Exact Diagonalization of Sums of Hamiltonians and Products of Unitaries

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.