Exact Evidence Maps for Quantum and Optical Image Classifiers: How BEM Opens the Black Box

When an image classifier makes a decision, we can usually only guess at the reason. But for quantum classifiers and all-optical classifiers—which act linearly on amplitude-encoded images, with no trained classical network before or after, up until measurement—this decomposition requires no approximation. Each class's probability above its uninformative share can be exactly decomposed into contributions from the encoded amplitudes and presented on the image as a Born Evidence Map (BEM), which can be measured directly on the hardware performing the classification.

Why Exact Evidence Maps Are So Rare

In classical deep learning, most methods for explaining model decisions rely on approximation. Tools such as Integrated Gradients, SHAP, and Grad-CAM estimate feature importance through gradients or sampling; they are computationally expensive and their results depend on the choice of baseline. For quantum classifiers and all-optical classifiers, the situation is different: because the model itself is a linear operation, the classification probability can be written as a quadratic form of the encoded amplitudes, and its decomposition is exactly closed-form.

The authors point out that for the quadratic form behind each class probability, its BEM is equivalent to Integrated Gradients from a zero baseline, and a closed-form solution exists. This means pixel-level exact contribution maps can be obtained without additional training or extensive sampling.

Amplitude-encoded images refer to an encoding scheme that maps image pixel values to the complex amplitudes of a quantum state or an optical field. In the quantum case, the brightness of each pixel corresponds to the amplitude of a qubit (e.g., encoded via rotation angles); in the optical case, pixel values modulate the complex amplitude of the light field. Because the classifier operates linearly on the amplitudes, the class probability can ultimately be expressed as a quadratic form of the encoded amplitudes (i.e., (p_c = \mathbf{a}^\dagger M_c \mathbf{a}), where (\mathbf{a}) is the amplitude vector and (M_c) is a class-dependent Hermitian matrix). This quadratic structure is the fundamental reason BEM can be decomposed in closed form: contributions can be exactly attributed to each amplitude component without gradient approximation or sampling.

Born Evidence Maps (BEM) are named after the Born rule in quantum mechanics: the probability of obtaining a certain measurement outcome equals the squared modulus of the state vector projection. In the classifier, each class probability is given by a quadratic form of the amplitude vector, and BEM exactly decomposes the excess of this probability relative to an uninformative baseline (uniform distribution) into the contribution of each pixel amplitude, overlaid on the original image as a heatmap. Because the decomposition is closed-form, BEM requires no additional training or sampling and can be measured directly on the hardware performing the classification (e.g., by estimating the expectation of each quadratic-form component through repeated measurements).

The Practical Power of BEM: Deletion Experiments

The authors verified BEM's faithfulness through deletion experiments. After removing the top 5% of pixels by evidence, classification accuracy dropped from 73% to 9%. In comparison, occlusion methods require an additional 1,024 forward runs per image to achieve a similar effect, while gradient saliency maps show a much smaller accuracy drop after deleting the same proportion of pixels.

These data show that BEM is not only exact but also significantly better than common approximation methods at identifying key pixels. For high-stakes applications requiring interpretability, this exactness has practical value.

One Architecture, Two Implementation Paths

The most engineering-elegant part of the research is that the same architecture can serve both quantum computing and optical computing. Specifically: after a Fourier-plane mask, a brick-wall circuit (acting on qubits) or a phase plate (acting on a lens bench) is applied, and finally a Born residual reader with no trainable parameters reads out the result.

Normalizing each image centering on its own mean is another key design choice. This gives dark and bright features equal weight and avoids brightness bias interfering with classification.

Fourier-plane masks refer to placing a trainable or fixed mask (such as a phase plate) at the Fourier plane (focal plane) of a lens system to modulate the spatial frequency components of the image. In optical implementations, the lens naturally performs the Fourier transform, and the mask linearly filters the light field in the frequency domain; in quantum circuits, equivalent frequency-domain modulation is simulated through unitary operations. This design transforms image classification into linear weighting of frequency-domain features, which together with the subsequent brick-wall circuit or phase plate forms a linear classifier.

Brick-wall circuits are a quantum circuit ansatz in which two-qubit gates are arranged in a staggered pattern like a brick wall: the first layer of gates acts on adjacent qubit pairs (0,1), (2,3)..., the second layer acts on (1,2), (3,4)..., and so on alternately. This structure is easy to implement on superconducting quantum chips and offers good expressivity and hardware friendliness. In the architecture of this paper, the brick-wall circuit acts on the amplitude-encoded qubits to achieve linear mixing between classes, with trainable (or fixed) parameters, and is finally measured by the Born reader.

Performance in Simulation and on Real Hardware

In simulation, four phase plates at a single wavelength achieved 72.3% accuracy on Caltech-10 color photos, 96.8% on MNIST, 86.7% on Fashion-MNIST, and 47.9% on color CIFAR-10. These results demonstrate the architecture's competitiveness on standard datasets, especially the high accuracy on simple datasets.

On IBM's ibm_kingston quantum computer, with the mask applied before approximate loading, classification accuracy on 300 Caltech-10 test images was 37.7%, compared with 10% for random guessing. The same configuration can reach 56.3% in exact simulation. In addition, for ten images that were correctly and confidently classified, the 2x2 evidence maps estimated on-chip were consistent with exact results, with a correlation coefficient r=0.77.

The accuracy drop on real hardware mainly comes from approximate loading errors, but the consistency of the evidence maps shows that BEM remains effective on-chip.

BEM Reveals Data Shortcuts

Beyond verifying the basis for classification, BEM also exposes shortcuts in the data. The authors mention that the evidence maps revealed a shortcut in the dataset—the classifier may rely on spurious features unrelated to the class to make judgments. Such findings usually require additional testing or manual review in classical models, whereas BEM can be read directly from the evidence maps.

Implications for Explainable AI

This work offers several noteworthy signals. First, for linear or near-linear models, exact explanation is feasible, and there is no need to settle for approximate attribution. Second, although quantum computing and optical computing follow different hardware paths, they can share the same interpretability framework at the algorithmic level, reducing the cost of cross-platform migration. Third, the ability to measure evidence maps directly on hardware means that interpretability can become part of the inference process rather than an analysis step added afterward.

For researchers concerned with AI safety and interpretability, BEM offers an approach that goes beyond approximation methods under specific architectures. Although the current experimental scale is limited, its exactness and hardware feasibility are worth further exploration.

One-line Summary

Linear quantum and optical classifiers can achieve closed-form exact pixel-level evidence maps that are directly measurable on hardware.

Paragraph Summary

This article introduces a method for providing exact interpretability for linear quantum classifiers and all-optical classifiers—Born Evidence Maps (BEM). Because these models perform linear operations on amplitude-encoded images, class probabilities can be expressed as quadratic forms of the encoded amplitudes, so their contribution decomposition is exactly closed-form and equivalent to zero-baseline Integrated Gradients. Deletion experiments show that removing the top 5% of pixels by evidence drops accuracy from 73% to 9%, far outperforming gradient saliency maps and avoiding the 1,024 additional runs per image required by occlusion methods. The same architecture (Fourier mask + brick-wall circuit/phase plate + Born readout) can be implemented on both quantum hardware and optical benches. On an IBM quantum chip, classification accuracy on 300 Caltech-10 images reached 37.7% (random is 10%), and the chip-estimated evidence maps had a correlation coefficient of r=0.77 with exact values. BEM also revealed shortcuts in the dataset, demonstrating the additional value of exact evidence maps for data review.