Abstract: A method and system for estimating the ground state energy of a quantum Hamiltonian. The disclosed algorithm may run on any hardware and is suited for early fault tolerant quantum computers. The algorithm employs low-depth quantum circuits with one ancilla qubit with classical post-processing. The algorithm first draws samples from Hadamard tests in which the unitary is a controlled time evolution of the Hamiltonian. The samples are used for evaluating the convolution of the spectral measure and a filter function, and then inferring the ground state energy from this convolution. Quantum circuit depth is linear in the inverse spectral gap and poly-logarithmic in the inverse target accuracy and inverse initial overlap. Runtime is polynomial in the inverse spectral gap, inverse target accuracy, and inverse initial overlap. The algorithm produces a highly-accurate estimate of the ground state energy with reasonable runtime using low-depth quantum circuits.