⚛️ quantum physics

Quantum machine learning geospatial time-series by seasonal encoding

This paper demonstrates that employing structure-aware adaptive seasonal encodings, rather than standard value-based methods, enables quantum machine learning models to effectively capture temporal patterns in geospatial time-series data and maintain forecasting accuracy as circuit depth increases, even outperforming or matching classical methods in noisy simulations.

Original authors: Devjyoti Tripathy, Reece Robertson, Josey Stevens, Catherine Nakalembe, Sebastian Deffner

Published 2026-11-02

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CC BY 4.0

Original authors: Devjyoti Tripathy, Reece Robertson, Josey Stevens, Catherine Nakalembe, Sebastian Deffner

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

Predicting the future of our planet's weather is a task that has long relied on spotting patterns in the past. Scientists have spent decades building tools to track how rainfall, temperature, and other environmental signals change over time, hoping to foresee droughts or floods before they strike. These signals are rarely random; they often follow a rhythm, rising and falling with the seasons in a way that repeats year after year, though never with perfect precision. In recent years, a new field called quantum machine learning has emerged, offering a different way to process information. Instead of using the standard bits of classical computers, these systems use quantum states, which can exist in complex combinations, to represent data. The hope is that by mapping real-world data into these quantum states, computers might find patterns that are too subtle for traditional methods to see. However, simply adding more layers to these quantum circuits does not automatically make them smarter. In fact, making these models deeper can sometimes confuse them, causing them to lose the very patterns they are trying to learn.

A team of researchers set out to solve this puzzle by focusing on a specific type of data: monthly rainfall records from a drought-prone region in East Africa. They wanted to understand why some quantum models fail as they grow larger and whether a specific way of feeding data into the machine could fix the problem. Their investigation revealed that the way data is translated into the quantum language matters more than the sheer size of the machine. They found that if you simply feed the raw numbers of rainfall into the system, the model gets worse at predicting the future as you add more processing layers. But if you teach the machine to recognize the time of year along with the amount of rain, it stays sharp and accurate, even as the model becomes more complex. This discovery suggests that for quantum computers to truly learn from time-based data, they need to be given the calendar as a guide, not just the numbers.

The researchers tested their ideas using a dataset of monthly precipitation from Napak, Uganda, a place where rainfall is critical for agriculture but highly variable. They took a window of fourteen months of rain data and asked the computer to predict the rainfall for the next month. To do this, they built a hybrid system that combines a classical computer with a quantum circuit. The quantum part of the system works by taking the input data and encoding it into the state of several quantum bits, or qubits. This encoding process is like translating a sentence from English into a different language; the choice of translation method determines how well the meaning is preserved. The team tried several different translation methods, or encodings. The first two methods were simple: they mapped the amount of rain directly onto the quantum bits, ignoring the date entirely. The other three methods were more sophisticated. They mapped the amount of rain to one coordinate and the month of the year to a different coordinate, effectively giving the machine a sense of seasonality.

To see how these methods held up, the researchers built models with varying depths, meaning they added more layers of processing steps, ranging from one layer up to six. In a perfect, noise-free simulation, the simple methods that ignored the date quickly fell apart. As the models got deeper, they lost their ability to recognize the connection between the current month and the same month in previous years. The predictions became erratic, and the model failed to capture the seasonal rhythm of the rain. In contrast, the models that explicitly included the month of the year remained stable. They kept the seasonal structure intact, preserving the relationship between the data points as the model grew deeper. The most successful approach was an adaptive one, where the model was allowed to slightly adjust how it represented the seasons during training, rather than sticking to a rigid, pre-set map. This flexible model maintained its accuracy and its ability to track the timing of the rain across all the different depths tested.

The team also checked if these results would hold up on real-world quantum hardware, which is currently noisy and prone to errors. They simulated the effects of hardware noise and found that the advantage of the seasonal encoding persisted, at least for models with a shallow to intermediate number of layers. The models that understood the seasons still outperformed the ones that did not. However, as the models became very deep, the noise from the hardware began to overwhelm the benefits of the encoding, and all models struggled to maintain accuracy. This suggests that while the encoding strategy is powerful, it must be paired with hardware that is quiet enough to let the deeper layers work. Even with these limitations, the best quantum models performed just as well as established classical methods, such as statistical models and artificial neural networks, which are the current standard for weather forecasting.

The core of this discovery lies in how the quantum machine sees the data. When the researchers looked closely at the mathematics of the system, they found that the depth of the circuit changes the geometry of the space in which the data lives. For the simple encodings, adding more layers distorted this space in a way that made it harder to tell the difference between a rainy month and a dry one, or between this year's rain and last year's. The seasonal encodings, however, arranged the data in a way that respected the natural cycle of the year. By mapping the month to a specific direction in the quantum space, the model could keep the seasonal patterns distinct, even as the layers of processing added complexity. This allowed the machine to learn the underlying rhythm of the climate without getting lost in the noise of the data.

This work does not claim that quantum computers have already solved the problem of weather forecasting or that they are superior to classical computers in every way. Instead, it offers a crucial lesson for the future of this technology. It shows that simply making quantum models bigger is not the answer. To make them work better, especially for data that changes over time, we must design them to understand the structure of the world they are studying. By building the knowledge of seasons directly into the way the data is presented, the researchers created a model that is more robust and more reliable. This approach could be applied to other types of time-based data, from tracking the spread of diseases to monitoring traffic patterns, suggesting that the key to unlocking the potential of quantum machine learning may lie in teaching the machine to see the world as it truly is: a place of repeating patterns and seasonal cycles.

Technical Summary: Quantum Machine Learning for Geospatial Time-Series via Seasonal Encoding

Problem Statement
Quantum machine learning (QML) offers a framework for embedding classical data into quantum states to learn complex patterns. A specific architecture, quantum data reuploading, repeatedly inserts classical inputs between trainable variational layers to increase the expressive capacity of the model. However, increasing the reuploading depth ( $L$ ) does not guarantee improved predictive performance and can introduce optimization challenges. This is particularly critical for time-series forecasting, where predictive information relies on preserving temporal relationships and recurring structures (e.g., seasonality). Standard quantum encodings often map data solely based on magnitude, ignoring the explicit temporal position of an observation within a cycle. The central question addressed is whether incorporating known temporal structures directly into the quantum data encoding, rather than relying on the variational circuit to infer them, improves the robustness and performance of deep quantum reuploading models.

Methodology
The authors investigate this problem using monthly precipitation data from Napak, Uganda, derived from the ERA5-Land reanalysis dataset. The dataset spans January 1981 to November 2025, exhibiting a dominant annual recurrence period ( $T_0 = 12$ months) but imperfect periodicity.

Model Architecture:
- A hybrid quantum-classical forecasting model is employed.
- Input: A lag window of $w=14$ months is used to predict the next month.
- Quantum Circuit: A data reuploading architecture with $L$ layers ( $L=1$ to $6$). Each layer consists of a data encoding block $S_\ell$ followed by a trainable variational block $V_\ell$ (a hardware-efficient ansatz with single-qubit rotations and nearest-neighbor CNOT entanglement).
- Readout: Pauli-Z measurements on all qubits are passed to a classical output head (a tanh-activated linear layer) to generate the forecast.
Encoding Strategies:
Five distinct encoding schemes are compared, keeping the variational circuit and readout fixed:
- Value-Based Encodings (Baselines):
  - $R_y$ : Maps scaled precipitation magnitude to a rotation about the Y-axis.
  - $R_xR_yR_z$ : Maps magnitude to rotations about all three axes.
  - Limitation: These ignore the seasonal phase; observations with similar magnitudes but different calendar months are mapped identically.
- Seasonal Encodings (Proposed):
  - Fixed Seasonal CDF: Maps precipitation magnitude to the polar coordinate (via the empirical Cumulative Distribution Function) and the calendar month to the azimuthal coordinate on the Bloch sphere. This explicitly encodes the 12-month cycle.
  - Learned Polar Seasonal: Starts with the fixed seasonal geometry but allows the polar coordinate mapping to deform via trainable parameters dependent on the month and layer.
  - Learned Polar + Azimuthal Seasonal: Allows both polar and azimuthal coordinates to adapt via trainable, bounded maps.
Theoretical Framework:
The authors develop a fidelity-based quantum geometric description. They analyze the fidelity $F_L(t, t+T_0)$ between quantum states representing lag windows separated by one seasonal period. Using a local expansion, they relate the distinguishability of these inputs to a Fubini-Study metric generated by the effective input generators of the reuploading circuit. This analysis demonstrates that increasing depth alters the geometry through which temporal differences are resolved.
Evaluation:
- Simulations: Performed using ideal state-vector simulations (PennyLane/PyTorch) and noisy simulations (Qiskit Aer with the IBM FakeAachen noise model).
- Metrics: Root Mean Squared Error (RMSE), Pearson correlation, Fourier power cosine similarity, and Amplitude variance ratio.
- Baselines: Results are benchmarked against classical methods: ARIMA, Random Forests, and LSTMs (with depth-matched architectures).

Key Results

Ideal Simulations:
- Geometric Representation: As depth $L$ increases, the "seasonal fidelity" (similarity between states of windows separated by 12 months) decreases for all encodings. However, value-based encodings degrade rapidly, dropping to the level of random states ( $F \approx 1/d$ ) by $L=6$ . In contrast, adaptive seasonal encodings maintain high fidelity relative to the random state baseline, preserving the representation of temporal recurrence.
- Forecasting Performance: Value-based encodings show a marked decline in forecasting accuracy (increasing RMSE) and loss of spectral/temporal structure as depth increases. Conversely, adaptive seasonal encodings (particularly the Learned Polar variant) remain stable across all depths. The best performance is achieved at $L=2$ (RMSE $\approx 52.81$ mm), significantly outperforming value-based baselines ( $\approx 63$ mm).
- Structure Preservation: The adaptive seasonal models preserve temporal correlation and spectral structure even as the quantum states of annually separated windows become more distinguishable. This indicates that the models retain predictive information despite the geometric separation of seasonal cycles.
Noisy Simulations:
- The hierarchy observed in ideal simulations persists at shallow and intermediate depths ( $L=1, 2, 3$ ) under the IBM FakeAachen noise model. Adaptive seasonal encodings continue to outperform fixed seasonal and value-based encodings.
- At larger depths ( $L > 3$ ), performance for all models degrades due to circuit noise and the increasing difficulty of optimizing the hybrid objective function (SPSA optimization becomes sensitive). However, the relative advantage of the adaptive seasonal encoding remains discernible before the noise dominates.
Classical Comparison:
- The best-performing quantum model (Learned Polar, $L=2$ ) achieves forecasting errors comparable to established classical methods (Random Forest, ARIMA, and LSTM).
- Specifically, the quantum model achieves an RMSE of $52.81 \pm 2.55$ mm, matching the best LSTM result ( $52.84 \pm 0.82$ mm) and outperforming ARIMA and Random Forests.
- Under noise, the quantum model's performance decreases but remains competitive with classical baselines.

Significance and Claims
The paper claims that structure-aware quantum data encoding is a general strategy for building deeper quantum reuploading models. The primary contribution is demonstrating that:

Simply increasing circuit depth is insufficient and often detrimental for time-series learning if the encoding does not respect the data's inherent structure.
Explicitly encoding known temporal structures (like seasonality) into the quantum feature map allows the model to preserve predictive temporal information as depth increases.
Adaptive encodings, which allow the reference geometry to deform during training while maintaining the known structure, offer the most robust performance.
Quantum models utilizing these encoding strategies can achieve performance competitive with classical statistical and machine learning methods, even in the presence of hardware noise at shallow to intermediate depths.

The authors conclude that while a definitive "quantum advantage" in terms of raw performance over classical methods is not yet established, the ability of structured quantum encodings to maintain performance in deep circuits suggests a viable path for applying QML to complex, structured spatiotemporal data. Future work is proposed to extend these principles to multivariate time series and real quantum hardware.

Technical Summary: Quantum Machine Learning for Geospatial Time-Series via Seasonal Encoding

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