Universal algorithm for transforming Hamiltonian eigenvalues

Abstract

Manipulating Hamiltonians governing physical systems has found a broad range of applications, from quantum chemistry to semiconductor design. In this work, we provide a new way of manipulating Hamiltonians, by transforming their eigenvalues while keeping their eigenstates unchanged. We develop a universal algorithm that deterministically implements any desired (suitably differentiable) function on the eigenvalues of any unknown Hamiltonian, whose positive-time and negative-time dynamics are given as a black box. Our algorithm uses correlated randomness to efficiently combine two subroutines -- namely controlization and Fourier series simulation -- exemplifying a general compilation procedure that we develop. The time complexity of our algorithm is significantly reduced via said compilation technique compared to a na{ï}ve concatenation of the subroutines and outperforms similar methods based on the quantum singular value transformation.

12+23 pages, 14 figures, close to published version. Note that a previous version of this article [v4] contains an algorithm to implement negative-time Hamiltonian dynamics without any auxilliary systems