Abstract
Recently, spatial photonic Ising machines (SPIM) have been demonstrated to compute the minima of Hamiltonians for largescale spin systems. Here we propose to implement an antiferromagnetic model through optoelectronic correlation computing with SPIM. Also we exploit the gauge transformation which enables encoding the spins and the interaction strengths in a single phaseonly spatial light modulator. With a simple setup, we experimentally show the groundstatesearch acceleration of an antiferromagnetic model with 40000 spins in numberpartitioning problem. Thus such an optoelectronic computing exhibits great programmability and scalability for the practical applications of studying statistical systems and combinatorial optimization problems.
Introduction
The spinglass models are widely used for investigations of interacting systems in both science and engineering^{1,2,3,4,5,6,7,8,9,10,11,12}. In the past decades, the developments in spin machines have generated tremendous interest due to the prospects of solving a large class of NPhard problems by searching the ground states of spin system Hamiltonians^{13}. Optimization problem solvers with remarkable performance have been demonstrated in various systems, e.g., trapped ions^{14,15,16}, atomic and photonic condensates^{17,18}, superconducting circuits^{19}, coupled parametric oscillators^{20,21,22,23,24,25,26,27,28,29,30}, injectionlocked or degenerate cavity lasers^{31,32,33,34}, integrated nanophotonic circuits^{35,36,37,38,39,40}, and polaritons^{41,42,43}.
Recently, like optical analog computations exploring the spatial degrees of freedom^{44,45,46,47,48,49,50,51,52,53,54,55}, the spatial photonic Ising machine (SPIM) has been proposed with reliable largescale Ising spin systems, even up to thousands of spins^{56}. With spatial light modulations, these spatial Ising spin setups benefit from the high speed and parallelism of optical signal processing^{56,57,58,59,60}. Although the modeling of ferromagnetic and spinglass systems has been demonstrated^{61}, how to implement antiferromagnetic models in SPIM has not been proposed yet. In particular, the antiferromagnetic Ising models are important and extensive in research fields like oxide materials^{62} and giant magnetoresistance^{63,64}. Also, the combinatorial optimization problems with antiferromagnetic Ising models have many realworld applications such as multiprocessor scheduling, minimization of circuit size and delay, cryptography, and logistics analysis^{65,66}.
In this work, we propose to implement the antiferromagnetic model through optoelectronic correlation computing. We show that an antiferromagnetic Hamiltonian can be evaluated through the correlation between a distribution function and the measured optical intensity with SPIM, which can neither be implemented through simply adopting the gauge transformation proposed in our previous work^{61}, nor by the targetintensity approach in the original SPIM paper^{56}. We experimentally demonstrate the accelerated searching for optimizing the numberpartitioning problem, where 40,000 spins are connected with random antiferromagnetic interaction strengths. Here we use the algorithm only to show that the SPIM with the measurementfeedback scheme efficiently accelerates the searching process with the reduced Hamiltonian, even when the system is associated with the optical aberrations and the measurement uncertainty. Our results show that the proposed antiferromagnetic model in SPIM can evolve toward the ground state, exhibiting an efficient approach by scalable degrees of freedom in spatial light modulation.
Results
Optoelectronic correlation computing for Mattistype Ising model
We first introduce the gauge transformation for SPIM, which encodes the spins and the interaction strengths in a single phaseonly spatial light modulator (SLM)^{61}. We consider a Mattistype spinglass system with the Ising model Hamiltonian \(H={\sum}_{jh}{J}_{jh}{\sigma }_{j}{\sigma }_{h}\), where the spin configuration is \({{{{{{{\bf{S}}}}}}}}\,{{\mbox{=}}}\,\{{\sigma }_{j}\}\) (j = 1, 2, ⋯ N) and σ_{j} takes binary value of either +1 or −1, representing the spinup or spindown state, respectively. The interaction strengths can be expressed as J_{jh} = ξ_{j}ξ_{h}G(j − h), where G(j − h) is the interaction strength as a function of the distance between two spins with the unit of energy, and the amplitude modulation ξ_{j} is limited as −1 ≤ ξ_{j} ≤ 1. By the gauge transformation shown in Fig. 1a, when rotating each original spin σ_{j} with the angle \({\alpha }_{j}=\arccos {\xi }_{j}\), the new spin vector \({\sigma }_{j}^{\prime}\) is projected on the zaxis to obtain the effective spin \({\sigma }_{j}^{^{\prime} z}={\xi }_{j}{\sigma }_{j}\). As a result, the gauge transformation keeps the Hamiltonian invariant \(H={\sum}_{jh}G(jh){\sigma }_{j}^{^{\prime} z}{\sigma }_{h}^{^{\prime} z}\), while the interactions between the z components of gaugetransformed spins are equal to the strength G(j − h).
The gauge invariance property promises that the experimental implementation only needs a single phaseonly SLM, with uniform illumination by a collimated uniform laser beam (Fig. 1b). This setup circumvents the difficulty of pixel alignment in the previously proposed SPIM^{56,57,58,59}, and therefore greatly improves the system stability and the computing fidelity. Since the spins are loaded through twodimensional spatial modulation, the jth spin is distributed in a square lattice at j = (m, n), where 1 ≤ m ≤ N_{x}, 1 ≤ n ≤ N_{y}. According to ref. ^{61} and Supplementary Note 1, after the gauge transformation, each spin is encoded by a macropixel with phase modulation φ_{m,n} such that \({\sigma }_{j}^{^{\prime} z}=\exp (i{\varphi }_{m,n})\) and
where \({\alpha }_{m,n}=\arccos {\xi }_{m,n}\). Then with lens L1 of the focal length f performing Fourier transformation, we detect the bandlimited intensity distribution I(u) confined within the first diffraction order zone A on the focal plane, and \(I({{{{{{{\bf{u}}}}}}}})={\sum}_{jh}{\sigma }_{j}^{^{\prime} z}{\sigma }_{h}^{^{\prime} z}{e}^{i\frac{2\pi }{f\lambda }({{{{{{{{\bf{x}}}}}}}}}_{j}{{{{{{{{\bf{x}}}}}}}}}_{h})\cdot {{{{{{{\bf{u}}}}}}}}}{{{\mbox{sinc}}}}^{2}(\frac{{{{{{{{\bf{u}}}}}}}}W}{f\lambda })\), where λ is the wavelength, f is the focal length of lens L1, W is the length of each macropixel, x_{j} = Wj is the center position of the jth pixel, u = (u, v) is the spatial coordinate in the focal plane, and \(\,{{\mbox{sinc}}}\,({{{{{{{\bf{u}}}}}}}})=\frac{\sin \pi u}{\pi u}\frac{\sin \pi v}{\pi v}\). Suppose that we preset a distribution function g_{c}(u) and evaluate the correlation function F as
Here
that is, G(k) is the Fourier transformation of \({g}_{{{{{{{{\rm{c}}}}}}}}}\left({{{{{{{\bf{u}}}}}}}}\right){{{\mbox{sinc}}}}^{2}(\frac{W{{{{{{{\bf{u}}}}}}}}}{f\lambda })\). Indeed, Eq. (2) shows that by presetting an appropriate g_{c}, a Mattistype Ising Hamiltonian can be evaluated as
We note that the distribution function g_{c}(u) is distinct from the target intensity I_{T}(u) proposed in ref. ^{56}. Here g_{c}(u) can be an arbitrary real function to guarantee an even function of the interaction strength [c.f. Eq. (3)], which has either positive or negative values, while the target intensity I_{T} always has nonnegative values. In particular, for the antiferromagnetic model, all the interaction strengths J_{jh} < 0. In the case that ξ_{j}s are positive, G(j−h) should be negative to ensure antiferromagnetic interactions between all the spins. It leads that g_{c} must be negative for some values of u, otherwise Eq. (3) shows G > 0 for j = h. Moreover, when g_{c} has both positive and negative values, the Hamiltonian can be evaluated through the correlation function as Eq. (2), while it cannot be implemented by the targetintensity approach in ref. ^{56}. Although Eq. (2) requires the numerical computation, as discussed in the last paragraph of the “Experimental ground state search” in the Results, the optical computation is dominant in our proposed spatial photonic Ising machine.
Numberpartitioning problem with the antiferromagnetic Hamiltonian
The antiferromagnetic model in SPIM provides a computation platform for studying the challenging combinatorial optimization problems. As a demonstration, here we present the groundstatesearch process of a combinatorial optimization problem, the NPhard numberpartitioning problem^{13,67}: One would like to divide a set Ξ = {ξ_{j}}, containing N real numbers (j = 1, 2, ⋯ N), into two subsets Ξ_{1} and Ξ_{2}, such that the difference between the summations of elements in two subsets \({\sum }_{1}={\sum}_{{\xi }_{j}\in {{{\Xi }}}_{1}}{\xi }_{j}\) and \({\sum }_{2}={\sum}_{{\xi }_{j}\in {{{\Xi }}}_{2}}{\xi }_{j}\) is as small as possible. Without loss of generality, we suppose all ξ_{j}s in the set Ξ are real numbers belonging to the range (0, 1]. Specifically, when the number set Ξ has parity symmetry, the spins of such models can be analytically zed. For instance, for the set with an even total number of ξ_{j}s, when ξ_{j} = ξ_{N+1−j}, the spin should be σ_{j} = − σ_{N+1−j} to ensure the equivalence of two subsets. In general, by labeling the elements belonging to two different subsets Ξ_{1} and Ξ_{2} with σ_{j} = 1 and −1, respectively, the optimization is equivalent to minimizing the antiferromagnetic Hamiltonian \(H={({\sum}_{j}{\xi }_{j}{\sigma }_{j})}^{2}={\sum}_{jh}{\xi }_{j}{\xi }_{h}{\sigma }_{j}{\sigma }_{h}\).
To implement such an antiferromagnetic Hamiltonian in SPIM, we explore the gauge transformation to search a spin configuration \({{{{{{{{\bf{S}}}}}}}}}^{\prime}=\{{\sigma }_{j}^{^{\prime} z}\}\) where \({\sigma }_{j}^{^{\prime} z}={\xi }_{j}{\sigma }_{j}\) while keeping the interaction strength between any two spins G(k) = −1. Due to the gauge invariance, the Hamiltonian is \(H={\sum}_{jh}{\sigma }_{j}^{^{\prime} z}{\sigma }_{h}^{^{\prime} z}\) and the optimized value of \(\left{\sum }_{1}{\sum }_{2}\right\) is the total magnetization strength of the gaugetransformed spins \(\leftm^{\prime} \right=\frac{1}{N}\left{\sum}_{j}{\sigma }_{j}^{^{\prime} z}\right\). During the experimental iterations, the spin configuration is updated gradually^{56}, and the system definitely evolves to the ground states, indicating the process of solving the optimization problem.
Experimental ground state search
We experimentally demonstrate the groundstatesearch acceleration with the numberpartitioning problem (see the “Experimental setup” in the Methods). Here each spin is encoded by a macropixel with 2by2 pixels on SLM with the length of W = 16 μm. As the beam size is much larger than that of the array, we assume that light illuminates each macropixel with a uniform amplitude.
In order to evaluate the correlation function F, we first numerically calculate the distribution function g_{c}(u) through Eq. (3). Since \(G\left({{{{{{{\bf{k}}}}}}}}\right)\) are known for specific optimization problems, \({g}_{{{{{{{{\rm{c}}}}}}}}}\left({{{{{{{\bf{u}}}}}}}}\right)\) can be numerically evaluated by the inverse Fourier transform, without numerical errors even for the discrete u. Given the interaction strength between any two spins G(k) = −1, Fig. 2a shows the calculated distribution function g_{c}(u). We note that g_{c}(u) have both positive and negative values, which means such a case cannot be implemented by the targetintensity approach as ref. ^{56}. We also need to calibrate the intensity measurement such that the distribution function g_{c}(u) has the same origin as the intensity distribution I(u). In order to reduce the impact of optical aberrations, we measure the intensity distribution on chargecoupled device (CCD) plane (Fig. 2b) by setting the SLM with uniform phase modulation. Therefore, the origin of I(u) is marked at the maximal intensity through the numerical fitting.
Next as an example, we demonstrate the searching process with the numberpartitioning problem of a set Ξ = {ξ_{j}} with N = 40,000 elements. The set Ξ is randomly generated that each element ξ_{j} is a real number randomly chosen from (0, 1], as presented in Fig. 3a in the form of a 200by200 array. We start with the initial state that all spins are uniformly distributed as σ_{j} = 1 and update the spin configuration for ground state search. Here we utilize the Markov chain Monte Carlo algorithm, where σ_{j}s are tentatively updated during each iteration, and the gaugetransformed spins \({\sigma }_{j}^{^{\prime} z}={\xi }_{j}{\sigma }_{j}\) are encoded on SLM following Eq. (1). Then we measure the intensity I(u) on the CCD and evaluate the system Hamiltonian through the correlation function F as Eq. (2). The updated spin configuration is accepted only when the Hamiltonian decreases.
Figure 3b shows the evolution of the Hamiltonian H and the amplitude of the gaugetransformed magnetization \(\leftm^{\prime} \right\) during the groundstatesearch process. In the experiment, four independent trials are conducted with the initial state that all spins are uniformly distributed. For all the four cases, the Hamiltonian H and the magnetization \(\leftm^{\prime} \right\) decrease rapidly at the beginning of the iteration, because the initial spin configuration strongly deviates from the ground state. As the number of iterations increases, the Hamiltonian tends to be stable, while the \(\leftm^{\prime} \right\) starts to fluctuate. We attribute it to the too weak intensity distribution I(u), which is strongly affected by the noise during the measurement. As a result, the spin configuration may be incorrectly updated with the distribution resulting in a larger \(\leftm^{\prime} \right\). The accuracy can be improved by using a wide dynamicrange detector for intensity measurement, or by adjusting the input light intensity in real time within a suitable range.
Here for clear visualization, corresponding to the red square in Fig. 3a, Fig. 3c and d present a part of the final configurations of the gaugetransformed spins \({{{{{{{{\bf{S}}}}}}}}}^{\prime}=\{{\sigma }_{j}^{^{\prime} z}\}\) and the resulting spin configuration \(\{{\sigma }_{j}\}\), respectively. Overall, for all these four trials, \(\leftm^{\prime} \right\) reaches lower than 1.7 × 10^{−3} within 100 iterations. Thus during the ground state search, the magnetization \(\leftm^{\prime} \right\) decreases by nearly three orders of magnitude, which indicates the validity of the gauge transformation method for antiferromagnetic model.
To evaluate the scalability of the ground state search for numberpartitioning problems, we define the computing fidelity as \(\left\frac{{\sum }_{1}\,\,{\sum }_{2}}{{\sum }_{1}\,+\,{\sum }_{2}}\right\), which is expected to be as small as possible like \(\leftm^{\prime} \right\), and investigate its performance as a function of the size N of the number set. We perform experiments for sizes N varying from 1600 to 40,000. For each N, we perform 10 independent experimental trials with different sets Ξ = {ξ_{j}}, and then the averaged computing fidelity is obtained by measuring the final states after 1000 iterations. From the results in Fig. 4, we can see that fidelity remains within 6.9 × 10^{−3}, demonstrating that the experimental setup works effectively for largescale number sets. The good scalability of SPIM with gauge transformation on numberpartitioning problems is inherited in the parallelism of the optical analog signal processing in the spatial domain.
We note that the stable variation of the fidelity with the number of spins is attributed to the background noise of CCD measurement. As the number of spins N grows, the measurement uncertainty of the Hamiltonian accumulates and the deviation between the subsets \(\left{{{\Sigma }}}_{1}{{{\Sigma }}}_{2}\right\) becomes larger. At the same time, the entire set \(\left{{{\Sigma }}}_{1}+{{{\Sigma }}}_{2}\right\) as the denominator is also increasing, while the fidelity remains stable with N. We add the details of a numerical simulation for the fidelity in Supplementary Note 2, which also agrees well with the experimental results shown in Fig. 4. It is worth noting that by reducing the impact of the background noise, the fidelity can be improved with lownoise CCD devices or dynamic adjustment of the intensity for laser source and the corresponding \({g}_{{{{{{{{\rm{c}}}}}}}}}\left({{{{{{{\bf{u}}}}}}}}\right)\).
We further investigate the system computing acceleration and analyze the fraction of the physical process of light in the total computation. We evaluate how many realnumber operations are required to simulate the physical process of light and the details are given in Supplementary Note 3, with the computational method utilized in ref. ^{68}. As a result, we show that the optical computation is dominant in the proposed SPIM, e.g., for the condition of N = 40,000, the fraction of the total computation by the physical process of light is more than 94%. We note that the optical computation process is ultrafast, highthroughput, and lowpowerconsumption in comparison with the digital one. Therefore, we believe our method can greatly improve speed and efficiency of the simulation for antiferromagnetic Hamiltonian.
Conclusion
We propose to implement antiferromagnetic model in SPIM. By gauge transformation, an antiferromagnetic Hamiltonian can be evaluated through the correlation between the distribution function and the measured optical intensity with SPIM. To improve the processing speed of the system, the ultrafast SLM and CCD at gigahertz rates with the most recent technologies^{69,70} is helpful and practical. Also, the computing accuracy can be improved with a more sensitive CCD camera. We note that our proposed method can be applied to the groundstatesearch process, e.g., adiabatic evolution and simulated annealing algorithms^{71,72}.
In summary, we optically demonstrate the groundstatesearch process of an antiferromagnetic Mattis model with thousands of spins, as well as the numberpartitioning problem. With the improved accuracy resulting from gauge transformation, we successfully reduce the total magnetization strength of the gaugetransformed spins \(\leftm^{\prime} \right\) by nearly three orders of magnitude. Thus for practical applications in modeling statistical systems and studying combinatorial optimization problems, such an optoelectronic computing exhibits great programmability and scalability in largescale systems.
Methods
Experimental setup
As shown in Fig. 1b, a collimated Gaussian beam (wavelength λ = 532 nm) is expanded by two confocal lenses L2 (50 mm focal length) and L3 (500 mm focal length). After expansion, the waist radius of the collimated beam is about 36 mm. Then a polarizer P is used to prepare the incident beam linearly polarized along the long display axis of the SLM (Holoeye PLUTONIR011). The SLM is calibrated through the twoshot method based on generalized spatial differentiator^{73}. Lens L1 with the focal length f = 100 mm performs Fourier transformation, where a CCD beam profiler (Ophir SP620) is used to detect the optical field intensity on the back focal plane.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Code availability
The code that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
The authors acknowledge funding through the National Natural Science Foundation of China (NSFC Grants Nos. 12174340, 91850108, and 61675179), the National Key Research and Development Program of China (Grant No. 2017YFA0205700), the Open Foundation of the State Key Laboratory of Modern Optical Instrumentation, and the Open Research Program of Key Laboratory of 3D Micro/Nano Fabrication and Characterization of Zhejiang Province.
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J.H. and Y.F. carried out numerical simulations, experiments, and data analysis. All the authors contributed to write the paper. Z.R. supervised the project.
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Huang, J., Fang, Y. & Ruan, Z. Antiferromagnetic spatial photonic Ising machine through optoelectronic correlation computing. Commun Phys 4, 242 (2021). https://doi.org/10.1038/s4200502100741x
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DOI: https://doi.org/10.1038/s4200502100741x
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