The variational quantum eigensolver (VQE) is a hybrid quantum-classical algorithm that operates through a feedback loop. State preparation and measurement are performed on a quantum computer, while a classical computer processes the measurement results and updates the quantum computer according to an optimization rule. This method has been shown to be effective in obtaining molecular spectra and preparing ground states of many-body Hamiltonians. However, VQE often requires deep quantum circuits, and the optimization landscape is highly complex, characterized by barren plateaus and numerous local minima.
Inspired by the Quantum Hamiltonian Descent (QHD) approach proposed by Leng [1], our goal is to improve the convergence of VQE to the global optimum by incorporating a "kinetic-energy-like" term into the optimization protocol. A kinetic-like term would allow the optimization process to avoid getting trapped in local minima by enabling tunneling out of these minima, akin to the effect of kinetic energy in quantum mechanics. Similarly to QHD, we aim to begin with a sufficiently strong kinetic term to promote exploration of the parameter space. Then, as the optimization progresses, the strength of the kinetic term should be gradually reduced, facilitating convergence to the ground state of the target Hamiltonian.
This approach offers an additional advantage: if the kinetic energy term facilitates transitions among all eigenstates of the target Hamiltonian, it provides a provable guarantee that the ground state can be efficiently prepared.
[1] Leng, J.; Hickman, E.; Li, J.; Wu, X. Quantum Hamiltonian Descent. arXiv:2303.01471. DOI: 10.48550/arXiv.2303.01471