Quantum approximation algorithms for many-body and electronic structure problems

Computing many-body ground state energies and resolving electronic structure calculations are fundamental problems for fields such as quantum chemistry or condensed matter. Several quantum computing algorithms that address these problems exist, although it is often challenging to establish rigorous bounds on their performances. Here we detail three algorithms that produce approximate ground states for many-body and electronic structure problems, generalizing some previously known results for 2-local Hamiltonians. Each method comes with asymptotic bounds on the energies produced. The first one produces a separable state which improves on random product states. We test it on a spinless Hubbard model, validating numerically the theoretical result. The other two algorithms produce entangled states via shallow or deep circuits, improving on the energies of given initial states. We demonstrate their performance via numerical experiments on a 2-dimensional Hubbard model, starting from a checkerboard product state, as well as on some chemistry Hamiltonians, using the Hartree-Fock state as reference. In both cases, we show that the approximate energies produced are close to the exact ones. These algorithms provide a way to systematically improve the estimation of ground state energies and can be used stand-alone or in conjunction with existing quantum algorithms for ground states.