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

This comprehensive benchmark study reveals that variational quantum machine learning algorithms generally fail to outperform simple classical counterparts in time series forecasting across various chaotic systems, even after extensive hyperparameter optimization.

Original authors: Tobias Fellner, David Kreplin, Samuel Tovey, Christian Holm

Published 2026-01-22

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Tobias Fellner, David Kreplin, Samuel Tovey, Christian Holm

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict the weather. You have a lot of historical data, but the patterns are messy, chaotic, and hard to pin down. For decades, scientists have used powerful "classical" computers (the kind in your laptop) to solve these puzzles. Recently, a new contender has entered the ring: Quantum Computers. The hope is that these machines, which operate on the strange laws of quantum physics, can solve these chaotic puzzles much faster and better than classical ones.

This paper is a comprehensive "boxing match" between the two. The researchers didn't just look at one fight; they set up 27 different challenges using three famous chaotic systems (think of them as complex, unpredictable dance routines) to see who actually wins.

Here is the breakdown of their findings in simple terms:

1. The Contenders

The Classical Boxers: These are the established champions. They include standard tools like MLPs (simple feed-forward networks), RNNs (networks that remember the past), and LSTMs (a smarter version of RNNs that remembers long-term patterns).
The Quantum Boxers: These are the new challengers using "Variational Quantum Algorithms." Think of these as quantum circuits that have been "trained" to learn. The paper tested five different types, including:
- Quantum Neural Networks (QNN): Basic quantum pattern recognizers.
- Quantum RNNs & QLSTMs: Quantum versions of the memory-heavy classical models.
- Hybrids: Some models that mix a tiny bit of quantum processing with a lot of classical processing.

2. The Arena (The Test)

The researchers didn't just ask, "Can you guess the next number?" They made it hard. They used chaotic systems (like the Lorenz system, which models weather, or the Hénon map). These are systems where a tiny change in the beginning leads to a massive difference later on.

The Challenge: They asked the models to predict the future at different distances:
- Step 1: What happens next? (Easy, almost linear).
- Halfway: What happens halfway to the point where chaos takes over? (Hard).
- Full Distance: What happens when the system has fully gone chaotic? (Very hard).

3. The Results: Who Won?

The verdict is clear: The Classical Boxers won almost every round.

The "Hybrid" Trap: The quantum models that performed the best were actually the ones that relied heavily on classical layers. Imagine a quantum car that has a quantum engine but is mostly held together by a classical chassis and driven by a classical driver. The paper found that these models did well, but it was likely because of the classical parts, not the quantum engine. If you stripped away the classical parts, the quantum performance dropped.
The Pure Quantum Struggle: The models designed specifically to be "purely quantum" for time series (like the Quantum RNN and Quantum LSTM) struggled significantly. They couldn't match the accuracy of even the simplest classical models.
The Complexity Gap: When the prediction task got harder (predicting further into the chaotic future), the classical models (especially the LSTM) pulled ahead by a wide margin. The quantum models simply couldn't keep up with the complexity.

4. The "Size" Factor

The researchers wanted to make sure the quantum models weren't just losing because they were too small. They checked the "muscle mass" (the number of trainable parameters) of both sides.

Even when they gave the quantum models a similar number of "parameters" (brain cells) as the classical models, the classical models still performed better or at least equal.
The only time a quantum model came close to beating a classical one was in a very specific, narrow scenario, but this didn't hold up across the board.

5. The Bottom Line

The paper concludes that, right now, variational quantum machine learning does not offer a practical advantage over classical methods for predicting time series.

The "Quantum" part isn't doing the heavy lifting: In the best-performing quantum models, the classical parts are doing most of the work.
The "Pure" quantum models aren't ready: The models built specifically to handle sequential data (like time series) using quantum mechanics are currently underperforming compared to their classical cousins.
No "Magic Bullet": There is no evidence yet that quantum computers can solve these chaotic prediction problems better than the super-computers we already have.

In a nutshell: If you need to predict a chaotic future today, stick with the classical tools. The quantum tools are still in the gym, training, but they haven't beaten the champions in the ring yet. The researchers suggest that if quantum computing is going to win at this, it will need to try a completely different strategy (like "Quantum Reservoir Computing") rather than just trying to mimic classical neural networks with quantum parts.

Technical Summary: Quantum vs. Classical Benchmark for Time Series Prediction

Problem Statement
While variational quantum machine learning (VQML) algorithms have been proposed as promising tools for time series prediction, their practical advantage over established classical methods remains uncertain. Existing demonstrations of quantum advantage in machine learning are largely restricted to artificial settings or simple one-step-ahead predictions, which often exhibit high linearity. There is a lack of comprehensive, minimally biased benchmarks that evaluate VQML models against classical counterparts on complex, non-linear forecasting tasks, particularly those involving long-term predictions in chaotic systems.

Methodology
The authors present a large-scale benchmark study comparing five variational quantum machine learning models against three classical machine learning models across 27 distinct time series prediction tasks.

Datasets: The study utilizes data from three chaotic dynamical systems known for their non-linear complexity: the one-dimensional Mackey-Glass equation, the two-dimensional Hénon map, and the three-dimensional Lorenz system.
Prediction Tasks: The models are evaluated on varying prediction horizons: one-step-ahead ( $k=1$ ), approximately half a Lyapunov time, and a full Lyapunov time. Sequence lengths ( $l$ ) of 4, 8, and 16 are tested.
Models:
- Quantum Models: Five architectures were selected based on literature with significant citation counts:
  1. d-QNN: A dressed quantum neural network where a parameterized quantum circuit (PQC) is sandwiched between trainable classical linear layers.
  2. ru-QNN: A re-uploading quantum neural network using sequential exponential angle encoding and variational blocks.
  3. QRNN: A quantum recurrent neural network (modified from the original proposal to omit qubit resets, which were found unnecessary).
  4. QLSTM: A quantum long short-term memory model replacing classical layers with PQCs.
  5. le-QLSTM: A linear-layer-enhanced QLSTM incorporating classical linear layers between PQCs.
- Classical Models: Three counterparts were used as baselines: Multi-Layer Perceptron (MLP), Recurrent Neural Network (RNN), and Long Short-Term Memory (LSTM).
Experimental Setup:
- All quantum models were classically simulated using the PennyLane library under ideal, noiseless conditions to establish an upper bound on performance.
- Hyperparameter Optimization: To ensure fairness, all models underwent extensive hyperparameter tuning via grid search. The search spaces were constrained to ensure comparable numbers of trainable parameters across models.
- Evaluation: Each configuration was trained ten times with different random initial weights. Performance was measured using the Median Mean Squared Error (MSE) on the test set, with variability reported via Median Absolute Deviation (MAD).

Key Results

Performance Comparison: In many cases, quantum models struggled to match the accuracy of simple classical counterparts of comparable complexity. The best-performing quantum models (d-QNN and le-QLSTM) achieved at most comparable prediction accuracies to classical models (RNN and LSTM) on short horizons.
Long-Term Prediction: For prediction horizons extending to a full Lyapunov time, classical models significantly outperformed their quantum counterparts.
Architecture Analysis:
- The d-QNN and le-QLSTM performed best among quantum models, but the authors note that these architectures rely heavily on classical linear layers. The authors suggest that the quantum components in these models may serve merely as additional layers without providing a distinct quantum advantage, as the classical layers likely dominate the learning process.
- The QLSTM and QRNN, which were specifically designed for temporal data processing, performed worse than the d-QNN and often underperformed compared to their classical analogs.
- The ru-QNN, designed as a baseline without heavy classical layer reliance, showed limited performance, particularly on multi-step tasks.
Parameter Scaling: When the number of trainable parameters was matched, the best classical and quantum models showed comparable performance on short horizons. However, for longer horizons, classical models (specifically LSTM) continued to improve with increased parameters, whereas quantum models did not demonstrate a similar scaling advantage.
Ranking: Across all 27 tasks, the classical LSTM and RNN achieved the best average rankings. The quantum models specifically designed for time series (QLSTM, QRNN) ranked among the worst.

Significance and Claims
The paper claims to provide the first large-scale, systematic benchmark comparing variational quantum models with classical counterparts for time series prediction. The primary significance of the work lies in its rigorous, fair comparison methodology, which includes extensive hyperparameter optimization and testing on complex chaotic systems.

The authors conclude that, under the current state of variational quantum algorithms, there is no discernible advantage to replacing classical neural layers with quantum neural networks for time series forecasting. The results suggest that the high-performing quantum models in this study rely heavily on classical components, raising questions about the actual contribution of quantum effects. Consequently, the study calls into question the utility of current VQAs for this specific application and highlights the need for novel approaches that better exploit quantum resources, such as quantum reservoir computing, rather than relying on variational methods that may be classically simulable.

1. The Contenders

2. The Arena (The Test)

3. The Results: Who Won?

4. The "Size" Factor

5. The Bottom Line

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