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

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## 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

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**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/).

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**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