Exact formula for geometric quantum complexity of cosmological perturbations

TL;DR

This work provides an exact geometric-quantum-complexity formula for cosmological perturbations by leveraging Nielsen's framework with a finite-dimensional su(1,1) representation. By treating the time-evolution operator as a product of two-mode squeezing and rotation within SU(1,1), the authors derive explicit complexity expressions for the rotation, squeezing, and their combination, and extend these to relative complexities. Applying the formalism to de Sitter and asymptotically static universes reveals significant deviations from prior upper-bound estimates, notably the dependence on the squeezing phase and the rotation in the evolution, and demonstrates saturation behavior in toy oscillator models. Overall, geometric complexity emerges as a nuanced, phase-sensitive diagnostic of cosmological dynamics, with clear implications for quantum information aspects of early-universe physics.

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.

Figures (7)

• Figure 1: Complexity of the time evolution operator of an inverted harmonic oscillator. The upper bound is based on the formula derived in Ref. Chowdhury:2024ntx.
• Figure 2: Log-Log plot of the absolute values of the Bogoliubov coefficients as a function of $|k\tau|$. For large values of $|k\tau|$, which corresponds to the early times, $|\beta_k| \rightarrow 0$ and $|\alpha_k| \rightarrow 1$, which is expected as we start from the Minkowski vacuum. The vertical dashed line separates the super-Hubble and the sub-Hubble regions.
• Figure 3: Log Log plot of the variation of the squeezing parameter as a function of $|k\tau|$. The vertical dashed line separates the super-Hubble and the sub-Hubble regions.
• Figure 4: Log-Log plot of the geometric complexity of evolution of a scalar field in de Sitter spacetime for different values of the initial angle $\phi(\infty)$. The black dashed curve shows the complexity result derived using the approximations used while dealing with the path ordered and the operator expansion. The dashed vertical line separates the sub-Hubble and the super-Hubble limits.
• Figure 5: Illustration of the scale factor for an asymptotically bounded expanding and contracting universe. The spacetime is flat in the past with $a=a_i$. The universe then expands or contracts for a finite interval of the conformal time $\tau$. In the future, the spacetime is again flat with $a=a_f$. In the first figure, we consider a situation where, in the expanding case, the universe expands from a scale factor $a_i$ to $a_f$. Then, in the contracting case, the values of $a_i$ and $a_f$ are interchanged. In the second case,we consider a situation where the final scale factor $a_f$ is fixed at . For the expanding case, we consider $a_i=0.5$, and for the contracting case, we take $a_i=1.5$.