# Computational Principle and Performance Evaluation of Coherent Ising Machine Based on Degenerate Optical Parametric Oscillator Network

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## Abstract

**:**

## 1. Introduction

## 2. Principle

#### 2.1. A Proposed Machine

#### 2.2. Quantum Search, Quantum Filtering, and Quantum-to-Classical Crossover

- Quantum parallel searchEach DOPO pulse is in a linear superposition state, but there is no correlation between different DOPO pulses established yet immediately after the pump pulse is switched-on:$$\begin{array}{ccc}& & \frac{1}{\sqrt{2}}{(\left|\uparrow \right.\u232a+\left|\downarrow \right.\u232a)}_{1}\otimes \frac{1}{\sqrt{2}}{(\left|\uparrow \right.\u232a+\left|\downarrow \right.\u232a)}_{2}\otimes \cdots \otimes \frac{1}{\sqrt{2}}{(\left|\uparrow \right.\u232a+\left|\downarrow \right.\u232a)}_{N}\hfill \\ & =& \frac{1}{\sqrt{{2}^{N}}}(\left(\right)open="|"\; close="\rangle ">\uparrow \uparrow \cdots \uparrow +\cdots +\left(\right)open="|"\; close="\rangle ">\downarrow \downarrow \cdots \downarrow )\hfill & .\end{array}$$
- Quantum filteringThe probability amplitudes of the two degenerate ground states are amplified, while those of the excited states are deamplified when the pump rate is approaching to the oscillation threshold:$$\begin{array}{c}\hfill \frac{1}{\sqrt{2}}(\left(\right)open="|"\; close="\rangle ">\uparrow \downarrow \cdots \downarrow +\left(\right)open="|"\; close="\rangle ">\downarrow \uparrow \cdots \uparrow )& +\u03f5(\mathrm{all}\phantom{\rule{4pt}{0ex}}\mathrm{the}\phantom{\rule{4pt}{0ex}}\mathrm{other}\phantom{\rule{4pt}{0ex}}\mathrm{states}),\end{array}$$
- Spontaneous symmetry breakingAt the critical point of the DOPO threshold, random spontaneous emission noise selects one of the two degenerate ground states with equal probabilities. For instance, the probability amplitude of $\left(\right)$ state is amplified but that for the $\left(\right)$ state is deamplified at this stage in one particular run, and vice versa in another run.
- Quantum-to-classical crossoverWith increasing the pump rate well above the threshold, the chosen spin state dominates over all the other spin states including the other ground state via pump depletion (cross-gain saturation). The quantum state of each DOPO approaches a high excited coherent state with 0-phase or π-phase, which is the closest analog for the classical electromagnetic field.

#### 2.3. Quantum Theory of Coherent Ising Machines

## 3. Working Equations

#### 3.1. c-Number Stochastic Differential Equations for Multiple-Pulse DOPO with Mutual Coupling

#### 3.2. Turn-on Delay Time

## 4. Numerical Simulations

## 5. Discussion

#### 5.1. Measurement-Feedback Control

#### 5.2. Application to Various Problems

- Via MAX-2-SATMAX-2-SAT is a problem to find an assignment for a given 2-CNF which satisfies as many clauses as possible. We can construct a CNF of MAX-2-SAT $\tilde{S}$ from a given 3-CNF S with using an auxiliary literal ${Y}^{(i)}$ for each clause ${C}_{i}={X}_{1}^{(i)}\vee {X}_{2}^{(i)}\vee {X}_{3}^{(i)}$ [1]:$$\tilde{S}=\underset{i=1,\dots ,m}{\bigwedge}{D}_{i},$$$$\begin{array}{ccc}\hfill {D}_{i}& =& {X}_{1}^{(i)}\wedge {X}_{2}^{(i)}\wedge {X}_{3}^{(i)}\wedge {Y}^{(i)}\hfill \\ & & \wedge (\neg {X}_{1}^{(i)}\vee \neg {X}_{2}^{(i)})\wedge (\neg {X}_{2}^{(i)}\vee \neg {X}_{3}^{(i)})\wedge (\neg {X}_{3}^{(i)}\vee \neg {X}_{1}^{(i)})\hfill \\ & & \wedge ({X}_{1}^{(i)}\vee \neg {Y}^{(i)})\wedge ({X}_{2}^{(i)}\vee \neg {Y}^{(i)})\wedge ({X}_{3}^{(i)}\vee \neg {Y}^{(i)}).\hfill \end{array}$$$$\begin{array}{c}\hfill {\mathcal{H}}_{\mathrm{QUBO}}=\sum _{i=1}^{m}\{4-{x}_{1}^{(i)}-{x}_{2}^{(i)}-{x}_{3}^{(i)}+2{y}_{1}^{(i)}\\ \hfill +{x}_{1}^{(i)}{x}_{2}^{(i)}+{x}_{2}^{(i)}{x}_{3}^{(i)}+{x}_{3}^{(i)}{x}_{1}^{(i)}\\ \hfill -{x}_{1}^{(i)}{y}^{(i)}-{x}_{2}^{(i)}{y}^{(i)}-{x}_{3}^{(i)}{y}^{(i)}\},\end{array}$$
- Via Maximum independent set (MIS)3-SAT can be described in terms of MIS on a graph with $3m$ vertices. The graph can be constructed to connect the contradicting literals in m triangle graphs (each triangle graph represent a clause). Then there exist independent sets of size m, if and only if the original 3-CNF is satisfiable. Since the MIS is mapped to an Ising problem with the equivalent number of vertices [41], 3-SAT becomes $3m$-spin Ising problem in this case. The Ising coupling term takes only ${J}_{ij}\in \{0,\pm 1\}$ in this case.
- Via 3-body problem3-SAT is easily interpreted as an optimization problem which has cost function with up to 3-body interactions$$\begin{array}{ccc}\hfill {\mathcal{H}}^{(i)}& =& -{J}_{1}^{(i)}{x}_{1}^{(i)}-{J}_{2}^{(i)}{x}_{2}^{(i)}-{J}_{1}^{(i)}{x}_{3}^{(i)}\hfill \\ & & +{J}_{12}^{(i)}{x}_{1}^{(i)}{x}_{2}^{(i)}+{J}_{23}^{(i)}{x}_{2}^{(i)}{x}_{3}^{(i)}+{J}_{31}^{(i)}{x}_{3}^{(i)}{x}_{1}^{(i)}\hfill \\ & & -{J}_{123}^{(i)}{x}_{1}^{(i)}{x}_{2}^{(i)}{x}_{3}^{(i)},\hfill \end{array}$$$$\begin{array}{ccc}\hfill {J}_{j}^{(i)}& =& \left(\right)open="\{"\; close>\begin{array}{c}\hfill 1\phantom{\rule{1.em}{0ex}}\mathrm{if}\phantom{\rule{4pt}{0ex}}{X}_{j}^{(i)}={x}_{k},\\ \hfill 0\phantom{\rule{1.em}{0ex}}\mathrm{if}\phantom{\rule{4pt}{0ex}}{X}_{j}^{(i)}={\overline{x}}_{k},\end{array}\hfill \end{array}$$$$\begin{array}{ccc}\hfill {J}_{{j}_{1}{j}_{2}}^{(i)}& =& {J}_{{j}_{1}}^{(i)}{J}_{{j}_{2}}^{(i)},\hfill \end{array}$$$$\begin{array}{ccc}\hfill {J}_{123}^{(i)}& =& {J}_{1}^{(i)}{J}_{2}^{(i)}{J}_{3}^{(i)},\hfill \end{array}$$$${E}_{\mathrm{p}}^{(i)}=b(3{y}^{(i)}+{x}_{1}^{(i)}{x}_{2}^{(i)}-2{y}^{(i)}{x}_{1}^{(i)}-2{y}^{(i)}{x}_{2}^{(i)}),$$$$\begin{array}{ccc}\hfill {\mathcal{H}}_{\mathrm{QUBO}}^{(i)}& =& -{J}_{1}^{(i)}{x}_{1}^{(i)}-{J}_{2}^{(i)}{x}_{2}^{(i)}-{J}_{1}^{(i)}{x}_{3}^{(i)}\hfill \\ & & +{J}_{12}^{(i)}{x}_{1}^{(i)}{x}_{2}^{(i)}+{J}_{23}^{(i)}{x}_{2}^{(i)}{x}_{3}^{(i)}+{J}_{31}^{(i)}{x}_{3}^{(i)}{x}_{1}^{(i)}\hfill \\ & & -{J}_{123}^{(i)}{y}^{(i)}{x}_{3}^{(i)}+{E}_{\mathrm{p}}^{(i)},\hfill \end{array}$$

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

CIM | Coherent Ising machine |

DOPO | Degenerate optical parametric oscillator |

NP | Non-deterministic polynomial-time |

MAX-CUT | Maximum cut problem |

GW | Goemans–Williamson |

SA | Simulated annealing |

SG | Sahni–Gonzalez |

BLS | Breakout local search |

CSDE | c-number stochastic differential equation |

SHG | Second harmonic generation |

PSA | Phase sensitive amplifier |

FPGA | Field programmable gate array |

ASIC | Application specific integrated circuit |

QUBO | Quadratic unconstrained binary optimization |

SAT | Boolean satisfiability problem |

CNF | Conjunctive normal form |

MIS | Maximum independent set |

TSP | Traveling salesman problem |

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**Figure 1.**A coherent Ising machine based on the time-division multiplexed DOPO with mutual coupling implemented by optical delay lines. A part of each pulse is picked off from the main cavity by the output coupler followed by an optical phase sensitive amplifier (PSA) which amplifies the in-phase amplitude ${\tilde{c}}_{i}$ of each DOPO pulse. The feedback pulses, which are produced by combining the outputs from $N-1$ intensity and phase modulators, are injected back to the main cavity by the injection coupler.

**Figure 2.**Quantum states of a DOPO at (

**a**) below; (

**b**) above; and (

**c**) critical point of its oscillation threshold.

**Figure 3.**Four steps of computation in a coherent Ising machine: (

**a**) quantum parallel search; (

**b**) quantum filtering; (

**c**) spontaneous symmetry breaking; and (

**d**) quantum-to-classical crossover.

**Figure 4.**Normalized DOPO signal amplitudes when CIM is solving $N=800$ MAX-CUT is shown. Each trajectory describes a DOPO.

**Figure 5.**Quantum measurement-feedback controlled CIM. Small portion of each signal pulse is out-coupled through the directional coupler I, and its in-phase component is measured by optical balanced homodyne detector, where LO pulse is directly obtained from the master laser. Two detector outputs are converted to digital signals and input into an electronic digital circuit, where a feedback signal for i-th signal pulse is computed. Independently obtained feedback pulse from the master laser is modulated in its intensity and phase to achieve ${\sum}_{j}{\xi}_{ij}{\tilde{c}}_{j}$ and coupled into i-th signal pulse by directional coupler II. Flows of optical fields and electrical signals are shown as solid and dashed lines, respectively.

**Table 1.**Computation time (sec.) on $\{\pm 1\}$-weighted complete graphs. The symbol “–” means it didn’t reach the desired accuracy (see also [39]).

Method/Algorithm | Graph Order $n\phantom{\rule{4pt}{0ex}}(=N)$ | ||||
---|---|---|---|---|---|

40 | 160 | 800 | 4000 | 20,000 | |

GW | 0.00345 | 0.170 | 22.5 | $6.65\times {10}^{3}$ | $1.81\times {10}^{6}$ |

SA | 0.00906 | 0.020 | 1.30 | 10.2 | 210 |

BLS | $7.92\times {10}^{-4}$ | 0.0255 | 0.0465 | 1.22 | 52.3 |

SG3 | – | – | – | 0.592 | 15.3 |

CIM | 0.00297 | 0.00256 | 0.00040 | 0.00055 | 0.00199 |

**Table 2.**Fiber length and pulse repetition frequency determine the computation time (100 round trips) and the maximum number of pulses. The computation time is repetition frequency independent but linearly depends on the cavity length of the fiber. Note that the phase stabilization in the fiber length of 20 km is challenging, and the 200 GHz repetition frequency is not yet practically available.

Fiber (km) | Repetition (GHz) | Time (s) | ${N}_{\mathrm{max}}$ |
---|---|---|---|

0.2 | 2 | ${10}^{-4}$ | 2000 |

2 | 2 | ${10}^{-3}$ | $2\times {10}^{4}$ |

20 | 2 | 0.01 | $2\times {10}^{5}$ |

20 | 20 | 0.01 | $2\times {10}^{6}$ |

20 | 200 | 0.01 | $2\times {10}^{7}$ |

**Table 3.**Mapping to the Ising Hamiltonian. MAX-CUT, 3-SAT, MIS, TSP stand for maximum cut, 3-satisfiability, maximum independent set, traveling salesman problem, respectively. The number of DOPO pulses N is determined by the number of nodes n and the number of edges m in the graph and the number of colors c or the number of variables n and the number of clauses m in CNF, or l is a bit length of an element of the ${\mathbb{R}}^{n}$ vector. Problems with ${}^{*}$ can be mapped to an Ising problem with only $\{0,\pm 1\}$ weights.

Original Problem | Complexity Class | N |
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MAX-CUT${}^{(*)}$ | NP-hard | n |

Graph coloring${}^{*}$ | NP-hard | $nc$ |

Maximum clique${}^{*}$ | NP-hard | n |

MIS${}^{*}$ | NP-hard | n |

Minimum vertex cover${}^{*}$ | NP-hard | n |

Maximum set packing${}^{*}$ | NP-hard | #subsets |

TSP | NP-hard | ${n}^{2}$ |

Hamiltonian path${}^{*}$ | NP-complete | ${n}^{2}$ |

Graph isomorphism${}^{*}$ | NP | n |

${l}_{0}$-norm regularization | NP-hard | $n(l+1)$ |

3-SAT via MAX-2-SAT | NP-complete | $m+n$ |

3-SAT via MIS${}^{*}$ | NP-complete | $3m$ |

3-SAT via 3-body | NP-complete | $m+n$ |

3-SAT (3-body in FPGA) | NP-complete | n |

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**MDPI and ACS Style**

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.
https://doi.org/10.3390/e18040151

**AMA Style**

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(4):151.
https://doi.org/10.3390/e18040151

**Chicago/Turabian Style**

Haribara, Yoshitaka, Shoko Utsunomiya, and Yoshihisa Yamamoto.
2016. "Computational Principle and Performance Evaluation of Coherent Ising Machine Based on Degenerate Optical Parametric Oscillator Network" *Entropy* 18, no. 4: 151.
https://doi.org/10.3390/e18040151