How well does the Variational Quantum Eigensolver algorithm perform for a random fermionic Hamiltonian?

Given a random q-local fermionic Hamiltonian, how well does the Variational Quantum Eigensolver (VQE) algorithm perform? In this talk, we propose field theory techniques as a novel tool to answer this question in the limit of large N, where N is the number of fermions. For typical quantum chemistry Hamiltonians, Hartree-Fock usually already recovers r~99% of the ground state energy, and the remaining 1% is partially recovered by more sophisticated techniques like coupled clusters. However, the situation is completely different for the random q-local fermionic Hamiltonians we consider (with q>2). In that case, we show that Hartree-Fock states have a vanishing approximation ratio r=0% in the large-N limit. This means the ground state energy is entirely attributed to strong correlations, and that achieving a non-zero approximation ratio is already an achievement. This prompts us to propose an ansatz inspired by the variational coupled cluster algorithm for which we demonstrate an approximation ratio of r~62%. Finally, we propose to use the same large-N techniques to benchmark other ansatzes which are tractable on a quantum computer, like the unitary coupled cluster states.