Quantum machine learning geospatial time-series by seasonal encoding

Abstract

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 $L$= 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.

The paper asks how data encoding affects depth-dependent performance in quantum data-reuploading models for seasonal forecasting. It derives a local fidelity expansion, $1-F_L\simeq\boldsymbol{\delta}^{T}g^{(L)}\boldsymbol{\delta}$, where the input-space Fubini–Study metric is the covariance of effective generators summed over reuploading layers. This describes how depth changes sensitivity to approximately recurrent inputs, without establishing that higher seasonal fidelity implies better forecasts. Five encodings are compared in a 14-qubit hybrid model at depths $L=1,\ldots,6$: two value-only rotation maps, a fixed seasonal map assigning precipitation rank and calendar month to polar and azimuthal coordinates, and two trainable seasonal variants. Evaluation uses 539 monthly ERA5-Land precipitation observations from Napak, Uganda, with 350 training, 70 validation, and 105 test targets; ideal results use five training seeds. Value-only models lose forecasting accuracy and temporal correlation with depth, whereas adaptive seasonal encodings remain comparatively stable. The learned-polar model achieves its best ideal RMSE at $L=2$, $52.81\pm2.55$ mm, versus approximately 63 mm for the best value-only models, with Pearson correlation $0.634\pm0.044$. Classical RMSEs are 55.28 mm for random forests, 55.50 mm for ARIMA, and 52.84 mm for the best small LSTM. Hardware-derived noisy simulations retain the seasonal-encoding hierarchy through $L=3$; the best noisy RMSE is 56.54 mm. Stated limitations include single-series univariate evaluation, underestimated forecast variability, noise and optimizer sensitivity at greater depth, and no demonstrated quantum advantage. Hardware execution, multivariate tests, and more closely matched classical comparisons remain future work.