hamiltonian (Equation 10) are set to $A=1$ and C = 0 for all the results.

**Figure 4.**Histograms of the interaction energies of the final states of CIM over 1000 runs. The blue bar is the simulation result and the red dot is the estimated Boltzmann distribution. The green line at the bottom of each figure shows the number of states for the given Hamiltonian. All the histograms are normalized to 1.

**Figure 5.**The histogram of finding the degenerate states on the same energy surface over 1000 runs. (

**a**) $E=[-3.854,-3.768)$; (

**b**) $E=[-4.626,-4.540)$.

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sakaguchi, H.; Ogata, K.; Isomura, T.; Utsunomiya, S.; Yamamoto, Y.; Aihara, K. Boltzmann Sampling by Degenerate Optical Parametric Oscillator Network for Structure-Based Virtual Screening. *Entropy* **2016**, *18*, 365.
https://doi.org/10.3390/e18100365

**AMA Style**

Sakaguchi H, Ogata K, Isomura T, Utsunomiya S, Yamamoto Y, Aihara K. Boltzmann Sampling by Degenerate Optical Parametric Oscillator Network for Structure-Based Virtual Screening. *Entropy*. 2016; 18(10):365.
https://doi.org/10.3390/e18100365

**Chicago/Turabian Style**

Sakaguchi, Hiromasa, Koji Ogata, Tetsu Isomura, Shoko Utsunomiya, Yoshihisa Yamamoto, and Kazuyuki Aihara. 2016. "Boltzmann Sampling by Degenerate Optical Parametric Oscillator Network for Structure-Based Virtual Screening" *Entropy* 18, no. 10: 365.
https://doi.org/10.3390/e18100365