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Quantum Modelling of Time Series: Expressivity vs. Trainability

Jacob L. Cybulski, Sebastian Zając

Published Aug 20, 2025• 0 citations
Publisher

Abstract

Abstract As time series feature unstructured data and unique processing requirements, they do not easily fit conventional approaches to quantum modeling. This paper therefore presents various quantum model architectures suitable for encoding and analysis of time series data. In particular, it investigates their architectural design aspects, such as methods of encoding and reuploading of temporal data, parameterizing quantum circuits, controlling the circuit qubit resources and entanglement, and dealing with issues of state evolution in the model Hilbert space, as well as navigability of the classical parameter space for an optimizer.Each approach enhances or impedes model expressivity, i.e. its ability to effectively represent time series data in quantum space, as well as its trainability, i.e. its capacity to learn and generalize for predictive accuracy and efficiency in the process of model optimization. The paper evaluates different quantum circuit configurations for their impact on the expressivity and trainability of a quantum time series analysis model, and analyzes how these two design qualities influence model performance in training and testing.

DOI
10.21203/rs.3.rs-7253212/v1

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Abstract As time series feature unstructured data and unique processing requirements, they do not easily fit conventional approaches to quantum modeling. This paper therefore presents various quantum model architectures suitable for encoding and analysis of time series data. In particular, it investigates their architectural design aspects, such as methods of encoding and reuploading of temporal data, parameterizing quantum circuits, controlling the circuit qubit resources and entanglement, and dealing with issues of state evolution in the model Hilbert space, as well as navigability of the classical parameter space for an optimizer.Each approach enhances or impedes model expressivity, i.e. its ability to effectively represent time series data in quantum space, as well as its trainability, i.e. its capacity to learn and generalize for predictive accuracy and efficiency in the process of model optimization. The paper evaluates different quantum circuit configurations for their impact on the expressivity and trainability of a quantum time series analysis model, and analyzes how these two design qualities influence model performance in training and testing.
When it comes to realizing immediate quantum advantages in practical applications like machine learning and optimization techniques, numerous methods have emerged as a frontrunner technique. Such algorithms often involve quantum circuits for data encoding and the training of Quantum Neural Networks (QNNs) to minimize target functions when applied to tasks involving classical data. Quantum Machine Learning (QML) is a promising area of quantum computing because it may effectively handle difficult learning tasks by taking advantage of the high dimensional Hilbert space to train better representations from limited data. Despite QML's rising popularity, few academic works have addressed the language's security features. There has been a lot of buzz about the promising new discipline of quantum machine learning. Training a parameterized quantum circuit is a common practice in contemporary QML for performing data analysis on classical or quantum datasets. To make accurate predictions on an independent test dataset, state-of-the-art QML techniques first variationally optimize a parameterized quantum circuit using the training dataset. Quantum modeling in QML has also been shown to offer an exponential benefit in sample complexity. In the realms of pattern recognition, machine learning, and data mining, feature dimensionality reduction as a crucial link in the process has become one hot and tough spot. Most academics have gravitated towards this topic because of the difficulty of its studies. The goal of this research is to learn how to implement low loss in the procedure of feature dimension reduction, preserve the integrity of the original data, locate the most effective mapping, and obtain the best possible low-dimensional data. Preliminary findings suggest that QML models may offer advantages over classical models for classical data analysis. In this paper, a thorough investigation into generalization performance in QML is performed. This paper presents a brief survey on Qubits Encoding using Quantum Machine Learning models that help researchers to identify the limitations of the existing models and to design new efficient models for better performance levels.
Abstract In this work, we explore the application of 1D quantum convolutional layers for time series forecasting, framing the problem within the context of machine learning and quantum circuit expressivity. By encoding multiple time steps as features into a quantum circuit, the forecasting task becomes intrinsically multidimensional. We leverage the recent theoretical results that describe Variational Quantum Circuits (VQCs) as multidimensional Fourier series, to design a more expressive 1D quantum convolutional neural network in which an enhanced performance is obtained by reuploading the data. We aim to relate the number of accessible coefficients in each architecture design with the obtained metrics. Our analysis incorporates key machine learning concepts such as expressivity and the presence of barren plateaus, evaluating their impact on generalization and training efficiency. Notably, we show that even with a limited number of trainable parameters, quantum circuits can approximate high-degree Fourier functions, underscoring their expressive capacity and computational efficiency. Our results demonstrate that more expressive ansatz, characterized by richer Fourier spectra, consistently yield improved predictive performance as the number of qubits increases. This study bridges theoretical insights from quantum physics with practical advances in machine learning model design for time series forecasting.
Abstract In this work, we apply 1D quantum convolution in the task of time series forecasting. By encoding multiple points into the quantum circuit to predict subsequent data, each point becomes a feature and the problem becomes a multidimensional one. Taking as basis previous theoretical works which demonstrated that Variational Quantum Circuits (VQCs) can be expressed as multidimensional Fourier series, the capabilities of different architectures and ansatz are explored. This analysis considers the concepts of circuit expressivity and the presence of barren plateaus. Analyzing the problem within the framework of the Fourier series enabled the incorporation of data reuploading in the architecture, resulting in enhanced performance. Rather than a strict requirement for the number of free parameters to exceed the degrees of freedom of the Fourier series, our findings suggest that even a limited number of parameters can produce Fourier functions of higher degrees. This highlights the remarkable expressive power of quantum circuits. This observation is also significant in reducing training times. The ansatz with greater expressivity and number of non-zero Fourier coefficients consistently delivers favorable results across different scenarios, with performance metrics improving as the number of qubits increases.
Quantum data reuploading models can increase the expressive capacity of variational quantum models, but do not necessarily improve predictive performance. In the present work, we investigate this behavior for seasonal time series forecasting and show that data encoding determines how well temporal structures are represented as circuit depth increases. We develop a fidelity-based quantum geometric description in which repeated encodings contribute to an effective input generator, so that reuploading depth changes the quantum geometry through which approximately recurrent inputs are distinguished. We construct fixed and adaptive seasonal encodings that represent observation magnitude and seasonal phase in distinct Bloch-sphere coordinates. We evaluate these encodings for monthly precipitation derived from ERA5-Land, a global reanalysis dataset, from a representative semi-arid, drought-prone tropical region in East Africa, using a 14-month lag window and reuploading depths LLL= 1-6. We find that in ideal simulations, standard value-based encodings degrade markedly with depth, whereas adaptive seasonal encodings preserve temporal correlation, spectral structure, and forecasting accuracy. Noisy simulations show that this encoding hierarchy persists at shallow and intermediate depths, while larger depths are limited by circuit noise and optimization difficulty. The seasonal quantum models remain competitive with established classical forecasting methods (ARIMA, random forests, and LSTMs). These results identify structure-aware quantum data encoding as a general strategy for building deeper quantum reuploading models.
We propose a single-qubit Quantum Machine Learning (QML) model for time series forecasting, built around the concept of a Quantum Reupload Unit (QRU), a hardwareefficient quantum circuit architecture with shallow depth. The proposed model demonstrates enhanced predictive power compared to variational methods such as quantum circuits (VQC), parameterized quantum circuits (PQC), and quantum residual blocks (QRB). The proposed QRU outperforms classical learning models such as Recurrent Neural Networks (RNNs) and Long-Short Term Memory (LSTM) with the same number of parameters. The novelty of this approach is its ability to model temporal patterns without relying on an extensive memory state, which reduces resource demands while preserving forecast accuracy. The expressivity of the model is evaluated through Fourier spectral decomposition. We analyze the trainability of our model using the absorption witness metric. We benchmarked the proposed model on the Mackey-Glass chaotic time series and the real-world river level dataset from TAIAO. The proposed model consistently exhibits enhanced expressivity over both of the datasets. These results highlight the significance of QRUs as promising candidates for learning models that can be conveniently deployed on noisy intermediate-scale quantum (NISQ) hardware.
Quantum Neural Networks (QNNs) offer a promising framework for integrating quantum computing principles into machine learning, yet their practical capabilities and limitations remain insufficiently studied. In this work, we systematically investigate the trainability and approximation properties of QNNs by benchmarking diverse circuit architectures and encoding strategies across synthetic and real-world datasets. We analyze several ansätze, including Hamming, binary, exponential, ternary, turnpike and Golomb, by evaluating their ability to learn synthetic data modeled as random finite Fourier series. To assess real-world applicability, we further evaluate QNNs on two time-series classification tasks: a Fischertechnik pneumatic leak detection dataset and the publicly available NASA bearing fault dataset. Our experiments show that while broader frequency spectra can theoretically enhance expressivity, practical trainability is strongly influenced by architectural factors such as qubit count and circuit depth. Notably, we find that QNNs perform best when the frequency spectrum is tailored to the target function's complexity but remains as compact as possible. Moreover, architectures with identical frequency spectra can differ in trainability, with configurations using more qubits and fewer layers generally performing better, except in the single-layer case. These findings provide guidelines for selecting QNN ansätze and offer new insights into the interplay between expressivity and trainability in quantum machine learning.