Ground-state energy estimation on early fault-tolerant quantum computers

The problem of estimating the ground-state energy of a quantum Hamiltonian is one of the most important and promising applications of early fault-tolerant quantum computers, which are expected to have a very limited number of logical qubits and may have difficulty in handling circuit beyond a certain maximal depth. We argue that algorithms based on the Hamiltonian evolution input model are suitable in the early-fault tolerant regime, can be as efficient as those based on the block encoding input model, and can achieve near-optimal complexities for estimating the ground-state energy and for preparing the ground state. We will also introduce new techniques to significantly reduce the preconstant of circuit depth, while maintaining Heisenberg-limited scaling.