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Seeing Why: Exact Evidence Maps for Quantum and Optical Image Classifiers

This paper introduces Born Evidence Maps (BEM), an exact, approximation-free method for interpreting linear quantum and all-optical image classifiers by decomposing class probabilities into measurable amplitude contributions, demonstrating their effectiveness in identifying critical features and revealing data shortcuts across both simulated and real IBM quantum hardware experiments.

Original author: Teerawat Tongloy, Rangsan Jomtarak

Published 2026-11-03

📖 6 min read🧠 Deep dive

CC BY 4.0

Original author: Teerawat Tongloy, Rangsan Jomtarak

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the author. For technical accuracy, refer to the original paper. Read full disclaimer

When a computer looks at a photograph and decides what it sees, the reasoning behind that choice is usually a mystery. Even the most advanced systems operate as black boxes: they take an image in and spit out a label, but the path between the two is hidden inside layers of complex calculations. Scientists have developed methods to guess which parts of an image mattered most, but these guesses are often approximations that do not add up to the final decision. This lack of transparency is a significant hurdle, especially as researchers begin to build classifiers using light or quantum particles instead of traditional silicon chips. If these new machines are to be trusted, we need to know exactly why they made a choice, not just a guess. The challenge is to find a way to see the reasoning inside the machine, to trace the decision back to the specific pixels that drove it, without losing accuracy or relying on rough estimates.

A team of researchers has now built a system that solves this problem for both optical and quantum image classifiers. They created a machine that can look at a picture and, at the same time, produce a map showing exactly which parts of that picture contributed to the answer. Unlike previous methods that offer only an estimate, this system provides an exact accounting. Every bit of evidence the machine gathers adds up perfectly to the final probability of its decision. The researchers demonstrated this on two very different types of hardware: a quantum computer chip and a simulated optical bench made of lenses and light. In both cases, they could point to the specific pixels that convinced the machine to identify a boat, a leopard, or a car, and they could prove that these pixels were the true reason for the decision.

The core of this work is a new way of encoding images so that the machine treats dark and bright features equally. In many systems, bright areas of a photo dominate the analysis, causing the machine to ignore dark but important details. The researchers solved this by shifting the image so that the average brightness becomes a neutral reference point. This allows a dark shadow on a boat's hull to carry just as much weight as a bright patch of sky. Once the image is prepared this way, it passes through a series of filters. In the optical version, light travels through a series of glass plates that bend and shape the beam. In the quantum version, the image is translated into the state of a few quantum bits, which are then manipulated by a specific sequence of operations. Both paths lead to a final measurement that determines the class of the object.

What makes this work unique is the ability to reverse-engineer the decision. The researchers showed that the math behind these linear systems allows them to break down the final probability into contributions from every single pixel. They call these "Born evidence maps." These maps act like a heat map, highlighting the parts of the image that pushed the machine toward a specific answer and showing which parts pulled it away. Because the math is exact, the sum of all these pixel contributions equals the machine's confidence in its answer. There is no approximation involved; the explanation is as precise as the decision itself. This is a significant departure from other methods, which often require running the image through the system thousands of times with different parts blocked out to guess what mattered. Here, the explanation comes directly from the single run of the image.

To test if these maps were actually useful, the researchers performed a rigorous experiment. They took a set of test images and systematically deleted the pixels that the map said were most important. When they removed just five percent of the pixels with the highest evidence, the machine's accuracy plummeted from seventy-three percent down to nine percent. This proved that the map was correctly identifying the critical features. In contrast, when they deleted pixels based on older, less precise methods, the machine's accuracy barely dropped. The maps also revealed a hidden flaw in the data they were using. The system noticed that pictures of leopards in their training set often had a dark strip at the very top edge, a quirk of how the photos were cropped. The machine had learned to use this dark strip as a shortcut to identify leopards, rather than looking at the animal itself. The researchers used the maps to spot this behavior, removed the dark strips from the images, and retrained the system. The result was a machine that finally learned to recognize the leopard's body and spots, rather than the empty space above it.

The researchers then took their optical design and ran it on a real quantum computer, specifically the IBM ibm_kingston chip. They loaded three hundred test images onto the chip and asked it to classify them. The machine achieved an accuracy of thirty-seven point seven percent, which is nearly four times better than random guessing. More importantly, they measured the evidence maps directly on the chip for a selection of images. These real-world measurements matched the exact theoretical maps with a correlation of zero point seven seven. This confirmed that the quantum hardware was not only making decisions but was also producing the exact same internal reasoning that the theory predicted. While the quantum chip was limited by the difficulty of loading the images into its memory, the optical simulation showed that a physical bench of lenses and mirrors could reach an accuracy of seventy-two point three percent on the same set of color photographs.

The study also established the physical requirements for building such an optical machine. The researchers simulated the effects of real-world imperfections, such as slight misalignments of the glass plates or fluctuations in the laser power. They found that the system is surprisingly robust; it can tolerate small errors in alignment and still perform well. However, the alignment must be precise to within about a micron, and the laser power must remain stable during the measurement. These findings provide a clear blueprint for engineers who wish to build these devices. The work demonstrates that it is possible to create image classifiers that are not only accurate but also fully transparent. By using light or quantum particles, and by designing the system to be linear, the researchers have removed the mystery from the decision-making process. The machine no longer just gives an answer; it shows its work, pixel by pixel, proving that the path to understanding how these new technologies think is now open.

Technical Summary: Seeing Why: Exact Evidence Maps for Quantum and Optical Image Classifiers

Problem Statement
Current explanations for image classifiers, such as gradients, class activation maps, Integrated Gradients, and Shapley values, are typically approximations. They often fail to sum exactly to the decision they explain or require numerical estimation via path integrals or feature coalitions. This issue persists in quantum and optical classifiers, which often rely on hybrid architectures with pretrained classical networks or require numerical estimation of attributions. The authors address the need for an image classifier architecture where the contribution of every input amplitude to every class probability can be decomposed exactly, without approximation, and where these explanations can be measured directly on the hardware performing the classification.

Methodology
The paper proposes a unified architecture for both qubit-based and all-optical classifiers that operates linearly on amplitude-encoded images up to the final measurement, with no trained classical network before or after the quantum/optical transformation.

Encoding (The Mean Plane): To ensure dark features carry as much signal as bright ones, the authors remove each picture's channel mean from the color planes. These means are moved to a dedicated "means plane" (the fourth plane), scaled to carry 50% of the picture's energy. This allows dark deviations to become large negative amplitudes with equal energy to positive ones.
Fourier-Plane Mask: Before the main processing, each color plane passes through a learned complex mask in its spatial-frequency domain. On a lens bench, this is a 4f system with a spatial light modulator; on qubits, it is a Quantum Fourier Transform (QFT) with a post-selected mask. This filter is shift-equivariant and acts as a convolution with a real kernel.
Main Processing:
- Qubits: A "brickwall" circuit of depth 11 on 12 qubits, consisting of single-qubit rotations and nearest-neighbor CNOTs.
- Optics: Four phase plates separated by 3.2 mm, propagating light via Fresnel diffraction.
Readout (Born Headroom): The system uses a "Born Headroom Readout" (BHR). Instead of a standard softmax, it calculates the "headroom" ( $u_g$ ) for each class $g$ , defined as the measured probability ( $P_g$ ) minus the class's "no-information share" ( $b_g$ , the mass a maximally mixed input would receive), normalized by the remaining room ( $1-b_g$ ). These headrooms are converted to probabilities via a fixed LeakyReLU and softmax with a pre-set scale parameter $\phi$ , requiring no trainable parameters in the readout layer.
Born Evidence Maps (BEM): The core theoretical contribution is the derivation of an exact additive decomposition of the class probability above its no-information share ( $P_g - b_g$ ). For a linear map $A$ and projection $\Pi_g$ , the evidence of an amplitude $x$ is given by $e_g(x) = a(x) \text{Re}[\langle Mg \rangle(x)] / Z$ , where $M_g = A^\dagger(\Pi_g - b_g)A$ . This decomposition is an algebraic identity, equivalent to Integrated Gradients from a zero baseline for the underlying quadratic form, and sums exactly to $P_g - b_g$ .

Key Contributions

Exact Decomposition: The paper provides a closed-form, exact decomposition of class probabilities for linear amplitude-encoded classifiers. Unlike previous methods, these maps sum exactly to the class probability above its baseline, avoiding numerical approximation.
Hardware-Measurable Maps: The authors demonstrate that these evidence maps can be estimated directly from hardware measurements.
- On qubits, the maps are derived from class probabilities using the polarization identity of quadratic forms, requiring only forward and adjoint passes (no Hadamard tests).
- On an optical bench, the maps are derived solely from camera intensities by measuring the light for the full image, the image with a block masked, and the image with the block kept.
Mean Plane Encoding: The introduction of the "mean plane" encoding is shown to be the most significant architectural improvement, shifting the explanation focus from the background to the object itself.
Unified Architecture: A single design (Fourier mask + linear transformation + BHR) serves both quantum (brickwall) and optical (phase plates) hardware, trained on datasets ranging from MNIST to color Caltech-10.
Data Shortcut Discovery: The maps were used to identify and remove a data shortcut (a dark frame on leopard images in the untrimmed dataset), demonstrating the utility of exact explanations in debugging data.

Results

Performance: On Caltech-10 (color, 32x32), the full optical model (four phase plates) achieved 72.3% accuracy. The qubit model (depth 11 brickwall) achieved 54.9% in simulation.
Hardware Execution: On IBM's ibm_kingston quantum chip, the qubit model achieved 37.7% accuracy on 300 test images (3.8 times chance), compared to 56.3% in exact simulation. The accuracy drop was primarily attributed to the state-loading circuit (fidelity ~0.77) rather than chip noise.
Explanation Quality:
- Deletion Tests: Deleting the top 5% of pixels by evidence dropped accuracy from 73% to 9% (optical) and 56% to 5% (qubit), matching the performance of occlusion methods but requiring far fewer runs.
- Correlation: On the quantum chip, 2x2 block evidence maps for 10 correctly classified images correlated with exact maps at r = 0.77.
- Object Focus: With the mean plane, evidence concentrated on the object (1.32–1.37 times the area share), whereas plain amplitude encoding spread evidence evenly.
Tolerances: Simulations of a physical optical bench indicate that the system is robust to 8-bit modulators but requires precise alignment (plate shifts < 0.8 µm, rotation < 0.25°) and stable laser power (< 0.1% drift per frame).

Significance and Claims
The paper claims to provide the first optical image classifier whose class probabilities decompose exactly into contributions of encoded amplitudes, and a quantum classifier with no trained classical network whose explanations were measured on a real quantum computer.

The authors are modest about the results, explicitly stating:

The all-optical model is a linear diffractive network; no quantum advantage is claimed, and classical light suffices.
The accuracies (e.g., 72.3% on Caltech-10) are below those of small electronic convolutional networks (85.4%) or large diffractive ensembles.
The decomposition is exact for the encoded amplitudes, not raw pixels, due to the mean-subtraction encoding.
The hardware results are limited by the current state of quantum state loading, not necessarily the classifier architecture itself.
The "shortcut" found in the data was identified using the test set, meaning the results on the trimmed set are exploratory and lack a completely independent holdout.

The primary significance lies in the interpretability of the system: the ability to measure exactly why a linear quantum or optical classifier made a decision directly on the hardware, without approximation, and to use these maps to validate the model's reliance on the object rather than background artifacts.

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