  Understanding quantum phase transitions in physical systems is fundamental to
characterize their behaviour at small temperatures. Achieving this requires
both accessing good approximations to the ground state and identifying order
parameters to distinguish different phases. Addressing these challenges, our
work introduces a hybrid algorithm that combines quantum optimization with
classical machine learning. This approach leverages the capability of near-term
quantum computers to prepare locally trapped states through finite
optimization. Specifically, we utilize LASSO for identifying conventional phase
transitions and the Transformer model for topological transitions, applying
these with a sliding window of Hamiltonian parameters to learn appropriate
order parameters and estimate the critical points accurately. We verified the
effectiveness of our method with numerical simulation and real-hardware
experiments on Rigetti's Ankaa 9Q-1 quantum computer. Our protocol not only
provides a robust framework for investigating quantum phase transitions using
shallow quantum circuits but also significantly enhances efficiency and
precision, opening new avenues in the integration of quantum computing and
machine learning.
