  The increasing focus on long-term time series prediction across various
fields has been significantly strengthened by advancements in quantum
computation. In this paper, we introduce a data-driven method designed for
long-term time series prediction with quantum dynamical embedding (QDE). This
approach enables a trainable embedding of the data space into an extended state
space, allowing for the recursive retrieval of time series information. Based
on its independence of time series length, this method achieves depth-efficient
quantum circuits that are crucial for near-term quantum computers. Numerical
simulations demonstrate the model's improved performance in prediction accuracy
and resource efficiency over existing methods, as well as its effective
denoising capabilities. We implement this model on the Origin ''Wukong''
superconducting quantum processor with a learnable error-cancellation layer
(LECL) for error mitigation, further validating the practical applicability of
our approach on near-term quantum devices. Furthermore, the theoretical
analysis of the QDE's dynamical properties and its universality enhances its
potential for time series prediction. This study establishes a significant step
towards the processing of long-term time series on near-term quantum computers,
integrating data-driven learning with discrete dynamical embedding for enhanced
forecasting capabilities.
