# Quantum Denoising in the Realm of Brain-Computer Interfaces: A Preliminary Study

Ahmed Mostafa, Yash Chauhan, Wasif Ahmed, Gregoire Cattan

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Ahmed Mostafa, Yash Chauhan, Wasif Ahmed, Gregoire Cattan. Quantum Denoising in the Realm of Brain-Computer Interfaces: A Preliminary Study. October 6 University, Egypt; National Institute of Science Education and Research, India; Lahore University of Management Sciences, Pakistan; IBM, Poland. 2024. [hal-04501908](https://hal.science/hal-04501908v1)

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# Quantum Denoising in the Realm of Brain-Computer Interfaces: A Preliminary Study

Ahmed Mostafa¹, Yash Chauhan², Wasif Ahmed³, and Gregoire Cattan⁴

¹October 6 University

²School of Physical Sciences, National Institute of Science Education and Research, India

³Lahore University of Management Sciences (LUMS) and

⁴IBM, Data and AI, Poland

Passive Brain-Computer interfaces based on electroencephalography (EEG) data require calibration-free classifiers that can extract features from the EEG, even when the signal-to-noise ratio is low. This research explored self-supervised learning with a particular focus on its attention mechanism for extracting features from EEG. We specifically investigated quantum autoencoders due to their originality and relevance to neural networks as well as their application for signal denoising. First, we trained a classifier on EEG data from a cross-session experiment in which participants performed tasks of varying difficulty levels. Second, we compared the performance of the classifier with and without the use of quantum autoencoders for feature extraction from a noisy signal. This step is called quantum denoising. Surprisingly, the findings indicated that there was no advantage to using quantum denoising for feature extraction. The pipeline without quantum autoencoders also struggled to generalize effectively, revealing inherent limitations. Notably, computational complexity led to extreme dimensional reduction. This study serves as a proof of concept for the technical feasibility of quantum autoencoding with real time-series data, which identifies avenues for future exploration.

**Keywords:** Self-Supervised Learning, Quantum Computing, Autoencoder, Brain-Computer Interface, Electroencephalography (EEG), Riemannian Optimization

## I. INTRODUCTION

Research on brain-computer interfaces (BCIs) started in the early 1970s with the work of Vidal et al.,¹ who designed an interface to control a cursor on a computer screen using only electroencephalography (EEG) signals.² This type of BCI, known as an active BCI, offers direct communication between the brain and a computer, bypassing the usual muscular pathway. For this reason, research has strived to adapt BCI technology for people suffering from severe motor disabilities. However, such research has faced several drawbacks, namely the BCI's cumbersomeness, hardware costs, a lack of system reactivity (i.e., low accuracy and/or low bit rate), and the need for calibration before use.³ In contrast to active BCIs, passive BCIs have tremendous potential for both clinical and public applications. They are responsive interfaces that monitor and adapt to the user's cognitive state without his/her active participation.⁴ For example, they are used to detect fatigue while driving, adapt a task's difficulty to the cognitive workload, or adapt a speech to the listener's emotional state. However, to achieve a passive BCI, it is necessary to train calibration-free classifiers that can extract features across different sessions without being retrained, even when the signal-to-noise ratio is low.

### A. Transformer Architecture

The Transformer, a revolutionary deep learning model, has left an indelible mark on diverse domains, such as natural language processing (NLP) and computer vi-

sion (CV).¹⁰ Initially crafted for machine translation, the Transformer has evolved into a dominant force in NLP, particularly with the prevalence of pretrained models (PTMs). Its impact extends seamlessly into CV, audio processing, and other fields. At the core of the Transformer lies the self-attention mechanism, a pivotal element that allows each word or token to comprehend the entirety of the input sequence, thereby capturing the global context. Multi-head attention operates in parallel, enabling attention to be focused on different aspects. Positional encoding injects vital positional information into word embeddings, while feed-forward neural networks (FFNs) apply independent nonlinear transformations at each position. Layer normalization ensures stable training by normalizing activities within each layer. The architectural framework comprises an encoder and a decoder (FIG. 1), with the former incorporating an embedding layer for converting input tokens into dense vectors. Stacked self-attention and FFN layers within the encoder capture the context and facilitate learning representations. Training the Transformer involves a two-step process - pre-training on a substantial corpus for unsupervised learning of general language representations, followed by fine-tuning to adapt the pre-trained model to specific tasks. This adaptability is exemplified in applications such as text classification, question answering, language modeling in NLP, as well as in computer vision and audio processing tasks. In essence, the Transformer's modular architecture, driven by the self-attention mechanism and bolstered by pre-training capabilities, has not only revolutionized deep learning but has also redefined the landscape across a spectrum of domains.
<figure><img src="image_2.png" /><figcaption>FIG. 1. The encoder-decoder structure of the Transformer architecture.²⁰</figcaption></figure>

## B. Attention Mechanisms and EEG Signal Denoising

Attention mechanisms have emerged as valuable tools for denoising EEG signals.¹¹ EEG records the brain's electrical activity using scalp electrodes; however, inherent noise from sources such as muscle activity (EMG), eye movements (EOG), and cardiac activity (ECG) necessitates effective denoising for accurate interpretation. Traditional filtering methods, such as bandpass filters, face limitations because while they may eliminate noise, they risk distorting the underlying EEG features. To further complicate matters, EEG signals exhibit nonstationary behavior by varying across time and frequency. To overcome these challenges, attention mechanisms offer a targeted approach to analyzing specific time-frequency components of EEG signals. Methods such as nonlocal means (NLM), bidimensional empirical mode decomposition (BEMD), and short-time Fourier transform (STFT) are used.¹¹ While BEMD breaks signals down into sub-time-frequency components to remove noise, NLM uses structural self-similarity to locally smooth pictures and successfully remove noise while maintaining important features. Furthermore, STFT reveals frequency content across time. Researchers have proposed a comprehensive denoising strategy that combines these techniques to produce smoother waveforms and enhance the fidelity of EEG signals. Validation through correlation coefficients between denoised single-trial EEG signals and reference signals has demonstrated the effectiveness of the proposed method, which surpasses the performance of other denoising techniques. In conclusion, attention mechanisms

significantly contribute to improving the quality of EEG data by effectively addressing the inherent challenges and noise in these signals. The combination of STFT, BEMD, and NLM presents a promising approach for EEG denoising and has showcased notable results in terms of the enhancement of signal fidelity.

# II. METHODOLOGY

## A. Data

This study's dataset comprised an open EEG dataset tailored for passive BCI applications.⁵,¹⁸ It encompassed EEG recordings from 15 participants (6 female, with an average age of 25 years). A total of 62 active Ag–AgCl electrodes were available in the dataset. The reference was to the right mastoid electrode. The participants engaged in 3 distinct experimental sessions, each of which was separated by 1 week. At the beginning of each session, the resting state of the participant (measured as 1 minute with eyes open) was recorded. Subsequently, participants undertook 3 tasks of varying difficulty levels (i.e., easy, medium, and difficult). The task assignments were randomized. A 2-second nonoverlapping epoch was obtained for each task.

<figure><img src="image_3.png" /><figcaption>FIG. 2. EEG acquisition process with different difficulty levels (adapted from Huang et al).²¹</figcaption></figure>

### 1. Preprocessing

The data in the dataset had already been preprocessed with a pass-band FIR filter between 1 and 40 Hz. Electrodes with an average amplitude above 2 sd across channels were rejected, and the rejected channels were further interpolated with spherical interpolation. An artifact rejection (ICLabel) and an average re-referencing (CAR) were also applied to the data. We downsampled the data to 128 Hz and applied an additional pass-band IIR filter between 1 and 35 Hz. FIG. 2 presents a schematic progression of the EEG acquisition. We extracted all easy (n = 149) and medium-difficulty (n = 149) epochs starting from 0 to 1 s after the onset of stimulation and applied spatial filtering using xDAWN (number of filters = 1).¹⁹
The use of a spatial filter allowed us to reduce the epochs' duration (and therefore the computation time) while improving the signal-to-noise ratio. The value of 1 filter was set in consideration of the computational limitations of the quantum emulator.

## 2. Quantum Denoising

We applied an additional quantum denoising technique to the data and proceeded to compare the performance of the pipeline with and without this denoising step. The quantum denoising process involved employing a quantum tum autoencoder, notable for its utilization of a variety of theoretical quantum circuits. A variational quantum circuit is a quantum circuit that contains parametric gates, where the weights are determined through the optimization of a loss function. Essentially, variational quantum circuits are very close to neural networks.

## B. Autoencoders

According to Qiskit's Quantum Autoencoder tutorial,³² a classical autoencoder (CAE) is a kind of neural network used for learning effective data representations and data reduction. It is composed of 3 main layers: the input layer, which receives data of a specific length n; the bottleneck layer, which is a dimension-reducing encoder; and the output layer, which is tasked with restoring the data to its original dimensions. This architecture aids in reducing the data's dimensionality while retaining important information from the input. CAEs are applied in various fields, such as image denoising, anomaly detection, and facial recognition. By contrast, a quantum autoencoder (QAE) serves a similar purpose to its classical counterpart but operates on quantum states instead. QAEs also feature 3 layers, starting with the input layer, where a quantum state $|\Psi\rangle$ consisting of n qubits is introduced. It then proceeds to an encoder, which is a parametrized quantum circuit that compresses the quantum state into fewer qubits, specifically n - k qubits, which results in $|\Psi_{comp}\rangle \otimes |0\rangle^{\otimes k}$. Here, $|\Psi_{comp}\rangle$ embodies the compressed quantum state. This setup is optimized through training to minimize a loss function.

Next, the bottleneck layer captures the essence of the compressed state, discarding the k superfluous qubits. These n - k qubits are often termed the latent space, while the discarded k qubits are known as the trash space. The final stage, namely the output layer, reintroduces k qubits in the $|0\rangle$ state and applies a decoding parametrized circuit to reconstruct the initial quantum state from its compressed form. In practical quantum circuits, such as those designed with Qiskit, the addition or removal of qubits within a circuit is not directly achievable. Thus, the initial setup must include

the input state alongside a reference state, an auxiliary qubit, and a classical register for measurements, all of which are established at the start of the circuit.

## C. Loss Function

The goal of the cost function is to maximize the fidelity between the output state and the input state. We use the same cost function as in the Qiskit tutorial.³² Let's define A and B the latent and trash spaces, respectively. $|\Psi_{AB}\rangle$ represents the input state, which consists of n qubits. We present the reference state $|a\rangle_{B'}$, which comprises the extra k qubits utilized in the decoder, in the reference space B'. Our encoder is the parameterized circuit, represented as $U(\theta)$. To achieve maximal state compression, the parameter $\theta$ must be adjusted. In the decoder, $U^{\dagger}(\theta)$ is used. As $\max F(\Psi_{AB}, \rho_{out})$, our goal is to maximize the fidelity between the input and output states.³² $\rho_{out}$ is defined as follows:

$$ \rho_{out} = U^{\dagger}(\theta)_{AB'} \mathrm{Tr}_B[U(\theta)_{AB} [\Psi_{AB} \otimes a_{B'}] U^{\dagger}(\theta)_{AB}] U(\theta)_{AB'} $$

The encoder's parametrized circuit must have its settings adjusted to maximize its fidelity. However, as the number of qubits increases, determining fidelity between 2 states can become complex as it requires more gates and leads to deeper circuits. To simplify the computation of fidelity, one can implement a swap gate between the garbage and reference states.¹⁵ Our cost function is now a function of the trash state:

$$ \max F(\mathrm{Tr}_A[U(\theta)AB\Psi ABU^{\dagger}(\theta)AB], aB'), $$

since the reference state is fixed. The encoder's parameters are changed during training, and the fidelity between the trash and reference states is evaluated using a swap test. The swap test and fidelity measurement are made easier by the inclusion of an additional qubit and a classical register during circuit initialization.

## D. SWAP Test

The swap test is frequently employed to evaluate the similarity between 2 quantum states using CNOT gates on their respective qubits. Implementing the circuit multiple times, with the number of iterations represented by M, and conducting the swap test allows for the measurement of the auxiliary qubit. The calculation of the frequency at which states in the $|1\rangle$ state are observed is given by the following equation:

$$ S = 1 - \frac{2L}{M} $$

where L is the number of occurrences of the $|1\rangle$ state. The objective is to enhance S, which indicates the equivalence of the 2 states being compared. Thus, the aim is to maximize this metric, thereby effectively minimizing the $\frac{2L}{M}$ term. This expression is therefore utilized as the cost function in the comparison process.
## E. Parametrization

EEG epochs (149 + 149 trials, 2 channels, 64 samples) were vectorized (one epoch = 128 samples [2 × 64]) and passed as input of a quantum variational autoencoder (QVAE). Our QVAE architecture consisted of a 7-qubit parametrized quantum circuit that used the *RealAmpli-tudes* ansatz from Qiskit as the encoder circuit. The number of qubits (i.e., 7) was dependent on the number of samples in an EEG vector (128 samples = 2⁷). We used 6 qubits for the latent space and 1 qubit for the trash space. We measured the fidelity of the reconstruc-tion using a swap test between the encoded input state and the decoded output state. This fidelity loss was op-timized using the COBYLA local optimization routine within Qiskit to train the parameters of the *RealAmpli-tudes* encoder array. In practice, we found a fidelity of approximately 0.3 for each training set. The output and, theoretically denoised, vectors of EEG data (one vector = 128 samples) was then reshaped into matrices of size 2 channels by 64 samples.

## F. Feature Extraction

After preprocessing and denoising, the epochs of the signal (2 channels, 64 time samples) were transformed into symmetric positive definite (SPD) sample covariance matrices (SCMs). These source signals can, in general, be represented in the form of a spatial covariance matrix in the following form:

$$C_i = \frac{1}{T-1} X_i X_i^T$$

where $X_i^T$ denotes the transpose of the matrix $X_i$. These spatial covariance matrices are semi-positive ma-trices that lie on a smooth Riemannian manifold. The matrices contained 4 (2 × 2) components. Considering the Riemannian geometry of SPD matrices, we vectorized these matrices by projection into the tangent space of the Riemannian manifold.⁷ All vectors contained 3 elements (2 [2 + 1] / 2) and were passed to a linear discriminant analysis (LDA) classifier.

## G. Evaluation

The LDA performance (with and without quantum denoising) was assessed with a 5-fold, cross-validation, cross-session evaluation and measured using the area under the curve (AUC) of the classifier. To avoid implemen-tation bias, performance was assessed within MOABB,⁸ a BCI benchmark for reproducible results.

<figure><img src="image_4.png" /><figcaption>FIG. 3. Performance of the LDA classifier with and without quantum denoising.</figcaption></figure>

## III. RESULTS

The performance of the LDA classifier without and with quantum denoising is depicted in FIG. 3. In the figure above, the pipeline with quantum denoising is denoted as “LDA denoised.” Compared with the conventional LDA, it revealed reduced variance, with a notable decrease in the receiver operating characteristic (ROC) AUC score’s variability. This reduction implied that, through employing quantum denoising (specifically using the QVAE), the classification outcomes became more consistent. However, the LDA denoised pipeline failed to generalize effectively, as indicated by the accuracy plateauing around 0.5. This suggests that the QVAE might not have captured sufficient information in its latent space. By contrast, the conventional LDA pipeline exhibited high variance, operating without quantum denoising, and demonstrated a broader range of ROC AUC scores. This variance can be attributed to the substantial noise ratio present in the dataset. Despite this noise influence, the LDA pipeline still achieved commendable classification accuracy, showcasing resilience in the face of simulation constraints.

<table><caption>TABLE I. Precise score and time required for the classification.</caption><thead><tr><td>Pipeline</td><td>Score</td><td>Time</td></tr></thead><tbody><tr><td>LDA</td><td>0.542521</td><td>0.093218</td></tr><tr><td>LDA_denoised</td><td>0.497668</td><td>8933.458008</td></tr></tbody></table>

The average time for a session evaluation was 8933.45s for LDA denoised and 0.09s for LDA only (TABLE I). This illustrates that quantum simulation requires extensive resources.

## IV. DISCUSSION AND CONCLUSION

Due to computational constraints, our study has several important limitations. We used an xDAWN
spatial filter for initial dimensionality reduction before inputting data into the quantum pipeline. Out of the 64 channels, only 2 EEG components were selected by the filter. The dimensionality of the data was further reinforced by projecting the sample covariance matrices into the tangent space of the Riemannian manifold. Selecting a higher number of components (e.g., 3-4) could have preserved more salient information, leading to better results for the 2 pipelines (quantum and classical). Moreover, our QVAE used a latent space of 6 qubits along with 1 trash qubit. This small latent space likely limited the representational capacity for encoding intricate signal features.

Additionally, our QVAE pipeline used the quantum swap test as the cost function. While the swap test provides a useful quantum-native cost evaluation, it lacks gradients. We relied on the gradient-free COBYLA optimization routine, but its performance lags behind gradient-based techniques. Using alternate cost functions that are compatible with quantum gradient estimation may improve the optimization. Crucially, even with the severe dimensionality reductions that we employed, our quantum pipeline simulations still required over 3 days to complete. Continued hardware and software advances will enable less constrained QVAE architectures to be similar. Future studies should allocate additional resources to modeling larger latent spaces for capturing intricate signal characteristics. Overall, this work faced significant dimensionality restrictions. Lifting some of those limitations, coupled with enhanced classical-quantum integration, presents many opportunities for improving mental workload classification. The incorporation of Riemannian optimization into EEG signal classification tasks is a potential direction for future research. This geometrically informed optimization framework could potentially enhance the efficiency of training variational quantum classifiers, thus leading to improved accuracy and robustness.

In conclusion, this research contributes to the emerging field of quantum computing applications in neuroscience, specifically within the realm of BCIs. While the preliminary results may not conclusively demonstrate quantum superiority, the study lays a solid foundation for future investigations. The identified research directions hold the potential to address the limitations encountered and push the boundaries of quantum-assisted EEG signal processing, thereby fostering advancements at the intersection of quantum computing and BCIs.

# ANNEX

## Riemannian Optimizer

In this study, we used a COBYLA optimizer for our quantum denoising. However, the Riemannian optimizer could be an interesting alternative. Leveraging the geometric structure of EEG signals, the Riemannian gradient flow optimizer introduced here may allow for more effective training of the variational classifier. Key search directions would be to apply the Riemannian optimization¹⁷ scheme to EEG classification benchmarks. We hypothesize that the geometrically aware optimizer can more efficiently traverse the optimization landscape and find improved minima, possibly leading to higher model accuracy and robustness. Additionally, incorporating differential geometry into the circuit arrangement itself through parameterizations that respect the manifold structure may further enhance the performance. The interplay between geometrically inspired circuit architecture and Riemannian optimization warrants a dedicated investigation for EEG classification models.

## Karcher Mean and Riemannian MDM

Since EEG signal measurements form a time series on a smooth manifold and thus have a differential geometric structure, we can implement different means or metrics for our optimization problem. Moreover, by analyzing alternative metrics to the one that we implemented, we can perform a comparison test on our results with different metrics to determine which one works best. The choice of the Riemannian manifold for studying a manifold is critical. A wide number of metrics have been proposed for studying the manifold of SPD matrices (e.g., log-Euclidean, Cholesky, and Procrustes). In the treatment of EEG signals, we generally employ Riemannian manifolds. A manifold becomes a Riemannian manifold when it is endowed with an affine invariant Riemannian metric (AIRM), where each covariance matrix becomes a point on this manifold. Because of its invariant nature, the metric is invariant to affine transformations; therefore, the sensor and source spaces are equivalent under this setting.

For our optimization problem, we consider convex sets on a smooth manifold; thus, it is a convex optimization problem that we seek to solve. A convex set on a smooth manifold is a set C in a Riemannian manifold, and (M,g) is called convex if for any pair of points x, y ∈ C any minimizing geodesic [x, y] lies in C.³³ A convex function on a Riemannian manifold is a real-valued function whose geodesic distance or arc is convex in nature. When treating EEG data, the manifold of SPD matrices denoted by

$$P(n) = \{p \in \mathbb{R}^{n \times n} | a^T p a > 0 \forall a \in \mathbb{R}^n \setminus \{0\}\}$$

and the tangent space at $T_p P(n)$

$$T_p P(n) = \{X \in \mathbb{R}^{n \times n} | X = X^T\},$$

(i.e. the set of symmetric matrices) is studied when the manifold is AIRM$^{6,31}$. Since we tend to optimize the
signals that we obtain, we find the geodesic distance between 2 covariance matrices. This can be described as:

$$ \delta(C_i C_j) = \|\log m C_i^{-\frac{1}{2}} C_j C_i^{-\frac{1}{2}}\| = \sqrt{\sum_{i=1}^{E} \log^2 \lambda_q} $$

where, E represents the number of vectors and λ represents the eigenvalues of the matrix $C_i^{-\frac{1}{2}} C_j C_i^{-\frac{1}{2}}$. One problem related to the source signals is that we do not necessarily know the nature of the sources of these signals, whether they are from a single or mixed source. If the recorded activity is from a mixed source, then the covariance matrix can be linearly approximated through a linear mixture of source signals described as follows:

$$ X_i = M S_i $$

where M represents the mixing matrix and $S_i$ represents the source signals. Substituting the above to find our covariant matrix $C_i$, it is given by the following equation:

$$ C_i = \frac{1}{T-1} M S_i S_i^T M^T $$

While we have described the covariance matrices in the SPD manifold, finding the distance between covariance matrices in the SPD manifold is a quadratic optimization problem. A quantum method can be useful for finding the best solution for such a problem. The approach involves transforming the problem into a constraint programming model, which can be solved by optimizers such as QAOA.34 Positive-definite matrices are pivotal in numerous fields, such as computer science, engineering, physics, and applied mathematics, and they play a crucial role in a wide range of applications. These matrices are notably present in covariance matrices within statistics, where they serve as fundamental components in convex and semidefinite programming's search spaces; act as kernels in machine learning algorithms; form the basis of density matrices within quantum information theory; represent data points in radar imaging; and constitute diffusion tensors that are crucial for medical imaging techniques, among other applications. A common task involving these matrices is their averaging, which leads to the exploration of the Karcher mean as a suitable method for metric computation. This approach seeks to provide a unified method for averaging matrices that preserves the properties essential for their diverse applications.

In normal flat geometry, the average or mean of points is found by simply adding them together and dividing by the number of points. However, when dealing with curved spaces, this process is not as straightforward because the concept of a "straight line" between points (which is used in calculating the mean in Euclidean space) does not apply in the same way. Thus, we use the Karcher mean as the equivalent in a manifold space in this context. The Karcher mean of N positive definite

matrices $A_1, A_2, \dots, A_n$ is defined as the minimizer of the sum of squares of the Riemannian trace metric distances to each of the $A_i$; that is,29

$$ \Lambda(A_1, A_2, \dots, A_n) = \arg \min_{X \in P} \sum_{i=1}^{n} \delta^2(X, A_i) $$

The Karcher mean is an example of an optimization problem that could be solved with a quantum optimizer, such as QAOA34 or CV-QAOA35, which in theory provide excellent results with quadratic optimization. This could be a viable choice as it considers the geometry and distances specific to that space. This is crucial for optimization problems where the underlying structure is non-Euclidean.

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