Heisenberg-limited ground state energy estimation for early fault-tolerant quantum computers

TL;DR

This paper introduces a resource-efficient quantum algorithm for estimating ground state energies with Heisenberg-limited precision, suitable for early fault-tolerant quantum computers, using minimal qubits and circuit complexity.

Contribution

It presents a novel method that achieves Heisenberg-limited energy estimation with only one ancilla qubit and classical post-processing, reducing implementation costs.

Findings

Achieves Heisenberg-limited precision with minimal quantum resources.

Provides an approximate spectral distribution of the Hamiltonian.

Requires only simple quantum circuits and classical computation.

Abstract

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 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 post-processing 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.