Exact Diagonalization of Sums of Hamiltonians and Products of Unitaries

TL;DR

This paper introduces new tools for analyzing eigenvalues and eigenvectors of sums of Hamiltonians and products of unitaries, with applications in quantum information and adiabatic quantum computing.

Contribution

The paper develops explicit non-perturbative methods for eigenvalue and eigenvector analysis applicable to sums of self-adjoint operators and products of unitaries.

Findings

Provides explicit formulas for eigenvalues and eigenvectors.

Applies tools to Shannon sampling in information theory.

Connects adiabatic quantum evolution with entanglement and computational speed.

Abstract

We present broadly applicable tools for determining the behavior of eigenvalues and eigenvectors under the addition of self-adjoint operators and under the multiplication of unitaries, in finite-dimensional Hilbert spaces. The new tools provide explicit non-perturbative expressions for the eigenvalues and eigenvectors. To illustrate the broad applicability of the new tools, we outline several applications, for example, to Shannon sampling in information theory. A longer companion paper applies the new tools to adiabatic quantum evolution, thereby shedding new light on the connection between an adiabatic quantum computation's usage of the resource of entanglement and the quantum computation's speed.