Adaptive quantum encoding preserves rainfall forecasts over multiple layers

Devjyoti Tripathy of the University of Maryland, Baltimore County, and colleagues are applying quantum machine learning to seasonal rainfall forecasting in a drought-prone region of East Africa. Their work demonstrates that the method of encoding data into a quantum model is more effective at preserving forecasting accuracy than simply increasing the model’s complexity. The researchers found that standard data encodings degrade as circuit depth increases, while adaptive seasonal encodings maintain temporal correlation and spectral structure. These results identify structure-aware quantum data encoding as a general strategy for building deeper quantum reuploading models.

Seasonal Time-Series Forecasting with Quantum Machine Learning

Monthly precipitation patterns in a drought-prone region of Uganda are serving as a testbed for a new approach to quantum machine learning, demonstrating that how data is encoded into a quantum model is critical for accurate seasonal forecasting. Researchers are evaluating the performance of quantum circuits designed to predict rainfall in Napak, Uganda, a location experiencing high climate variability and frequent rainfall deficits. This focus on a specific, climate-vulnerable region establishes a baseline for applying the architecture to similar areas globally, according to the study.

The team constructed both fixed and adaptive seasonal encodings, representing observation magnitude and seasonal phase in distinct coordinates on the Bloch sphere, a fundamental concept in quantum computing. Evaluations used monthly precipitation data from ERA5-Land, a global reanalysis dataset, spanning from January 1981 through November 2025, with a 14-month lag window and reuploading depths ranging from one to six layers. The results demonstrate that while adaptive seasonal encodings preserve forecasting accuracy in ideal simulations, circuit noise and optimization difficulties limit performance at larger depths.

The study emphasizes the importance of encoding structure. The team also benchmarked their quantum approach against classical statistical and machine learning methods, including ARIMA, random forest and LSTM, providing a comparative assessment of performance.

Quantum Data Encoding Impacts Temporal Structure Representation

The geometry of how inputs are represented within a quantum circuit is demonstrably altered by the method of data encoding, influencing the system’s ability to distinguish between similar observations as computational layers increase. Researchers found that the similarity between time-series inputs, measured by quantum-state fidelity, is directly linked to the data encoding map, revealing how depth modifies the representation of seasonal structure within the model. This connection between encoding and quantum geometry provides a basis for analyzing how increasing the number of reuploading layers impacts the preservation of temporal information.

The study demonstrates that an encoding solely based on instantaneous observation values lacks information about its position within a recurring temporal cycle, suggesting that directly embedding predictive temporal structure into the quantum data encoding can be advantageous. This approach allows the model to better represent recurring observations, rather than requiring the circuit to deduce the patterns from repeated value-based reuploads. The team quantified this relationship using fidelity, a measure of quantum-state similarity, where a value of 1 indicates identical states and lower values signify greater distinguishability.

Each monthly observation from a 364-observation dataset was mapped to a point on the sphere, colored according to its calendar month, visually demonstrating the encoding’s ability to capture temporal organization. This visualization highlights how the encoding transforms raw data into a quantum representation where seasonal relationships are geometrically encoded, potentially enhancing the model’s ability to process time-series data.

Fixed vs. Adaptive Seasonal Encodings for Precipitation Data

The team’s analysis related the distinguishability of these inputs to a Fubini-Study metric generated by each reuploading layer, explicitly showing that circuit depth modifies the geometry used to resolve temporal differences in the data. This led to the construction of season-aware quantum encodings, mapping precipitation magnitude to a polar coordinate and calendar month to an azimuthal coordinate, effectively positioning observations on the Bloch sphere. A fixed seasonal representation was created alongside adaptive variants allowing the model to learn without discarding inherent temporal organization within the signal.

Conventional value-based angle encodings were compared with the fixed and adaptive seasonal encodings in ideal simulations, followed by an examination of the seasonal models under noisy conditions. The encoding operation at each reuploading layer combines a variational circuit with a seasonal encoding, defined as A_ℓ(x_t,θ^(enc)_ℓ,θ^(var)_ℓ) = V_ℓ(θ^(var)_ℓ)S_ℓ(x_t,θ^(enc)_ℓ).

The depth-L circuit is then constructed by sequentially applying these encoding-variational blocks, with the reuploading depth, L, representing the total number of these combined layers. For fixed data encodings, the encoding component contains no trainable parameters, while adaptive encodings optimize these parameters alongside the variational circuit. The resulting quantum state, representing the lag window after L reuploading layers, is denoted as =, where trainable parameters are omitted for clarity.

Two initial encodings depended solely on precipitation magnitude, serving as value-based baselines, while the remaining three explicitly accounted for the observation’s position within the annual cycle. The final two adaptive encodings allowed this seasonal geometry to evolve during training. Researchers identified a dominant annual frequency of 12 months, and constructed encodings that progressively incorporate this temporal structure.

The team defined $r_{tq}$ as the observation’s position within the full time series and $m_{tq}$ as its position within the recurring annual cycle, with $m_{tq}$ absent for value-based encodings and becoming explicit only for seasonal encodings. The analysis showed that the geometry of approximately recurring inputs depends on the data encoding map, while a dominant annual frequency with a period of 12 months was identified.

The fixed seasonal encoding maps the empirical training data’s cumulative distribution to the polar coordinate and assigns the calendar month to the azimuthal coordinate, mapping the data onto 12 equally spaced seasonal meridians on the Bloch sphere.

ERA5-Land Precipitation Results with Reuploading Depths L=1-6

The findings demonstrate that while standard value-based encodings experience a marked degradation in performance with increasing depth, adaptive seasonal encodings successfully preserve both temporal correlation and spectral structure, maintaining forecasting accuracy. These results indicate that simply increasing the complexity of a quantum circuit, adding more layers, does not guarantee improved performance in time-series forecasting. In ideal simulations, the adaptive seasonal encodings proved substantially more stable across varying depths compared to their value-based counterparts.

This stability persisted in noisy simulations at shallow and intermediate depths, though circuit noise and optimization challenges ultimately limited performance at larger values of L. The team benchmarked the resulting forecasts against established classical methods, ARIMA, random forests and LSTMs, demonstrating competitive performance from the seasonal quantum models. This analysis explicitly shows how circuit depth alters the geometry through which temporal differences in the data are resolved.

Precipitation magnitude determined the polar coordinate, while position within the annual cycle, or calendar month, determined the azimuthal coordinate. The monthly precipitation record was extracted from the ECMWF ERA5-Land Monthly Aggregated dataset via Google Earth Engine, yielding 539 consecutive monthly observations. These observations were then used to create supervised examples consisting of 14 consecutive monthly observations predicting the subsequent month’s precipitation.

The data was divided into 350 training targets spanning March 1982-April 2011, 70 validation targets spanning May 2011-February 2017, and 105 test targets spanning March 2017-November 2025. The study’s results suggest that increasing circuit depth alone is insufficient to improve quantum time-series learning, and that the method of encoding data is critical for effectively representing temporal structures as circuit depth increases.

Quantum Model Performance Compared to Classical Forecasting Methods

Quantum models employing structure-aware data encoding maintain forecasting accuracy across increasing circuit depth, a feat not consistently achieved by standard encoding methods. Unlike many quantum machine learning approaches where increased model complexity automatically improves performance, this work demonstrates that the method of data representation is paramount for seasonal time-series forecasting.

The team’s findings challenge the assumption that simply adding layers to a quantum circuit will yield better predictions, particularly when dealing with data exhibiting inherent temporal organization. This suggests that quantum approaches are not immediately eclipsed by classical techniques, even with current limitations in quantum hardware and optimization.

The research highlights how the choice of data encoding dictates how well temporal structures are represented as the number of reuploading layers increases within the quantum circuit. The architecture used repeatedly inserts classical input between trainable quantum transformations, a technique that allows for increasingly complex functions to be represented, but also introduces optimization challenges. The team’s analysis revealed that the ability of quantum feature maps and entangling operations to construct correlated representations is not enough to guarantee improved predictive performance.

Instead, the way relationships in the classical input space are translated into the geometry of Hilbert space, through data encoding, strongly influences the model’s learning capacity. This preservation of temporal information is critical because forecasting accuracy relies on maintaining the relationships between observations across time.

The team’s hybrid quantum-classical machine learning model uses a data encoding block followed by a variational block, repeated for each reupload layer, and concludes with Pauli-Z measurement and classical post-processing to obtain the prediction. The forecasting model is trained by minimizing the empirical prediction loss over training windows. The temporal structure within the precipitation data is the basis for examining how recurrent information is represented by the quantum model. This approach moves beyond simply increasing circuit depth, focusing instead on how the input data is shaped within the quantum system to enhance predictive capabilities.

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