Adaptive variational preparation of the Fermi-Hubbard eigenstates

Approximating the ground states of strongly interacting electron systems in quantum chemistry and condensed matter physics is expected to be one of the earliest applications of quantum computers. In this paper, we prepare highly accurate ground states of the Fermi-Hubbard model for small grids of up to six sites (12 qubits) by using an interpretable, adaptive variational quantum eigensolver (VQE) called ADAPT-VQE [H. R. Grimsley et al., Nat. Commun. 10, 3007 (2019)2041-172310.1038/s41467-019-10988-2]. In contrast with nonadaptive VQE, this algorithm builds a system-specific ansatz by adding an optimal gate built from one-body or two-body fermionic operators at each step. We show this adaptive method outperforms the nonadaptive counterpart in terms of fewer variational parameters, short gate depth, and scaling with the system size. The fidelity and energy of the prepared state appear to improve asymptotically with ansatz depth. We also demonstrate the application of adaptive variational methods by preparing excited states and Green's functions using a proposed ADAPT-SSVQE algorithm. Lower depth, asymptotic convergence, noise tolerance of a variational approach [A. Peruzzo et al., Nat. Commun. 5, 4213 (2014)2041-172310.1038/ncomms5213; J. R. McLean et al., New J. Phys. 18, 023023 (2016)NJOPFM1367-263010.1088/1367-2630/18/2/023023; K. Sharma et al., New J. Phys. 22, 043006 (2020)NJOPFM1367-263010.1088/1367-2630/ab784c], and a highly controllable, system-specific ansatz make the adaptive variational methods particularly well suited for noisy intermediate-scale quantum devices.