Quantum-Enhance Signal Processing via VQE for Improved Biomechanical Feedback Control

Digit. Signal Process. 166 (2025) 105357 Available online 3 June 2025 1051-2004/© 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/bync-nd/4.0/). Contents lists available at ScienceDirect Digital Signal Processing journal homepage: www.elsevier.com/locate/dsp Quantum-enhanced signal processing via VQE for improved biomechanical feedback control Javier Villalba-Díeza,b, ,∗, Joaquín Ordieres-Meréc, aFakultät Wirtschaft, Hochschule Heilbronn, Max-Planck-Str.39, Heilbronn, 74081, Baden-Württemberg, Germany bDepartment of Mechanical Engineering, Universidad de La Rioja, Edificio Departamental, c/ San José de Calasanz, 31, Logroño, 26004, La Rioja, Spain cEscuela Técnica Superior de Ingenieros Industriales, Universidad Politécnica de Madrid, C. José Gutierrez Abascal, 2, Madrid, 28006, Madrid, Spain A R T I C L E I N F O A B S T R A C T Keywords: Variational Quantum Eigensolver (VQE) Quantum-classical hybrid algorithm Biomechanical sensor analysis Feedback control system Wearable technology The Variational Quantum Eigensolver (VQE) is a hybrid quantum-classical algorithm that has demonstrated significant potential for solving quantum chemistry problems, particularly in determining the ground state energy of small molecules like H2. In this paper, we extend the application of VQE beyond quantum chemistry by utilizing it to analyze sensor data from engineered socks equipped with an accelerometer, and gyroscope sensors. Our goal is to explore the sensitivity of accelerometer and gyroscope signals to specific motion frequencies by encoding their data into the quantum states of the H2molecule’s qubits. We introduce an automatic control mechanism based on a classical feedback loop, where the output of the VQE is compared to the desired input, and corrective actions are applied using a constant 𝐾to ensure the output follows the input closely. This feedback loop is designed to assist the algorithm in managing local minima, noise, and computational challenges. Using this feedback-controlled VQE system, we optimize sensor signal analysis and determine which sensor exhibits higher sensitivity to specific biomechanical frequencies. Our experimental results indicate the potential flexibility of VQE in analyzing specific biomechanical data, providing preliminary insights into the broader applications of quantum algorithms in wearable technology. As quantum hardware advances, VQE may offer applications in complex systems and diverse fields, including personalized healthcare and motion capture. 1. Introduction The increasing intersection of quantum computing with real-world applications is opening new avenues to address complex data-driven problems [1–3], beyond the traditional domains of quantum chemistry [4–6]. An area of growing interest is the application of quantum algorithms to wearable technology [7,8], particularly in the field of healthcare care, where motion analysis and biomechanical data play a crucial role in patient monitoring and diagnostics [9,10]. Wearable devices, such as sensor-equipped socks, provide continuous, high-resolution data on a patient’s movement, gait, and balance, allowing real-time tracking of motor function [11,12]. However, analyzing these data in a timely and precise manner remains a significant challenge for classical algorithms due to the complexity and variability inherent in human motion [13]. In this paper, we propose leveraging the distinctive and transformative advantages of quantum information, specifically through the Variational Quantum Eigensolver (VQE) [14], to process and interpret biomechanical data from wearable sensors. *Corresponding author at: Fakultät Wirtschaft, Hochschule Heilbronn, Max-Planck-Str.39, Heilbronn, 74081, Baden-Württemberg, Germany. E-mail address: [email protected] (J. Villalba-Díez). The true strength of quantum information lies in its fundamentally different nature, enabling a more natural and efficient representation of problems and simplifying the mapping between the problem space and the algorithm. This capability offers novel pathways for analyzing complex datasets that are challenging for classical methods. Unlike traditional classical systems, VQE allows for an intuitive encoding of sensor data into quantum states, taking advantage of quantum mechanics’ representational richness. In principle, the two-qubit VQE framework can be extended to an 𝑁-qubit system for more complex sensor inputs. However, scaling up involves increased circuit depth and more elaborate state preparation routines as 𝑁grows, which we discuss as follows: (i) the data-embedding step may require 𝑂(2𝑁)gates in the worst case, motivating the use of structured or block-encoding techniques; (ii) the Hamiltonian measurement overhead grows with the number of Pauli terms, requiring grouping strategies; and (iii) variational Ansatz complexity and the risk of barren plateaus increase, necessitating problem–informed Ansätze and subsequent initialization. We also outline a modular vision using multiple small ‘H2-based chiplets’ https://doi.org/10.1016/j.dsp.2025.105357 Digital Signal Processing 166 (2025) 105357 2 J. Villalba-Díez and J. Ordieres-Meré to parallelize larger workloads, in alignment with emerging hardware paradigms [15]. These characteristics ensure robustness and reliability in the analysis process. To further enhance the system’s performance, we incorporate a classical feedback mechanism into the VQE framework. This feedback loop dynamically adjusts the algorithm, optimizing its ability to identify subtle patterns in real–time sensor data, particularly for distinguishing between healthy and impaired movement patterns. The feedback-controlled VQE system analyzes data collected from sensor-equipped socks embedded with accelerometers, and gyroscopes [16,17], which capture detailed biomechanical metrics such as foot movement distribution during activities like walking. By encoding these signals into quantum states, we utilize the quantum system’s sensitivity to explore specific frequencies and motion characteristics. The feedback mechanism not only mitigates discrepancies and noise but also enhances the precision of biomechanical data analysis. This approach demonstrates the versatility of quantum algorithms like VQE beyond their traditional applications in quantum chemistry, offering promising insights into wearable sensor data analysis and underscoring the potential of quantum information to address real-world challenges. The hybrid nature of VQE, combining the strengths of quantum and classical computation, aligns well with the requirements of the noisy intermediate-scale quantum (NISQ) era [18,19], where quantum hardware is still limited by decoherence and gate errors. The reliance of VQE on classical optimization [20] shows to be more noise tolerant compared in this case to fully quantum algorithms, such as Quantum Phase Estimation (QPE) [21], which require deeper circuits and more qubits [22]. Furthermore, the flexibility of VQE allows it to be applied to a wide range of quantum systems, from small molecules such as H2to more complex materials and chemical reactions [23–25], provided the Hamiltonian  𝐻and are tailored accordingly. It should be emphasized that our choice of a two-qubit VQE realized on the simplest diatomic molecule, H2, is a strategic step toward a new quantum computational framework tailored for emerging hydrogen–based hardware [26]. Hydrogen is the most abundant and inexpensive element in nature, and hydrogenic qubits [27], whether realized as proton-spin registers in H2complexes [28], hydrogen-terminated silicon quantum dots [29], or H2absorbed on two-dimensional materials—offer the promise of compact, photonic quantum modules [15]. By demonstrating the full VQE pipeline on H2, we achieve conceptual clarity, exact validation against known ground-state energies, and seamless compatibility with current simulators and prototype devices. This foundational demonstration establishes the design principles for “H2chiplets” that can be tiled or integrated into classical digital signal processing architectures, thus enabling specialized quantum co–processors without the need for large, monolithic quantum computers. However, VQE also faces challenges, particularly in the selection of an effective [30]. The expressiveness must be sufficient to capture the ground state while remaining feasible for implementation on noisy quantum hardware [31]. Moreover, the quantum measurements required to evaluate the Hamiltonian  𝐻can introduce significant overhead, as large numbers of shots may be needed to achieve the desired precision [32]. Finally, the optimization landscape in VQE can be fraught with local minima, making it difficult for classical optimizers to find the true global minimum, especially for larger, more complex systems [33,34]. To address some of the challenges outlined above we propose the introduction of an automatic control system with a classical feedback loop [35]. In such a system, the output of the VQE optimization process is continuously compared to the desired input (the true ground state energy or target optimization value), and corrective actions are taken in real-time to minimize deviations. This feedback mechanism operates by applying a constant correction factor 𝐾to the system, dynamically adjusting the parameters to better align the quantum output with the classical optimization goals. We introduce the feedback loop in order to reducing the risk of becoming trapped in local minima by continuously guiding the system toward the global minimum [36]. Furthermore, this automatic control system can mitigate the effects of quantum noise and measurement uncertainty by enforcing stability in the optimization process, thereby improving the convergence rate and precision of the energy estimation. By integrating this feedback-based approach into the VQE framework, we aim to create a more robust algorithm that can handle the noise and complexity inherent to quantum devices, while ensuring that the optimization process consistently follows the desired trajectory toward the ground state energy. We found a practical application of the feedback-controlled VQE system, and have engineered a wearable device in the form of socks equipped with distinct sensors on each foot: one accelerometer, one gyroscope, and one magnetometer. Each sensor captures specific biomechanical data related to foot movement distribution. Our goal is to use the quantum variational eigensolver to analyze sensor data in real time, leveraging the quantum system’s sensitivity to identify patterns that may be undetectable only through classical methods. Specifically, we will feed the accelerometer and gyroscope signals from each foot into the VQE by encoding the data in the quantum states of a simplified hydrogen (H2) molecule, which consists of two qubits. The accelerometer and gyroscope data are mapped to the quantum states of these qubits, allowing us to evaluate the quantum response to different frequencies and motion patterns. In doing so, our objective is to determine which sensor, accelerometer or gyroscope, is most sensitive to specific frequencies or motion-related variables that are crucial for understanding gait dynamics and balance control. The feedback loop previously discussed will ensure that the VQE system is automatically adjusted to optimize the analysis, dynamically correcting for noise or discrepancies in the sensor signals. The results of this experiment will provide valuable insight into sensor sensitivity, ultimately enhancing the understanding of real-world motion capture and sensor data fusion through quantum methods. This approach not only demonstrates the versatility of VQE beyond quantum chemistry but also offers a novel perspective on how quantum computing can be applied to the fields of biomechanical and wearable technology. It should be emphasized that we do not claim a quantum advantage for the H2two-qubit case. Rather, our goal is to establish a quantum computational signal processing framework. The two-qubit H2example was chosen for its tractability on current hardware and simulators, allowing full–state control and validation against known analytical results. While a classical solver can diagonalize the 4×4 Hamiltonian instantly and classical digital signal processing methods handle the same sensor data more efficiently at this scale, the quantum approach lays the groundwork for future integration with quantum sensors and chiplet– based architectures, where classical methods may not suffice. By combining the computational capabilities of quantum and classical representations of the information, VQE allows us to tackle complex, classically intractable problems, such as analyzing biomechanical sensor data in real time. As our experiment with the engineered sensor-equipped socks shows, VQE can extend beyond its traditional applications in chemistry, offering new possibilities in fields like wearable technology and motion analysis. With continuous advancements in quantum hardware, the flexibility and scalability of VQE will expand, opening the door to more sophisticated applications in both scientific and technological domains, from quantum chemistry to personalized healthcare and sensor fusion. 1.1. Advantages of our hardware and software solutions The integration of a feedback-controlled VQE with wearable technology, specifically socks embedded with biomechanical sensors, offers several distinct advantages in both hardware and software. These advantages are particularly evident in the real-time analysis and adaptive control of sensor data, which addresses critical challenges in motion analysis and wearable health technologies. One of the key strengths of our hardware solution lies in the sophisticated design of the socks, which are equipped with multiple sensors, including accelerometers, and gyroscopes. These sensors collect high- Digital Signal Processing 166 (2025) 105357 3 J. Villalba-Díez and J. Ordieres-Meré resolution detailed data on foot movements, and balance distribution. The ability to monitor biomechanical patterns from both feet in real time provides a significant advantage over traditional motion analysis tools, which are often limited to indirect or less granular data collection methods. Using direct sensor data, we can capture a more comprehensive view of gait dynamics, making our hardware particularly suitable for analyzing subtle differences in movement patterns between healthy individuals and patients with motor impairments. On the software side, the integration of VQE into this system introduces a powerful tool for processing and interpreting complex, dynamic datasets. The quantum algorithm is well-suited because of the ease of embedding periodic functions in the amplitude of qubits. VQE’s quantum parallelism allows it to efficiently handle this complexity, while the classical optimizer iteratively refines the parameters to produce more accurate results. This hybrid approach promisses that the system can analyze data in a way that maximizes both accuracy and computational efficiency. By encoding the accelerometer, and gyroscope signals from each foot into the quantum states of a simplified hydrogen (H2) molecule, we harness the computational strengths of quantum systems to reveal patterns in the sensor data that may not be apparent by classical means. A particularly notable advantage of our solution is the implementation of an automatic control system through a classical feedback loop. This feedback loop continuously monitors the quantum output, comparing it to the desired target values, and applies corrections in real time. The use of a feedback correction factor 𝐾allows the system to dynamically adjust the parameters of the quantum circuit, ensuring that deviations from the optimal path are minimized. This is particularly useful in real-time applications where the sensor data can fluctuate, as the feedback mechanism helps stabilize the quantum algorithm’s performance by reducing noise sensitivity and improving the convergence of the optimization process [37]. This ability to adapt and correct in real time is a major advantage, particularly when analyzing motion data in unpredictable or variable conditions, such as in patients with fluctuating motor function. Additionally, the flexibility of our quantum-classical hybrid system allows it to be adaptable to other applications. Although the current implementation focuses on the analysis of biomechanical data from wearable sensors, the framework can be extended to other forms of dynamic data analysis, such as in medical monitoring or industrial process control, where real-time feedback and adaptability are crucial. The use of quantum computing in this context provides a forward-looking solution, with the potential to outperform classical methods as quantum hardware continues to advance. Moreover, the noise-tolerant nature of VQE [38], when paired with the feedback control, provides resilience against the inherent imperfections of current quantum hardware. As quantum devices still face limitations due to decoherence and gate errors [39], our solution leverages the hybrid nature of VQE to overcome these obstacles by relying on classical optimization to handle the aspects that are most sensitive to noise, while still exploiting quantum advantages where possible. There are several practical challenges in scaling our two–qubit VQE signal processing framework to larger systems. First, state initialization for high-dimensional sensor data may require 𝑂(2𝑁)gate operations or ancillary qubits, imposing significant overhead. Second, as the qubit count grows, circuit depth and measurement overhead increase, risking decoherence and incurring greater shot counts to estimate expectation values of increasingly many Pauli terms. Third, the current two–qubit demonstration provides conceptual clarity rather than performance gain: classical methods currently outperform our VQE pipeline at this scale. Finally, for moderate systems (2–4 qubits), one can still leverage efficient encodings (e.g., basis or amplitude encoding of few features) and parallelized chiplet architectures, tiling small H2-based modules, each processing a subset of the data to mitigate initialization and noise challenges. These strategies, together with anticipated improvements in qubit fidelity and algorithmic error mitigation, outline a path forward for maintaining efficiency as the system scales. The real-time adaptability, precision in motion analysis, and scalability of the system position it as a valuable tool in the fields of healthcare and biomechanics. By leveraging quantum computing, our solution not only addresses current limitations of classical systems in handling large, complex datasets, but also sets the stage for future improvements as quantum hardware evolves. The integration of a classical feedback loop further enhances the reliability and performance of the system, making it a robust solution for real-world applications in dynamic environments. 1.2. Organization of the paper This paper is structured as follows: Section 2introduces the case study that demonstrates the integration of a feedback-controlled VQE with wearable sensor technology. This section details the system architecture, including the hardware and software components, and explains how the quantum-classical hybrid control framework is applied to analyze sensor data from the engineered socks. Section 2.1 establishes the scope of the study, focusing on the analysis of biomechanical data collected from healthy and sick patients to evaluate gait dynamics. In Section 2.2, we provide a comprehensive overview of hardware implementation, including the design and functionality of the sensorembedded socks and the data acquisition process. Section 2.3 describes the implementation of the software, with particular attention to the VQE algorithm, the automatic feedback control system, and the integration of quantum computing techniques to process and interpret sensor data. Data collection procedures are discussed in Section 2.4, where we outline the methodology used to gather and preprocess sensor signals from accelerometers, and gyroscopes sensors. The results of the study, including the performance of the system in analyzing motion data and identifying sensor sensitivity to specific frequencies, are presented in Section 2.5. Finally, Section 3offers a detailed discussion of the findings, highlighting the advantages of our hardware and software solutions, identifying limitations, and suggesting potential future applications of the quantum-enhanced control system in wearable health technologies and other dynamic data-driven fields. In line with the recommendations of Eisenhardt [40], we follow a clear case study roadmap. 2. Case study 2.1. Scope establishment Gait disturbances arise as a symptom of underlying neurological damage caused by Multiple Sclerosis (MS), reflecting the progression of disability, affecting more than 90% of patients [41,42]. These disturbances are often subtle in the early stages of the disease, making them difficult to detect through traditional clinical methods, which rely heavily on subjective assessments. The diagnosis and monitoring of disease progression typically rely on the Expanded Disability Status Scale (EDSS) and the timed 25-foot walk (T25FW) tests. While these tools are widely used, they suffer from low sensitivity and limited reproducibility, which often results in delayed diagnosis and suboptimal treatment strategies. This lack of precision can be particularly problematic for tracking the early, nuanced changes in gait that may indicate disease progression. A promising solution lies in the integration of wearable sensor technologies to provide objective, continuous, and high-resolution data on gait disturbances. In this study, we utilized Sensoria smart socks, equipped with accelerometers, and gyroscopes sensors, to capture detailed gait dynamics. These sensors monitor linear and angular movement at a sampling frequency of 45-55 Hz. Data were collected during participants’ daily activities, synchronized, normalized, and filtered to remove noise. The processed signals were then analyzed for features like step frequency, and length variability distribution. This hardware system ensures compliance with GDPR and enables real-time, high-fidelity monitoring of gait impairments in MS. Digital Signal Processing 166 (2025) 105357 4 J. Villalba-Díez and J. Ordieres-Meré Wearable sensor systems offer an objective method for assessing gait disturbances and hold potential as tools for monitoring clinical progression in both controlled laboratory environments and during daily activities [43]. These devices not only record physical condition but also utilize real-time data processing to analyze and interpret movement patterns, offering insights that could support timely clinical decisionmaking. For example, the system could alert clinicians to minor gait disturbances that precede more significant mobility issues, allowing for earlier intervention and more personalized treatment plans. Moreover, the continuous nature of data collection means that these systems provide a dynamic, real-time picture of the patient’s condition, offering much more detail than the static snapshots provided by conventional tests like the EDSS or T25FW [44]. Despite the potential benefits, the application of wearable sensor systems in MS is still relatively limited [43,44]. Current studies have primarily focused on laboratory-based gait assessments, which, while useful, do not fully capture the everyday functional mobility of patients. These laboratory assessments have demonstrated that wearable sensors can detect subtle deficits in gait quality during the early stages of MS, even before patients notice subjective functional impairments [45]. Such early detection is critical, as it may allow clinicians to intervene before the disease significantly impacts the patient’s daily life. However, the challenge lies in extending these systems to continuous, real-world monitoring, which could provide a more comprehensive view of the disease’s impact on mobility over time. In addition to capturing step counts and walking time, which are useful for quantifying mobility in clinical settings, wearable sensor systems can also monitor more complex gait parameters, such as step length variability, stance and swing phase duration distribution across the foot [46]. These metrics offer a richer dataset for clinicians, enabling a more nuanced understanding of how MS affects gait and balance over time. The real-time collection and analysis of such data are crucial for tracking disease progression and tailoring therapeutic interventions to the individual needs of the patient. Longitudinal studies leveraging these technologies could provide invaluable insights into how gait metrics evolve in MS patients and help refine predictive models of disease progression. The scope of our approach involves transforming the raw signals captured from the wearable sensors into a format suitable for quantum computation, specifically for input into the VQE algorithm. Our system captures high-resolution data from the accelerometers, and gyroscopes, and magnetometers sensors embedded in the wearable socks, which monitor the foot’s linear and angular movement during walking and other daily activities. These signals are crucial for understanding gait dynamics, particularly in patients with MS, where early and precise detection of subtle gait disturbances can significantly affect clinical decisions and treatment optimization. The accelerometer records the linear acceleration of the foot along three axes (x, y, z), providing data on changes in velocity. Moreover, the gyroscope captures angular velocity, detecting rotational movements along the same axes. In addition, the magnetometer captures comparative elevation signals, providing additional context for analysis. Time alignment for signals from different sensors is provided through pivoting technique in such a way that signals from all devices are provided within a common time base. These time series signals are inherently complex, with high variability depending on the patient’s gait patterns and the stage of the disease. To process these data for input into the VQE quantum circuit, we first preprocess the signals to filter out noise and normalize the data, ensuring that the quantum system can handle the inputs effectively. In the following we will use the accelerometer and gyroscope signals. Once preprocessed, the signals are encoded in the parameters of the quantum states used in the VQE. Specifically, the signals from the accelerometers and gyroscopes are mapped to the angles of the rotation gates applied to the qubits in the quantum circuit. For example, accelerometer data along the x-axis can be encoded as a parameter Fig. 1. Image of the IoT socks. (Source courtesy of Sensoria Health Inc.) 𝜃𝑥, which is used to rotate a qubit around the Y-axis via the 𝑅𝑦(𝜃𝑥) gate, while gyroscope data can similarly be encoded into 𝜃𝑦for rotation around the Z-axis using the 𝑅𝑧(𝜃𝑦)gate. These rotations initialize the quantum state to reflect the movement dynamics captured by the sensors, creating a quantum representation of the patient’s gait. Our system involves two qubits, corresponding to the left and right feet, with the accelerometer and gyroscope signals from each foot determining the initial state of each qubit. This setup enables us to explore the quantum system’s sensitivity to various frequencies and motion patterns, potentially identifying subtle distinctions between healthy and impaired gait. To ensure that the quantum circuit adapts dynamically to the sensor inputs, we employ an automatic feedback control system. The VQE algorithm iteratively adjusts the parameters of the quantum circuit to minimize the expectation value of the system’s Hamiltonian  𝐻, which represents the energy of the system. However, to further optimize the process, we incorporate a classical feedback loop that continuously monitors the quantum output and compares it to the desired optimization target. This feedback loop applies a corrective factor 𝐾, which dynamically adjusts the qubit rotation angles based on discrepancies between the quantum output and the actual sensor signals. In this way, the system is automatically controlled to ensure that the quantum output remains aligned with the true physical motion being measured. The feedback mechanism is crucial for handling noise and ensuring that the system converges toward an optimal solution, even when the sensor signals exhibit variability or subtle disturbances. As the VQE circuit processes each set of input data, the feedback loop fine-tunes the system in real-time, continuously improving the accuracy of the quantum state representation of the patient’s gait. This dynamic adjustment significantly improves the precision of the analysis, allowing the system to detect subtle gait abnormalities that might go unnoticed with classical analysis techniques. In summary, the scope of our approach extends from capturing raw accelerometer and gyroscope signals to encoding these data into a quantum circuit through parameterized gate operations. By employing an automatically controlled VQE framework, we ensure that the quantum system dynamically adapts to the sensor inputs, providing a highly sensitive tool for analyzing gait patterns in real-time. The integration of feedback control with quantum computation represents a novel approach to monitoring motor function in patients with MS, offering the potential for earlier detection of gait disturbances and more personalized treatment strategies. 2.2. Hardware implementation Wearable motion sensors are inertial measurement units (IMU) made up of accelerometers and/or gyroscopes that measure linear acceleration and angular velocity and record body movement in the three axes of space. They are shown to be reliable, sensitive and inexpensive assessment tools that have a growing application in gait analysis [47] and other movement disorders [48] in the field of neurology. After intense analysis of the available wearable devices, we have selected two instrumented smart socks from Sensoria Inc. (Sensoria Health Inc. Seattle, WA, USA) [49], having as advantage against other providers the integrated information from accelerometer, and gyroscope (see Fig. 1). Digital Signal Processing 166 (2025) 105357 5 J. Villalba-Díez and J. Ordieres-Meré Fig. 2. Proposed IT framework integrating wearable data sources. 2.2.1. Data acquisition and processing hardware The hardware design makes available technical specifications that allow us to develop our own data capture application running on an Android mobile phone [50] that captures 12 data channels (three components of the acceleration vector, three for the gyroscope vector) with a frequency between 45 Hz and 55 Hz. This sampling frequency enables one to accurately describe the walking structure. Consequently, multiple physical layers are anticipated, where all pertinent data converge at a personal hub. This hub is responsible for preliminary data processing and subsequent submission to the relevant cloud platform, depending on the device manufacturer or the specific data collection application employed. According to the proposed framework (Fig. 2), the hub layer adds relevant information, since it is in charge of continuously querying the IoT devices attached to socks at the established sampling rate, to integrate the GPS information coming to the phone and, potentially other relevant sensors such as heart rate, leg providing data, etc., while special attention was paid to the General Data Protection Regulation (GDPR) adopted by the EU. In this way, no specific information from people is directly collected, and all the dataset is referred to specific id-tokens provided by neurologists, and they are study based. Once the data are collected, the setup of the records is prepared to be stored remotely. Then data compression, encryption, and delivery are performed in the cloud, where an Influx DB time series database [51] was selected to store the collected data. Our approach seeks to replace traditional gait analysis and biomechanical modeling, which are typically conducted in specialized laboratory settings using optical motion capture to monitor segments of body movements, with an unsupervised, nondevoted gait monitoring of the patient’s daily activities. The aim is to reduce invasive effects and still maintain reproducibility with a limited level of supervision, as requested by [52]. Another relevant component of the framework is the semantic layer. It allows us to understand the behavior at higher level. In fact, until now, we do have stream of data related to the 12 channels per leg around 50 times per second, but there is no vertical integration. Such a layer, completely processed at the cloud level, will be in charge of finding the timing that specific id-tokens were providing data. In addition, it is relevant to identify the period of time when the person was walking effectively, which means that the two legs were moving synchronously with the appropriate timing, and the accelerometer and gyroscope were reflecting the appropriate signals. Therefore, moving up through the semantic layer, progress is made between the layers, from raw data to activity time windows per person, to effective gait and key performance indicators (KPIs) of movement. From there, with the help of the differential GPS, the type of territory (walk complexity) and the KPI values can be raised, adding additional knowledge related to the interpretability of the data in the context of the person. With the help of the information at the different layers of integration, convenient links can be further defined to integrate it with other sources and elaborate additional analyses. 2.3. Software The system implemented here integrates classical and quantum components to analyze motion data captured from accelerometer and gyroscope sensors embedded in socks worn by healthy and sick patients. The goal is to assess differences in movement patterns by feeding the sensor data into a VQE and optimizing it using classical feedback. The data from the accelerometers and gyroscopes on the left and right feet are processed, mapped to qubit states, and iteratively optimized. The VQE is a hybrid quantum-classical algorithm, widely regarded as one of the most promising approaches to solving eigenvalue problems in quantum chemistry. The principal goal of VQE is to approximate the ground state energy of a quantum system, particularly in molecular structures, by efficiently leveraging the capabilities of both quantum and classical computation. In quantum chemistry, identifying the ground state energy of molecules such as H2(the hydrogen molecule) is a fundamental task that becomes exponentially harder as the size of the system increases. Classical algorithms struggle with these challenges, especially for strongly correlated electron systems, because of the vast size of the Hilbert space. VQE provides a path forward by exploiting quantum parallelism for state preparation and measurement while offloading optimization tasks to classical processors. In the VQE, the quantum system, such as the hydrogen molecule H2, is described by a Hamiltonian  𝐻, which encodes the total energy of the system. The Hamiltonian  𝐻is typically expressed as a sum of Pauli operators, derived through a fermion-to-qubit transformation such as the Jordan-Wigner or Bravyi-Kitaev mappings. For a molecular system, the electronic Hamiltonian  𝐻in second quantization is given by:  𝐻=∑ 𝑝𝑞 ℎ𝑝𝑞𝑎† 𝑝𝑎𝑞+1 2∑ 𝑝𝑞𝑟𝑠 ℎ𝑝𝑞𝑟𝑠𝑎† 𝑝𝑎† 𝑞𝑎𝑟𝑎𝑠, where 𝑎† 𝑝and 𝑎𝑞are fermionic creation and annihilation operators, and ℎ𝑝𝑞 and ℎ𝑝𝑞𝑟𝑠 represent oneand two-electron integrals, respectively. This Hamiltonian  𝐻is mapped to a qubit representation, resulting in a sum of Pauli operators:  𝐻=∑ 𝑖 𝑐𝑖𝑃𝑖, where 𝑐𝑖are coefficients, and 𝑃𝑖are tensor products of Pauli matrices 𝜎𝑥,𝜎 𝑦,𝜎 𝑧. For the H2molecule, this qubit Hamiltonian  𝐻can be simplified and efficiently represented using only a few qubits. The objective of VQE is to find the eigenvalue of the Hamiltonian  𝐻that corresponds to the ground state energy of the system. Mathematically, this is expressed as finding the lowest eigenvalue 𝐸0, where:  𝐻|𝜓0⟩=𝐸0|𝜓0⟩, with |𝜓0⟩being the ground state wavefunction. The quantum computer prepares and measures a trial wavefunction |𝜓(𝜃)⟩, parameterized by Digital Signal Processing 166 (2025) 105357 6 J. Villalba-Díez and J. Ordieres-Meré 𝜃, and iteratively adjusts these parameters to minimize the expectation value ⟨𝜓(𝜃)| 𝐻|𝜓(𝜃)⟩, thereby approximating the ground state energy. The algorithm begins by preparing a trial wavefunction known as the Ansatz, which is a parameterized quantum state dependent on a set of classical parameters 𝜃. This can take various forms, but for quantum chemistry applications, the Unitary Coupled Cluster (UCC) is often employed due to its ability to capture electron correlations accurately. The quantum computer constructs this by applying a sequence of quantum gates to an initial state, encoding the complexity of the system’s behavior into a compact quantum state. The choice of is crucial, as it must balance between expressivity (the ability to represent the true ground state) and feasibility (the resources required to prepare the state on noisy quantum hardware). Once the ansatz is prepared, the quantum computer performs a series of measurements to evaluate the expectation value of the Hamiltonian  𝐻with respect to the state, ⟨𝜓(𝜃)|𝐻|𝜓(𝜃)⟩. This expectation value corresponds to the energy of the system for the current set of parameters 𝜃. Quantum measurements are inherently probabilistic, and thus multiple measurements (often referred to as shots) are required to obtain an accurate estimate of the energy. The accuracy of these measurements is influenced by the number of shots, the noise present in the quantum hardware, and the structure of the Hamiltonian  𝐻being evaluated. After obtaining the energy estimate from the quantum computer, a classical optimizer is used to update the parameters 𝜃in a way that minimizes the energy. Common classical optimization algorithms used in VQE include gradient-based methods such as gradient descent and gradient-free methods like COBYLA (Constrained Optimization BY Linear Approximations). The classical optimizer adjusts the parameters iteratively, in order to find the set of 𝜃that minimizes the expectation value of the Hamiltonian  𝐻, thereby approximating the energy of the ground state of the system. This hybrid loop—quantum state preparation, measurement, and classical optimization—continues until the energy converges to a minimum value that corresponds to the approximate ground-state energy of the system. When applied to a simple molecular system like H2, VQE begins by constructing the electronic Hamiltonian  𝐻, which takes into account the interactions between the two electrons and the two protons in the molecule. The Hamiltonian  𝐻for H2is relatively small, and the problem can often be mapped to as few as 2-4 qubits, depending on the chosen base set and the level of approximation used. A straightforward, such as a hardware-efficient, is often sufficient to capture the electronic structure of H2, although more sophisticated forms may be required for larger and more complex molecules. The quantum computer then adjusts the parameters, refining the representation of the electronic wavefunction of the molecule to minimize the energy. The range of feedback gains 𝐾ranges from 0.1 to 0.9, with input sensor signals dynamically adjusted over several iterations. Each value of 𝐾represents the proportional control used in a feedback loop to refine the response of the quantum system to errors. The feedback loop iteratively updates the input signals based on the computed errors between expected and observed quantum-state measurements. For each iteration, the quantum circuit is constructed with two qubits, each representing the data from one foot’s sensor inputs (accelerometer and gyroscope signals). The input sensor data is first converted into sinusoidal signals, which are expressed as: 𝑠(𝑡)=𝐴⋅sin(2𝜋⋅𝜈⋅𝑡), where 𝐴is the amplitude and 𝜈is the frequency derived from the sensor data. These sinusoidal signals serve as the rotation angles for the quantum gates in the circuit. The accelerometer data from the socks (represented as modA) is applied to qubit 1, and the gyroscope data (modG) is applied to qubit 2. The quantum circuit is initialized with Hadamard gates 𝐻on both qubits to create a superposition of states. This is followed by a series of rotation gates applied to each qubit, with the rotation angles derived from the sensor input. Specifically, the gates 𝑅𝑥(𝜃), 𝑅𝑦(𝜃), and 𝑅𝑧(𝜃) are applied, where 𝜃is the sensor input modulated to a value within the range of [0,2𝜋]. The circuit also includes a controlled NOT (CNOT) gate that entangles the two qubits: 𝑞𝑐 =𝐻0𝑅𝑥(𝜃1)0𝐻1𝑅𝑦(𝜃2)1𝑅𝑧(𝜃2)1𝐶𝑋(0,1), where 𝜃1and 𝜃2are the sensor input data for qubit 1 and qubit 2, respectively. This entanglement allows the system to capture the correlation between the signals from the left and right foot sensors. This circuit shows the application of Hadamard 𝐻gates to both qubits, followed by rotation gates parameterized by the sensor data and a CNOT gate to entangle the qubits, concluding with a measurement on both qubits. The following is a representation of the quantum circuit used: 𝐻𝑅𝑥(𝜃1)∙? ? 𝐻𝑅𝑦(𝜃2)𝑅𝑧(𝜃2)? ? After constructing the circuit, measurements are performed on both qubits, which gives the probabilities that each qubit is in state |0⟩. These probabilities are recorded, and the error is calculated as the squared difference between the measured probability and the expected input signal: error_qubit1 =(𝑝qubit1 −inputqubit1)2 and error_qubit2 =(𝑝qubit2 −inputqubit2)2. The input signals are then updated using a feedback mechanism that subtracts the product of the feedback gain 𝐾and the error of the initial input. This ensures that the system adapts to minimize errors and improve the accuracy of the quantum state measurements over multiple iterations. To further analyze the data, the sinusoidal signals for each qubit are fitted to the measured probabilities, and the resulting mathematical functions are expressed symbolically as: sinusoidal_qubit1 =𝐴1sin(𝐵1𝑥+𝐶1)+𝐷1 and sinusoidal_qubit2 =𝐴2sin(𝐵2𝑥+𝐶2)+𝐷2, where 𝐴1,𝐵 1,𝐶 1,𝐷 1and 𝐴2,𝐵 2,𝐶 2,𝐷 2are the fitted parameters for the accelerometer and gyroscope data of each qubit, respectively. These sinusoidal fits allow us to assess how well the quantum circuit can represent the sensor inputs and to evaluate the differences between healthy and sick patients based on their sensor data. To gain deeper insights into the frequency response of the system, sinusoidal functions are transformed into the Laplace domain. The Laplace transform of the sinusoidal functions is given by: {𝐴sin(𝐵𝑥 +𝐶)+𝐷}=𝐴𝐵𝑒𝐶𝑠 𝑠2+𝐵2+𝐷 𝑠 , which is used to construct Bode plots for both qubits. The Bode plots show the magnitude and phase of the system’s response to varying frequencies, offering a frequency-domain analysis of the quantum system’s sensitivity to different motions. Finally, the results of the Laplace transform are plotted in Bode plots for each value 𝐾. The magnitude and phase responses are displayed for both qubits, providing a clear visualization of how the quantum system’s dynamics change in response to different feedback gains and sensor inputs. These insights are crucial for understanding the system’s behavior and for identifying motion discrepancies between healthy and sick patients. 2.4. Data collection By using the technology described in the hardware section (see Section 2.2) different people were monitored. Data were collected from Digital Signal Processing 166 (2025) 105357 7 J. Villalba-Díez and J. Ordieres-Meré a diverse cohort, including healthy individuals and MS patients (with EDSS levels ranging from 0 to 6). Participants wore the Sensoria socks during daily activities for up to eight hours per day over two days. The raw signals were preprocessed to remove noise, normalize values, and ensure alignment across sensors. This dataset formed the basis for subsequent quantum analysis, offering insights into gait dynamics and disease progression. A diverse group of individuals was selected to represent the patient population. This group included a representative from healthy individuals and patients with varying levels of disability as measured by the EDSS. Specifically, representatives were included for EDSS scores of 3, 4.5, and 5.5. This selection ensured a comprehensive representation of different stages of the condition under study. Based on the semantic anaysis of the data, time segments where user’s walking happened. On such time segments, data extraction for both legs and duration for between three and four mins have been considered for the analysis. With this method, we collected significant amount of records from people with different levels of disease, as well as from healthy adults as well. This data samples have been used in this research as support for research questions. Information taken for the analysis involved modulus of acceleration and modulus of gyroscope per IMU, therefore the specific components from data vectors were integrated in a more consistent way. IMUs provide measurements of: •Accelerometer: Linear acceleration (m/s2) along the x, y, and z axes. •Gyroscope: Angular velocity (rad/s) around the x, y, and z axes. Obviously, calibrate the IMU to remove biases, scale factors, and misalignments is required as part of the procedure, accessing raw data streams from the accelerometer, and gyroscope, including timestamps for precise integration. This information allows to describe the orientation of an object in 3D space, through the Euler angles (roll, pitch, yaw), by using specific algorithms, such as: •Complementary Filter: where the gyroscope provides short-term orientation changes, while the accelerometer corrects long-term orientation. •Kalman Filter: which use a more complex probabilistic model for state estimation, providing accurate orientation even with noise and drift. • Madgwick-Mahony Filter: which uses an algorithm specifically optimized for IMUs, integrating accelerometer and gyroscope data. The data provided by IMUs is sufficient to represent movement trajectories, so it was exclusively used for the analysis conducted in this research. To ensure compliance with GDPR regulations, patient reference was anonymized using coded identifiers. These identifiers were assigned to patients undergoing treatment at Hospital Universitario de Getafe in Madrid, Spain. All protocols and procedures were reviewed and approved by the hospital’s ethics committee. The mapping of patient details to their assigned codes is securely maintained by neurological doctors. For this study, since the analysis relies solely on IMU data and corresponding labels with EDSS values, no additional personal information was required. This research is part of a broader collaboration agreement between the research team from the Technical University of Madrid, led by one of the coauthors, and the Neurology Service of Hospital Universitario de Getafe in Madrid, Spain [16,53]. The overarching goal of this partnership is to explore how technology can enhance the identification of disease progression over time and address the management barriers and enablers that health systems encounter in extending the potential benefits of eHealth solutions. This specific study contributes to the first area of development within this collaboration, focusing on the use of technology to monitor and analyze disease progression. Through this work, we aim to provide insights and tools that can ultimately support improved healthcare outcomes. 2.5. Results In this study, we begin by conducting a detailed investigation of the Bode magnitude diagrams to identify significant frequency peaks that characterize the system’s resonance behavior. Resonant frequencies, indicated by these peaks, reveal points at which the system exhibits the highest sensitivity to external stimuli. Identifying these frequencies is crucial for understanding the inherent dynamics of our system, particularly in the context of motor coordination impairment in MS, where frequency response characteristics can illuminate aspects of motor control deterioration. To explore the system’s sensitivity to different signal types, we feed inputs from two types of sensors—an accelerometer and a gyroscope— into two qubits. By examining the resultant Bode magnitude diagrams for each signal type, we assess the system’s responsiveness to each, as evidenced by the peak heights in the frequency response. The Bode magnitude peak heights correspond directly to the system’s sensitivity, with higher peaks indicating a stronger system response at specific frequencies. As shown in Fig. 3, this phase of analysis reveals that the accelerometer input produces markedly higher peaks in the Bode magnitude response compared to the gyroscope. Thus, the system exhibits greater sensitivity to the accelerometer signals, underscoring its suitability as the primary sensor for subsequent stages of our analysis. Following this, we proceed with a focused study utilizing accelerometer data, isolating signals from both the left and right feet. By feeding these accelerometer signals into the qubits, we aim to analyze how gait asymmetry—particularly across varying levels of illness severity, as measured by the EDSS—is manifested in the system’s response. As exemplary shown in Fig. 4for values of EDSS of 3, this methodology allows us to directly relate the Bode magnitude response characteristics to the degree of motor dysfunction in MS, providing a nuanced understanding of how specific frequencies correspond to critical motor coordination deficits. This initial Bode analysis enables us to refine our approach to studying gait dynamics in MS by pinpointing relevant frequencies and determining sensor sensitivity. The identification of the accelerometer as the primary responsive sensor allows us to focus on the most impactful data source, setting the stage for a deeper investigation into gait asymmetry and motor coordination loss across various stages of MS. By leveraging these insights, we aim to uncover a frequency-based marker of motor dysfunction progression, which can serve as a potential biomarker for tracking MS severity and guiding therapeutic intervention. The distance between the peaks of stepping frequency for the left and right feet is calculated by discretizing the Bode-plot and identifying the points that present and abnormal gradient in the magnitude. These points are marked in red in Fig. 5. As shown in Fig. 6, our analysis suggests a possible relationship between the EDSS score and the degree of gait asymmetry in individuals with MS, as measured by the distance between the peaks of stepping frequency for the left and right feet. The blue dots in the figure represent individual measurements of this distance for different patients. Each dot corresponds to a specific patient, reflecting the variability in the stepping pattern and the degree of asymmetry at a given EDSS level. Consistency is observed through the EDSS level when differences in step frequency depending on the leg increased with the intensity of disability, which the observed gait disabilities found in practice. The observed relationship follows a sigmoid form, described by the equation: 𝑑(𝑥)= 0.05 1+𝑒−1.471(𝑥−3.524) +0.009 where 𝑑(𝑥)represents the average distance between the peaks of stepping frequency for the left and right feet, and 𝑥is the EDSS level, Digital Signal Processing 166 (2025) 105357 8 J. Villalba-Díez and J. Ordieres-Meré Fig. 3. Bode magnitude response for accelerometer and gyroscope inputs, illustrating greater sensitivity in the accelerometer signal. Fig. 4. Bode magnitude response for left and right foot inputs, illustrating the distance between the peaks in the frequency response. Digital Signal Processing 166 (2025) 105357 9 J. Villalba-Díez and J. Ordieres-Meré Fig. 5. Discretization of Bode plot for distance between peaks calculation. Fig. 6. EDSS level vs. distance between left and right foot magnitude peaks for different patients. an established measure of disability in MS. The EDSS scale quantifies impairment, with higher scores indicating greater mobility and coordination challenges. The sigmoid curve captures the nonlinear progression of gait asymmetry, showing subtle changes in early disability stages and more pronounced shifts as motor dysfunction intensifies. The blue dots not only illustrate the spread of individual patient data but also highlight the alignment of these real-world measurements with the predicted curve. This visualization underscores the practical relevance of the model and offers valuable insights into critical transitions in motor control degradation associated with MS progression. At lower EDSS levels (approximately 𝑥<3.0), the curve remains relatively flat, and the distance between the peaks of stepping frequency is minimal, hovering around 9⋅10−3 seconds. This indicates that individuals with early-stage MS experience only minimal gait asymmetry, maintaining relatively coordinated and balanced foot movements. Clinically, this suggests that during these mild stages of the disease, motor coordination is largely preserved, and patients may not exhibit noticeable gait disturbances. As the EDSS level increases to around 𝑥=3.5, corresponding to moderate disability, the curve begins to rise sharply. The distance between the frequency peaks increases rapidly, reaching approximately 30 ⋅10−3 seconds by the time the EDSS level reaches 𝑥=4.5. This steep middle section of the curve suggests that gait asymmetry worsens significantly as MS progresses to moderate stages. The increase in asymmetry indicates a substantial loss of coordination between the left and right feet, reflecting an accelerated deterioration of motor control. In clinical terms, this transition marks a critical point in the disease’s progression, where gait abnormalities become more pronounced, potentially leading to balance issues, difficulty in walking, and a greater risk of falls. This phase is crucial for intervention, as it signals the onset of more severe motor dysfunction that may require more intensive therapeutic strategies to preserve mobility. At higher EDSS levels (approximately 𝑥>5.0), the curve plateaus again, with the distance between the frequency peaks stabilizing at around 59⋅10−3 seconds. Although the asymmetry remains high, further increases in the EDSS level lead to only marginal changes in the distance between peaks. This final plateau suggests that once motor control has deteriorated to this degree, additional neurological damage results in only minimal further loss of coordination. The patient’s gait has reached a point where both feet are moving in a highly uncoordinated manner, but the functional asymmetry has reached its maximum possible expression. Clinically, this plateau could indicate that the disease has progressed to a stage where neurological damage is so advanced that further treatment may focus primarily on symptom management rather than functional restoration. The sigmoid form of the curve reflects the non-linear progression of motor impairment in MS, with a period of relatively slow deterioration followed by a rapid decline during the moderate stages and a subsequent stabilization as the disease becomes more severe. This pattern provides