Heisenberg-Limited Ground-State Energy Estimation for Early Fault-Tolerant Quantum Computers

Under suitable assumptions, the quantum-phase-estimation (QPE) algorithm is able to achieve Heisenberg-limited precision scaling in estimating the ground-state energy. However, QPE requires a large number of ancilla qubits and a large circuit depth, as well as the ability to perform inverse quantum Fourier transform, making it expensive to implement on an early fault-tolerant quantum computer. We propose an alternative method to estimate the ground-state energy of a Hamiltonian with Heisenberg-limited precision scaling, which employs a simple quantum circuit with one ancilla qubit, and a classical postprocessing procedure. Besides the ground-state energy, our algorithm also produces an approximate cumulative distribution function of the spectral measure, which can be used to compute other spectral properties of the Hamiltonian.

Automated summary

The study proposes an alternative method for estimating the ground-state energy of a Hamiltonian with Heisenberg-limited precision scaling using a simple quantum circuit and classical postprocessing. The algorithm also generates an approximate cumulative distribution function of the spectral measure, which can be used to compute other spectral properties of the Hamiltonian.