Geometric measure of quantum complexity in cosmological systems

TL;DR

This paper develops a geometric approach to quantify quantum complexity in cosmological systems, deriving bounds for harmonic oscillators and applying them to scalar fields in de Sitter space, revealing a logarithmic complexity growth on large scales.

Contribution

It introduces an explicit upper bound formula for quantum complexity of time-dependent harmonic oscillators in cosmological backgrounds, extending Nielsen's geometric complexity to quantum fields in curved spacetime.

Findings

Complexity grows logarithmically on super-Hubble scales.

Derived an explicit upper bound for quantum complexity in cosmological settings.

Confirmed consistency with previous de Sitter complexity studies.

Abstract

In Nielsen's geometric approach to quantum complexity, the introduction of a suitable geometrical space, based on the Lie group formed by fundamental operators, facilitates the identification of complexity through geodesic distance in the group manifold. Earlier work has shown that the computation of geodesic distance can be challenging for Lie groups relevant to harmonic oscillators. Here, this problem is approached by working to leading order in an expansion by the structure constants of the Lie group. An explicit formula for an upper bound on the quantum complexity of a harmonic oscillator Hamiltonian with time-dependent frequency is derived. Applied to a massless test scalar field on a cosmological de Sitter background, the upper bound on complexity as a function of the scale factor exhibits a logarithmic increase on super-Hubble scales. This result aligns with the gate complexity and earlier studies of de Sitter complexity. It demonstrates the consistent application of Nielsen complexity to quantum fields in cosmological backgrounds and paves the way for further applications.