Amplitude-Only Rank-Free Photonic Ising Machine

• AR-SPIM is an amplitude-only optical computing system that encodes and solves full, rank-free Ising Hamiltonians without eigen-decomposition bottlenecks.
• The system utilizes amplitude SLMs and DMDs to dynamically implement binary masks, achieving high throughput (200 Hz) and precise optical intensity measurements.
• It enables advanced applications such as combinatorial optimization, phase transition studies, and machine learning through scalable, high-speed optical workflows.

An amplitude-only modulated rank-free spatial photonic Ising machine (AR-SPIM) is an optical computing system capable of directly encoding and solving general Ising Hamiltonians—including those with arbitrary spin–spin coupling ranks and external fields—using only amplitude modulation. AR-SPIM architectures exploit the spatial mapping of Ising variables onto the pixels of spatial light modulators (SLMs) or digital micromirror devices (DMDs), and crucially enable scalable, high-speed optical optimization workflows without the eigen-decomposition bottlenecks of conventional photonic Ising machines. These systems enable rapid, high-precision combinatorial optimization, large-scale ground-state searches, phase transition studies, and machine-learning tasks by leveraging novel mask-based decomposition methods and incoherent or coherent optical field processing (Zheng et al., 25 Dec 2025, Sakellariou et al., 2024, Yamashita et al., 2023, Stroev et al., 24 Aug 2025).

1. Ising Hamiltonian, Rank-Free Decomposition, and Optical Encoding

The core computational problem is minimizing the Ising Hamiltonian

$$H(\sigma) = -\sum_{i<j} J_{ij} \sigma_i \sigma_j - \sum_i h_i \sigma_i$$

where $J\in\mathbb{R}^{N\times N}$ (symmetric coupling matrix), $h\in\mathbb{R}^N$ (external field), and $\sigma_i \in \{\pm 1\}$. Traditional SPIMs are limited to low-rank (typically rank-1) $J$ due to optical hardware constraints; AR-SPIM overcomes this by allowing every nonzero $J_{ij}$ and $h_i$ to be encoded directly, yielding rank-free operation (Zheng et al., 25 Dec 2025, Yamashita et al., 2023).

The key theoretical innovation is the decomposition of $J$ and $h$ as sums of Hadamard (element-wise) products:

• For $R$ nonzero levels (bits), write $J = \sum_{k=1}^R \alpha_k H^{(k)}$, $h = \sum_{k=1}^R \beta_k v^{(k)}$, where $H^{(k)}$ and $v^{(k)}$ are binary masks, and $\alpha_k$, $\beta_k$ are real weights captured in the amplitude SLM mask.
• Practically, two-term decompositions suffice for signed 8/9-bit couplings (Zheng et al., 25 Dec 2025):
  • Positive and negative contributions are handled by separate masks and weights ($+1$ or $-1$), allowing compact, high-precision encoding.

In the optical domain, the amplitude SLM stores the absolute values of $J$ and $h$, while the DMD or binary SLM loads the dynamically computed mask for each spin configuration. The total optical intensity measured for a configuration encodes $H(\sigma)$ linearly (up to a global constant) (Zheng et al., 25 Dec 2025).

2. Optical Implementation and Workflow

The AR-SPIM experimental system comprises:

• Amplitude SLM (A-SLM): Loaded once with a full 8/9-bit grayscale mask representing $|J_{ij}|$ and $|h_i|$.
• DMD/Binary SLM: Dynamically loaded per spin configuration with binary pass-masks $D_J(\sigma)$ and $D_h(\sigma)$, reflecting current $\sigma$.
• 4-f Relay and Photodetector: Ensures diffraction cleanliness and integrates the light for intensity measurement.
• Incoherent LED Source: Eliminates coherent interference, ensuring intensity linearity and high encoding accuracy (Zheng et al., 25 Dec 2025).

The iteration loop is as follows:

1. Initialize spin state $\sigma^0$ (random or fixed).
2. For each Metropolis–Hastings iteration (200 Hz):
   • Compute and update binary masks $D_J$, $D_h$ for current $\sigma$.
   • Measure intensity $I(\sigma)$; compute energy $H(\sigma) = C - I(\sigma)$.
   • Propose spin flip, update only the affected rows/columns of the mask, repeat measurement for new configuration, accept or reject based on standard Metropolis criterion.

A critical detail is that only $O(N)$ mask elements need to update per spin flip, enabling high throughput (Zheng et al., 25 Dec 2025).

3. Rank-Free Architecture and Methods Across Modalities

Three principal AR-SPIM modalities are distinguished in the literature:

• Hadamard-masked amplitude-only (incoherent) AR-SPIM: As deployed in (Zheng et al., 25 Dec 2025), this approach leverages amplitude SLMs and DMDs for direct intensity-based encoding without requiring auxiliary spins, interference fringes, or coherent sources.
• Paired pixel phase/amplitude encoding (coherent) AR-SPIM: Each coupling $J_{ij}$ is mapped to two SLM pixels with opposite phases or amplitudes, and an ancilla subtraction isolates the quadratic term, supporting arbitrary interaction graphs at linear pixel cost (Sakellariou et al., 2024).
• Low-rank amplitude factorization: $J$ is decomposed as a sum of rank-1 terms via eigenmode or SVD decomposition; $K$ SLM patterns encode these low-rank factors, with one optical measurement per term (Yamashita et al., 2023). Practical for problems with low or moderate rank, such as knapsack or structure-learning in Boltzmann machines.
• Programmable $k$-local (higher-order) amplitude-only AR-SPIM: By extending the mask/fan-out strategy with folded 4f relays and window-specific amplitude gratings, arbitrary $k$-local (beyond pairwise) Ising interactions—fully rank-free in the tensor sense—are physically implemented (Stroev et al., 24 Aug 2025).

A summary table of architectural variants:

Scheme	Modulator(s)	Key Innovation
Hadamard mask (incoherent)	A-SLM + DMD	Elementwise amplitude + sign masks, full J/h direct
Paired-pixel (coherent)	SLM (phase/amplitude)	Two-pixel encoding, ancilla subtraction
Low-rank amplitude	Dual amplitude SLM	Optical SVD, time/space multiplexed rank-1 terms
$k$-local folded relay	Single amplitude SLM	Per-clique univariate amplitude polynomials

4. Quantitative Performance and Experimental Results

AR-SPIM systems demonstrate precision, speed, and scalability improvements over prior SPIMs:

• Encoding accuracy: Linear coefficient of determination $R^2 > 0.9997$, Pearson $r > 0.9998$ globally; near ground state region $R^2 > 0.9800$, $r > 0.9899$ (Zheng et al., 25 Dec 2025). Comparable agreement observed for phase-encoded coherent variants (Sakellariou et al., 2024).
• Optimization benchmark: Error rates $<$0.3% for ground-state search on 797-spin biased Max-cut problems (up to 9-bit $(\pm255)$ couplings), after 5000 steps at 200 Hz (Zheng et al., 25 Dec 2025).
• Phase transitions: Replica-method studies on 797-spin Sherrington–Kirkpatrick model resolve ferromagnetic–paramagnetic crossover and symmetry breaking under uniform field (Zheng et al., 25 Dec 2025).
• Graph partitioning: On 100-spin (sparse) GPP and weighted GPP, AR-SPIM matches or slightly outperforms METIS ground-truths, preserving statistical efficiency up to graph densities $p\simeq0.2$ (Sakellariou et al., 2024).
• Learning and sampling: Amplitude-only, low-rank AR-SPIMs efficiently train Boltzmann machines for classification (MNIST, $N$ up to 794) and generative sampling ($K$ up to 50) (Yamashita et al., 2023).

Performance and scaling metrics:

Metric	Reported Value/Range
Max spin count	797 (amplitude/DMD), 100 (coherent/SLM)
Encoding dynamic range	8/9-bit ($\pm255$)
Iteration/step rate	200 Hz (amplitude/DMD), 30 Hz (phase SLM)
Hamiltonian linear error	$<$0.02 for $I<4.2$ V; $<$0.003 otherwise
MCMC sweep time	5 ms/spin-flip (amplitude AR-SPIM)

5. Comparison with Prior SPIM Architectures and Limitations

Previous SPIM approaches were hampered by bottlenecks:

• Rank limitation: Only Mattis- or eigenmode-based (rank-1/low-rank) $J$ couplings;
• Time–spin-count trade-off: Time/space/wavelength multiplexing for dense $J$ increases step time or reduces scale (Sakellariou et al., 2024, Yamashita et al., 2023);
• Coherence artifacts: Phase SLMs introduce interference, nonlinearities, stricter calibration, and limit dynamic range.

AR-SPIM resolves these by:

• Direct, elementwise mask encoding: Full $J$, $h$ mapped without auxiliary spins or decompositions (Zheng et al., 25 Dec 2025).
• Rank-free scalability: No fundamental barrier to arbitrary coupling patterns, including $k$-local ($k>2$) (Stroev et al., 24 Aug 2025).
• Amplitude-only operation: Removes coherent interference artifacts; enhances linearity, encoding precision, and compatibility with high-speed SLMs/DMDs (Zheng et al., 25 Dec 2025).

Limitations and technical considerations:

• SLM/DMD resolution: Dense couplings require $O(N^2)$ pixels—scaling remains challenging for large, dense instances. Sparse graphs mitigate this, and omitting zero masks can free pixel budget (Zheng et al., 25 Dec 2025, Sakellariou et al., 2024).
• Hardware throughput: Camera/SLM update rates set the ultimate annealing bandwidth; amplitude SLMs remain slower than DMDs, with MHz-range devices on the near horizon (Zheng et al., 25 Dec 2025).
• Calibration and noise: Optical uniformity and flat-field response must be controlled; two-point energy calibration (fit parameters $\alpha,\beta$) corrects residual errors (Sakellariou et al., 2024).

6. Extensions, Applications, and Outlook

The AR-SPIM framework generalizes beyond pairwise Ising models:

• $k$-local interactions: Folded 4f relays with per-window amplitude gratings natively encode arbitrary $k$-body Ising terms and higher-order combinatorial objectives (Stroev et al., 24 Aug 2025).
• Physical gradient-based learning: In situ training of mask coefficients via two-frame (forward/adjoint) protocols enables direct photonic implementation and training of Kolmogorov–Arnold network layers (Stroev et al., 24 Aug 2025).
• Discrete optimization: Arbitrary NP-hard problems (SAT, coloring, partition) are directly encodable via Ising mappings, without eigen-decomposition (Sakellariou et al., 2024, Zheng et al., 25 Dec 2025).
• Boltzmann machine learning: Scalable AR-SPIM training for generative and discriminative tasks, with observably low-rank factorization learned on data (Yamashita et al., 2023).

Given the modularity of amplitude-only mask encoding, no auxiliary multiplexing, and programmable reconfigurability, AR-SPIMs are positioned as experimental platforms for quantum many-body simulation, machine learning, and large-scale discrete optimization (Zheng et al., 25 Dec 2025, Stroev et al., 24 Aug 2025).

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References

• (Zheng et al., 25 Dec 2025) Incorporating rank-free coupling and external field via an amplitude-only modulated spatial photonic Ising machine
• (Sakellariou et al., 2024) Encoding arbitrary Ising Hamiltonians on Spatial Photonic Ising Machines
• (Yamashita et al., 2023) Low-rank combinatorial optimization and statistical learning by spatial photonic Ising machine
• (Stroev et al., 24 Aug 2025) Programmable k-local Ising Machines and all-optical Kolmogorov-Arnold Networks on Photonic Platforms