Abstract
Optical Ising machines with twobody interactions have shown potential in solving combinatorial optimization problems which are extremely hard to solve with digital computers. Yet, some physical systems cannot be properly described by only twobody interactions. Here, we propose and demonstrate a nonlinear optics approach to emulate Ising machines containing many spins (up to a million in the absence of optical imperfections) and with tailored alltoall two and fourbody interactions. Our approach employs a spatial light modulator to encode and control the spins in the form of the binaryphase values, and emulates the highorder interaction with frequency conversion in a nonlinear crystal. By implementing adaptive feedback, the system can be evolved into effective spin configurations that wellapproximate the groundstates of Ising Hamiltonians with alltoall connected manybody interactions. Our technique could serve as a tool to probe complex, manybody physics and give rise to exciting applications in bigdata optimization, computing, and analytics.
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Introduction
A wide range of modern applications across biology^{1}, medicine^{2}, finance^{3}, and social networks^{4} benefit from efficient processing and optimization of big data with complex structures and correlations. However, many such tasks are nondeterministic polynomialtime (NP) hard, which could take existing supercomputers years to solve^{5}. In this challenge, intense research efforts are underway to pave alternative approaches for computing and information processing. Among them, Ising machines have been shown to offer viable solutions for important NP hard problems such as cut with maximum number of edges (MAXCUT) in a graph^{6}, protein folding^{7}, and traveling salesman^{8}, among others^{9,10,11,12,13,14,15}. To this end, a variety of Ising machines have been demonstrated in (effective) spin systems of trapped atoms^{16}, polariton condensates^{17}, and superconducting circuits^{18}. Some other artificial Ising machines have also been implemented using coupled oscillators^{19,20,21}, nanophotonic waveguides^{22,23}, randomly coupled lasers^{24,25}, and timemultiplexed optical parametric oscillators^{26,27}.
For the tasks of finding the ground states of manybody Hamiltonians, photonic systems enjoy the distinct advantages of high connectivity and speed^{28,29,30}. For example, a fast coherent Ising machine can be realized in a looped optical parametric oscillator with temporally multiplexed pulses^{31,32}, albeit with limited spin numbers^{31} or relying on photodetection and electronic feedback to emulate the spin–spin interaction^{33,34}. In contrast, a linearoptical Ising machine based on spatial light modulation was shown to subtend about 80,000 spins by coding them as the binary phases of pixels on a spatial light modulator (SLM)^{35}. The farfield optical power of a modulated beam gives the expected energy of spin–spin interaction. The relatively simple setup, yet high connectivity and scalability make this approach attractive to Ising machines with fully connected twobody interaction.
Yet, there are physical systems and numeric models whose dynamics cannot be fully captured by twobody interactions, and proper descriptions of multibody interaction are required^{36,37,38,39,40}. Some of the NP hard problems are rooted in the multibody interactions present in the spin system. For example, the Boolean satisfiability (kSAT) problem is NP hard when k ≥ 3, where k prescribes the order of interactions or constrains^{41}. This poses a significant computational challenge, whose complexity and volume exceed by far that of Ising problems with only twobody interaction, even for a moderate number of spins^{42,43}. Although a small class of manybody interaction can be decomposed onto a series of twobody interactions via some recursive or algebraic means^{44,45,46}, they often subject to strict constraints^{47,48} or require tedious error corrections^{49,50}. For simulating complex systems and processing data with highorder correlation, suitable Ising machines remain desirable that support simultaneously high connectivity, multibody interaction, and a large number of spins.
In this study, we propose and experimentally demonstrate such an Ising machine hosting adjustable twobody interaction, fourbody interaction, and alltoall connections over a large number of spins. It realizes the spins as the binary phases of wavelets in a coherent laser beam, and implements effective multibody interaction through nonlinear frequency conversion. Using SLM’s (or equivalently, digital micromirror devices (DMDs)), one million spins are possible at present, although optical imperfections must be taken into account and could limit the actual accessible spin number in practice. The fully connected twobody interaction is emulated with the optical power of the modulated light in the Fourier plane. The fourbody interaction, also fully connected, is realized effectively by passing the modulated light through a lithiumniobate crystal in the Fourier plane for second harmonic (SH) generation. By simultaneously measuring the optical powers of the modulated light and its SH coupling into a fiber, complex Hamiltonians with alltoall connected twobody and fourbody interactions can be emulated over a large number of spins. Through feedback control, the system can be evolved into the vicinity of the ground state of its Hamiltonian, exhibiting ferromagnetic, paramagnetic, spin glass, and other novel nonlinear susceptibility phase transitions.
The present Ising emulator could pave a pathway to otherwise inaccessible territories of big data analytics and quantum simulation^{39,40}. The highorder, manybody interaction can also serve as powerful activation functions for all optical machine learning^{51,52}. For example, an immediate application is to use this machine as the qstate Potts model with twobody and fourbody interactions on a square lattice^{53}. Finally, while the current setup uses SH generation, even richer physics and controllability can be achieved by using other nonlinear optical processing like sumfrequency generation^{54}, and fourwave mixing^{55}, where other types of spin interaction and connection can be engineered.
Results
Theoretical analysis
The basic idea of the present Ising machine is illustrated in Fig. 1, which emulates chemical potential, twobody interaction, and fourbody interaction over a large number of spins. Each spin is encoded as the binary phase of a pixel on a SLM. The total chemical potential energy is represented by the weighted sum of all spins. To realize the interactions, a coherent Gaussian pump beam is reflected off the SLM, focused using a Fourier lens to a nonlinear crystal for SH generation. The resulting beams at the original fundamental wavelength and the new SH wavelength are then separated at a dichroic mirror and captured by optical fibers. The pump power in the fiber is then measured to emulate the energy associated with spin–spin interaction, and that of the SH beam is to capture the fourbody interaction among spins. Incorporating all three, a Hamiltonian describing the chemical potential, twobody interaction, and fourbody interaction can be effectively constructed.
To derive the effective Hamiltonian, we consider a Gaussian input pump beam of wavelength λ_{p}, peak amplitude E_{0}, and beam waist w_{p}. It shines a SLM whose phase mask consists of pixels (m, n) centered around \(({x}_{m}^{\prime},{y}_{n}^{\prime})\), each giving 0 or π phase modulation. The transverse electric field immediately after the SLM is approximately^{35}
Here \({\xi }_{mn}={E}_{0}\exp \left[({x}_{m}^{^{\prime} 2}+{y}_{n}^{^{\prime} 2})/{{\rm{w}}}_{{\rm{p}}}^{2}\right]\) is the amplitude at pixel (m, n), Π is the unity rectangular function, and σ_{mn} = ±1 for the 0/π binary phase modulation.
The electric field is then transformed using a Fourier lens of focal length F and coupled into a periodicpoled lithium niobate (PPLN) crystal of length L, so that it reads at the center of the crystal (z = 0):
Here, \({\eta }_{mn}=\exp \left[2\pi i(x{x}_{m}^{\prime}+y{y}_{n}^{\prime})/{\lambda }_{{\rm{p}}}F\right]\), κ_{p} = (2πn_{p})/λ_{p}, and n_{p} is the refractive index of the pump in the PPLN crystal. For simplification, we introduce contracted notations ξ_{i} and σ_{i}, with ξ_{i=m+(n−1)N} ≡ ξ_{mn}, and σ_{i=m+(n−1)N} ≡ σ_{mn}, with i = 1, 2, . . . N^{2} to index the N × N spins (pixels). In our setup, only nearaxis light is fiber coupled and measured, so that sinc(axπ/λ_{p}F), sinc(ayπ/λ_{p}F) ≈ 1, giving
In the PPLN crystal, the pump creates its SH, whose transverse electricfield at the output reads
where χ^{(2)} is the secondorder nonlinear susceptibility, κ_{h} = (2πn_{h})/λ_{h} is the wave number, n_{h} is refractive index, and ω_{h} is the angular frequency of the SH wave. The nonlinear dynamics inside the crystal is shown in the Supplementary Note 1. At the crystal output, the pump and SH waves are each coupled into a singlemode fiber for detection, whose optical power is given by
where
are the normalized backpropagated fiber modes of beam waist \({w}_{{\rm{f}}}^{{\rm{p}}}\) and \({w}_{{\rm{f}}}^{{\rm{h}}}\) for the pump and SH waves, respectively.
Substituting Eqs. (3), (4), and (6) in Eq. (5), the detected power for the pump and SH waves is given in the form of
and
where J_{ij} and J_{ijsr} prescribe the strength of the twobody and fourbody interactions, respectively. As an example, Supplementary Note 2 presents the analytic results of J_{ij} and J_{ijsr} under the approximation of Eq. (4).
With Eqs. (7) and (8), we can define a single parameter E to characterize the energy of the whole system, as
where \(C=\mathop{\sum }\nolimits_{i = 1}^{{N}^{2}}{\mu }_{i}{\sigma }_{i}\) is the weighted sum of spins that represents their total chemical energy, with the local chemical potential μ_{i} ∈ [−1, 1]. In this equation, α, β, and γ are free parameters defining the contribution of the chemical potential, twobody, and fourbody interaction energy, respectively, to the total energy.
The total energy E can be minimized by optimizing the binary phase mask on the SLM through adaptive feedback control. This is equivalent to finding the groundstate solutions of the effective Hamiltonian
where \({\hat{H}}_{1}\) is the Hamiltonian describing the chemical potential, \({\hat{H}}_{2}\) for the twobody, and \({\hat{H}}_{4}\) for the fourbody interaction, respectively, with
and
The twobody and fourbody interactions can be tailored by modulating the input pump wave, varying the fiber optical modes, and modifying the phase matching conditions for the nonlinear process. As an example, Fig. 2 considers a toy system of four spins, with the input pump partially blocked and the output pump and SH light across the remaining pixels coupled equally into the fiber mode. We can use a DMD, which can flexibly control no light and bright light regions in the input pump wave. The resulting interaction coefficients J_{ij} and J_{ijsr} are shown in Fig. 2a, b. For α = β = γ = −1, the system’s ground state is simply with all spin up. For all the possible spin configurations, we plot the probability distribution function of the energy (E) and magnetization, defined as \(M=\mathop{\sum }\nolimits_{i = 1}^{{N}^{2}}{\sigma }_{i}/{N}^{2}\), in Fig. 2c, d, respectively. As shown, the manybody interactions can be tailored into complex forms by simple linear optics operations.
Experimental setup
The experimental setup for the present nonlinear optical Ising machine is shown in Fig. 3. We use an optical pulse train at 1551.5 nm as the pump. Each pulse has 5 ps fullwidth at halfmaximum (FWHM) and 50 MHz repetition rate. The pump’s average power is about 40 mW and its pulse energy is ~0.8 nJ. We use a portion of the input pump beam to monitor the power fluctuation which is <5% during a whole day operation. The transverse FWHM of the pump beam is 2.6 mm incident on the SLM (Santec SLM100, 1440 × 1050 pixels, pixel pitch 10.4 × 10.4 μm) at a 50° incidence angle^{54,56}. Initially, a random binary phase mask with phase value 0 or π is uploaded onto the SLM. Half waveplate is used to ensure the vertically polarized light parallel to the crystal’s optical axis. A lens focused the beams (focus length F = 200 mm) inside a temperaturestabilized PPLN crystal with a poling period of 19.36 μm (5 mol.% MgOdoped PPLN, 10 mm length, 3 mm width, and 1 mm height from HC Photonics) for SH generation. The pump beam waist inside the crystal is 45 μm. The output is then filtered with a dichroic mirror to separate the SH and pump light^{57}. Each arm is coupled into a singlemode fiber (SMF28) using a fiber collimator consisting of aspheric lens (Thorlabs C220TMDC and A375TMB) and then detected by power meters (Thorlabs PM100D with sensors S132C and S130C). The measurement results are sent to a computer through a MATLAB interface for feedback control, which updates the phase mask on the SLM to find the ground state of the customized Ising Hamiltonian.
To visualize the evaluation of the optimization process, we can split the SH and pump light in two parts using flipable beam splitters (BS). Lenses placed at one arm of the flipable BS are used to image the pump on a nearinfrared (NIR) camera (FINDRSCOPE Model No. 85700 with a pixel resolution of 17.6 μm) and the SH light on a charged coupled device (CCD) camera (Canon Rebel T6 with a pixel pitch of 4.3 μm). The present optical Ising machine is susceptible to the noises from a laser power fluctuation, beam misalignment, phase mismatching, etc. Those noises can be minimized by real time monitoring and periodically recalibrating the input laser power, SH generation efficiency, temperature variation, optical alignment, fiber coupling efficiency, and so on.
Feedbackcontrol optimization
As shown in Eq. (11), the chemical potential of each spin is flexibly defined by μ_{i} and their collective contribution to the total energy is controlled by α. This provides the knob of studying the magnetization under a variety of local and global singlespin parameters. For this paper, however, we will focus on the manybody interaction and consider only μ_{i} = 1 in all of the following results. Meanwhile, we will leave fine tuning two and fourbody interaction to our future work, but only control each’s aggregated contribution to the total energy by varying β and γ, respectively. One can tailor these interactions by amplitude modulation of the input Gaussian beam using, e.g., SLMs or DMDs, varying the numerical aperture of the output fiber couplers, slightly changing the quasiphase matching conditions, and so on.
To find the ground states of the total Hamiltonian, the SLM’s initial phase mask is prepared in small clusters with randomly chosen 0 or π phases. The resulting pump and SH waves are coupled separately into singlemode fibers, whose optical power is measured for feedback control. To minimize the total energy, we adaptively flip the spins within a randomly chosen cluster, following the standard Monte Carlo approach^{58}. A flow chart of this procedure is shown in Fig. 4, where the spin flipping during each iteration is accepted or rejected according to a Boltzmann’s probability function \(P=\exp [U/\tau ]\), with U = E_{new} − E_{old} being the change in energy and τ the thermal energy. To avoid trapping into a local minimum, we vary both the cluster size s and τ during iterations. It is noteworthy that this algorithm is not necessarily the most efficient, but nevertheless adequate for our current demonstrations as the first case study of this Ising platform. A machinelearningbased Monte Carlo method could be utilized in the future to speed up this optimization^{59}.
Figure 5 illustrates the process of optimization for 800 × 800 spins with eight initial random phase masks. Figure 5a shows how the optical pump power is increased; thus, the decrease of total energy E to approach the ground state of the system. With α = 0, there is spontaneous symmetry breaking, as the system energy remains unchanged if all spins are flipped. As such, the feedback control will optimize the spins toward either positive or negative magnetic states with equal probability^{35}. This is clear in Fig. 5b, where the magnetization trends both ways. For all initial phase trials, the absolute value of average magnetization can reached to ~ 0.75.
In Fig. 6, we show the two sets of measurements (a) and (b) for 400 × 400 spins and (c) and (d) for 800 × 800 spins, respectively, with purely fourbody interactions, i.e., α = 0 and β = 0. Figure 6a, c plot the SH power evolution for γ = 1, for which the system ground states correspond to the minimum SH power. In both figures, different initial spins are optimized to give similar minimum SH power, which indicates the robustness of our optimization method. For γ = 1, the system evolves into a paramagneticlike state that minimizes the SH power in the fiber. In Fig. 6c, the values are close to the minimum detectable power of our optical sensor (~5 nW). Because of a smaller pixel size in Fig. 6c, the spin disorder is stronger to give lower SH power in Fig. 6a. In opposite, Fig. 6b, d are for γ = −1, where the ground states are obtained at the highest SH power. In this case, the system exhibits a ferromagneticlike behavior. For 400 × 400 and 800 × 800 spins, the optimization leads to similar maximum SH power despite different initial spin conditions. The convergence is slower for the latter case, as there are four times more spins to be optimized. Overall, our system can reliably and efficiently evolve into the vicinity of its ground state.
To further understand the optimization mechanism, we take images of the pump and SH beams at the crystal output by splitting them using flipping beam splitters, as shown in Fig. 3. Through 4F systems, the pump is imaged on a NIR camera, and the SH on a CCD camera. Figure 7a, b show the pump and SH images, respectively, under different iteration numbers as they minimize the energy of 800 × 800 spin system with α = 0, β = 1, and γ = 1. As shown, both light beams become scattering and show speckle patterns as the optimization goes. The resulting fibercoupled SH and pump power is shown in Fig. 7c. Both decrease with the iteration numbers, dropping by orders of magnitude to minimum values after 150 iterations. It is noteworthy that the final SH power level is very close to the detection level of the sensors, which prohibits further reduction via the present feedback control.
As seen in Eq. (10), the twobody and fourbody interaction energies are dependent only on the relative alignment of the spins, but not on each’s absolute orientation. Thus, spontaneous symmetry breaking could occur during the optimization, leading to bifurcation^{35}. To avoid this symmetry breaking, one could set a nonzero α to control the convergence direction of spin optimization. As an example, in Fig. 8, we compare the results with α = 1 in (a, b) and α = −1 in (c, d), both with β = −0.5 and γ = −1. As shown, α can indeed dictate the spin alignment to result in either positive or negative magnetization states. In both cases, starting from a rather randomized phase mask shown in the inset of Fig. 8a, c, the spins become relatively aligned to increase the SH and pump power, but magnetization can have positive or negative orientation as can be seen Fig. 8b, d. The inset of Fig. 8b, d shows the final phase mask of the optimum solution, where black and white colors represents the positive and negative orientation of the spins, respectively. Similar results are shown in Supplementary Note 3 with other input phase masks. It is noteworthy that in our setup, the computing tasks such as matrix multiplication, Fourier transformation, and fourbody interactions, are realized through linear optics, nonlinear optics, and optical detection. The relaxation of the spin system to its ground state, on the other hand, is achieved through feedback control. To speed it up, fieldprogrammable gate arrays (FPGAs) or fieldprogrammable photonic gate arrays (FPPGAs) can be employed in the future^{60,61,62}.
We last consider the cases where twobody and fourbody interactions contribute oppositely to the total energy of the system. For instance, the twobody interaction can be attractive but fourbody be repulsive, or vice versa. Such systems can be conveniently configured by defining the prefactors β and γ in Eq. (10). The optimization will maximize one, while minimizing the other. As an example, Fig. 9 plots the optimization trajectory for opposite twobody and fourbody interactions. In Fig. 9a, β = 1, and γ = −1, so that the system energy is maximized by reducing the pump power, while increasing the SH power. Its opposite configuration is in Fig. 9b, where β = −1, and γ = 1 so that the same optimization increases the pump power, while reducing the SH power. This example suggests that the nature of twobody and fourbody interactions can be conveniently maneuvered in our Ising machine, which makes it versatile for simulating various systems in solidstate physics^{63,64,65}, chemical engineering^{66}, and so on. This Ising machine is efficient, because it calculates the manybody interaction energy during a single pass through a nonlinear crystal, realizing simultaneously matrix multiplication, Fourier transformation, etc., of largesize data.
Discussion
Our nonlinear optical Ising machine can be applied to some of the NP hard problems involving multibody interactions, such as kSAT problems^{41}, and spinglass models^{67}. Yet, the main limitation with our current setup comes from the slow response of the SLM’s. To establish a practical advantage, we will need to replace the SLM’s with fast digital micromirrors controlled by FPGA, use alloptical feedback, or design hybrid opticalFPPGA hardwares^{60,61,62}. Meanwhile, although the present optical Ising machine only supports two and fourbody interactions, other processes of nonlinear optics, such as sum frequency, fourwave mixing, and highorder harmonic generation, can be employed to realize even higherorder interactions, whose computational complexity and intensity could quickly grow beyond the capability of existing computing platforms.
Using spatial phase modulation and SH generation, we have constructed Ising emulators with alltoall connections and tailored chemical potential, twobody interaction, and fourbody interaction. Their groundstate solutions can be effectively and reliably approximated by adaptive feedback control, whose speed is currently limited by the processing time of the SLM. A significant speedup is achievable by using ferroelectric liquidcrystal based SLMs or programmable plasmonic phase modulators^{68,69}. At the present, the maximum number of accessible pixels is about 1 million and can be further increased by using modulators with more pixels or combining multiple modulators. The total supported spin number, on the other hand, is less in practice, due to the optical perfections. Although this study considers only SH generation, it can be straightforwardly extended to other nonlinear processes, such as sumfrequency generation, differencefrequency generation, fourwave mixing, for other interesting Ising machines^{54,55,56,57,70}. Such nonlinear optical realizations of Ising machines could contribute as supplements for big data optimization and analyses that remain challenging for classical supercomputers or forthcoming quantum machines but with a limited number of qubits.
Data availability
The data that support the plots and other findings of this paper are available from the corresponding author upon a reasonable request.
Code availability
The code is available from the corresponding author upon a reasonable request.
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Acknowledgements
We thank Stephanie Marcua and Prajnesh Kumar for helpful discussion. This research was supported in part by National Science Foundation (Award numbers 1806523 and 1842680).
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Y.H. conceptualized and supervised the project. S.K. and H.Z. carried out the experiments and analyzed the data. All authors contributed to the scientific discussion and writing of this paper.
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Kumar, S., Zhang, H. & Huang, YP. Largescale Ising emulation with four body interaction and alltoall connections. Commun Phys 3, 108 (2020). https://doi.org/10.1038/s4200502003765
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DOI: https://doi.org/10.1038/s4200502003765
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