Quantum Heaviside Eigen Solver

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.