  The paper investigates the techniques of quantum computation in metrological
predictions, with a particular emphasis on enhancing prediction potential
through variational parameter estimation. The applicability of quantum
simulations and quantum metrology techniques for modelling complex physical
systems and achieving high-resolution measurements are proposed. The impacts of
various parameter distributions and learning rates on predictive accuracy are
investigated. Modelling the time evolution of physical systems Hamiltonian
simulation and the product formula procedure are adopted. The time block method
is analyzed in order to reduce simulation errors, while the Schatten-infinite
norm is used to evaluate the simulation precision. Methodology requires
estimation of optimized parameters by minimizing loss functions and resource
needs. For this purpose, the mathematical formulations of Cramer Rao Bound and
Fischer Information are indispensable requirements. The impact of learning
rates on regulating the loss function for various parameter values. Using
parameterized quantum circuits, the article outlines a four-step procedure for
extracting information. This method involves the preparation of input states,
the evolution of parameterized quantum states, the measurement of outputs, and
the estimation of parameters based on multiple measurements. The study analyses
variational unitary circuits with optimized parameter estimation for more
precise predictions. The findings shed light on the effects of normal parameter
distributions and learning rates on attaining the most optimal state and
comparison with classical Long Short Term Memory (LSTM) predictions, providing
valuable insights for the development of more appropriate approaches in quantum
computing.
