Exact formula for geometric quantum complexity of cosmological perturbations

TL;DR

This paper derives an exact formula for the geometric quantum complexity of cosmological perturbations by using a finite-dimensional matrix representation, enabling precise calculations in de Sitter and other cosmological models.

Contribution

It introduces an exact method for computing quantum complexity in cosmological settings using a matrix representation of the Lie algebra, overcoming previous perturbative limitations.

Findings

Exact geodesic distance formula for $rak{su}(1,1)$ algebra

Application to de Sitter spacetime and contracting/expanding models

Enhanced precision in quantum complexity calculations for cosmology

Abstract

Nielsen's geometric approach offers a powerful framework for quantifying the complexity of unitary transformations. In this formulation, complexity is defined as the length of the minimal geodesic in a suitably constructed geometric space associated with the Lie group of relevant operators. Despite its conceptual appeal, determining geodesic distances on Lie group manifolds is generally challenging, and existing treatments often rely on perturbative expansions in the structure constants. In this work, we circumvent these limitations by employing a finite-dimensional matrix representation of the generators, which enables an exact computation of the geodesic distance and hence a precise determination of the complexity. We focus on the $\mathfrak{su}(1,1)$ Lie algebra, relevant for quantum scalar fields evolving on homogeneous and isotropic cosmological backgrounds. The resulting expression for the complexity is applied to de Sitter spacetime as well as to asymptotically static cosmological models undergoing contraction or expansion.