Quantum vs. classical: A comprehensive benchmark study for predicting time series with variational quantum machine learning

TL;DR

This study conducts a large-scale, fair benchmark comparing variational quantum algorithms to classical baselines for time-series forecasting across three chaotic systems and 27 tasks, with thorough hyperparameter optimization. It reveals that quantum models rarely surpass simple classical models of similar complexity, and the top performers often rely heavily on classical parameters, calling into question practical quantum advantage in this setting. The findings point to the need for new quantum strategies—such as quantum reservoir computing—to better exploit quantum resources for time-series analysis, while providing an upper-bound reference by simulating noiseless quantum models. Overall, the work establishes rigorous expectations for VQAs in time-series tasks and guides future research toward approaches that can leverage quantum dynamics more effectively.

Abstract

Variational quantum machine learning algorithms have been proposed as promising tools for time series prediction, with the potential to handle complex sequential data more effectively than classical approaches. However, their practical advantage over established classical methods remains uncertain. In this work, we present a comprehensive benchmark study comparing a range of variational quantum algorithms and classical machine learning models for time series forecasting. We evaluate their predictive performance on three chaotic systems across 27 time series prediction tasks of varying complexity, and ensure a fair comparison through extensive hyperparameter optimization. Our results indicate that, in many cases, quantum models struggle to match the accuracy of simple classical counterparts of comparable complexity. Furthermore, we analyze the predictive performance relative to the model complexity and discuss the practical limitations of variational quantum algorithms for time series forecasting.

Figures (15)

• Figure 1: The key features of the benchmark study.
• Figure 2: Overview of the different models part of this study. $E$ and $V(\theta)$ denote the encoding layer and the variational layer, respectively. (a): In the d-QNN a is squeezed between two linear classical layers. (b): The ru-QNN utilizes the data re-uploading scheme by sequentially encodes data points. (c): The quantum circuit of the QRNN is divided into two registers: a data register, where sequential data encoding and measurement are performed, and a hidden register, which stores the hidden quantum state and passes it along through the network. (d): In the QLSTM model, the hidden states $h_i$ and cell states $c_i$ are passed between different cells. Each cell consists of connected by a specific LSTM structure. In the case of the le-QLSTM, additional classical linear layers are incorporated within the cell.
• Figure 3: Benchmark results across combinations of data sets, prediction steps, and sequence lengths. The left column shows the results for the Mackey-Glass data set, the middle column shows the results for the Hénon data set, and the right column shows the results for the Lorenz data set. The top row shows the results for one-step ahead prediction, the middle shows the results for about half a Lyapunov time, and the bottom shows the results for about a full Lyapunov time. Within each subplot, the median MSEs for different sequence lengths are shown for each model, with different marker types indicating different sequence lengths. Error bars represent the MAD over ten random initializations. The results reflect the best performance of all model architectures and hyperparameters tested.
• Figure 4: Here we show the median MSE over the number of parameters for different hyperparameter configurations of the d-QNN, le-QLSTM, QRNN, and LSTM models. The results are obtained for training on the Lorenz data set with a sequence length of 16. The upper plot shows the prediction errors for one-step prediction, while the lower plot shows the prediction errors for a prediction horizon of 25.
• Figure 5: Ranking of the nine different models based on their best median MSE across all 27 learning problems. For each learning problem, the models are ranked according to their best median MSE after selecting the optimal architecture and hyperparameter set. Colors represent ranks, from dark blue (1st) to dark red (8th). Models are ordered according to the average rank obtained.