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Squeezed vacuum state of light is an important concept of quantum optics which has an uncertainty reduction in a specific quadrature compared to the coherent vacuum state. The coherent Ising machines (CIMs) based on the squeezed state are capable of searching the ground state of the Ising model, which can be used to solve combina‑ torial optimization problems and have been experimentally demonstrated to have excellent computational perfor‑ mance. This review introduces the recent progress of a CIM hardware solver based on optical parametric oscillators, including the delayed optical path scheme and a measurement feedback scheme. Also, the basic principles, unique advantages, and potential challenges are described. We expect that the applications of large‑ scale CIM hardware solv‑ ers will have a huge impact on the acceleration of the computation power. The combinatorial optimization problem requires find - 1 Introduction ing an optimal solution that minimizes the loss function In the past decade, the development of integrated elec- among the combinations of all options, which belongs to tronic circuits characterized by Moore’s law has slowed the non-deterministic polynomial time (NP)-hard prob- down, and various forms of non-silicon physical com- lem. There are many combinatorial optimization prob - puting have been highly anticipated [1–4]. Unlike digi- lems in various important domains, such as financial tal computers of von Neumann architecture, computers portfolios [18], drug discovery [19], integrated circuit in non-silicon forms take the task of simulating physical design [20], and Max-Cut problems [21]. Ising models are systems., such as superconducting quantum computers mathematical abstractions of spin glasses that describe [5–8], quantum annealers [9, 10], physical neural net- disordered magnetic systems. The Hamiltonian of Ising works [3], and coherent Ising machines (CIMs) [11–13]. machine could be expressed as H =− J σ σ , Among them, CIMs have been extensively studied due Ising ij i j i,j to their excellent performance on combinatorial optimi- where J denote the coupling coefficients between the i ij zation problems and have been implemented by various th and j th spins and σ ∈ {−1, 1} . The Max-Cut problem physical systems, such as optics [11–13], nanomagnetic- is equivalent to searching for the ground state energy of net arrays [14, 15], single-atom [16], and complementary an Ising model [21] oxide semiconductor devices [17]. Edges = J − J σ σ . ij ij i j (1) i,j i,j *Correspondence: Chuan Wang Therefore, the solution of the ground states of the Ising wangchuan@bnu.edu.cn School of Artificial Intelligence, Beijing Normal University, model is also an NP-hard problem. Beijing 100875, China Quantum computers should be able to simulate quan- Beijing QBoson Quantum Technology Co., Ltd., Beijing 100015, China tum systems more efficiently than classical computers Applied Optics Beijing Area Major Laboratory, Beijing Normal University, Beijing 100875, China [22, 23], but achieving large-scale quantum computers is still challenging at this stage. Many approximation © The Author(s) 2023. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. Lu et al. AAPPS Bulletin (2023) 33:7 Page 2 of 8 algorithms, such as Monte Carlo-based simulated solved. We believe this review can provide a reference annealing, can be applied to solve these combinato- for CIM research in the future. rial optimization problems. However, the solution time increases exponentially on conventional computers 2 Coherent Ising machines based on squeezed depending on the scale of the problem [24]. Therefore, states it is necessary to consider whether physical simulators Photons have been seen as a good medium for comput- can be used to replace traditional computers. There - ing and are often used in the field of quantum computing fore, the CIM hardware solver is the best choice for [27–29] due to their inherent advantages in information implementing a physical simulator to solve combinato- processing, such as highly parallel computing due to no rial optimization problems [25]. interference between different bands [2, 30–34], wide The noise of squeezed states exhibits the squeezing bandwidth from additional orbital angular momentum and anti-squeezing of the quadratures in the orthogo- and polarization [35, 36, 36, 37], and low loss over long nal components through nonlinear effects, respec - distances [38, 39]. tively. The squeezed light that anti-squeezed in-phase Compared with coherent light, the squeezed vacuum noise can help us to realize CIM with a quantum par- state of light has reduced uncertainty in a specific quad - allel search process. As a hardware solver, the experi- rature and can be widely used in quantum information mentally demonstrated excellent performance of CIM processing [40–42], gravitational wave detection 43, in combinatorial optimization has aroused extensive 44], quantum sensing [45–47], and optical parametric research interests [11, 12, 26]. In this brief review, we amplification [48]. Figure 1a [48] shows an experimental focus on the recent progress of squeezed-light-based scheme for generating squeezed light. The experiment CIMs, comparing their differences and the unique produces squeezed noise below the shot noise limit and advantages of their special structures. The main struc - anti-squeezed noise above the shot noise limit, as shown ture of the review is as follows: Section 2 mainly intro- in Fig. 1b [49]. Noise above the shot noise limit helps duces the principle and optical implementation of CIM to achieve the quantum parallel search. degenerate optical parametric oscillator (DOPO)-based In addition, the mutual injection between the optical CIM. Section 3 summarizes some challenges to be pulse networks can simulate the interaction relationship Fig. 1 Experimental scheme for generating squeezed light. a Experimental scheme for generating bright squeezed light and squeezed vacuum with an optical parametric oscillator (OPA). EOM, electro‑ optic modulator; DM, dichroic mirror; SHG, second harmonic generator; HR, high reflector. Figure 1a reproduced from Ref. [48] with permission of the Springer Nature. b Typical vacuum squeezing noise measurement (red) compared to the shot‑noise limit (black). Figure 1b reproduced from Ref. [49] with permission of the American Physical Society. c The anti‑ferromagnetic computing process of CIM (N = 2) and the success rate of finding two degenerate ground states varies with time Lu et al. AAPPS Bulletin (2023) 33:7 Page 3 of 8 in the Ising model. Therefore, squeezed light implemen - nonlinear crystals [60–66], as shown in Fig. 2a. OPO is an tation of CIM based on degenerate optical parametric open dissipative system with a second-order phase tran- oscillator (DOPO) is an effective way. We present a cal - sition at the oscillation threshold, with gain competition culation process of CIM based on a degenerate paramet- among the multiple modes in the laser [67]. Moreover, ric oscillator in anti-ferromagnetic 2-spin by numerical it has been experimentally proved that an Ising Ham- simulation in Fig.1c. The probabilities of both degener - iltonian can be simulated in a mutual injection-locked ate ground states(|↑↓� and|↓↑�) at the beginning of the laser network [68]. A DOPO, under coherent excitation † −iω t iω t d d calculation are both rising under the influence of noise ˆ ˆ , the interacting Hamilto- H = iǫ(a e − a e ) d p †2 2 † until the collective symmetry-breaking stage. Then, the nian is H = i (aˆ aˆ − aˆ aˆ ) , and a ˆ (a ˆ ) is the pump int p p s s s p system randomly selects a ground state, and a quantum- (signal) mode. Considering the linear dissipation of the to-classical crossover occurs. Recently, the experimental pump (signal) mode as γ (γ ) , the Heisenberg-Langevin p s progress of the optical CIM has been listed as follows. equations of the system are [69] daˆ κ p √ 2.1 CIM based on DOPO =ǫ − γ aˆ − aˆ + γ F , p p p 1 dt 2 The Ising model is used to simulate the random process (2) daˆ κ of the phase transition of matter. It is a binary spin sys- † =− γ aˆ − aˆ aˆ + γ F ,. s s p s 2 dt 2 tem, and each bit is randomly flipped under the influence of the potential energy field [50]. Therefore, the artifi - cially analog spin in the Ising machine that simulates the Here, F is the time-dependent noise operators for 1(2) evolution of the Ising model should also have this prop- † ′ ′ ˆ ˆ the pump (signal) field, and F (t)F (t ) = δ δ(t − t ) . i ij erty. Coincidentally, such a binary value occurs when the 3 When γ is large, we can adiabatically eliminate the pump evolution of nonlinear dynamics satisfies x˙ = αx − x . operator a ˆ by daˆ /dt = 0 . Therefore, the Heisenberg- p p Nonlinear optics, which has been widely studied in opti- Langevin equation for the signal mode becomes cal frequency combs [51, 52], optomechanics [53, 54], squeezed light [48, 55, 56], and optical frequency conver- daˆ √ † † 2 † ˆ ˆ sion [57–59]. Wang Zhe et al. proposed that CIM can be = −γ aˆ + Raˆ − K aˆ aˆ + 2K aˆ F + γ F . s s 1 s 2 s s s s dt implemented using optical parametric oscillators (OPO) (3) [21]. Here, R = κǫ/γ is gain coefficient and K = κ /(2γ ) p p Optical parametric oscillation (OPO) describes the is saturation coefficient. Its equivalent master equation is nonlinear optical process that converts a high-energy [69] (2) photon into two low-energy photons through χ Fig. 2 Schematic diagram of the analog spin system based on DOPO. a DOPO process. A high‑ energy photon (indicated in blue) incident on the PPLN crystal is converted into two identical low‑ energy photons (indicated in red). PPLN: Periodically polarized lithium niobate crystal. b Quantum noise distribution of DOPO, b is below the threshold, and c is above the threshold. d The trend of the potential function does not oscillate below the threshold (red) and a bistable potential above the threshold (gray line). The pump energy rises, and the analog spin selects a state with lower potential energy Lu et al. AAPPS Bulletin (2023) 33:7 Page 4 of 8 principle of CIM [73]. The CIM based on DOPO has dρˆ γ † †2 2 =γ aˆ , ρaˆ + H .c. + p aˆ − aˆ , ρˆ s s s s s been proven to have excellent performance in dealing dt 2 (4) with combinatorial optimization problems, such as Max- 2 2 †2 + g aˆ , ρˆaˆ + H .c. . s s Cut, in numerical simulation [21]. Based on DOPO, these schemes successfully imple- where parametric gain p = K /γ , and two-photon ment a CIM, which implements the coupling between absorption coefficient is g = R/γ . Then, we extend the different pulses through a delayed optical path [73–75]. total field density operator ρ using the positive-P repre- The earliest scheme shown in Fig. 3a to implement opti- sentation P(α, β) and get the Fock-Planck equation [70]. cal CIM is based on a femtosecond laser-driven DOPO We ignore the third- and higher-order terms of the Fock- [73]. A fixed-length delay optical path can implement Planck equation and obtain the c-number SDEs of i − th different pulse couplings, and different coupling struc - DOPO by the Ito rules tures can be implemented by different length delay opti - cal paths, as shown in Fig. 3c. Later, the scheme was dα 2 2 2 2 ˆ =− α + pβ − g β α + p − g α f , i i i i,α i i extended to the 16-spin system to solve the more com- dt (5) plex combinatorial optimization problem, as shown in dβ 2 2 2 2 ˆ =− β + pα − g β α + p − g β f . i i i i,β Fig. 3d. T. Inagaki et al. used a 10,000-OPO CIM to simu- i i dt late a 1D anti-ferromagnetic Ising chain. They observed Here, f is real random noise number and the formation of spin domains in Fig. 3e, indicating that i,α(β ) ′ ′ ˆ ˆ f (t)f (t ) = δ δ δ(t − t ) . Under the positive-P the DOPO network can simulate the behavior of low- i,a ij ab j,b temperature Ising spins [75]. representation, α, β is generally a complex number, but In addition, the optical CIM provides a new solution to the parametric oscillations ensure that α, β can be the combinatorial optimization problem and a reference restricted to real numbers [21]. By ignoring the quantum for other ways to implement CIM. noise term, the amplitude evolution of the ith DOPO can be expressed as 2.2 CIM based on the measurement feed‑back dµ 3 Although the optical CIM based on the optical delay path = (p − 1)µ − µ + εJ µ . i ij j (6) dt performs well in some of these optimization problems, j=1,j�=i the fixed-length delay optical path can only map the com - Here,µ is the amplitude of i th DOPO pulse, andε binatorial optimization problem with a single structure is the coupling strength between pulses. The evolu - [74]. However, the combinatorial optimization problem dV (µ) tion of the amplitude of DOPO satisfies µ ˙ = − , is complex, and the CIM that can be connected arbitrar- dµ 2 4 andV (µ) = (1 − p)µ /2 + µ /4 is the potential func- ily becomes a new challenge. tion [71]. The evolution of DOPO is shown in Fig.2b and Therefore, a measurement feedback system (MFB) is c. Below the pump threshold, the pulses are amplified proposed by first measuring the pulse amplitude and then and unamplified in the x−axis andp−axis in a squeezed modulating the pulse through the optical modulator and vacuum state. When the parametric oscillation gain is injecting it into the target pulse at the appropriate time greater than the dissipation of the cavity, the pulse can [11, 12]. CIM with MFB adds a field programmable gate have a stable non-zero amplitude on the x-axis. How- array (FPGA) based on the DOPO fiber-ring cavity and ever, the probability of selecting phase 0 or phaseπ is implements the measurement-calculation-feedback pro- 50% each. This phenomenon is the collective symmetry cess through FPGA [26], as shown in Fig. 4a. McMahon breaking [72]. P. L. et al. first implemented a fully programmable opti - In addition, the relationship between V(x) and the cal CIM that can easily map arbitrarily connected Ising pump threshold is given in Fig.2d. The system is only models, even including the case of asymmetric coupling stable near the amplitude zero point when the pump J = J [11], as shown in Fig. 4b. CIM with MFB is imple- ij ji intensity is below the oscillation threshold. Moreover, mented by taking parts (10% ) of each pulse for measure- by increasing pump intensity to the oscillation thresh- ment and then feedback injection. The machine can find old, the potential energy of the system becomes a para- exact solutions, or good approximate solutions, to vari- digmatic bistable potential. The system chooses positive ous difficult combinatorial optimization problems under amplitude or negative amplitude with equal probability. 100-spins. In Fig. 4c, the scheme has been extended to By adjusting the pump gain, the existence of the coupling 2000-spins to solve more complex combinatorial optimi- term makes the depth of the potential difference. The zation problems by increasing the fiber loop length and system will select the ground state with lower energy, the firing rate of the pulses [12]. The experimental results the optimal spin configuration, which is the operating show that the machine can achieve the same calculation Lu et al. AAPPS Bulletin (2023) 33:7 Page 5 of 8 Fig. 3 Example of CIM based on DOPO. a Schematic diagram of 4‑ OPO composition coherent Ising machine from Ref. [73]. b The interval of the output OPO pulse and the coupling of the injection optical path is realized at c, from Ref. [73]. Figure 3a, b and c reproduced from Ref. [73] with permission of the Springer Nature. d Schematic diagram of 16‑ OPO composition coherent Ising machine reproduced from Ref. [74] with permission of the Springer Nature. e 1D Ising model simulation observed defect densityn and correlation lengthx as a function of normalized 1551 nm pump d 0 amplitude reproduced from Ref. [75] with permission of the Springer Nature Fig. 4 Example of CIM with MFB and applications. a The schematic diagram of CIM with MFB reproduced from Ref. [26] with permission of the American Association for the Advancement of Science. b 100‑spin CIM realizes full connection reproduced from Ref. [11] with permission of the American Association for the Advancement of Science. c 2000‑spin CIM machines for solving combinatorial optimization problems reproduced from Ref. [12] with permission of the American Association for the Advancement of Science. d CIM simulation of 2D Ising problem evolution versus low‑temperature Monte Carlo simulation reproduced from Ref. [76] with permission of the Springer Nature. e CIM to simulate spiking neuron dynamics reproduced from Ref. [77] with permission of the Springer Nature. f CIM simulates the Potts model schematic and solves the clustering and coloring problems reproduced from Ref. [78] with permission of the Springer Nature. Lu et al. AAPPS Bulletin (2023) 33:7 Page 6 of 8 result in 1% of the time of the simulated annealing (SA) Moreover, the CIM-inspired algorithm combined with implemented in the state-of-the-art central processing the FPGA-implemented machine also showed excellent unit (CPU), showing the accelerated calculation effect of computational performance in solving combinatorial optical CIM [12]. The latest research about 100,000-spins optimization problems, even better than optical CIM [87, CIM shows that by optimizing the parallel FPGA, the 88]. Implementing CIM with better computing perfor- computation time of CIM with MFB only depends on the mance is also a challenge for simulating quantum systems time which the light pulse runs in the fiber-ring [79]. using a physical system. CIM with MFB shows computational performance The introduction of nonlinear filtering to CIM reduces advantages over current digital computer [80–82]. R. amplitude heterogeneity [89–91]. Also, it connects CIM Hamerly et al. also compared the performance of CIM with quantum neural networks and quantum perceptrons with MFB and quantum annealer. They found that CIM due to the wide application of nonlinear functions in arti- with MFB is weaker than quantum annealer in the cal- ficial intelligence. The neural network and quantum per - culation of simple graphs (sparse) while having obvious ceptron based on the time multiplexing optical fiber-ring advantages over quantum annealer in more complex of CIM is worth looking forward to [92, 93]. For example, graphs (dense) [26]. Furthermore, to understand the evo- the spiking neural network has been simulated in CIM lution process of CIM searching for the ground state, F. [77]. Meanwhile, CIMs based on nonlinear optics are also Böhm et al. used CIM with MFB to simulate the evolu- expected to promote the research of nonlinear optics due tion process of 2D anti-ferromagnetic Ising networks. to their fully connected properties and system scalability. They found that CIMs behave like low-temperature spin In summary, we review the CIM based on squeezed systems, making them suitable for solving combinato- light in this brief review, summarizing the advantages, rial optimization tasks, as shown in Fig. 4d [76]. Spik- challenges, and potential applications of the existing ing neurons, specialized neurons that process signals in schemes. Combined with the recent rapid development human brain, have been shown to have better comput- of electro-optic modulators and photonic chips [94–96], ing performance than traditional artificial neurons and it is reasonable to expect CIMs with better computing are regarded as the next computational neural network performance. [83, 84]. T. Inagaki et al. studied the dynamic behavior of spiking neurons in neural clusters simulated by CIM. Authors’ contributions They found that it helps to improve the computational C.W. is the lead author in organizing this paper. B.L. and L.L. wrote the draft of performance of CIM, as shown in Fig. 4e [77]. CIM is this paper. All authors contributed to writing and polishing the manuscript. All the authors read and approved the final manuscript. also considered to be able to simulate the Potts model of multi-valued spins, as shown in Fig. 4f [78], and it is Funding shown, through experiments, that clustering and coloring This research was funded by the National Natural Science Foundation of China (Grants Nos. 62131002, 62071448) and the Fundamental Research Funds for problems can be achieved. the Central Universities (BNU). Availability of data and materials 3 Discussion and outlook Not applicable. The optically implemented CIM shows the advantages of physical simulation machines in simulating physical Declarations systems. This novel form of optical computing provides Competing interests a way to develop new computer structures, but there are The authors declare that they have no competing interests. also potential challenges that we need to address. 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AAPPS Bulletin – Springer Journals
Published: Mar 1, 2023
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