Fig. 1: Hybrid quantum algorithm.
From: A hybrid quantum-classical method for electron-phonon systems
a The hybrid quantum algorithm iterates between a variational quantum eigensolver (VQE) for the electronic and a non-Gaussian solver (NGS) for the phonon part of the many-body ground state. The quantum circuit structure for a 4-site example at half filling contains Givens rotations G, on-site gates P, and hopping gates H. The P and H layers are repeated n times to express the ground state wavefunction. Within each layer, gates share the same variational parameters θi, which are optimized on a classical computer inside each VQE iteration. b Convergence of the NGS-VQE algorithm, reflected by the total energy as a function of inner-loop (NGS or VQE) iteration steps for a 4-site Hubbard-Holstein model with u = 10, λ = 10, and ω = 1. VQE steps were performed with quantum circuit statevector simulations and a circuit depth of n = 5. Alternative outer-loop iterations are colored in red (for VQE) and blue (for NGS) and the data points are compressed after NGS #1, for illustration purposes. c Convergence of the ground state infidelity 1 − F during each iteration. The reference state chosen for each outer-loop iteration was obtained by exact diagonalization on classical computers.
