# Polynomial filter diagonalization of large Floquet unitary operators

### Submission summary

 As Contributors: David J. Luitz Preprint link: scipost_202106_00028v1 Date accepted: 2021-07-09 Date submitted: 2021-06-16 21:54 Submitted by: Luitz, David J. Submitted to: SciPost Physics Academic field: Physics Specialties: Quantum Physics Approach: Computational

### 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 $\geq 10^6$.

Published as SciPost Phys. 11, 021 (2021)

Resubmission reflecting changes suggested by the referees.

### Submission & Refereeing History

#### Published as SciPost Phys. 11, 021 (2021)

Resubmission scipost_202106_00028v1 on 16 June 2021
Submission 2102.05054v1 on 12 February 2021

## Reports on this Submission

### Report

The article now meets all acceptance criteria and my other remarks were addressed appropriately, so I recommend the new version for publication.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -

### Report

All of the requested changes were taken into account and a variety of points clarified. I believe that the manuscript is now suitable for publication.

• validity: high
• significance: high
• originality: high
• clarity: top
• formatting: perfect
• grammar: perfect