Medical Imaging Classification with Cold-Atom Reservoir Computing using Auto-Encoders and Surrogate-Driven Training

# Medical Imaging Classification with Cold-Atom Reservoir Computing using Auto-Encoders and Surrogate-Driven Training
Source: [https://arxiv.org/html/2605.06727](https://arxiv.org/html/2605.06727)
Nuno Batista1, Ana Morgado1, Oscar Ferraz1, Sagar Silva Pratapsi3, Jorge Lobo2, and Gabriel Falcao1 1Instituto de Telecomunicações, Dept\. of Electrical and Computer Engineering, University of Coimbra, Portugal 2ISR \- Institute of Systems and Robotics, Dept\. of Electrical and Computer Engineering, University of Coimbra, Portugal 3CFisUC, Department of Physics, University of Coimbra, Portugal Email: \{nuno\.batista, ana\.morgado, oscar\.ferraz, gff\}@co\.it\.pt, sagar\.pratapsi@uc\.pt, jlobo@isr\.uc\.ptThis work was supported by FCT \- Fundação para a Ciência e Tecnologia, I\.P\. by project reference UID/50008/2023 IT, UIDB/04564/2020, UIDP/04564/2020, 2022\.06780\.PTDC, and 2023\.14860\.PEX \(Q\-Bet\) with DOI identifiers 10\.54499/UID/50008/2023, 10\.54499/UIDB/04564/2020, 10\.54499/UIDP/04564/2020, 10\.54499/2022\.06780\.PTDC, and 10\.54499/2023\.14860\.PEX, respectively\. It was also funded by the Open Quantum Institute \(OQI\), and conducted in collaboration with Centro de Estudos Sociais \(CES\), University of Coimbra \(UC\), and Instituto de Telecomunicações \(IT\)\. OQI itself is an initiative hosted by CERN, born at GESDA, supported by UBS\)\.\(Corresponding author: Nuno Batista\)

###### Abstract

We introduce a hybrid quantum\-classical pipeline, based on neutral\-atom reservoir computing, for medical image classification, focusing on the binary classification task of polyp detection\. To deal effectively with the high dimensionality, we integrate a guided auto\-encoder\. This pipeline learns compact and discriminative representations of image data that are also well\-suited for quantum reservoir computing\. A key challenge in such systems is the non\-differentiable nature of quantum measurements, which creates a ‘gradient barrier’ for standard training\. We overcome this barrier by incorporating a differentiable surrogate model that emulates the quantum layer, enabling end\-to\-end backpropagation through the entire system\. This guided training process is jointly optimized for classification accuracy and for faithful image recovery from the auto\-encoder\. The learned latent representations are encoded as pulse detuning parameters within a Rydberg Hamiltonian, and quantum embeddings are subsequently obtained through expectation values\. These embeddings are then passed to a linear classifier\. Our simulations show that this method outperforms some traditional approaches that use PCA or unguided autoencoders\. We also conduct ablation studies to assess the impact of various quantum and training parameters, demonstrating the robustness and flexibility of our proposed pipeline for real\-world medical imaging applications, even in the current NISQ era\.

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## IIntroduction

Advances in medical imaging have significantly improved disease diagnosis and treatment planning\. For conditions like colorectal cancer, early detection of polyps through colonoscopy image analysis is critical for reducing mortality\[[1](https://arxiv.org/html/2605.06727#bib.bib1)\]\. Deep learning techniques, especially autoencoders, are widely used to extract compressed, informative features from high\-dimensional images for classification\[[2](https://arxiv.org/html/2605.06727#bib.bib2)\]\. However, classical neural networks may struggle to capture intricate correlations in complex medical data\[[3](https://arxiv.org/html/2605.06727#bib.bib3)\]\.

Quantum computing offers novel opportunities for machine learning, particularly through quantum reservoir computing \(QRC\), where a physical quantum system processes classical inputs into high\-dimensional nonlinear embeddings\[[4](https://arxiv.org/html/2605.06727#bib.bib4),[5](https://arxiv.org/html/2605.06727#bib.bib5)\]\. Recent works show that analog quantum systems, such as neutral\-atom platforms, can serve as untrained reservoirs with rich dynamics for temporal and pattern recognition tasks\[[6](https://arxiv.org/html/2605.06727#bib.bib6),[7](https://arxiv.org/html/2605.06727#bib.bib7)\]\.

In this work, we propose a quantum\-guided autoencoder architecture that integrates a classical image encoder with a neutral\-atom quantum reservoir\.

In hybrid approaches, a classical encoder compresses image data, and a quantum reservoir expands the encoded features into a higher\-dimensional Hilbert space, potentially boosting classification performance, as these expanded features might capture high\-order correlations and nonlinearities that classical methods cannot tractably represent\.

A major challenge in such hybrid quantum\-classical models is the non\-differentiability of quantum measurements, which obstructs gradient\-based optimization\. Additionally, tuning quantum parameters can suffer from barren plateaus, where gradients vanish in high\-dimensional Hilbert spaces\[[8](https://arxiv.org/html/2605.06727#bib.bib8)\]\. To address this, we introduce a classical neural surrogate that emulates the quantum reservoir’s input\-output behavior\. This surrogate enables end\-to\-end training via backpropagation, while the quantum system remains fixed and non\-trainable\. It is only used to train the surrogate model and to create embeddings for downstream classifiers\.

Critically, the physical reservoir’s exponentially large state‐space \(and associated many\-body correlations\) cannot be classically emulated, hence the surrogate is merely a differentiable proxy during training, not a replacement of the real quantum dynamics at inference\.

Our results illustrate the viability of QRC for real\-world medical tasks and offer a scalable path to hybrid quantum\-classical learning, even in the noisy intermediate\-scale quantum \(NISQ\) era\.

## IIBackground and Related Work

This section provides an overview of the foundational concepts and prior research relevant to this work\.

### II\-APrinciples of Reservoir Computing

Reservoir computing is a computational framework derived from recurrent neural networks \(RNNs\)\. It involves a fixed, high\-dimensional dynamical system \(the reservoir\) that projects input data into a rich feature space\. Only the output layer is trained, simplifying the learning process and reducing computational overhead\. This approach is particularly effective for time\-series prediction and pattern recognition tasks\.

Mathematically, letu​\(t\)∈ℝmu\(t\)\\in\\mathbb\{R\}^\{m\}be the input at timett, withx​\(t\)∈ℝnx\(t\)\\in\\mathbb\{R\}^\{n\}being the reservoir state, andy​\(t\)∈ℝky\(t\)\\in\\mathbb\{R\}^\{k\}the output\. The reservoir dynamics and output are given by

x​\(t\)\\displaystyle x\(t\)=f​\(Wi​n​u​\(t\)\+Wr​e​s​x​\(t−1\)\)\\displaystyle=f\(W\_\{in\}u\(t\)\+W\_\{res\}x\(t\-1\)\)\(1\)y​\(t\)\\displaystyle y\(t\)=Wo​u​t​x​\(t\),\\displaystyle=W\_\{out\}x\(t\),\(2\)whereffis a nonlinear activation function,Wi​nW\_\{in\}andWr​e​sW\_\{res\}are fixed input and reservoir weight matrices, andWo​u​tW\_\{out\}is the trained output weight matrix\. A diagram of a typical reservoir computing architecture is shown in Fig\.[1](https://arxiv.org/html/2605.06727#S2.F1)\.

Input LayerReservoir LayerOutput Layeru​\(t\)u\(t\)Wr​e​sW\_\{res\}y​\(t\)y\(t\)WinW\_\{\\text\{in\}\}x​\(t\)x\(t\)WoutW\_\{\\text\{out\}\}

Figure 1:Reservoir computing architecture showing input nodes, a recurrent reservoir network with internal dynamics, and an output layer\.
### II\-BQuantum Reservoir Computing with neutral atoms

Quantum Reservoir Computing \(QRC\) extends the reservoir computing paradigm into the quantum domain\. QRC aims to enhance computational capabilities by accessing larger state spaces and different types of non\-linearities\. Notably, large\-scale experiments utilizing neutral\-atom analog quantum computers have demonstrated the scalability and effectiveness of QRC in various machine learning applications\[[7](https://arxiv.org/html/2605.06727#bib.bib7)\]\.

In QRC, classical input datau​\(t\)u\(t\)is encoded into quantum states\|ψ​\(t\)⟩\|\\psi\(t\)\\rangle, which evolve under a fixed HamiltonianHH:

\|ψ​\(t\+Δ​t\)⟩=U​\|ψ​\(t\)⟩=e−i​H​Δ​t​\|ψ​\(t\)⟩,\\ket\{\\psi\(t\+\\Delta t\)\}=U\\ket\{\\psi\(t\)\}=e^\{\-iH\\Delta t\}\\ket\{\\psi\(t\)\},\(3\)whereUUis the unitary evolution operator andΔ​t\\Delta tis the difference between consecutive timestamps\. Measurements of observablesO^\\hat\{O\}yield the outputs

y​\(t\)=⟨ψ​\(t\)\|O^\|ψ​\(t\)⟩\.y\(t\)=\\langle\\psi\(t\)\|\\hat\{O\}\|\\psi\(t\)\\rangle\.\(4\)The output weights are trained classically, while the quantum reservoir remains fixed\.

Neutral atom platforms, particularly those utilizing Rydberg states, have emerged as promising candidates for implementing QRC due to their scalability and controllable interactions\.

In the work by M\. Kornjača et al\.\[[7](https://arxiv.org/html/2605.06727#bib.bib7)\], a large\-scale, gradient\-free QRC algorithm was developed and experimentally implemented on a neutral\-atom analog quantum computer\. This system achieved competitive performance across various machine learning tasks, including classification and time\-series prediction, demonstrating effective learning with increasing system sizes up to108108qubits\.

The dynamics of such laser\-driven neutral atoms, working in the Rydberg state regime, is given by the Hamiltonian\[[7](https://arxiv.org/html/2605.06727#bib.bib7)\]

H​\(t\)\\displaystyle H\(t\)=Ω​\(t\)2​∑j\(\|gj⟩⟨rj\|\+\|rj⟩⟨gj\|\)\\displaystyle=\\frac\{\\Omega\(t\)\}\{2\}\\sum\_\{j\}\\left\(\\outerproduct\{g\_\{j\}\}\{r\_\{j\}\}\+\\outerproduct\{r\_\{j\}\}\{g\_\{j\}\}\\right\)\+∑j<kVj​k​nj​nk−∑j\[Δg​\(t\)\+αj​Δl​\(t\)\]​nj,\\displaystyle\\quad\+\\sum\_\{j<k\}V\_\{jk\}n\_\{j\}n\_\{k\}\-\\sum\_\{j\}\\left\[\\Delta\_\{\\mathrm\{g\}\}\(t\)\+\\alpha\_\{j\}\\Delta\_\{\\mathrm\{l\}\}\(t\)\\right\]n\_\{j\},\(5\)whereΩ​\(t\)\\Omega\(t\)is the global Rabi drive amplitude between a ground state\|gj⟩\\ket\{g\_\{j\}\}and a highly\-excited Rydberg state\|rj⟩\\ket\{r\_\{j\}\}of an atomjj,nj=\|rj⟩⟨rj\|n\_\{j\}=\\outerproduct\{r\_\{j\}\}\{r\_\{j\}\}, andVj​k=C6/‖rj−rk‖6V\_\{jk\}=C\_\{6\}/\\\|r\_\{j\}\-r\_\{k\}\\\|^\{6\}describes the van der Waals interactions between atoms, and the detuning is split into a global termΔg​\(t\)\\Delta\_\{\\mathrm\{g\}\}\(t\)and a site\-dependent termΔl​\(t\)\\Delta\_\{\\mathrm\{l\}\}\(t\), with site modulationαj∈\[0,1\]\\alpha\_\{j\}\\in\[0,1\]\.

By initializing the system in a specific state and allowing it to evolve under this Hamiltonian, the resulting quantum state encodes information about the input data\. Measurements of observables on this state yield outputs that can be used for tasks such as classification or prediction, with only the final readout layer requiring training\.

### II\-CDimensionality Reduction for Image Data

Dimensionality reduction is a crucial preprocessing step in machine learning and data analysis, aiming to reduce the number of input variables in a dataset while preserving as much information as possible\. This process enhances computational efficiency and facilitates data visualization\.

#### II\-C1Principal Component Analysis

Principal Component Analysis \(PCA\)\[[9](https://arxiv.org/html/2605.06727#bib.bib9)\]is a linear dimensionality reduction technique that transforms a set of correlated variables into a set of uncorrelated variables called principal components\. The goal is to capture the maximum variance in the data with the fewest number of components\.

PCA is effective for datasets where the principal components align with the directions of maximum variance, but it may not capture complex, nonlinear relationships in the data\[[10](https://arxiv.org/html/2605.06727#bib.bib10)\]\.

#### II\-C2Autoencoder Architectures

Autoencoders are a class of artificial neural networks designed to learn efficient codings of input data in an unsupervised manner\. They consist of two main parts: an encoder that compresses the input into a latent\-space representation, and a decoder that reconstructs the input from this representation\.

Given an input𝒙∈ℝd\\bm\{x\}\\in\\mathbb\{R\}^\{d\}, the encoder maps𝒙\\bm\{x\}to a latent representation𝒛∈ℝk\\bm\{z\}\\in\\mathbb\{R\}^\{k\}\(wherek<dk<d\):

𝒛=ε𝝎​\(𝒙\)\.\\bm\{z\}=\\varepsilon\_\{\\bm\{\\omega\}\}\(\\bm\{x\}\)\.\(6\)The decoder then reconstructs the input𝒙^∈ℝd\\bm\{\\hat\{x\}\}\\in\\mathbb\{R\}^\{d\}:

𝒙^=𝒟𝝆​\(𝒛\),\\bm\{\\hat\{x\}\}=\\mathcal\{D\}\_\{\\bm\{\\rho\}\}\(\\bm\{z\}\),\(7\)where we follow the notation from\[[11](https://arxiv.org/html/2605.06727#bib.bib11)\]\. The network is trained to minimize the reconstruction loss:

ℒ​\(𝒙,𝒙^\)=‖𝒙−𝒙^‖2\.\\mathcal\{L\}\(\\bm\{x\},\\bm\{\\hat\{x\}\}\)=\\\|\\bm\{x\}\-\\bm\{\\hat\{x\}\}\\\|^\{2\}\.\(8\)This architecture of an autoencoder is illustrated in Fig\. 2 a\)\.

Autoencoders can capture complex, nonlinear relationships in the data, making them suitable for tasks like image compression, denoising, and anomaly detection\[[12](https://arxiv.org/html/2605.06727#bib.bib12),[1](https://arxiv.org/html/2605.06727#bib.bib1)\]\.

#### II\-C3Quantum\-Guided Autoencoding

Quantum\-Guided Autoencoders integrate quantum computing principles into the autoencoder framework to leverage quantum advantages in processing and representing data\. These models aim to perform dimensionality reduction and classification within a single architecture, enhancing performance on complex datasets\.

In the Quantum\-Guided Autoencoder \(QGA\) model, a classical encoder first reduced the dimensionality of the input data\. The compressed data is then processed by a parametrized quantum circuit, which acts as the decoder and classifier\. The quantum circuit transforms the input state\|ψin⟩\|\\psi\_\{\\text\{in\}\}\\rangleinto an output state\|ψout⟩\|\\psi\_\{\\text\{out\}\}\\rangleusing a unitary operation:

\|ψout⟩=U​\(𝜽\)​\|ψin⟩\.\|\\psi\_\{\\text\{out\}\}\\rangle=U\(\\bm\{\\theta\}\)\|\\psi\_\{\\text\{in\}\}\\rangle\.\(9\)The parameters𝜽\\bm\{\\theta\}are optimized to minimize a loss function that combines reconstruction error and classification accuracy\. Measurements on\|ψout⟩\|\\psi\_\{\\text\{out\}\}\\rangleyield the final classification result\.

This approach has demonstrated an improved performance over traditional methods in tasks such as identifying the Higgs boson in particle collision data, showcasing the potential of quantum\-guided models in handling high\-dimensional, complex datasets\[[11](https://arxiv.org/html/2605.06727#bib.bib11)\]\.

## IIIMethodology

### III\-AProposed System Architecture

Our proposed pipeline is a Quantum\-Guided Autoencoder with a Reservoir Surrogate \(QGARS\)\. It is an hybrid architecture that synergizes classical autoencoding techniques with quantum reservoir processing to achieve efficient learning of features\. These are expected to better match with the reservoir computing layer, whose embeddings are used to perform a classification task\.

The system is designed to overcome the inherent non\-differentiability of quantum operations, which traditionally impede gradient\-based training methods\. The overall architecture is illustrated in Fig\. 2 and thearchitectural componentsare the following:

- •Classical Autoencoder: - –Encoder:transforms high\-dimensional input data𝒙\\bm\{x\}into a lower\-dimensional latent representation𝒛\\bm\{z\}\. - –Decoder:The decoder network𝒟ρ​\(𝒛\)\\mathcal\{D\}\_\{\\rho\}\(\\bm\{z\}\)reconstructs the input data from the latent representation, producing𝒙^\\bm\{\\hat\{x\}\}\.
- •Quantum Reservoir Layer: - –Parameter Mapping:g​\(𝒛\)g\(\\bm\{z\}\)maps the latent representation𝒛\\bm\{z\}to the local detuning frequenciesΔi\\Delta\_\{\\mathrm\{i\}\}of the Rydberg atoms\. - –Quantum Evolution:Evolves the system under a HamiltonianH​\(𝚫\)H\(\\bm\{\\Delta\)\}, resulting in a quantum state\|ψ​\(𝚫\)⟩\|\\psi\(\\bm\{\\Delta\}\)\\rangle\. - –Measurement:Performs measurements on the quantum state to obtain embeddings𝒒∈ℝk\\bm\{q\}\\in\\mathbb\{R\}^\{k\}, which serve as training targets for the surrogate model or as inputs for the classification task later on\.
- •Surrogate Model:A differentiable classical neural networkfsur​\(𝒛;𝜽sur\)f\_\{\\text\{sur\}\}\(\\bm\{z\};\\bm\{\\theta\}\_\{\\text\{sur\}\}\), where𝜽sur\\bm\{\\theta\}\_\{\\text\{sur\}\}are parameters tuned to approximate the mapping from𝒛\\bm\{z\}to𝒒\\bm\{q\}, effectively emulating the quantum layer’s behaviour\.
- •Linear Classifier:A classical neural network that maps the quantum embeddings𝒒\\bm\{q\}to predicted class labels𝒚^\\bm\{\\hat\{y\}\}\.

The following sections give a more detailed explanation of each component\.

![Refer to caption](https://arxiv.org/html/2605.06727v1/x1.png)

Figure 2:Overview of the QGARS pipeline\. \(a\) Classical autoencoder:ε𝝎​\(𝒙\)\\varepsilon\_\{\\bm\{\\omega\}\}\(\\bm\{x\}\)is the encoder network that maps input𝒙\\bm\{x\}to latent code𝒛\\bm\{z\}, and𝒟𝝆​\(𝒛\)\\mathcal\{D\}\_\{\\bm\{\\rho\}\}\(\\bm\{z\}\)is the decoder network reconstructing𝒙^\\hat\{\\bm\{x\}\}\. \(b\) Rydberg atom quantum reservoir: the latent code𝒛\\bm\{z\}is mapped to detuning parametersΔi\\Delta\_\{i\}, the system evolves under the Hamiltonian in Eq\. \([5](https://arxiv.org/html/2605.06727#S2.E5)\), and measurements of observables⟨Zi⟩,⟨Zi​Zj⟩,⟨Zi​Zj​Zk⟩\\langle Z\_\{i\}\\rangle,\\langle Z\_\{i\}Z\_\{j\}\\rangle,\\langle Z\_\{i\}Z\_\{j\}Z\_\{k\}\\rangleyield quantum embeddings\. \(c\) Surrogate model: a feedforward neural networkfsur​\(𝒛;𝜽sur\)f\_\{\\mathrm\{sur\}\}\(\\bm\{z\};\\bm\{\\theta\}\_\{\\mathrm\{sur\}\}\)trained everyNNepochs to approximate the quantum embeddings\. \(d\) Linear classifier: maps surrogate outputs𝒒′\\bm\{q\}^\{\\prime\}to class predictions, closing the loop with total lossℒ=\(1−λ\)​ℒR\+λ​ℒC\\mathcal\{L\}=\(1\-\\lambda\)\\mathcal\{L\}\_\{R\}\+\\lambda\\mathcal\{L\}\_\{C\}\.
### III\-BGuided Autoencoder

The Guided Autoencoder combines classical with quantum processing to leverage the strengths of both paradigms\.

Ideally, the latent space representation created by the encoderεω​\(𝒙\)\\varepsilon\_\{\\omega\}\(\\bm\{x\}\)would be passed through two sections of the model:

- •The Decoder Network𝒟ρ​\(z\)\\mathcal\{D\}\_\{\\rho\}\(\\bm\{z\}\),which tries to reconstruct the original input data\.
- •The Quantum Reservoir,which extracts highly dimensional quantum embeddings and passes them to a linear classifier to perform a classification task and map the embeddings to the image labels\.

However, some problems emerge when using a real quantum reservoir directly during the autoencoder training stage, which will be discussed in a later section\.

#### III\-B1Loss Function Design

The QGA is trained to minimize a composite loss function that balances reconstruction fidelity and classification accuracy:

ℒ=\(1−λ\)⋅ℒR\+λ⋅ℒC,\\mathcal\{L\}=\(1\-\\lambda\)\\cdot\\mathcal\{L\}\_\{R\}\+\\lambda\\cdot\\mathcal\{L\}\_\{C\},\(10\)where:

- •ℒR=‖𝒙−𝒙^‖2\\mathcal\{L\}\_\{R\}=\\\|\\bm\{x\}\-\\hat\{\\bm\{x\}\}\\\|^\{2\}is the reconstruction loss, measuring the mean squared error between the input𝒙\\bm\{x\}and its reconstruction𝒙^\\bm\{\\hat\{x\}\}\.
- •ℒC=−∑iyi​log⁡\(y^i\)\\mathcal\{L\}\_\{C\}=\-\\sum\_\{i\}y\_\{i\}\\log\(\\hat\{y\}\_\{i\}\)is the classification loss, computed as the cross\-entropy between the true labelsyiy\_\{i\}and the predicted probabilitiesy^i\\hat\{y\}\_\{i\}, which are obtained through a linear mapping of the approximated embeddings𝒒′\\bm\{q\}^\{\\prime\}\.
- •0<λ<10<\\lambda<1is a hyperparameter that controls the trade\-off between reconstruction and classification objectives\.

### III\-CThe Gradient Barrier Problem

Hybrid quantum\-classical models, such as the Quantum\-Guided Autoencoder \(QGAE\), aim to leverage quantum computational advantages within classical machine learning frameworks\. However, a significant challenge arises due to the non\-differentiable nature of quantum operations, we refer to this as the ‘gradient barrier’\.

In classical neural networks, training relies on backpropagation, which requires the computation of gradients through all components of the network\. In hybrid models, the inclusion of quantum layers introduces operations that are inherently non\-differentiable:

- •Quantum State Preparation: Encoding classical data into quantum states often involves operations that are not smoothly differentiable with respect to the input data\.
- •Quantum Measurements: Observing quantum states collapses them, introducing stochasticity and breaking the deterministic gradient flow required for backpropagation\.

These aspects hinder the direct application of gradient\-based optimization methods across the quantum\-classical boundary, obstructing end\-to\-end training of hybrid models\. The surrogate model solution, proposed in the next section[III\-D](https://arxiv.org/html/2605.06727#S3.SS4), addresses this problem\.

### III\-DSurrogate Modeling for Quantum Layers

To circumvent the gradient barrier, surrogate models have been proposed\. These are classical, differentiable models trained to approximate the input\-output behavior of quantum circuits\. By replacing the non\-differentiable quantum components with their surrogate counterparts during training, gradients can be propagated through the entire network, enabling end\-to\-end optimization\.

Recent studies have demonstrated the efficacy of surrogate models in mitigating the effects of barren plateaus and facilitating the training of hybrid quantum\-classical models\. These approaches leverage classical optimization techniques while still capturing quantum computational advantages\[[13](https://arxiv.org/html/2605.06727#bib.bib13)\]\.

#### III\-D1Surrogate Model Architecture

In our pipeline, the surrogate modelfsur​\(𝒛;𝜽sur\)f\_\{\\text\{sur\}\}\(\\bm\{z\};\\bm\{\\theta\}\_\{\\text\{sur\}\}\)is implemented as a feedforward neural network designed to learn the mapping𝒒​´\\bm\{q\}´from the autoencoder’s latent space to the quantum embeddings𝒒\\bm\{q\}produced by the quantum reservoir:

fsur​\(𝒛;𝜽sur\)=𝒒′≈𝒒=⟨ψ​\(g​\(𝒛\)\)\|O^\|ψ​\(g​\(𝒛\)\)⟩f\_\{\\text\{sur\}\}\(\\bm\{z\};\\bm\{\\theta\}\_\{\\text\{sur\}\}\)=\\bm\{q\}^\{\\prime\}\\approx\\bm\{q\}=\\langle\\psi\(g\(\\bm\{z\}\)\)\|\\hat\{O\}\|\\psi\(g\(\\bm\{z\}\)\)\\rangle\(11\)whereO^\\hat\{O\}represents the observables whose expectation values are to be determined\.

Its architecture comprises multiple fully connected layers with nonlinear activation functions, facilitating the approximation of the complex, non\-linear transformations inherent in quantum dynamics\.

#### III\-D2Training Procedure

- •Data Input: Feed a batch of latent data𝒛\\bm\{z\}from the autoencoder into the real quantum reservoir, recording the resulting quantum embeddings\.
- •Surrogate Training: Train the surrogate model by minimizing the loss function: ℒsur=‖𝒒′−𝒒‖2,\\mathcal\{L\}\_\{\\text\{sur\}\}=\\\|\\bm\{q\}^\{\\prime\}\-\\bm\{q\}\\\|^\{2\},\(12\)
- •Regular Updates: Periodically update the surrogate model during training to ensure it accurately reflects the quantum reservoir’s behavior\.

#### III\-D3Gradient Flow Through Surrogate Models

By integrating the surrogate model into the training pipeline, we establish a continuous computational graph from the output layer back to the input layer, enabling the use of gradient\-based optimization techniques\. The surrogate model serves as a differentiable proxy for the quantum reservoir, allowing the classification loss to influence the autoencoder’s parameters effectively\.

### III\-ERydberg Hamiltonian and Quantum Dynamics

While a fixed classical reservoir can nonlinearly mix inputs, its capacity is polynomially bounded\. In contrast, our neutral\-atom reservoir leverages many\-body quantum dynamics to encode data in a space whose dimension grows exponentially with atom number, unlocking patterns that classical reservoirs would need unreasonably complex architectures to approximate\. This means that the quantum reservoir is a crucial part of our pipeline and that the surrogate is just an approximation to aid in the autoencoder training process\.

#### III\-E1Data Encoding Schemes

The input data is encoded into the quantum system by mapping the latent representations from the autoencoder to the detuning parameters of the Rydberg Hamiltonian\. This encoding process involves adjusting the detuning parametersΔl​\(t\)\\Delta\_\{\\mathrm\{l\}\}\(t\)for each atom in the chain, allowing the system to represent the input data in a way that is compatible with the quantum dynamics of the reservoir\.

#### III\-E2Quantum Readout Methods and Embedding Dimensionality

The quantum reservoir’s output is obtained by probing, in successive time steps, the state of the atoms after the systems evolution up to a certain timett\. The readout is performed by measuring specific observables, such as⟨Zi⟩\\langle Z\_\{i\}\\rangle,⟨Zi​Zj⟩\\langle Z\_\{i\}\\,Z\_\{j\}\\rangle, and⟨Zi​Zj​Zk⟩\\langle Z\_\{i\}\\,Z\_\{j\}\\,Z\_\{k\}\\rangle, which measure respectively one\-, two\- and three\-qubit correlations\.

This means that the dimensionality of the QRC embeddings generated in our pipeline is a direct consequence of the number of qubits \(NN\) employed in the neutral\-atom chain, the specific quantum mechanical observables measured, and the number of discrete time steps \(TT\) over which these measurements are collected everyΔ​t\\Delta t\.

If we measure all of the three proposed observables, the total count of the values measured at a single time stepONO\_\{N\}is the sum of the counts for each type of observable:

ON\\displaystyle O\_\{N\}=N\+\(N2\)\+\(N3\)\.\\displaystyle=N\+\\binom\{N\}\{2\}\+\\binom\{N\}\{3\}\.\(13\)This set ofONO\_\{N\}measurements is repeated atTTsuccessive time steps during the evolution of the quantum reservoir\. Consequently, the total dimensionality,\|𝒒\|\\lvert\\bm\{q\}\\rvert, of the final QRC embedding vector is:

\|𝒒\|=T​ON\.\\lvert\\bm\{q\}\\rvert=TO\_\{N\}\.\(14\)This formulation shows that the embedding dimension scales polynomially with the number of qubitsNNand the number of timestepsTTas𝒪​\(T​N3\)\\mathcal\{O\}\(TN^\{3\}\)for the considered observables\.

## IVExperimental Setup

### IV\-ADatasets

We evaluate our proposed architecture on three datasets:

1. 1\.Synthetic Polyps:A synthetic dataset of polyp images generated to simulate realistic medical imaging scenes\.
2. 2\.CVC\-ClinicDB:Real image patches extracted from the CVC\-ClinicDB dataset, a well\-known benchmark for polyp detection\.
3. 3\.MNIST:A reduced version of the MNIST dataset, containing only the digits 0 and 1, suitable for binary classification tasks\.

### IV\-BFeature Reduction Methods

We implemented and compared three distinct feature reduction approaches:

#### IV\-B1Principal Component Analysis \(PCA\)

We extracted the top\-k principal components \(typically 4\-12\) from the flattened image data, which were then used as inputs to both classical and quantum\-enhanced classifiers\.

#### IV\-B2Autoencoder

We implemented a simple neural network\-based autoencoder with a encoder network that compresses the input to a lower dimensional latent space and a decoder network that reconstructs the original input from the latent representation\. To improve training stability and prevent overfitting, we also added batch normalization and dropout layers\.

The autoencoder was trained to minimize reconstruction loss using the Adam optimizer with learning rates between0\.0010\.001–0\.010\.01and weight decay for regularization\.

#### IV\-B3Quantum\-Guided Autoencoder

We introduced a quantum\-guided autoencoder that incorporates feedback from the quantum reservoir during training\. This approach combines:

- •A standard autoencoder architecture\.
- •A linear classifier that maps quantum embeddings to class labels\.
- •A classification loss computed on quantum embeddings\.
- •A weighting parameterλ\\lambdato control the trade\-off between reconstruction and classification objectives \([10](https://arxiv.org/html/2605.06727#S3.E10)\)\.
- •A differentiable surrogate model that approximates the quantum reservoir’s behavior, enabling end\-to\-end backpropagation through the entire pipeline\. The frequency of quantum updates is controlled by a hyperparameter, allowing for flexible integration of quantum dynamics into the training process\.

### IV\-CQuantum Reservoir Computing Layer

Our quantum reservoir computing implementation uses a Rydberg atom simulator with the following parameterizable components of an atomic chain:

- •Atom Chain Length \(number of qubits\):The number of atoms in the chain\.
- •Rabi Frequency:The global Rabi drive amplitude, which influences the strength of interactions between atoms\.
- •Lattice Spacing:The empty space between two consecutive atoms\.
- •Measurement Strategy:The specific observables measured to obtain quantum embeddings\.
- •Time steps:The number of time steps where the reservoir will be probed\.
- •Evolution Time:The time we let the quantum reservoir evolve under the Hamiltonian \([5](https://arxiv.org/html/2605.06727#S2.E5)\)\.

Unless explicitly stated, the results that we show in the next section[V](https://arxiv.org/html/2605.06727#S5)were obtained with the following hyperparameter values:1212qubits,10​μ​m10\\mu mbetween atoms,⟨Zi⟩\\langle Z\_\{i\}\\rangle,⟨Zi​Zj⟩,\\langle Z\_\{i\}Z\_\{j\}\\rangle,and⟨Zi​Zj​Zk⟩\\langle Z\_\{i\}Z\_\{j\}Z\_\{k\}\\rangleobservables, Rabi FrequencyΩ=π\\Omega=\\ \\pi,T=16T=16time steps and evolved over4\.0​μ​s4\.0\\mu s\(which means that the system was probed every4\.0T=Δ​t=0\.25​μ​s\\frac\{4\.0\}\{T\}=\\Delta t=0\.25\\mu s\.

We can use \([14](https://arxiv.org/html/2605.06727#S3.E14)\) to determine the dimension of our quantum embeddings\|𝒒\|\\lvert\\bm\{q\}\\rvertwhen using the mentioned parameters:

\|𝒒\|=16×123\+5×16×126=4768\\lvert\\bm\{q\}\\rvert=\\frac\{16\\times 12^\{3\}\+5\\times 16\\times 12\}\{6\}=4768\(15\)

### IV\-DLinear Classifier for the QRC Embeddings

After the traning process of the QGARS model, the following quantum embeddings are passed to a linear classifier\. The classifier is a simple feedforward neural network without hidden layers with the following architecture:

- •An input layer with the size of the embeddings\.
- •An output layer with two neurons\.
- •Cross\-entropy loss for training, optimized using the Adam optimizer\.

In total, this linear classifier has95389538parameters when the reservoir contains1212qubits\.

### IV\-EComparison Methods

We benchmarked the prediction accuracy of our model against the following classical methods:

- •PCA/Autoencoder \+ Linear Classifier:A linear classifier trained on the PCA\-reduced features or the latent representations from the autoencoder\. Essentially, this model is the same as the one used to classify the quantum embeddings described in section[IV\-D](https://arxiv.org/html/2605.06727#S4.SS4), but it only receives the classically extracted features as input\.
- •PCA/Autoencoder \+ Neural Network:A fully connected 4\-layer neural network with two100100\-neuron hidden layers, trained on the features from the same reduction methods as the linear classifier\. This neural network has1160211602parameters when training on1212dimensional inputs\.

We also benchmarked our QGARS encoding strategy against the performance of the Linear Mapping of QRC embeddings extracted from the latent representation of data using other feature reduction methods, such as PCA and the classical autoencoder\.

The accuracy results were measured with 2000 training images and 400 testing images over 5 different seeds\.

### IV\-FParameter Sweep Strategy

To optimize the performance of our quantum\-guided autoencoder, we conducted a systematic parameter sweep over the following hyperparameters:

- •Guided Lambda Parameter \(λ\\lambda\): To analyse the effect that the trade\-off between reconstruction and classification objectives in the loss function \([10](https://arxiv.org/html/2605.06727#S3.E10)\) affects the QRC
- •Quantum Update Frequency:The frequency at which the quantum reservoir is updated during training\. We experimented with update frequencies of 5, 7, 10 and 25 epochs\.
- •Number of Features:The number of features is the same as the number of qubits \(atoms\), as each feature encoded in each atom’s local detuning parameter\.

The results from these experiments are discussed in the next section[V](https://arxiv.org/html/2605.06727#S5)\.

## VResults and Discussion

### V\-ADimensionality Reduction Method Comparison

We first compare classification accuracy of the linear classifier on the QRC embeddings across the different dimensionality reduction methods PCA, vanilla autoencoder, and our QGARS pipeline on each dataset\. Table[I](https://arxiv.org/html/2605.06727#S5.T1)summarizes mean accuracy and standard deviation over five runs, using 12 features/qubits\. These results show that the QGARS pipeline is able to outperform traditional feature reduction methods commonly used as preprocessing steps on QRC pipelines\.

TABLE I:Mean ± standard deviation of the classification accuracy \(%\) QRC with different dimensionality reduction methods![Refer to caption](https://arxiv.org/html/2605.06727v1/x2.png)Figure 3:Accuracy of the different dimensionality reduction methods for QRC as a function of the number of qubits with the synthetic polyp dataset\. \(green\) QGARS, \(blue\) PCA and \(red\) Autoencoder\. The dashed lines represent the training accuracy, while the whole lines are the test accuracies\.![Refer to caption](https://arxiv.org/html/2605.06727v1/x3.png)

![Refer to caption](https://arxiv.org/html/2605.06727v1/x4.png)

![Refer to caption](https://arxiv.org/html/2605.06727v1/x5.png)

Figure 4:two\-dimensional t\-SNE projections of the 4768\-dimensional QRC embeddings for \(left\) PCA \+ QRC, \(center\) AE \+ QRC, \(right\) QGARS \+ QRCWe also evaluate the effect of the number of qubitsNNon the performance with the different dimensionality reduction methods as shown in Fig\.[3](https://arxiv.org/html/2605.06727#S5.F3)in general, it’s easier to train the model as we increaseNN, the slight performance drop as we go fromN=10N=10toN=12N=12might just be a byproduct due to the higher dimensionality of𝒒\\bm\{q\}, we expect that with a larger amount of input images, the accuracy would keep improving\. We hope to better understand this effect with future experiments\.

In order to have a visual demonstration of how well the quantum reservoir embeddings separate classes, we project the high\-dimensional embeddings down to three dimensions via t\-distributed Stochastic Neighbour Embedding \(t\-SNE\)\. Fig\.[4](https://arxiv.org/html/2605.06727#S5.F4)shows side\-by\-side plots for PCA → QRC, AE → QRC, and QGARS → QRC pipelines\. The QGARS embeddings exhibit the tightest and most distinct clusters, indicating superior discriminative structure\.

### V\-BAblation Studies

We verified that varying the guidance weightingλ\\lambdain \([10](https://arxiv.org/html/2605.06727#S3.E10)\) has significant impact on the performance of our accuracy\. To assess its influence, Fig\.[5](https://arxiv.org/html/2605.06727#S5.F5)plots test accuracy vs\.λ\\lambda\.

![Refer to caption](https://arxiv.org/html/2605.06727v1/x6.png)Figure 5:Accuracy as a function of the reconstruction/classification trade\-off parameterλ\\lambda\.We find optimal performance aroundλ=0\.5\\lambda=0\.5; too little guidance \(λ<0\.1\\lambda<0\.1\) approaches a point where it is equivalent to a traditional AE, so it yields under\-trained classifiers\.

A strange effect that we verified was that even using the valueλ=1\\lambda=1, the performance of the classifier is still competitive\. Although we still need to study this phenomenon, it suggests that the classical part of the autoencoder might not be as important as the surrogate component when producing a latent space that is appropriate for QRC classification\.

In future work, we plan to to conduct more ablation studies, mainly on the effect that the frequency with which we update the surrogate model and the influence of the quantum parameters in the reservoir layer\.

### V\-CComparison with Classical Methods

To benchmark the benefit of our QGARS pipeline, we compare against purely classical feature\-reduction approaches followed by standard classifiers\. Table[II](https://arxiv.org/html/2605.06727#S5.T2)reports test accuracy on each dataset for the different methods\.

TABLE II:Classification accuracy \(%\) for classical baselines \(2000 training images and 400 testing images\)In contrast, the QGARS pipeline achieves a maximum of92\.0%\\mathbf\{92\.0\\%\},90\.5%\\mathbf\{90\.5\\%\}ans99\.0%\\mathbf\{99\.0\\%\}on the same datasets \(see Table[I](https://arxiv.org/html/2605.06727#S5.T1)\)\.

#### V\-C1Training Loss Dynamics

Fig\.[6](https://arxiv.org/html/2605.06727#S6.F6)shows the training loss curves for the classical baselines and QGARS on the Synthetic Polyp dataset\. We plot the reconstruction loss for PCA/AE methods combined with classification loss for the AE\+NN and QGARS pipelines\.

Although the loss function of the QGARS pipeline shows signs of instability, which can be attributed to the high dimensionality of the quantum embeddings, it reaches a lower loss in comparison to the classical counterparts\.

## VILimitations and Future Work

### VI\-ACurrent Limitations

While QGARS demonstrates strong performance, several limitations remain:

- •Classical overhead:Frequent surrogate retraining and batch quantum simulations introduce non\-trivial classical compute overhead, potentially offsetting some of the quantum advantage\.
- •Data and Task Specificity:The current architecture has been evaluated primarily on specific medical imaging tasks\. Its generalization to other domains or types of data remains to be thoroughly investigated\.

![Refer to caption](https://arxiv.org/html/2605.06727v1/x7.png)Figure 6:Training loss vs\. epoch for PCA\+Linear, AE\+Linear, AE\+NN, and QGARS withλ=0\.7\\lambda=0\.7on the Synthetic Polyp dataset\.
### VI\-BPotential Extensions

Several avenues can further enhance QGARS:

- •Different Hamiltonians:Explore alternative interaction Hamiltonians \(such as the one from the Heisenberg XXZ model\) to better tailor reservoir dynamics, as experimented by A\. D\. Lorenzis et al\.\[[14](https://arxiv.org/html/2605.06727#bib.bib14)\]\.
- •Convolutional Autoencoders:Replace the fully connected autoencoder with a convolutional AE\[[15](https://arxiv.org/html/2605.06727#bib.bib15)\]to learn spatially localized features\. That would not only allow the model to better manage input size but also provide a richer structure for the reservoir\.
- •Surrogate Model Complexity:The surrogate model required to emulate the complex quantum reservoir has a very large number of parameters\. We plan to investigate alternative architectures, such as Physics\-Informed Neural Networks \(PINNs\)\[[16](https://arxiv.org/html/2605.06727#bib.bib16)\], which use physical laws to potentially improve model fidelity\.
- •Experiment on Real Quantum Hardware:Although the experiments we conducted on simulators showed promising results, to validate the usefulness of our pipeline in the NISQ era, we still need to run experiences on real quantum processing units \(QPUs\) with more qubits\.

## VIIConclusion

We have presented Quantum\-Guided Autoencoder with Reservoir Surrogate \(QGARS\), a hybrid quantum\-classical architecture that integrates a classical autoencoder with a neutral\-atom quantum reservoir and a differentiable surrogate model to enable end‐to‐end training\. By jointly optimizing reconstruction and classification losses, QGARS learns compact and discriminative features tailored for quantum reservoir computing\. Our experiments on synthetic polyp, CVC\-ClinicDB, and binary MNIST datasets demonstrate that QGARS outperforms classical PCA‐ and autoencoder‐based baselines that are commonly used for quantum reservoir computing pipelines\. Ablation studies highlight the importance of guidance weightλ\\lambda, surrogate update frequency, and reservoir parameters\. Although challenges remain—particularly regarding high‐dimensional embeddings, surrogate fidelity, and classical overhead, our proposed extensions and hardware considerations chart a clear path toward robust, scalable quantum‐enhanced medical imaging pipelines in the NISQ era\.

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