Exact formula for geometric quantum complexity of cosmological perturbations

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.

💡 Research Summary

The paper presents an exact analytical treatment of operator‑complexity for quantum scalar perturbations evolving on homogeneous and isotropic cosmological backgrounds, using Nielsen’s geometric approach. Traditional applications of Nielsen’s framework rely on solving geodesic equations on high‑dimensional unitary groups such as SU(2ⁿ). Because the structure constants of the underlying Lie algebra appear in the Euler‑Arnold equations, most previous works resort to perturbative expansions and obtain only upper bounds on the complexity.

The authors circumvent these difficulties by focusing on the su(1,1) algebra, which naturally describes the two‑mode squeezing and rotation operators that constitute the time‑evolution operator of a free scalar field mode in a Friedmann‑Robertson‑Walker (FRW) universe. They adopt a concrete 2 × 2 matrix representation of the su(1,1) generators (essentially the Pauli matrices up to a factor) and equip the corresponding group manifold with a right‑invariant metric G_{IJ}=δ_{IJ} (or a more general penalty matrix). In this representation the Euler‑Arnold equation reduces to a linear differential equation whose solution can be written explicitly in terms of the target unitary’s squeezing parameter r and rotation angle θ.

The central result is a closed‑form expression for the geometric complexity: \