Quantum Heaviside Eigen Solver

TL;DR

This paper introduces a universal quantum algorithm called the quantum Heaviside eigen solver, which efficiently computes eigenvalues and eigenstates of Hamiltonians in quantum many-body physics and chemistry, demonstrating quadratic speedup over classical methods.

Contribution

The paper presents a novel quantum algorithm that combines a quantum judge and selector to efficiently find eigenvalues and eigenstates with quadratic speedup, applicable to general Hamiltonians.

Findings

Achieves quadratic speedup over classical diagonalization methods.

Successfully tested on quantum simulators for physical models.

Provides a universal approach for eigen solving in quantum systems.

Abstract

Solving Hamiltonian matrix is a central task in quantum many-body physics and quantum chemistry. Here we propose a novel quantum algorithm named as a quantum Heaviside eigen solver to calculate both the eigen values and eigen states of the general Hamiltonian for quantum computers. A quantum judge is suggested to determine whether all the eigen values of a given Hamiltonian is larger than a certain threshold, and the lowest eigen value with an error smaller than $\varepsilon $ can be obtained by dichotomy in $O\left( {{{\log }}{1 \over \varepsilon }} \right)$ iterations of shifting Hamiltonian and performing quantum judge. A quantum selector is proposed to calculate the corresponding eigen states. Both quantum judge and quantum selector achieve quadratic speedup from amplitude amplification over classical diagonalization methods. The present algorithm is a universal quantum eigen solver for Hamiltonian in quantum many-body systems and quantum chemistry. We test this algorithm on the quantum simulator for a physical model to show its good feasibility.