Noise-enhanced spatial-photonic Ising machine

1 Introduction

Solving large combinatorial problems is crucial for widespread applications in fields such as artificial intelligence, cryptography, biophysics, and complex networks. However, finding the optimal solution to many of these tasks causes the required resources to grow exponentially with the problem size, a reason why such problems are considered as computationally intractable for traditional computing architectures [1]. A promising approach to efficiently solve these problems is to recast them in terms of an Ising model [2], [3], which describes a system of classical interacting spins, and searching its ground state by an artificial network of spins evolving according to an Ising Hamiltonian. Ising machines are physical platforms made of electronic or photonic elements that can be programmed to encode Ising problems with known coupling values, and the ground state obtained after the system's relaxation provides the optimal solution. They have been realized in a variety of quantum and classical systems including cold atoms [4], [5], single photons [6], [7], superconducting [8], [9] and magnetic junctions [10], electromechanical [11] and CMOS circuits [12], polariton and photon condensates [13], [14], or lasers and nonlinear waves [15], [16], [17], but with practical difficulties in scalability, connectivity, or in engineering the spin interaction.

Photonic Ising machines encode the spin state in the phase or amplitude of the optical field. Realized photonically, such Ising machines hold the prospect of processing data in parallel at high speed through active optical components and hence be much faster than those based on other encoding schemes [18]. Various prototypes have recently been realized with sizes spanning from few to thousands of spins. In the class of photonic optimizers known as coherent Ising machines (CIMs), the nonlinear dynamics of time-multiplexed optical parametric oscillators [19], [20], [21], [22], [23], fiber lasers [24], or simple opto-electronic oscillators [25], is exploited to solve NP-hard optimization problems with notable performance [26]. CIMs are dissipative optical networks in which the ground-state search is performed in reverse direction by slowly raising the gain, according to a general non-equilibrium bifurcation mechanism that currently inspires novel algorithms and settings [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37]. On the other hand, optimization platforms based on waveguides circuits [38], [39] and integrated nanophotonic processors [40], [41], [42], [43] operate as optical recurrent neural networks [44] converging to Ising energy minima.

Spatial-photonic Ising machines are a different class of optical devices for Ising problems that have been demonstrated very recently [45]. They make use of spatial light modulation for encoding an unprecedented number of spins [46] and the programmed Hamiltonian is optically evaluated by measuring the intensity distribution after propagation in free-space [45] or through nonlinear media [47]. These devices take advantage of optical vector-matrix multiplications and by the large pixel density of spatial light modulators (SLMs), thus enabling the implementation of large-scale neuromorphic computing , [48], [49], [50], [51], [52], [53].

Noise is an unavoidable ingredient in any hardware. In CIMs it represents one of the main error sources and can induce dynamics beyond the regime of Ising spins [55]. On the contrary, in recurrent algorithms and artificial neural networks for Ising problems, noise furnishes a finite effective temperature, and it is expected to facilitate and speed up convergence to the ground state if noise amplitude is adequately leveraged [40], [54]. The effect has been recently observed in a platform with few photonic spins and various competing interactions [41]. Noise-tolerant settings are especially important when scaling the device to solve systems with many units. Nevertheless, the impact of noise in large-scale photonic Ising machines remains mainly unexplored.

In the present article, we investigate the effect of experimental noise on a spatial-photonic Ising machine with hundreds of spins, proving the existence of an optimal noise level. The precise impact of noise depends on each problem's particular features. Specifically, we found that it enhances the machine's success probability on problems with both positive and negative interactions, while it is detrimental for models having only positive couplings. By providing a mechanism to escape from local minima, noise represents an additional parameter with beneficial properties for our optical computing device. Our findings demonstrate noise as a valuable resource in large-scale photonic computing.

2 Ising machine by spatial light modulation

In our Ising machine the spin variables are encoded on a coherent laser wavefront via binary values of the optical phase and processed by spatial light modulation [45]. As schematically illustrated in Figure 1(a), our optical setting employs an optical path in which an SLM encodes spins σi=±1 by 0, π phase delays over an amplitude-modulated beam. Linear interaction between spins occurs by interference on the detection plane and its strength is controlled by spatial modulation of the input intensity. The optical machine works via a measurement and feedback scheme. Once initialized to a given problem, feedback from the detected intensity allows the phase distribution on the SLM to converge towards a state minimizing an Ising Hamiltonian

with couplings Jij=ξiξjI˜T, where ξi is the amplitude illuminating the ith spin [45]. I˜T is the Fourier transform of a pre-determined target image, and the difference between IT and the image detected on the camera is the cost function. At each machine iteration, we measure the intensity on the CCD modes [Figure 1(b)] and the spins are updated in order to minimize the cost function. Importantly, and other than for CIMs [20], [21], [25], our machine avoids electronically computing the energy as well as the field acting on each spin.

Figure 1:

Photonic Ising machine by spatial light modulation. (a) The spins σi=±1 are encoded by a spatial light modulator (SLM) into binary optical phases ϕ=0, π in separated spatial points of the optical wavefront. Intensity modulation is employed to set the spin interaction via the amplitude distribution ξi. Recurrent feedback from the far-field camera allows evolution of the phase configuration towards the Ising ground state. (b) Intensity detected on the camera (CCD) modes for a random spin state and for the ground state of a ferromagnetic Ising model. Insets are illustrations of the corresponding spin state in energy landscape. Experimental fluctuations allow spontaneous changes in the spin configuration.

Due to the intrinsic noise of each experimental setup, the machine always behaves as coupled to a thermal bath. Noise therefore provides an effective temperature for the final spin ground state, as reported in Ref. [45]. Sources of noise come from intensity discretization and processing in each CCD mode, as well as from the imperfect spatial light modulation. As detailed below, we here control the noise level by means of a tunable error rate in the machine's measurement and feedback scheme.

2.2 Optimization with spontaneous noise

We first quantify the solutions found by our Ising machine for ρ=0, a condition analogous to previously reported experiments [45]. We optically implement two different classes of Ising systems: mean-field and Mattis spin glasses. In a mean-field Ising model, also known as infinite-range Ising model, the spins are all-to-all coupled with the same positive interaction strength. We realize such coupling using a plane wave of constant amplitude ξi=E0 and maximizing the intensity detected on a single CCD mode, such that I˜T=c, being c an arbitrary constant and J¯=cE02. In Mattis spin glasses the pairwise interaction is given as a product of two independent variables Jij∝ξiξj [56], [57]. In this case, since the couplings are both positive and negative, a minimal amount of noise introduces frustration [57]. In our spatial optical setting, pairs of negatively-coupled spins correspond to points of the optical field that give destructive interference on a fixed CCD mode [see bottom panel in Figure 2(d)]. Sparsity is implemented all-optically by decoupling a random subset of spins via amplitude modulation (i. e., ξi=0 for blocks of spins). Since the interaction matrix is given in any case as a product of separate amplitude values, the sparse Ising Hamiltonian still maintains the form of a Mattis model.

Figure 2:

Ising machine's ground states without noise control. (a–c) Results for ferromagnetic and (d–f) frustrated models with 100 spins (ρ=0). Problem graphs for (a) mean-field (infinite-range) Ising model and (d) Mattis spin glass with positive and negative couplings. For clarity, only links starting from the two black nodes are drawn. Insets show the corresponding target images IT. (b, e) Contour maps showing the evolution of the spin configuration during a single run of the machine (green regions are domain walls). (c, f) Normalized histograms of the ground-state energy obtained for the instances in (a) and (d), respectively. The red lines in (c) and (f) indicate the minimum energy known from exact solution of the models.

Optical ground states found by the device for a mean-field system of N=100 spins are reported in Figure 2(a–c). The evolution of a spin configuration initialized to a random distribution is shown in Figure 2(b) for a single machine run; after thousand iterations, a low-temperature ferromagnetic state is measured. The energy probability distribution of these ground states is reported in Figure 2(c); a peak close to the known minimum energy value [red line in Figure 2(c)] indicates that ground states of the Ising Hamiltonian are successfully found. In Figure 2(d) we show the graph for a Mattis spin glass instance along with the corresponding target image. In agreement with the programmed Hamiltonian [57], the spin evolution in Figure 2(e) exhibits the formation of two domains with opposite magnetization and equal size. However, from the ground-state energies in Figure 2(f), we observe a reduced ability of the machine in solving such frustrated models. As we demonstrate hereafter, increasing the noise level is an effective way to improve the machine performance on problems affected by trapping in local energy minima.

3 Effect of the noise level on the Ising machine performance

To introduce controllable noise, we exploit the detection process previously described. The Ising machine is made to operate under different noise levels. To quantify the performance when the setup is initialized to different parameters, we use two distinct and complementary quantities: the success probability ps and the Hamming distance h. To evaluate the success of the computation, we consider the correlation between the measured spin configuration and the known optimal solution, being C = ±1 for the zero-temperature spin system in the lowest energy state [57]. A machine run is defined as successful if its final state gives |C|>a, being a any fixed accuracy; the success probability ps is the fraction of repeated experiments with random initial conditions that converged successfully. The Hamming distance is a metric used in information theory. Here, h indicates the number of units (spins) that need to be inverted to reach the minimum energy configuration [9]. We would like to point out that using this quantity to characterize the quality of Ising machines is significantly more accurate than the ground-state energy. In fact, spin configurations very far from the known solution can still have energies comparable with the optimal one.

Figure 3 illustrates the performance of the spatial-photonic Ising machine as we vary the noise level. For the infinite-range Ising model, we found a success probability that decays as noise increases, a behavior independent of the selected accuracy [Figure 3(a)]. The measured h in Figure 3(b) has a growing trend, thus indicating that for emulating such Hamiltonians the best performance is obtained for minimum noise. A completely different picture emerges when solving frustrated Ising models. As shown in Figure 3(c) for dense Mattis spin glass instances, the low success probability observed under spontaneous noise rapidly increases as additional fluctuations are introduced. ps has a maximum value and then decays to zero for large noise levels. Moreover, the mean Hamming distance as a function of the noise level exhibits a pronounced minimum [Figure 3(d)], which indicates an optimal noise level, ρopt≈0.05. A similar behavior is found on sparse Mattis spin glasses, as reported in Figure 3(e–f), with ρopt≈0.07. This demonstrates that an optimal noise level promotes the exploration of energy landscapes with many minima during optimization on the optical platform.

Figure 3:

Optimal noise level in spatial-photonic Ising machines. (a, c, e) Success probability and (b, d, f) mean Hamming distance varying the noise level for various N = 100 Ising models: (a–b) mean-field Ising model, (c–d) dense and (e–f) sparse Mattis spin glasses. The problem graphs are inset. Different colors in (a, c, e) indicate data obtained at the specified accuracy level. Shaded regions indicate statistical error intervals. The existence of an optimal noise level for frustrated models is signaled by a minimum in the Hamming distance and a corresponding maximum in the success probability [dotted line in (d) and (f)].

3.1 Scaling of the optimal noise level

We investigate how the noise-enhanced machine operation depends on the system size. While keeping the SLM's active area constant, we vary the total number of spins and, for each system size N, we perform the experiments at different noise levels. The results obtained on dense Mattis spin glasses are shown in Figure 4. We observe an optimal noise level that significantly depends on the spin number, with values that grow and saturate as the system size increases. For small-scale systems (N = 16) additional noise yields only limited advantages due to finite size effects and the small number of frustrated configurations. However, a constant optimal level ρopt≈0.07 enhances the optimization for large scale Hamiltonians. This specific value is not a general recipe to improve the Ising machine, but guarantees improved performance on specific problem graphs in our setup. Ising instances with distinct properties (graph type, connection density etc.) would have diverse optimal levels, having landscapes with different features. These findings establish the noise level as a hyperparameter of the photonic computing device.

Figure 4:

Scaling properties. Optimal noise level (blue circles) as a function of the number of spins for dense Mattis spin glasses (inset graph). Purple dots indicate the measured Hamming fraction (h/N) for various system sizes.

Another important fact is that residual errors do not increase rapidly with N. To prove this property, Figure 4 shows the Hamming fraction h/N (i. e., the mean Hamming distance normalized to the spin number) as a function of N. Residual errors shows a sublinear increase with minor oscillations as N varies. This preliminary evidence suggest a smooth dependence of the machine accuracy on the system size, although larger systems naturally require more machine iterations to converge to their ground state. According to thermodynamic considerations, this is consistent with the presence of an effective thermal bath for the spin state. It suggests that lowering the effective temperature of the machine by improving the experimental setup can lead to larger ground-state probabilities even for large-scale models.

4 Conclusions

Understanding the role of noise in optical Ising machines and neuromorphic devices is crucial for their application to large-scale computational tasks. In particular, noise-tolerant settings are attractive candidates for developing unconventional computing architectures. We have reported the first evidence that spatial-photonic Ising machines can take advantage of noise in solving large-scale optimization problems. Devices based on spatial light modulation are scalable to larger sizes and can potentially host systems consisting of millions of spins. In particular, our computing setting can exploit the potential of nanophotonic light-modulation devices. Tunable dielectric metasurfaces, which allow to control both phase and polarization of the optical wavefront with subwavelength spatial resolution [58], can act as high-density phase modulators, enabling the integration of SLM-based Ising machines on a photonic chip. For example, more than 10⁶ optical spins over square millimeter could be obtained through the development of novel SLM technologies that integrate nanoantennas into liquid crystal cells [59]. Alternative nanophotonic platforms that employ electro-optic microcavity arrays would allow to achieve high fill factors along with phase-only modulation at GHz speeds [60]. At present, the iteration time of our Ising machine can be reduced to a few milliseconds by exploiting the most recent microelectromechanical SLM technologies [61], [62]. Spatial-photonic Ising machines are thus a promising approach for large-scale ultra-fast optical computing.

In conclusion, introducing a noisy-feedback mechanism in an SLM-based scheme, we have demonstrated the existence of an optimal noise level enhancing the machine performance on frustrated Ising models. Noise can hence be exploited as a tunable parameter to improve the exploration of energy landscapes with many minima, an interesting property that has been identified also in neural-network-based nanophotonic Ising samplers [40], [41]. Photonic Ising machines with controllable noise represent a route to realize photonic simulations of phase transition and finite-temperature phenomena. Our results show that noise is a valuable resource for optical computing, opening important possibilities for realizing classical and quantum annealing.