Polynomial filter diagonalization of large Floquet unitary operators

TL;DR

This paper introduces an efficient polynomial filter diagonalization method for large Floquet unitary operators, enabling the study of eigenvalues and eigenvectors in large quantum systems relevant to nonequilibrium phenomena.

Contribution

A novel Krylov space diagonalization technique using polynomial spectral transformation for exact eigenpair computation of large Floquet unitaries.

Findings

Method outperforms shift invert in speed and memory

Enables analysis of systems with over 20 qubits

Achieves eigenpair computations in Hilbert spaces over 10^6

Abstract

Periodically driven quantum many-body systems play a central role for our understanding of nonequilibrium phenomena. For studies of quantum chaos, thermalization, many-body localization and time crystals, the properties of eigenvectors and eigenvalues of the unitary evolution operator, and their scaling with physical system size $L$ are of interest. While for static systems, powerful methods for the partial diagonalization of the Hamiltonian were developed, the unitary eigenproblem remains daunting. In this paper, we introduce a Krylov space diagonalization method to obtain exact eigenpairs of the unitary Floquet operator with eigenvalue closest to a target on the unit circle. Our method is based on a complex polynomial spectral transformation given by the geometric sum, leading to rapid convergence of the Arnoldi algorithm. We demonstrate that our method is much more efficient than the shift invert method in terms of both runtime and memory requirements, pushing the accessible system sizes to the realm of 20 qubits, with Hilbert space dimensions $\geq 10^6$.