Abstract

As a solver for non-deterministic polynomial time (NP)-hard combinatorial optimization problems, the coherent Ising machine (CIM) is in the early stages of research, and the potential of this innovative physical system will be developed. Here, we propose a speed-up coherent Ising machine with a squeezed feedback system, which we call S-CIM. We couple squeezed feedback pulses generated by the squeezed feedback system into the degenerate optical parametric oscillator (DOPO) network. Simulations indicate that quantum inseparability of the coupled DOPO network is further enhanced during the whole optimization process, and quantum fluctuations are significantly smaller around the oscillation threshold. Computation experiments are performed on MAX-CUT problems of order between 4 and 20000. Numerical results demonstrate that S-CIM increases the optimal normalized output by 2.27% and significantly reduces the optimal computation time by 75.12%.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Many combinatorial optimization problems in computer science [1], drug research [2], protein folding [3], and big data processing [4] belong to the non-deterministic polynomial time (NP)-hard class [5]. The traditional von-Neumann computer has no enough capability to solve these problems. It can't find the exact solution with satisfying efficiency. Recently, some interesting attempts [69], such as simulated annealing generated by thermal annealing program in simulated metallurgical process and quantum annealing based on quantum tunneling theory, are proposed to tackle the NP-hard problems approximately. However, how to control the tight connection between quantum bits more efficiently needs to be explored [10]. These methods have inspired us to find alternatives from other reliable physical schemes.

Recently, Stanford University proposed and implemented a coherent Ising machine (CIM) based on degenerate optical parametric oscillators (DOPOs) [11,12] for seeking the optimal solutions of the NP-hard problems. The basic operating mechanism of CIM is derived from mapping the global photon decay rate of the DOPO network to the Ising Hamiltonian [5]

$$H ={-} \mathop \sum \limits_{1 \le j < l \le N} {J_{jl}}{\sigma _j}{\sigma _l},$$
where Jjl indicates the coupling coefficient between the jth and the lth spins, and σj = ±1 denotes two spin states, |↑> and |↓>. Due to the inherent characteristics of CIM, this experimental setup can implement an artificial Ising spin system, where the Ising couplings are realized by injecting feedback pulses into the network of DOPO. Furthermore, Ising spin states are suitably represented with binary phase states, |0 > and |π>, of the above-threshold DOPO pulses. When the gain of CIM gradually approaches the global minimum loss by increasing the external pump, the DOPO network will begin to oscillate and the phase state (spin state) configuration will be in the exact or approximate ground state of the Ising Hamiltonian. Finding the ground state of the Ising Hamiltonian is classified as the Ising problem, which also belongs to the NP-hard class [13]. CIM is used as an efficient optimization solver because of the reducibility of the NP problems [14]. Recently, Stanford University and NTT improved the scheme of the CIM by adding quantum measurement feedback control system, and successfully reduced the size of the machine [15]. Shortly afterward, NTT used dual-pump four-wave mixing in a highly nonlinear fiber [16] to generate more than 50,000 time-multiplexed DOPOs [17], which indicates that the DOPO network can be used as a stable large-scale artificial spin system. However, there are still some factors that reduce the optimization efficiency of the CIM. The signal-to-noise ratio (SNR) of the measurement by balanced homodyne detector was disturbed due to the antisqueezing effect generated in the fiber ring cavity [18] and the vacuum fluctuations introduced by the beam splitter [15]. In addition, the turn-on-delay oscillation effect postpones the end of the optimization process [19].

In this paper, a speed-up coherent Ising machine with a squeezed feedback system (S-CIM) is proposed. We formulate a calculation model of the coupled DOPO network based on squeezed feedback process by using truncated Wigner representation. Then we obtain the c-number stochastic differential equations (CSDEs) for the network consisting of N DOPOs. To make the results clear, a simple model composed of two coupled DOPOs is taken to study the quantum properties of the S-CIM. When the squeezed feedback pulses are injected into the 2-DOPO network, the variances of the quadrature amplitude component for DOPOs decreases, and quantum inseparability is enhanced throughout the optimization process. The simulations about the evolutions of the spin state (phase state) configurations and the average photon number in the fiber ring cavity have consistent results, suggesting that S-CIM oscillates earlier than CIM to get optimization results. Finally, the computational performance of the S-CIM is numerically evaluated by solving large-scale MAX-CUT problems of order up to 20000.

2. Calculation model

The schematic of the S-CIM shown in Fig. 1(a) consists of three parts: a N-DOPO network as an artificial Ising spin system; a squeezed feedback system consisting of a subthreshold optical parametric oscillator, a phase sensitive amplify (PSA), a balanced homodyne detector, an analog-to-digital converter (ADC), a field-programmable gate array (FPGA), a digital-to-analog converter (DAC), an intensity modulator (IM) and a phase modulator (PM); and an injection coupler for realizing Ising couplings.

 figure: Fig. 1.

Fig. 1. (a). The schematic of the speed-up coherent Ising machine with a squeezed feedback system (S-CIM). The PSA connected to BS1 amplifies the quadrature amplitude component of each extracted DOPO pulse, while the BS2 couples the modulated squeezed feedback pulse with the target DOPO pulse to implement the given Ising Hamiltonian. A: the network of N degenerate optical parametric oscillators (DOPOs). B: the squeezed feedback system. C: the injection coupler. (b) The corresponding calculation model of the coupled DOPO network based on squeezed feedback process.

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A portion of the S-CIM external pump is injected in the fiber ring cavity to form N signal DOPO pulses, and another portion is pumped into the subthreshold optical parametric oscillator to generate squeezed vacuum states. In the squeezed feedback process, the jth signal fields are extracted from the beam splitter 1 (BS1) to measure its quadrature amplitude component ${\tilde{x}_j}$. The output of the balanced homodyne detector is converted into a digital signal and input into the FPGA. Then we use an electronic digital circuit to compute the lth feedback pulse corresponding to the jth DOPO pulse. The output signal drives the IM and PM to accurately achieve the lth feedback pulse $\mathop \sum \nolimits_l {\xi _{jl}}{\tilde{x}_l}$, where ${\xi _{jl}}$ is proportional to the coupling coefficient Jjl, and Jjl is a $N \times N$ symmetric matrix, with Jjj = 0. Afterward the squeezed vacuum state is injected into the feedback pulse, the squeezed feedback pulse with a given Ising coupling Jjl emerge. Finally, the squeezed feedback pulses are coupled into the ring cavity through the beam splitter 2 (BS2), which produces a coupled DOPO network. Each part of the calculation model will be analyzed in detail, and the truncated Wigner representation will be used to eventually obtain the c-number stochastic differential equations (CSDEs) of the S-CIM containing N-coupled DOPOs.

2.1 Degenerate optical parametric oscillators

The S-CIM is driven by a master laser at frequency ωs. The external pump Fp with a frequency of ωp = 2ωs is excited by the second harmonic generation (SHG). The external pump Fp is divided into two optical paths via a beam splitter. One enters the fiber ring cavity and the other injects the subthreshold optical parametric oscillator. Due to the second-order nonlinear effect of the crystal, the pump entering the ring cavity generates signal pulses sequence at frequency ωs. N DOPOs run along the fiber ring cavity at regular time intervals t′ = T′/N, where T′ is the roundtrip time when a DOPO pulse propagates through the cavity. Therefore, the number of independent DOPO can be added by increasing the repetition frequency of the pump pulse or by extending the length of the ring cavity. Assuming that only two DOPOs are coupled in the cavity, the total Hamiltonian [11,20] of the network is obtained:

$$H = {H_{\textrm{free}}} + {H_{\textrm{int}}} + {H_{\textrm{pump}}} + {H_{\textrm{sr}}},$$
$${H_{\textrm{free}}} = \hbar {\omega _s}\mathop \sum \limits_{j = 1}^2 \hat{a}_{sj}^{\dagger} {\hat{a}_{sj}} + \hbar {\omega _s}\mathop \sum \limits_{j = 1}^2 \hat{a}_{pj}^{\dagger} {\hat{a}_{pj}},$$
$${H_{\textrm{int}}} = \frac{{i\hbar \kappa }}{2}\mathop \sum \limits_{j = 1}^2 ({\hat{a}{{_{sj}^{\dagger} }^2}{{\hat{a}}_{pj}} - \hat{a}{{_{pj}^{\dagger} }^2}{{\hat{a}}_{sj}}} ),$$
$${H_{\textrm{pump}}} = i\hbar \sqrt {{\gamma _p}} \mathop \sum \limits_{j = 1}^2 ({\epsilon \hat{a}_{pj}^{\dagger} {e^{ - i{\omega_p}t}} - \epsilon {{\hat{a}}_{pj}}{e^{i{\omega_p}t}}} ),$$
$${H_{\textrm{sr}}} = i\hbar \mathop \sum \limits_{j = 1}^2 {\hat{a}_{sj}}{\hat{\Gamma }}_{Rsj}^{\dagger} \sqrt {{\gamma _s}} + {{\hat{\Gamma }}_{Rsj}}\hat{a}_{sj}^{\dagger} \sqrt {{\gamma _s}} + {\hat{a}_{pj}}{\hat{\Gamma }}_{Rpj}^{\dagger} \sqrt {{\gamma _p}} + {{\hat{\Gamma }}_{Rpj}}\hat{a}_{pj}^{\dagger} \sqrt {{\gamma _p}} ,$$
Here, Hfree describes the free field Hamiltonian of signal fields and pump fields in the 2-DOPO network inside the cavity, where $\hat{a}_{sj}^{\dagger} $, ${\hat{a}_{sj}}$ are the creation operator and annihilation operator of the jth signal field and $\hat{a}_{pj}^{\dagger} $, ${\hat{a}_{pj}}$ are the correspondences for the pump field. Hint represents interaction Hamiltonian, where к is the parametric gain derived from a second-order nonlinear crystal. Hpump denotes pump field Hamiltonian excited by the external pump fields, where $\epsilon $ is pump rate. At last, Hsr shows the interaction between reservoir and the cavity fields. It consists of two signal and two pump fields, where ${\gamma _p}$, ${\gamma _s}$ are the pump and signal optical decay rates and ${\hat{\Gamma }}_{Rsj}^{\dagger} $, ${\hat{\Gamma }}_{Rpj}^{\dagger} $ are reservoir operators.

2.2 Squeezed feedback pulse

A small portion of the jth signal DOPO pulse is extracted from the fiber ring cavity by the BS1. Its quadrature amplitude component is measured by the balanced homodyne detector. The transmittance rate of BS1 is defined as T1 = sin2θ1. Since the measured intensity (1- T1) is relatively small, the quantum coherence between different amplitudes can be improved [18]. When the pulse propagates through the BS1, it combines with the vacuum fluctuations of the external interference, resulting in measurement errors of the detector. The annihilation operator of the vacuum fluctuation is defined as avac, and the output coupled field operator is

$$\hat{a}_{sj}^{\prime} = \sqrt {{T_1}} {\hat{a}_{sj}} - \sqrt {1 - {T_1}} {\hat{a}_{vac}},$$
The PSA [21,22] in front of the balanced homodyne detector is used to amplify the quadrature amplitude components of the out-coupled field:
$${\hat{a}_{sj,out}} = G\left( {\sqrt {{T_1}} {{\hat{a}}_{sj}} - \sqrt {1 - {T_1}} {{\hat{a}}_{vac}}} \right) + \sqrt {G - 1} \left( {\sqrt {{T_1}} \hat{a}_{sj}^ +{-} \sqrt {1 - {T_1}} \hat{a}_{vac}^ + } \right),$$
where G is the gain of the PSA. the SNR of the measurement is improved and a large coupling constant ξ [11] is achieved.

Then the measurement result of homodyne detector is transmitted to FPGA, we can calculate the feedback pulse for ${\hat{a}_{sj}}$. After the calculation, the output electrical signal drives the intensity and phase modulator to modulate the extracted pulse ${\hat{a}_{sj,out}}$ to feedback pulse ${{\xi }_{jl}}{\hat{a}_{sl}}$.

At the same time, the external pump generated by the master laser propagates into the subthreshold optical parametric oscillator to produce squeezed vacuum states [23,24]. After the squeezed vacuum state is injected [25,26] into the measurement feedback circuit, the obtained annihilation operator of the squeezed feedback pulse satisfies

$${\xi _{jl}}{\hat{b}_{sl}} = \frac{{{\xi _{jl}}}}{{G\sqrt {{T_1}} }}({{{\hat{a}}_{sj,out}}\cosh (r )- \hat{a}_{sj,out}^{\dagger} {e^{i\theta }}\sinh (r )} ),$$
where r is squeeze parameter. Since the evolution of the network only depends on the quadrature amplitude components of the signal DOPO pulses, we suppress the quantum noise of the quadrature amplitude component at the cost of the increased the quantum noise of the quadrature phase component.

2.3 Squeezed feedback process and the coupled DOPO network

In the squeezed feedback process, the signal DOPO pulse and squeezed feedback pulse, the former of which is prepared in a coherent state, are combined by the BS2. The transmittance rate of the BS2, defined as T2 = sin2θ2, is set very high (T2 ≈ 1). In this parameter region, the quantum noise is nearly completely suppressed because of the injected squeezed vacuum state. Carrying out this coupling procedure requires three valid modes [27] of the pulses as shown in Fig. 1(b): ${\hat{a}_{sj}}$ (Black arrow) is Mode 1+, ${\hat{b}_{sl}}$ (yellow arrow) is Mode 2+, and ${\hat{c}_{sjl}}$ (red arrow) is Mode 1. Mode 1+ describes the DOPOs in the main cavity which are not extracted by the BS1. Mode 2+ represents squeezed feedback pulses which are modulated by the squeezed feedback system. We assume that the state of the Mode 1 corresponding to the coupled DOPO pulse is |ψ> = S2(ξ2)D1(αsj)S1(r1)|0>, where αsj is the complex amplitude of the jth signal DOPO pulse. ${D_1}$ is the displacement operator of Mode 1+, S1 is the squeezed operator of Mode 1+, and S2 is the squeezed operator of Mode 2+. Here, let r1 = 0, so Mode1+ and Mode 2+ are coherent state and squeezed state, respectively. Then, the creation and annihilation operator of Mode1 are obtained as follows:

$$\begin{array}{l} \hat{c}_{sjl}^{\dagger} = \sqrt {{T_2}} \hat{b}_{sl}^{\dagger} + \sqrt {1 - {T_2}} \hat{a}_{sj}^{\dagger} \\ {{\hat{c}}_{sjl}} = \sqrt {{T_2}} {{\hat{b}}_{sl}} + \sqrt {1 - {T_2}} {{\hat{a}}_{sj}}, \end{array}$$
In this 2-DOPO network, the master equation for the fields consisting of two signal modes and two pump modes can be obtained by the standard technique [28]. Then, we use the Wigner function W(α) [29] for the four modes to expand the field density operator ρ:
$$\rho = \int {{e^{{\lambda ^{\ast }}\hat{c} - \lambda \hat{c}}}^{^{\dagger} }} \left\{ {\int {{e^{\lambda {\alpha^{\ast }} - {\lambda^{\ast }}\alpha }}} W(\alpha )d\alpha } \right\}d\lambda ,$$
where $\hat{c} = {({{{\hat{c}}_{s12}},{{\hat{c}}_{s21}},{{\hat{a}}_{p12}},{{\hat{a}}_{p21}}} )^T}$ and ${\lambda} = ({{\lambda_{s12}},{\lambda_{s21}},{\lambda_{p12}},{\lambda_{p21}}} )$. Substituting Eq. (11) into the master equation to obtain the Fokker–Planck equation, and the third and higher-order terms of the Fokker–Planck equations are truncated to obtain a set of CSDEs by the Ito rules [29]:
$$\begin{array}{l} \textrm{d}{\alpha _{s12}} = ({ - {\gamma_s}{\alpha_{s12}} + \kappa {\alpha_{p12}}\alpha_{s12}^{\ast }} )dt + \sqrt {{\gamma _s}} d{W_{s12}}(t )\\ \textrm{d}{\alpha _{s21}} = ({ - {\gamma_s}{\alpha_{s21}} + \kappa {\alpha_{p21}}\alpha_{s21}^{\ast }} )dt + \sqrt {{\gamma _s}} d{W_{s21}}(t )\\ \textrm{d}{\alpha _{p12}} = \left( { - {\gamma_p}{\alpha_{p12}} - \frac{\kappa }{2}\alpha_{s12}^2 + \epsilon } \right)dt + \sqrt {{\gamma _p}} d{W_{p12}}(t )\\ \textrm{d}{\alpha _{p21}} = \left( { - {\gamma_p}{\alpha_{p21}} - \frac{\kappa }{2}\alpha_{s21}^2 + \epsilon } \right)dt + \sqrt {{\gamma _p}} d{W_{p21}}(t ), \end{array}$$
where ${W_X}(t )$ is the Wiener increment. Under the condition of ${\gamma _p} \gg {\gamma _s}$ [11], the pump modes are adiabatically eliminated. Then, the CSDEs are reduced to normalized signal stochastic differential equations:
$$\begin{array}{l} \textrm{d}{A_{s12}} = \{{ - {A_{s12}} + ({E - A_{s12}^2} )A_{s12}^{\ast} - {\xi_{21}}{A_{s21}}} \}d\tau + gdW_{s12}^{\prime}(\tau )\\ \textrm{d}{A_{s21}} = \{{ - {A_{s21}} + ({E - A_{s21}^2} )A_{s21}^{\ast} - {\xi_{12}}{A_{s12}}} \}d\tau + gdW_{s21}^{\prime}(\tau ), \end{array}$$
where As12 = s12 is the normalized complex amplitude, $g = \frac{\kappa }{{\sqrt {2{\gamma _s}{\gamma _p}} }}$ is the saturation parameter, $E = \frac{\kappa }{{{\gamma _s}{\gamma _p}}}\epsilon $ is the normalized pump rate, ξjl is the effective coupling coefficient from lth squeezed feedback pulse to jth DOPO pulse, τ = γst is the normalized time, and the noise term is
$$\begin{array}{l} dW_{s12}^{\prime} = \frac{1}{4}{e^{ - \textrm{E}{\gamma _S}}}\textrm{d}{W_{s12}}(\tau )+ {A_{s12}}\textrm{d}{W_{p1}}(\tau )\\ dW_{s21}^{\prime} = \frac{1}{4}{e^{ - \textrm{E}{\gamma _S}}}\textrm{d}{W_{s21}}(\tau )+ {A_{s21}}\textrm{d}{W_{p2}}(\tau ), \end{array}$$
Finally, Eqs. (13) and (14) can be extended to an N-coupled DOPO network:
$$\begin{array}{l} d{A_{sjl}} = \left\{ { - {A_{sjl}} + ({E - A_{sjl}^2} )A_{sjl}^{\ast} - \mathop \sum \limits_{m = 1}^N \mathop \sum \limits_{n = 1}^N {\xi_{mn}}{A_{smn}}} \right\}d\tau + gdW_{sjl}^{\prime}(\tau )\\ dW_{sjl}^{\prime}(\tau )= \frac{1}{4}{e^{ - \textrm{E}{\gamma _S}}}\textrm{d}{W_{sjl}}(\tau )+ {A_{sjl}}\textrm{d}{W_{pj}}(\tau )({j \ne l,m \ne n} ). \end{array}$$
where m = j and n = l cannot be established at the same time to avoid double calculation of the coupled pulse.

3. Quantum properties

The quantum inseparability and fluctuations of 2-coupled DOPO network are discussed based on the truncated Wigner function [29] because of its convenience of evaluating the expectation value of a symmetrically ordered operator:

$${\left\langle {\hat{c}_{s12}^{{\dagger} j}\hat{c}_{s21}^{{\dagger} k}\hat{c}_{s12}^l\hat{c}_{s21}^m} \right\rangle _s} = \int {\alpha _{s12}^{{\ast}j}\alpha _{s21}^{{\ast }k}\alpha _{s12}^l\alpha _{s21}^m} W({\{\alpha \}} )d\alpha ,$$
here, {α} = (αs12, αs21)T for the 2-coupled DOPO network. In the present case, the quadrature amplitude component and the quadrature phase component of the coupled DOPO pulse are ${\hat{x}_{sjl}} = \frac{{({\hat{c}_{jl}^{\dagger} + {{\hat{c}}_{jl}}} )}}{2}$ and ${\hat{p}_{sjl}} = \frac{{({{{\hat{c}}_{jl}} - \hat{c}_{jl}^{\dagger} } )}}{{2i}}$, respectively. The EPR-type operators ${\hat{u}_ + } = {\hat{x}_{s12}} + {\hat{x}_{s21}}$ and ${\hat{v}_ - } = {\hat{p}_{s12}} + {\hat{p}_{s21}}$ are used to evaluate quantum inseparability. The evaluation criteria of quantum inseparability is ${\Delta }\hat{u}_ + ^2 + {\Delta }\hat{v}_ - ^2 < 1$ [30]. As shown in Fig. 2(a), the total variances of the EPR-type operators for the CIM with rapidly increasing pump rate, the CIM with gradually increasing pump rate and the S-CIM with gradually increasing pump rate are compared which are calculated by the truncated Wigner representation. The pumping schedule of the rapidly increasing pump rate is that the pump rate linearly increases from zero to 1.5 times DOPO oscillation threshold within the normalized time τ = 200, i.e. E = 1.5(τ/200), and the pumping schedule of the gradually increasing pump rate is E = 1.5(τ/800). Here the oscillation threshold refers to the threshold pump rate $E_{th}^{(0 )} = 1$ of a solitary DOPO. As shown in Fig. 2(a), the CIM with rapidly increasing pump rate has a weak quantum inseparability below the coupled oscillation threshold Eth = 1-ξ [11]. Here, we set the coupling constant ξ = 0.6 and the saturation parameter is g = 0.01 for better results. When the coupled DOPO network evolves above the coupled oscillation threshold, the quantum inseparability gradually disappears, i.e. ${\Delta }\hat{u}_ + ^2 + {\Delta }\hat{v}_ - ^2 \approx 1$. There is a small peak from E = 0.54 to E = 0.72 due to the turn-on-delay oscillation effect, which leads to the reduction of the optimization efficiency. Note that the small peak disappears in the CIM with gradually increasing pump rate, which indicates that the slower pumping schedule can weaken the turn-on-delay oscillation effect. As for the S- CIM with gradually increasing pump rate, it can be seen that the turn-on-delay oscillation effect is weakened, and the quantum inseparability is further deepened over the entire pumping schedule. Moreover, we compare the quantum inseparability of coupled DOPO network based on two different feedback process models: the valid coupling modes in this paper (Fig. 2(a)) and the unitary displacement operator in the Heisenberg picture $D(\beta {\theta _2}) = exp(\beta {\theta _2}{a^ + } - {\beta^\ast }{\theta _2}a)$ [18] (Fig. 2(b)), where $|\beta > $ is the feedback pulse prepared in a squeezed state. The numerical values confirm that the difference in the total variances $\left\langle {\Delta \widehat u_ +^2} \right\rangle + \left\langle {\Delta \widehat v_ -^2} \right\rangle $ computed by two methods is within the statistical error.

 figure: Fig. 2.

Fig. 2. Total variances of the EPR operators ${\Delta }\hat{u}_ + ^2 + {\Delta }\hat{v}_ - ^2$ versus normalized pump rate E for three different CIM schemes. The numerical values are computed from two feedback process models: the valid coupling modes approach (Fig. 2(a)) and the unitary displacement operator approach (Fig. 2(b)).

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The variances of the quadrature amplitude component represent quantum fluctuations which include the anti-squeezed effect caused by PSA inside the cavity and the vacuum fluctuations introduced from the BS1. Large fluctuations will reduce the SNR of detection, cause errors in the squeezed feedback system and affect the accuracy of the final optimization result. As shown in Fig. 3(a), the variances of the quadrature amplitude component for CIM and S-CIM are compared under the slowly pumping schedule E = 1.5(τ/800). Below the coupled oscillation threshold, the variances begin to increase from the vacuum state (0.5) with the rise of the pump rate. Once the coupled network oscillates, the variance gradually decreases to the original value. Comparing with CIM, S-CIM further reduces the quantum fluctuations in the coupled DOPO network. Similarly, we compared the quantum fluctuations of the coupled DOPO network obtained by the two feedback process models. As shown in Figs. 3(a) and 3(b), the difference in the variances of the quadrature amplitude component ${\widehat x_{s12}}$ computed by two methods is still within the statistical error. Therefore, the simulation results obtained by the valid coupling modes also have acceptable accuracy.

 figure: Fig. 3.

Fig. 3. Variances of the quadrature amplitude component ${\hat{x}_{s12}}$ versus normalized pump rate E for CIM and S-CIM with gradually increasing pump rate, respectively. The numerical values are computed from two feedback process models: the valid coupling modes approach (Fig. 3(a)) and the unitary displacement operator approach (Fig. 3(b)). Since the curves of ${\hat{x}_{s12}}$ and ${\hat{x}_{s21}}$ are almost identical, only the former is given here.

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4. Optimization process

Through the analysis of the optimization output statistics, the optimization process of the CIM is divided into four stages [19]: quantum parallel search, quantum filtering, spontaneous symmetry breaking, and quantum-to-classical crossover. Then in the 2-coupled DOPO network, the four specific optimization stages of the S-CIM are compared with the CIM.

The spin states (phase states) of two coupled DOPOs are mapped to four possible spin configurations |↑↓>, |↑↑>, |↓↓> and |↓↑>. During the optimization process of S-CIM, when the coupled DOPO network oscillates, the ground state of the Ising Hamiltonian is found, and the spin configuration is the solution of the corresponding Ising problem. Figures 4(a)–4(d) present the post-selected probabilities of four spin states during the S-CIM evolution as the final selected configuration is |↑↓>. As Fig. 4(a) shows, two DOPOs are independent of each other, and four spin configurations have the same probability of 0.25 which implies that the S-CIM system is at an early stage of ‘quantum parallel search’. Next, as shown in Fig. 4(b), the probability amplitudes of the spin configurations corresponding to the ground state, |↑↓> and |↓↑>, are amplified, and the probability amplitudes of the spin configurations corresponding to the excited state, |↑↑> and |↓↓>, are deamplified. The network evolves from linear superposition to quantum inseparability, which is called ‘quantum filtering’. Then, as shown in Fig. 4(c), the S-CIM system has a 50% probability of selecting one of the two ground states to amplify its probability amplitude. Here, the selected state is |↑↓>. Meanwhile, system deamplifies the probability amplitude of another unselected state |↓↑>. This process is called ‘spontaneous symmetry breaking’. Figure 4(d) shows the last step ‘quantum-classical crossover’. The probability amplitude of the selected state |↑↓> is increased to the maximum, while the probabilities of other spin states are reduced to the minimum. The optimization process ends, and the final optimization outputs are obtained by using S-CIM.

 figure: Fig. 4.

Fig. 4. In the 2-coupled DOPO network, the probabilities of finding |↑↓>, |↑↑>, |↓↓> and |↓↑> states at four different stages of the optimization process when the final selected ground state is |↑↓>. The gain of the coupled DOPO network (yellow dotted line) gradually increases from below the coupled oscillation threshold to reach the minimum global loss (blue curve) corresponding to the ground state of the Ising Hamiltonian.

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The probability evolutions of the two ground-state |↑↓> and |↓↑> are used to characterize the specific optimization stage, where the final selected state is |↑↓> and the unselected state is |↓↑>. Here, the CIM and the S-CIM have the same pumping schedule E = 1.5(τ/200). Firstly, let us focus on the CIM. When τ = 0, the two ground states |↑↓> and |↓↑> have the same probability of 0.25, corresponding to ‘quantum parallel search’ (green dashed line in Fig. 5). When 3 ≤ τ < 28, the probabilities of the two ground states gradually increases because of ‘quantum filtering’ (red dashed line in Fig. 5). When 28 ≤ τ < 84, the probability of the selected state |↑↓> continues to increase, but the probability of the unselected state |↓↑> starts to decrease, which are caused by ‘spontaneous symmetry breaking’ (blue dashed line in Fig. 5). Finally, the vivid result emerges at τ = 84, which indicates the stage of ‘quantum-classical crossover’. After calculation, the coupled oscillation threshold of the CIM is Eth= 0.4, and the corresponding normalization time is τs = 53. However, the CIM actually oscillates at τd = 84, which is due to the turn-on-delay oscillation effect. It can be clearly seen that in the three stages of ‘quantum filtering’, ‘spontaneous symmetry breaking’ and ‘quantum-classical crossover’, the time consumption of the S-CIM is significantly shorter than that of the CIM. The coupled DOPO network starts to oscillate at τ = 71.

 figure: Fig. 5.

Fig. 5. In the 2-coupled DOPO network, the probability evolutions of two ground-state spin configurations versus normalized time τ for CIM and S-CIM. The green, red, blue and pink portions in each curve represent the four procedures of the optimization process of quantum parallel search, quantum filtering, spontaneous symmetry breaking, and quantum-to-classical crossover, respectively. The optimization process of CIM is divided by yellow dotted lines, and the optimization process of S-CIM is divided by yellow solid lines.

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The dynamic oscillation thresholds Ed is accurately defined as the value of E that maximize $dlog(n )/dlog(E )$, where n is the average photon number in the fiber ring cavity. Figure 6 illustrates that S-CIM oscillates earlier than CIM according to the evolution of the average photon number $\langle{n}\rangle$, which is consistent with the analysis of the ground-state spin configurations (Fig. 5).

 figure: Fig. 6.

Fig. 6. In the 2-coupled DOPO network, the average photon number n versus normalized pump rate $p = \frac{E}{{{E_d}}}$ for CIM and S-CIM, respectively.

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5. Computational experiments

The success probability and the computation time of CIM and S-CIM in solving the NP-hard MAX-CUT problems are used to evaluate their performance. Firstly, the simplest MAX-CUT-3 problem is calculated, which consists of four vertices and six edges with identical weight ${J_{ij}} ={-} 1$. The reason for using a small instance is that the accuracy of the device is verified by exhausting all possible outcomes. The optimized outputs of CIM and S-CIM are shown in Fig. 7. When the network has four coupled DOPOs, there will be sixteen possible outputs. Considering the degeneracy of the phase states (spin states), three of the eight spin configurations correspond to the ground state of the Ising Hamiltonian, {|↑↓↓↑>,|↑↑↓↓>,|↑↓↑↓>}, i.e. the solution of the simplest MAX-CUT-3 problem. In order to make the comparison between CIM and S-CIM more obvious, the feedback rate (The rate at which squeezed feedback pulse and signal DOPO pulse are coupled) is reduced, and the maximum success rate does not reach 100%. Figure 7 shows that S-CIM can increase the success rate of computation by 5.4% in the 4-DOPO network.

 figure: Fig. 7.

Fig. 7. In the 4-coupled DOPO network, the comparison of spin configuration distributions in 1000 trials of numerical simulations for CIM and S-CIM, respectively.

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Next, large-scale instances with 800-20000 vertices and 0.02%-6% edge density are explored. The high order MAX-CUT problems were obtained from G-set graphs, which were randomly constructed by using a machine-independent graph generator created by G. Rinaldi [31]. The normalized optimization results and the average computation time are indicators for evaluating the performance of S-CIM. The computation time is defined as the evolution time of the quadrature amplitude component for DOPO from the initial state to the steady state.

When solving the same sample G1, Fig. 8 shows the evolution time of the quadrature amplitude component under two different schemes in one trial of numerical simulations. The bifurcation of quadrature amplitude components for CIM at 45 ≤ τ < 396 is caused by the spontaneous symmetry breaking. In the S-CIM, the stage of the spontaneous symmetry breaking occupies a shorter time range at 38 ≤ τ < 92. Finally, CIM evolves to steady state at τ = 396 [in Fig. 8(a)], while S-CIM reaches steady state at τ = 92 [in Fig. 8(b)]. This means that S-CIM completes the optimization earlier than CIM, which is in good agreement with the theoretical analysis.

 figure: Fig. 8.

Fig. 8. Quadrature amplitude components of DOPOs versus normalized time τ for CIM and S-CIM against the MAX-CUT problem on the G1 graph.

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Finally, the performance evaluation indicators of S-CIM are recorded in Table 1. V denotes the number of vertices in sample G-set graph, E denotes the number of edges, CGW denotes the optimal output of the Goemans-Williamson (GW) algorithm based on semi-definite programming (SDP) [32], USDP denotes the solution to the SDP upper bound of the MAX-CUT problem, CCIM and < CCIM> denote the best and average normalized outputs of running the CIM 500 times, respectively. Similarly, CS-CIM and < CS-CIM> represent the best and average normalized outputs of running the S-CIM 500 times, respectively. In order to compare these schemes, every output is normalized by (C + Eneg)/(USDP+Eneg), where Eneg denotes the number of negative edges in sample G-set graph, C denotes the cut value of three schemes. <τCIM> and <τS-CIM> are the average computation time of running CIM and S-CIM 500 times, respectively. The average normalized solutions of S-CIM are about 0.18%-2.27% higher than that of CIM. Because the vacuum fluctuations in the ring cavity of S-CIM are smaller than that of CIM, the detection efficiency is improved, and the optimization outputs are more satisfactory. The reason why the improvement is not significant is that, as the number of vertices increases, the number of suboptimal solutions increases at a faster rate, greatly reducing the accuracy of the final results. In addition, the average computation time of CIM and S-CIM is compared in Table 1. For all large-scale instances, the computation time of S-CIM is 17.93%-75.12% less than that of CIM. Quantum inseparability accelerates the coupled DOPO network oscillating, which improves the optimization efficiency of the S-CIM system.

Tables Icon

Table 1. Performance comparison of CIM and S-CIM in computing large-scale MAX-CUT problems on sample G-set graphs.

6. Conclusion

Squeezed feedback pulses are combined with signal DOPO pulses to generate an N-coupled DOPO network, which is used as a speed-up coherent Ising machine to solve the NP-hard MAX-CUT problem with shorter computation time and better optimization results. The calculation model of the S-CIM is formulated by using truncated Wigner representation, and its quantum properties are simulated. There is a turn-on-delay oscillation effect in the CIM. It leads to a difference between the dynamic threshold and the static threshold of the system. The coupled DOPO network actually takes more time to start oscillating. However, in the S-CIM with the schedule of the gradually increasing pump rate, the squeezing parameter of squeezed vacuum injection increases with increasing pump rate. Then the small peak indicating the turn-on-delay oscillation effect disappears, and the quantum inseparability of the system is enhanced. As a result, the time required for the network to search for the ground-state spin configuration is shortened, and the evolution of the average photon number in the fiber ring cavity is accelerated. All these suggest that S-CIM can get the optimization outputs earlier. In addition, the squeezed vacuum injection deamplifies the noise of the quadrature amplitude component at the cost of amplifying the noise of the quadrature phase component, which weakens the quantum fluctuations of the coupled DOPO network. Computation experiments of the large-scale MAX-CUT problems (N = 800-20000) demonstrate that the normalized optimization result for S-CIM is 2.27% higher than that of CIM in the best case, and the computation time is 75.12% less than that of CIM. These numerical results are consistent with the findings of the quantum properties, indicating that the introduction of the squeezed feedback system improves the optimization efficiency of CIM and gives a great push to the application of the CIM.

Funding

National Natural Science Foundation of China (11604377, 61775234); Qingdao National Laboratory for Marine Science and Technology (QNLM2016ORP0111); CAS “Light of West China” Program (XAB2017B18); Natural Science Basic Research plan in Shaanxi Province of China (2018JQ6067); The Independent Project of State Key Laboratory of Transient Optics and Photonics.

Disclosures

The authors declare no conflicts of interest.

References

1. F. Barahona, “On the computational complexity of Ising spin glass models,” J. Phys. A: Math. Gen. 15(10), 3241–3253 (1982). [CrossRef]  

2. M. Marc, G. Parisi, and M. Virasoro, Spin Glass Theory and Beyond: An Introduction to the Replica Method and Its Applications (World Scientific Publishing Company, 1987).

3. D. B. Kitchen, H. Decornez, J. R. Furr, and J. Bajorath, “Docking and scoring in virtual screening for drug discovery: methods and applications,” Nat. Rev. Drug Discovery 3(11), 935–949 (2004). [CrossRef]  

4. J. D. Bryngelson and P. G. Wolynes, “Spin glasses and the statistical mechanics of protein folding,” Proc. Natl. Acad. Sci. 84(21), 7524–7528 (1987). [CrossRef]  

5. I. H. Witten, E. Frank, M. A. Hall, and C. J. Pal, Data Mining: Practical Machine Learning Tools and Techniques (Morgan Kaufmann, 2016).

6. S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220(4598), 671–680 (1983). [CrossRef]  

7. Z. Su, D. Xue, and Z. Ji, “Designing LED array for uniform illumination distribution by simulated annealing algorithm,” Opt. Express 20(S6), A843–855 (2012). [CrossRef]  

8. P. Honzatko, J. Kanka, and B. Vrany, “Retrieval of the pulse amplitude and phase from cross-phase modulation spectrograms using the simulated annealing method,” Opt. Express 12(24), 6046–6052 (2004). [CrossRef]  

9. T. Kadowaki and H. Nishimori, “Quantum annealing in the transverse Ising model,” Phys. Rev. E 58(5), 5355–5363 (1998). [CrossRef]  

10. E. G. Rieffel, D. Venturelli, B. O’Gorman, M. B. Do, E. M. Prystay, and V. N. Smelyanskiy, “A case study in programming a quantum annealer for hard operational planning problems,” Quantum Inf. Process. 14(1), 1–36 (2015). [CrossRef]  

11. Z. Wang, A. Marandi, K. Wen, R. L. Byer, and Y. Yamamoto, “Coherent Ising machine based on degenerate optical parametric oscillators,” Phys. Rev. A 88(6), 063853 (2013). [CrossRef]  

12. A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent Ising machine,” Nat. Photonics 8(12), 937–942 (2014). [CrossRef]  

13. M. Gu and Á Perales, “Encoding universal computation in the ground states of Ising lattices,” Phys. Rev. E 86(1), 011116 (2012). [CrossRef]  

14. M. R. Garey and D. S. Johnson, Computers and Intractability: a Guide to the Theory of NP-Completeness (Freeman, 2009).

15. Y. Haribara, S. Utsunomiya, and Y. Yamamoto, “Computational principle and performance evaluation of coherent Ising machine based on degenerate optical parametric oscillator network,” Entropy 18(4), 151–166 (2016). [CrossRef]  

16. T. Inagaki, K. Inaba, R. Hamerly, K. Inoue, Y. Yamamoto, and H. Takesue, “Large-scale Ising spin network based on degenerate optical parametric oscillators,” Nat. Photonics 10(6), 415–419 (2016). [CrossRef]  

17. H. Takesue and T. Inagaki, “10 GHz clock time-multiplexed degenerate optical parametric oscillators for a photonic Ising spin network,” Opt. Lett. 41(18), 4273–4276 (2016). [CrossRef]  

18. A. Yamamura, K. Aihara, and Y. Yamamoto, “Quantum model for coherent Ising machines: Discrete-time measurement feedback formulation,” Phys. Rev. A 96(5), 053834 (2017). [CrossRef]  

19. D. Maruo, S. Utsunomiya, and Y. Yamamoto, “Truncated Wigner theory of coherent Ising machines based on degenerate optical parametric oscillator network,” Phys. Scr. 91(8), 083010 (2016). [CrossRef]  

20. K. Takata, A. Marandi, and Y. Yamamoto, “Quantum correlation in degenerate optical parametric oscillators with mutual injections,” Phys. Rev. A 92(4), 043821 (2015). [CrossRef]  

21. D. F. Walls, “Squeezed states of light,” Nature 306(5939), 141–146 (1983). [CrossRef]  

22. C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26(8), 1817–1839 (1982). [CrossRef]  

23. H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13(6), 2226–2243 (1976). [CrossRef]  

24. G. J. Milburn and D. F. Walls, “Squeezed states and intensity fluctuations in degenerate parametric oscillation,” Phys. Rev. A 27(1), 392–394 (1983). [CrossRef]  

25. L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57(20), 2520–2523 (1986). [CrossRef]  

26. Z. Dutton, J. H. Shapiro, and S. Guha, “LADAR resolution improvement using receivers enhanced with squeezed-vacuum injection and phase-sensitive amplification,” J. Opt. Soc. Am. B 27(6), A63–A72 (2010). [CrossRef]  

27. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23(8), 1693–1708 (1981). [CrossRef]  

28. H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations (Springer-Verlag, 2002).

29. D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 2007).

30. L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84(12), 2722–2725 (2000). [CrossRef]  

31. C. Helmberg and F. A. Rendl, “spectral bundle method for semidefinite programming,” SIAM J. Control 10(3), 673–696 (2000). [CrossRef]  

32. M. X. Goemans and D. P. Williamson, “Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming,” J. Assoc. Comput. Mach. 42(6), 1115–1145 (1995). [CrossRef]  

References

  • View by:

  1. F. Barahona, “On the computational complexity of Ising spin glass models,” J. Phys. A: Math. Gen. 15(10), 3241–3253 (1982).
    [Crossref]
  2. M. Marc, G. Parisi, and M. Virasoro, Spin Glass Theory and Beyond: An Introduction to the Replica Method and Its Applications (World Scientific Publishing Company, 1987).
  3. D. B. Kitchen, H. Decornez, J. R. Furr, and J. Bajorath, “Docking and scoring in virtual screening for drug discovery: methods and applications,” Nat. Rev. Drug Discovery 3(11), 935–949 (2004).
    [Crossref]
  4. J. D. Bryngelson and P. G. Wolynes, “Spin glasses and the statistical mechanics of protein folding,” Proc. Natl. Acad. Sci. 84(21), 7524–7528 (1987).
    [Crossref]
  5. I. H. Witten, E. Frank, M. A. Hall, and C. J. Pal, Data Mining: Practical Machine Learning Tools and Techniques (Morgan Kaufmann, 2016).
  6. S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220(4598), 671–680 (1983).
    [Crossref]
  7. Z. Su, D. Xue, and Z. Ji, “Designing LED array for uniform illumination distribution by simulated annealing algorithm,” Opt. Express 20(S6), A843–855 (2012).
    [Crossref]
  8. P. Honzatko, J. Kanka, and B. Vrany, “Retrieval of the pulse amplitude and phase from cross-phase modulation spectrograms using the simulated annealing method,” Opt. Express 12(24), 6046–6052 (2004).
    [Crossref]
  9. T. Kadowaki and H. Nishimori, “Quantum annealing in the transverse Ising model,” Phys. Rev. E 58(5), 5355–5363 (1998).
    [Crossref]
  10. E. G. Rieffel, D. Venturelli, B. O’Gorman, M. B. Do, E. M. Prystay, and V. N. Smelyanskiy, “A case study in programming a quantum annealer for hard operational planning problems,” Quantum Inf. Process. 14(1), 1–36 (2015).
    [Crossref]
  11. Z. Wang, A. Marandi, K. Wen, R. L. Byer, and Y. Yamamoto, “Coherent Ising machine based on degenerate optical parametric oscillators,” Phys. Rev. A 88(6), 063853 (2013).
    [Crossref]
  12. A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent Ising machine,” Nat. Photonics 8(12), 937–942 (2014).
    [Crossref]
  13. M. Gu and Á Perales, “Encoding universal computation in the ground states of Ising lattices,” Phys. Rev. E 86(1), 011116 (2012).
    [Crossref]
  14. M. R. Garey and D. S. Johnson, Computers and Intractability: a Guide to the Theory of NP-Completeness (Freeman, 2009).
  15. Y. Haribara, S. Utsunomiya, and Y. Yamamoto, “Computational principle and performance evaluation of coherent Ising machine based on degenerate optical parametric oscillator network,” Entropy 18(4), 151–166 (2016).
    [Crossref]
  16. T. Inagaki, K. Inaba, R. Hamerly, K. Inoue, Y. Yamamoto, and H. Takesue, “Large-scale Ising spin network based on degenerate optical parametric oscillators,” Nat. Photonics 10(6), 415–419 (2016).
    [Crossref]
  17. H. Takesue and T. Inagaki, “10 GHz clock time-multiplexed degenerate optical parametric oscillators for a photonic Ising spin network,” Opt. Lett. 41(18), 4273–4276 (2016).
    [Crossref]
  18. A. Yamamura, K. Aihara, and Y. Yamamoto, “Quantum model for coherent Ising machines: Discrete-time measurement feedback formulation,” Phys. Rev. A 96(5), 053834 (2017).
    [Crossref]
  19. D. Maruo, S. Utsunomiya, and Y. Yamamoto, “Truncated Wigner theory of coherent Ising machines based on degenerate optical parametric oscillator network,” Phys. Scr. 91(8), 083010 (2016).
    [Crossref]
  20. K. Takata, A. Marandi, and Y. Yamamoto, “Quantum correlation in degenerate optical parametric oscillators with mutual injections,” Phys. Rev. A 92(4), 043821 (2015).
    [Crossref]
  21. D. F. Walls, “Squeezed states of light,” Nature 306(5939), 141–146 (1983).
    [Crossref]
  22. C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26(8), 1817–1839 (1982).
    [Crossref]
  23. H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13(6), 2226–2243 (1976).
    [Crossref]
  24. G. J. Milburn and D. F. Walls, “Squeezed states and intensity fluctuations in degenerate parametric oscillation,” Phys. Rev. A 27(1), 392–394 (1983).
    [Crossref]
  25. L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57(20), 2520–2523 (1986).
    [Crossref]
  26. Z. Dutton, J. H. Shapiro, and S. Guha, “LADAR resolution improvement using receivers enhanced with squeezed-vacuum injection and phase-sensitive amplification,” J. Opt. Soc. Am. B 27(6), A63–A72 (2010).
    [Crossref]
  27. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23(8), 1693–1708 (1981).
    [Crossref]
  28. H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations (Springer-Verlag, 2002).
  29. D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 2007).
  30. L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84(12), 2722–2725 (2000).
    [Crossref]
  31. C. Helmberg and F. A. Rendl, “spectral bundle method for semidefinite programming,” SIAM J. Control 10(3), 673–696 (2000).
    [Crossref]
  32. M. X. Goemans and D. P. Williamson, “Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming,” J. Assoc. Comput. Mach. 42(6), 1115–1145 (1995).
    [Crossref]

2017 (1)

A. Yamamura, K. Aihara, and Y. Yamamoto, “Quantum model for coherent Ising machines: Discrete-time measurement feedback formulation,” Phys. Rev. A 96(5), 053834 (2017).
[Crossref]

2016 (4)

D. Maruo, S. Utsunomiya, and Y. Yamamoto, “Truncated Wigner theory of coherent Ising machines based on degenerate optical parametric oscillator network,” Phys. Scr. 91(8), 083010 (2016).
[Crossref]

Y. Haribara, S. Utsunomiya, and Y. Yamamoto, “Computational principle and performance evaluation of coherent Ising machine based on degenerate optical parametric oscillator network,” Entropy 18(4), 151–166 (2016).
[Crossref]

T. Inagaki, K. Inaba, R. Hamerly, K. Inoue, Y. Yamamoto, and H. Takesue, “Large-scale Ising spin network based on degenerate optical parametric oscillators,” Nat. Photonics 10(6), 415–419 (2016).
[Crossref]

H. Takesue and T. Inagaki, “10 GHz clock time-multiplexed degenerate optical parametric oscillators for a photonic Ising spin network,” Opt. Lett. 41(18), 4273–4276 (2016).
[Crossref]

2015 (2)

K. Takata, A. Marandi, and Y. Yamamoto, “Quantum correlation in degenerate optical parametric oscillators with mutual injections,” Phys. Rev. A 92(4), 043821 (2015).
[Crossref]

E. G. Rieffel, D. Venturelli, B. O’Gorman, M. B. Do, E. M. Prystay, and V. N. Smelyanskiy, “A case study in programming a quantum annealer for hard operational planning problems,” Quantum Inf. Process. 14(1), 1–36 (2015).
[Crossref]

2014 (1)

A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent Ising machine,” Nat. Photonics 8(12), 937–942 (2014).
[Crossref]

2013 (1)

Z. Wang, A. Marandi, K. Wen, R. L. Byer, and Y. Yamamoto, “Coherent Ising machine based on degenerate optical parametric oscillators,” Phys. Rev. A 88(6), 063853 (2013).
[Crossref]

2012 (2)

M. Gu and Á Perales, “Encoding universal computation in the ground states of Ising lattices,” Phys. Rev. E 86(1), 011116 (2012).
[Crossref]

Z. Su, D. Xue, and Z. Ji, “Designing LED array for uniform illumination distribution by simulated annealing algorithm,” Opt. Express 20(S6), A843–855 (2012).
[Crossref]

2010 (1)

2004 (2)

P. Honzatko, J. Kanka, and B. Vrany, “Retrieval of the pulse amplitude and phase from cross-phase modulation spectrograms using the simulated annealing method,” Opt. Express 12(24), 6046–6052 (2004).
[Crossref]

D. B. Kitchen, H. Decornez, J. R. Furr, and J. Bajorath, “Docking and scoring in virtual screening for drug discovery: methods and applications,” Nat. Rev. Drug Discovery 3(11), 935–949 (2004).
[Crossref]

2000 (2)

L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84(12), 2722–2725 (2000).
[Crossref]

C. Helmberg and F. A. Rendl, “spectral bundle method for semidefinite programming,” SIAM J. Control 10(3), 673–696 (2000).
[Crossref]

1998 (1)

T. Kadowaki and H. Nishimori, “Quantum annealing in the transverse Ising model,” Phys. Rev. E 58(5), 5355–5363 (1998).
[Crossref]

1995 (1)

M. X. Goemans and D. P. Williamson, “Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming,” J. Assoc. Comput. Mach. 42(6), 1115–1145 (1995).
[Crossref]

1987 (1)

J. D. Bryngelson and P. G. Wolynes, “Spin glasses and the statistical mechanics of protein folding,” Proc. Natl. Acad. Sci. 84(21), 7524–7528 (1987).
[Crossref]

1986 (1)

L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57(20), 2520–2523 (1986).
[Crossref]

1983 (3)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220(4598), 671–680 (1983).
[Crossref]

D. F. Walls, “Squeezed states of light,” Nature 306(5939), 141–146 (1983).
[Crossref]

G. J. Milburn and D. F. Walls, “Squeezed states and intensity fluctuations in degenerate parametric oscillation,” Phys. Rev. A 27(1), 392–394 (1983).
[Crossref]

1982 (2)

F. Barahona, “On the computational complexity of Ising spin glass models,” J. Phys. A: Math. Gen. 15(10), 3241–3253 (1982).
[Crossref]

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26(8), 1817–1839 (1982).
[Crossref]

1981 (1)

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23(8), 1693–1708 (1981).
[Crossref]

1976 (1)

H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13(6), 2226–2243 (1976).
[Crossref]

Aihara, K.

A. Yamamura, K. Aihara, and Y. Yamamoto, “Quantum model for coherent Ising machines: Discrete-time measurement feedback formulation,” Phys. Rev. A 96(5), 053834 (2017).
[Crossref]

Bajorath, J.

D. B. Kitchen, H. Decornez, J. R. Furr, and J. Bajorath, “Docking and scoring in virtual screening for drug discovery: methods and applications,” Nat. Rev. Drug Discovery 3(11), 935–949 (2004).
[Crossref]

Barahona, F.

F. Barahona, “On the computational complexity of Ising spin glass models,” J. Phys. A: Math. Gen. 15(10), 3241–3253 (1982).
[Crossref]

Bryngelson, J. D.

J. D. Bryngelson and P. G. Wolynes, “Spin glasses and the statistical mechanics of protein folding,” Proc. Natl. Acad. Sci. 84(21), 7524–7528 (1987).
[Crossref]

Byer, R. L.

A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent Ising machine,” Nat. Photonics 8(12), 937–942 (2014).
[Crossref]

Z. Wang, A. Marandi, K. Wen, R. L. Byer, and Y. Yamamoto, “Coherent Ising machine based on degenerate optical parametric oscillators,” Phys. Rev. A 88(6), 063853 (2013).
[Crossref]

Carmichael, H. J.

H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations (Springer-Verlag, 2002).

Caves, C. M.

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26(8), 1817–1839 (1982).
[Crossref]

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23(8), 1693–1708 (1981).
[Crossref]

Cirac, J. I.

L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84(12), 2722–2725 (2000).
[Crossref]

Decornez, H.

D. B. Kitchen, H. Decornez, J. R. Furr, and J. Bajorath, “Docking and scoring in virtual screening for drug discovery: methods and applications,” Nat. Rev. Drug Discovery 3(11), 935–949 (2004).
[Crossref]

Do, M. B.

E. G. Rieffel, D. Venturelli, B. O’Gorman, M. B. Do, E. M. Prystay, and V. N. Smelyanskiy, “A case study in programming a quantum annealer for hard operational planning problems,” Quantum Inf. Process. 14(1), 1–36 (2015).
[Crossref]

Duan, L. M.

L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84(12), 2722–2725 (2000).
[Crossref]

Dutton, Z.

Frank, E.

I. H. Witten, E. Frank, M. A. Hall, and C. J. Pal, Data Mining: Practical Machine Learning Tools and Techniques (Morgan Kaufmann, 2016).

Furr, J. R.

D. B. Kitchen, H. Decornez, J. R. Furr, and J. Bajorath, “Docking and scoring in virtual screening for drug discovery: methods and applications,” Nat. Rev. Drug Discovery 3(11), 935–949 (2004).
[Crossref]

Garey, M. R.

M. R. Garey and D. S. Johnson, Computers and Intractability: a Guide to the Theory of NP-Completeness (Freeman, 2009).

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220(4598), 671–680 (1983).
[Crossref]

Giedke, G.

L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84(12), 2722–2725 (2000).
[Crossref]

Goemans, M. X.

M. X. Goemans and D. P. Williamson, “Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming,” J. Assoc. Comput. Mach. 42(6), 1115–1145 (1995).
[Crossref]

Gu, M.

M. Gu and Á Perales, “Encoding universal computation in the ground states of Ising lattices,” Phys. Rev. E 86(1), 011116 (2012).
[Crossref]

Guha, S.

Hall, J. L.

L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57(20), 2520–2523 (1986).
[Crossref]

Hall, M. A.

I. H. Witten, E. Frank, M. A. Hall, and C. J. Pal, Data Mining: Practical Machine Learning Tools and Techniques (Morgan Kaufmann, 2016).

Hamerly, R.

T. Inagaki, K. Inaba, R. Hamerly, K. Inoue, Y. Yamamoto, and H. Takesue, “Large-scale Ising spin network based on degenerate optical parametric oscillators,” Nat. Photonics 10(6), 415–419 (2016).
[Crossref]

Haribara, Y.

Y. Haribara, S. Utsunomiya, and Y. Yamamoto, “Computational principle and performance evaluation of coherent Ising machine based on degenerate optical parametric oscillator network,” Entropy 18(4), 151–166 (2016).
[Crossref]

Helmberg, C.

C. Helmberg and F. A. Rendl, “spectral bundle method for semidefinite programming,” SIAM J. Control 10(3), 673–696 (2000).
[Crossref]

Honzatko, P.

Inaba, K.

T. Inagaki, K. Inaba, R. Hamerly, K. Inoue, Y. Yamamoto, and H. Takesue, “Large-scale Ising spin network based on degenerate optical parametric oscillators,” Nat. Photonics 10(6), 415–419 (2016).
[Crossref]

Inagaki, T.

T. Inagaki, K. Inaba, R. Hamerly, K. Inoue, Y. Yamamoto, and H. Takesue, “Large-scale Ising spin network based on degenerate optical parametric oscillators,” Nat. Photonics 10(6), 415–419 (2016).
[Crossref]

H. Takesue and T. Inagaki, “10 GHz clock time-multiplexed degenerate optical parametric oscillators for a photonic Ising spin network,” Opt. Lett. 41(18), 4273–4276 (2016).
[Crossref]

Inoue, K.

T. Inagaki, K. Inaba, R. Hamerly, K. Inoue, Y. Yamamoto, and H. Takesue, “Large-scale Ising spin network based on degenerate optical parametric oscillators,” Nat. Photonics 10(6), 415–419 (2016).
[Crossref]

Ji, Z.

Johnson, D. S.

M. R. Garey and D. S. Johnson, Computers and Intractability: a Guide to the Theory of NP-Completeness (Freeman, 2009).

Kadowaki, T.

T. Kadowaki and H. Nishimori, “Quantum annealing in the transverse Ising model,” Phys. Rev. E 58(5), 5355–5363 (1998).
[Crossref]

Kanka, J.

Kimble, H. J.

L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57(20), 2520–2523 (1986).
[Crossref]

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220(4598), 671–680 (1983).
[Crossref]

Kitchen, D. B.

D. B. Kitchen, H. Decornez, J. R. Furr, and J. Bajorath, “Docking and scoring in virtual screening for drug discovery: methods and applications,” Nat. Rev. Drug Discovery 3(11), 935–949 (2004).
[Crossref]

Marandi, A.

K. Takata, A. Marandi, and Y. Yamamoto, “Quantum correlation in degenerate optical parametric oscillators with mutual injections,” Phys. Rev. A 92(4), 043821 (2015).
[Crossref]

A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent Ising machine,” Nat. Photonics 8(12), 937–942 (2014).
[Crossref]

Z. Wang, A. Marandi, K. Wen, R. L. Byer, and Y. Yamamoto, “Coherent Ising machine based on degenerate optical parametric oscillators,” Phys. Rev. A 88(6), 063853 (2013).
[Crossref]

Marc, M.

M. Marc, G. Parisi, and M. Virasoro, Spin Glass Theory and Beyond: An Introduction to the Replica Method and Its Applications (World Scientific Publishing Company, 1987).

Maruo, D.

D. Maruo, S. Utsunomiya, and Y. Yamamoto, “Truncated Wigner theory of coherent Ising machines based on degenerate optical parametric oscillator network,” Phys. Scr. 91(8), 083010 (2016).
[Crossref]

Milburn, G. J.

G. J. Milburn and D. F. Walls, “Squeezed states and intensity fluctuations in degenerate parametric oscillation,” Phys. Rev. A 27(1), 392–394 (1983).
[Crossref]

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 2007).

Nishimori, H.

T. Kadowaki and H. Nishimori, “Quantum annealing in the transverse Ising model,” Phys. Rev. E 58(5), 5355–5363 (1998).
[Crossref]

O’Gorman, B.

E. G. Rieffel, D. Venturelli, B. O’Gorman, M. B. Do, E. M. Prystay, and V. N. Smelyanskiy, “A case study in programming a quantum annealer for hard operational planning problems,” Quantum Inf. Process. 14(1), 1–36 (2015).
[Crossref]

Pal, C. J.

I. H. Witten, E. Frank, M. A. Hall, and C. J. Pal, Data Mining: Practical Machine Learning Tools and Techniques (Morgan Kaufmann, 2016).

Parisi, G.

M. Marc, G. Parisi, and M. Virasoro, Spin Glass Theory and Beyond: An Introduction to the Replica Method and Its Applications (World Scientific Publishing Company, 1987).

Perales, Á

M. Gu and Á Perales, “Encoding universal computation in the ground states of Ising lattices,” Phys. Rev. E 86(1), 011116 (2012).
[Crossref]

Prystay, E. M.

E. G. Rieffel, D. Venturelli, B. O’Gorman, M. B. Do, E. M. Prystay, and V. N. Smelyanskiy, “A case study in programming a quantum annealer for hard operational planning problems,” Quantum Inf. Process. 14(1), 1–36 (2015).
[Crossref]

Rendl, F. A.

C. Helmberg and F. A. Rendl, “spectral bundle method for semidefinite programming,” SIAM J. Control 10(3), 673–696 (2000).
[Crossref]

Rieffel, E. G.

E. G. Rieffel, D. Venturelli, B. O’Gorman, M. B. Do, E. M. Prystay, and V. N. Smelyanskiy, “A case study in programming a quantum annealer for hard operational planning problems,” Quantum Inf. Process. 14(1), 1–36 (2015).
[Crossref]

Shapiro, J. H.

Smelyanskiy, V. N.

E. G. Rieffel, D. Venturelli, B. O’Gorman, M. B. Do, E. M. Prystay, and V. N. Smelyanskiy, “A case study in programming a quantum annealer for hard operational planning problems,” Quantum Inf. Process. 14(1), 1–36 (2015).
[Crossref]

Su, Z.

Takata, K.

K. Takata, A. Marandi, and Y. Yamamoto, “Quantum correlation in degenerate optical parametric oscillators with mutual injections,” Phys. Rev. A 92(4), 043821 (2015).
[Crossref]

A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent Ising machine,” Nat. Photonics 8(12), 937–942 (2014).
[Crossref]

Takesue, H.

T. Inagaki, K. Inaba, R. Hamerly, K. Inoue, Y. Yamamoto, and H. Takesue, “Large-scale Ising spin network based on degenerate optical parametric oscillators,” Nat. Photonics 10(6), 415–419 (2016).
[Crossref]

H. Takesue and T. Inagaki, “10 GHz clock time-multiplexed degenerate optical parametric oscillators for a photonic Ising spin network,” Opt. Lett. 41(18), 4273–4276 (2016).
[Crossref]

Utsunomiya, S.

D. Maruo, S. Utsunomiya, and Y. Yamamoto, “Truncated Wigner theory of coherent Ising machines based on degenerate optical parametric oscillator network,” Phys. Scr. 91(8), 083010 (2016).
[Crossref]

Y. Haribara, S. Utsunomiya, and Y. Yamamoto, “Computational principle and performance evaluation of coherent Ising machine based on degenerate optical parametric oscillator network,” Entropy 18(4), 151–166 (2016).
[Crossref]

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220(4598), 671–680 (1983).
[Crossref]

Venturelli, D.

E. G. Rieffel, D. Venturelli, B. O’Gorman, M. B. Do, E. M. Prystay, and V. N. Smelyanskiy, “A case study in programming a quantum annealer for hard operational planning problems,” Quantum Inf. Process. 14(1), 1–36 (2015).
[Crossref]

Virasoro, M.

M. Marc, G. Parisi, and M. Virasoro, Spin Glass Theory and Beyond: An Introduction to the Replica Method and Its Applications (World Scientific Publishing Company, 1987).

Vrany, B.

Walls, D. F.

D. F. Walls, “Squeezed states of light,” Nature 306(5939), 141–146 (1983).
[Crossref]

G. J. Milburn and D. F. Walls, “Squeezed states and intensity fluctuations in degenerate parametric oscillation,” Phys. Rev. A 27(1), 392–394 (1983).
[Crossref]

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 2007).

Wang, Z.

A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent Ising machine,” Nat. Photonics 8(12), 937–942 (2014).
[Crossref]

Z. Wang, A. Marandi, K. Wen, R. L. Byer, and Y. Yamamoto, “Coherent Ising machine based on degenerate optical parametric oscillators,” Phys. Rev. A 88(6), 063853 (2013).
[Crossref]

Wen, K.

Z. Wang, A. Marandi, K. Wen, R. L. Byer, and Y. Yamamoto, “Coherent Ising machine based on degenerate optical parametric oscillators,” Phys. Rev. A 88(6), 063853 (2013).
[Crossref]

Williamson, D. P.

M. X. Goemans and D. P. Williamson, “Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming,” J. Assoc. Comput. Mach. 42(6), 1115–1145 (1995).
[Crossref]

Witten, I. H.

I. H. Witten, E. Frank, M. A. Hall, and C. J. Pal, Data Mining: Practical Machine Learning Tools and Techniques (Morgan Kaufmann, 2016).

Wolynes, P. G.

J. D. Bryngelson and P. G. Wolynes, “Spin glasses and the statistical mechanics of protein folding,” Proc. Natl. Acad. Sci. 84(21), 7524–7528 (1987).
[Crossref]

Wu, H.

L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57(20), 2520–2523 (1986).
[Crossref]

Wu, L. A.

L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57(20), 2520–2523 (1986).
[Crossref]

Xue, D.

Yamamoto, Y.

A. Yamamura, K. Aihara, and Y. Yamamoto, “Quantum model for coherent Ising machines: Discrete-time measurement feedback formulation,” Phys. Rev. A 96(5), 053834 (2017).
[Crossref]

D. Maruo, S. Utsunomiya, and Y. Yamamoto, “Truncated Wigner theory of coherent Ising machines based on degenerate optical parametric oscillator network,” Phys. Scr. 91(8), 083010 (2016).
[Crossref]

Y. Haribara, S. Utsunomiya, and Y. Yamamoto, “Computational principle and performance evaluation of coherent Ising machine based on degenerate optical parametric oscillator network,” Entropy 18(4), 151–166 (2016).
[Crossref]

T. Inagaki, K. Inaba, R. Hamerly, K. Inoue, Y. Yamamoto, and H. Takesue, “Large-scale Ising spin network based on degenerate optical parametric oscillators,” Nat. Photonics 10(6), 415–419 (2016).
[Crossref]

K. Takata, A. Marandi, and Y. Yamamoto, “Quantum correlation in degenerate optical parametric oscillators with mutual injections,” Phys. Rev. A 92(4), 043821 (2015).
[Crossref]

A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent Ising machine,” Nat. Photonics 8(12), 937–942 (2014).
[Crossref]

Z. Wang, A. Marandi, K. Wen, R. L. Byer, and Y. Yamamoto, “Coherent Ising machine based on degenerate optical parametric oscillators,” Phys. Rev. A 88(6), 063853 (2013).
[Crossref]

Yamamura, A.

A. Yamamura, K. Aihara, and Y. Yamamoto, “Quantum model for coherent Ising machines: Discrete-time measurement feedback formulation,” Phys. Rev. A 96(5), 053834 (2017).
[Crossref]

Yuen, H. P.

H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13(6), 2226–2243 (1976).
[Crossref]

Zoller, P.

L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84(12), 2722–2725 (2000).
[Crossref]

Entropy (1)

Y. Haribara, S. Utsunomiya, and Y. Yamamoto, “Computational principle and performance evaluation of coherent Ising machine based on degenerate optical parametric oscillator network,” Entropy 18(4), 151–166 (2016).
[Crossref]

J. Assoc. Comput. Mach. (1)

M. X. Goemans and D. P. Williamson, “Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming,” J. Assoc. Comput. Mach. 42(6), 1115–1145 (1995).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Phys. A: Math. Gen. (1)

F. Barahona, “On the computational complexity of Ising spin glass models,” J. Phys. A: Math. Gen. 15(10), 3241–3253 (1982).
[Crossref]

Nat. Photonics (2)

T. Inagaki, K. Inaba, R. Hamerly, K. Inoue, Y. Yamamoto, and H. Takesue, “Large-scale Ising spin network based on degenerate optical parametric oscillators,” Nat. Photonics 10(6), 415–419 (2016).
[Crossref]

A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent Ising machine,” Nat. Photonics 8(12), 937–942 (2014).
[Crossref]

Nat. Rev. Drug Discovery (1)

D. B. Kitchen, H. Decornez, J. R. Furr, and J. Bajorath, “Docking and scoring in virtual screening for drug discovery: methods and applications,” Nat. Rev. Drug Discovery 3(11), 935–949 (2004).
[Crossref]

Nature (1)

D. F. Walls, “Squeezed states of light,” Nature 306(5939), 141–146 (1983).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. A (5)

A. Yamamura, K. Aihara, and Y. Yamamoto, “Quantum model for coherent Ising machines: Discrete-time measurement feedback formulation,” Phys. Rev. A 96(5), 053834 (2017).
[Crossref]

Z. Wang, A. Marandi, K. Wen, R. L. Byer, and Y. Yamamoto, “Coherent Ising machine based on degenerate optical parametric oscillators,” Phys. Rev. A 88(6), 063853 (2013).
[Crossref]

H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13(6), 2226–2243 (1976).
[Crossref]

G. J. Milburn and D. F. Walls, “Squeezed states and intensity fluctuations in degenerate parametric oscillation,” Phys. Rev. A 27(1), 392–394 (1983).
[Crossref]

K. Takata, A. Marandi, and Y. Yamamoto, “Quantum correlation in degenerate optical parametric oscillators with mutual injections,” Phys. Rev. A 92(4), 043821 (2015).
[Crossref]

Phys. Rev. D (2)

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26(8), 1817–1839 (1982).
[Crossref]

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23(8), 1693–1708 (1981).
[Crossref]

Phys. Rev. E (2)

M. Gu and Á Perales, “Encoding universal computation in the ground states of Ising lattices,” Phys. Rev. E 86(1), 011116 (2012).
[Crossref]

T. Kadowaki and H. Nishimori, “Quantum annealing in the transverse Ising model,” Phys. Rev. E 58(5), 5355–5363 (1998).
[Crossref]

Phys. Rev. Lett. (2)

L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57(20), 2520–2523 (1986).
[Crossref]

L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84(12), 2722–2725 (2000).
[Crossref]

Phys. Scr. (1)

D. Maruo, S. Utsunomiya, and Y. Yamamoto, “Truncated Wigner theory of coherent Ising machines based on degenerate optical parametric oscillator network,” Phys. Scr. 91(8), 083010 (2016).
[Crossref]

Proc. Natl. Acad. Sci. (1)

J. D. Bryngelson and P. G. Wolynes, “Spin glasses and the statistical mechanics of protein folding,” Proc. Natl. Acad. Sci. 84(21), 7524–7528 (1987).
[Crossref]

Quantum Inf. Process. (1)

E. G. Rieffel, D. Venturelli, B. O’Gorman, M. B. Do, E. M. Prystay, and V. N. Smelyanskiy, “A case study in programming a quantum annealer for hard operational planning problems,” Quantum Inf. Process. 14(1), 1–36 (2015).
[Crossref]

Science (1)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220(4598), 671–680 (1983).
[Crossref]

SIAM J. Control (1)

C. Helmberg and F. A. Rendl, “spectral bundle method for semidefinite programming,” SIAM J. Control 10(3), 673–696 (2000).
[Crossref]

Other (5)

H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations (Springer-Verlag, 2002).

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 2007).

M. R. Garey and D. S. Johnson, Computers and Intractability: a Guide to the Theory of NP-Completeness (Freeman, 2009).

I. H. Witten, E. Frank, M. A. Hall, and C. J. Pal, Data Mining: Practical Machine Learning Tools and Techniques (Morgan Kaufmann, 2016).

M. Marc, G. Parisi, and M. Virasoro, Spin Glass Theory and Beyond: An Introduction to the Replica Method and Its Applications (World Scientific Publishing Company, 1987).

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Figures (8)

Fig. 1.
Fig. 1. (a). The schematic of the speed-up coherent Ising machine with a squeezed feedback system (S-CIM). The PSA connected to BS1 amplifies the quadrature amplitude component of each extracted DOPO pulse, while the BS2 couples the modulated squeezed feedback pulse with the target DOPO pulse to implement the given Ising Hamiltonian. A: the network of N degenerate optical parametric oscillators (DOPOs). B: the squeezed feedback system. C: the injection coupler. (b) The corresponding calculation model of the coupled DOPO network based on squeezed feedback process.
Fig. 2.
Fig. 2. Total variances of the EPR operators ${\Delta }\hat{u}_ + ^2 + {\Delta }\hat{v}_ - ^2$ versus normalized pump rate E for three different CIM schemes. The numerical values are computed from two feedback process models: the valid coupling modes approach (Fig. 2(a)) and the unitary displacement operator approach (Fig. 2(b)).
Fig. 3.
Fig. 3. Variances of the quadrature amplitude component ${\hat{x}_{s12}}$ versus normalized pump rate E for CIM and S-CIM with gradually increasing pump rate, respectively. The numerical values are computed from two feedback process models: the valid coupling modes approach (Fig. 3(a)) and the unitary displacement operator approach (Fig. 3(b)). Since the curves of ${\hat{x}_{s12}}$ and ${\hat{x}_{s21}}$ are almost identical, only the former is given here.
Fig. 4.
Fig. 4. In the 2-coupled DOPO network, the probabilities of finding |↑↓>, |↑↑>, |↓↓> and |↓↑> states at four different stages of the optimization process when the final selected ground state is |↑↓>. The gain of the coupled DOPO network (yellow dotted line) gradually increases from below the coupled oscillation threshold to reach the minimum global loss (blue curve) corresponding to the ground state of the Ising Hamiltonian.
Fig. 5.
Fig. 5. In the 2-coupled DOPO network, the probability evolutions of two ground-state spin configurations versus normalized time τ for CIM and S-CIM. The green, red, blue and pink portions in each curve represent the four procedures of the optimization process of quantum parallel search, quantum filtering, spontaneous symmetry breaking, and quantum-to-classical crossover, respectively. The optimization process of CIM is divided by yellow dotted lines, and the optimization process of S-CIM is divided by yellow solid lines.
Fig. 6.
Fig. 6. In the 2-coupled DOPO network, the average photon number n versus normalized pump rate $p = \frac{E}{{{E_d}}}$ for CIM and S-CIM, respectively.
Fig. 7.
Fig. 7. In the 4-coupled DOPO network, the comparison of spin configuration distributions in 1000 trials of numerical simulations for CIM and S-CIM, respectively.
Fig. 8.
Fig. 8. Quadrature amplitude components of DOPOs versus normalized time τ for CIM and S-CIM against the MAX-CUT problem on the G1 graph.

Tables (1)

Tables Icon

Table 1. Performance comparison of CIM and S-CIM in computing large-scale MAX-CUT problems on sample G-set graphs.

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

H = 1 j < l N J j l σ j σ l ,
H = H free + H int + H pump + H sr ,
H free = ω s j = 1 2 a ^ s j a ^ s j + ω s j = 1 2 a ^ p j a ^ p j ,
H int = i κ 2 j = 1 2 ( a ^ s j 2 a ^ p j a ^ p j 2 a ^ s j ) ,
H pump = i γ p j = 1 2 ( ϵ a ^ p j e i ω p t ϵ a ^ p j e i ω p t ) ,
H sr = i j = 1 2 a ^ s j Γ ^ R s j γ s + Γ ^ R s j a ^ s j γ s + a ^ p j Γ ^ R p j γ p + Γ ^ R p j a ^ p j γ p ,
a ^ s j = T 1 a ^ s j 1 T 1 a ^ v a c ,
a ^ s j , o u t = G ( T 1 a ^ s j 1 T 1 a ^ v a c ) + G 1 ( T 1 a ^ s j + 1 T 1 a ^ v a c + ) ,
ξ j l b ^ s l = ξ j l G T 1 ( a ^ s j , o u t cosh ( r ) a ^ s j , o u t e i θ sinh ( r ) ) ,
c ^ s j l = T 2 b ^ s l + 1 T 2 a ^ s j c ^ s j l = T 2 b ^ s l + 1 T 2 a ^ s j ,
ρ = e λ c ^ λ c ^ { e λ α λ α W ( α ) d α } d λ ,
d α s 12 = ( γ s α s 12 + κ α p 12 α s 12 ) d t + γ s d W s 12 ( t ) d α s 21 = ( γ s α s 21 + κ α p 21 α s 21 ) d t + γ s d W s 21 ( t ) d α p 12 = ( γ p α p 12 κ 2 α s 12 2 + ϵ ) d t + γ p d W p 12 ( t ) d α p 21 = ( γ p α p 21 κ 2 α s 21 2 + ϵ ) d t + γ p d W p 21 ( t ) ,
d A s 12 = { A s 12 + ( E A s 12 2 ) A s 12 ξ 21 A s 21 } d τ + g d W s 12 ( τ ) d A s 21 = { A s 21 + ( E A s 21 2 ) A s 21 ξ 12 A s 12 } d τ + g d W s 21 ( τ ) ,
d W s 12 = 1 4 e E γ S d W s 12 ( τ ) + A s 12 d W p 1 ( τ ) d W s 21 = 1 4 e E γ S d W s 21 ( τ ) + A s 21 d W p 2 ( τ ) ,
d A s j l = { A s j l + ( E A s j l 2 ) A s j l m = 1 N n = 1 N ξ m n A s m n } d τ + g d W s j l ( τ ) d W s j l ( τ ) = 1 4 e E γ S d W s j l ( τ ) + A s j l d W p j ( τ ) ( j l , m n ) .
c ^ s 12 j c ^ s 21 k c ^ s 12 l c ^ s 21 m s = α s 12 j α s 21 k α s 12 l α s 21 m W ( { α } ) d α ,

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