Efficient ground-state energy estimation and certification on early fault-tolerant quantum computers

A major thrust in quantum algorithm development over the past decade has been the search for the quantum algorithms that will deliver practical quantum advantage first. Today's quantum computers - and even early fault-tolerant quantum computers - are limited in the number of operations they can implement per circuit. We introduce quantum algorithms for ground-state energy estimation (GSEE) that accommodate this design constraint. The first algorithm estimates ground-state energies, offering a quadratic improvement on the ground state overlap parameter compared to other methods in this regime. The second algorithm certifies that the estimated ground-state energy is within a specified error tolerance of the true ground-state energy, addressing the issue of gap estimation that beleaguers several ground state preparation and energy estimation algorithms. We note, however, that the scaling of this certification technique is currently less favorable than that of the GSEE algorithm. To develop the certification algorithm, we propose a novel use of quantum computers to facilitate rejection sampling. After a classical computer generates initial samples, the quantum computer is used to accept or reject these samples, resulting in a set of accepted samples that approximate draws from a target distribution. Although we apply this technique specifically for ground-state energy certification, it may find broader applications. Our work pushes the boundaries of what operation-limited quantum computers can achieve, bringing the prospect of quantum advantage closer to realization.