Quantum algorithm for finding periodicities in the spectrum of a black-box Hamiltonian or unitary transformation

TL;DR

This paper introduces a quantum algorithm that estimates eigenvalues and detects periodicities in a system's energy spectrum without requiring controlled operations, enabling analysis of black-box Hamiltonians.

Contribution

It presents a novel method to measure eigenvalues of U⊗U† without controlled-U gates, facilitating spectral analysis of unknown Hamiltonians on quantum computers.

Findings

Allows eigenvalue estimation without controlled-U gates

Enables detection of spectral periodicities in quantum systems

Applicable to systems with unknown Hamiltonians

Abstract

Estimating the eigenvalues of a unitary transformation U by standard phase estimation requires the implementation of controlled-U-gates which are not available if U is only given as a black box. We show that a simple trick allows to measure eigenvalues of U\otimes U^\dagger even in this case. Running the algorithm several times allows therefore to estimate the autocorrelation function of the density of eigenstates of U. This can be applied to find periodicities in the energy spectrum of a quantum system with unknown Hamiltonian if it can be coupled to a quantum computer.