Geometric measure of quantum complexity in cosmological systems

TL;DR

The work applies Nielsen's geometric approach to quantum complexity to cosmological quantum dynamics, deriving a leading-order upper bound on the complexity of time evolution for time-dependent oscillators and applying it to a massless scalar field in de Sitter space. By expressing the target unitary as a squeezing-rotation product and using an $\mathfrak{su}(1,1)$-based generator set, the authors obtain explicit relations between geometric parameters and Bogoliubov coefficients, yielding a bound that grows as $\sim \ln a$ in the super-Hubble IR limit. They validate the framework by comparing geometric complexity with gate complexity, showing agreement in UV/IR and highlighting a horizon-crossing peak in the geometric bound that is sensitive to the transition region. The results demonstrate the viability of Nielsen complexity as a tool for quantum fields in cosmological backgrounds and point to future refinements beyond leading order, with potential implications for simulating cosmological dynamics and probing spacetime structure.

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.

Figures (7)

• Figure 1: A sample frequency function $\omega(t)$ of the oscillator with constant frequency $\omega_{in}$ and $\omega_{out}$ in the "in" and "out" regimes.
• Figure 2: Diagrammatic representation of the frequency profile considered for the transition (\ref{['inout']}).
• Figure 3: Behavior of the upper bound on complexity for an oscillator with switched frequency as a function of $\omega_{\rm in}/\omega_{\rm out}$.
• Figure 4: Log-log plot of the absolute values of the Bogoliubov coefficients (\ref{['bogoliubovn']}) as functions of $|k\tau|$. Conformal time $\tau$ takes negative values, but since it appears in an absolute value, evolution for an expanding universe happens from right to left in this plot. Large values of $|k\tau|$, therefore, correspond to initial time scales and vice versa. At early times, $\tau \rightarrow -\infty$ or $|k\tau|\rightarrow \infty$, $|\beta| \rightarrow 0$ and $|\alpha| \rightarrow 1$. At late times, when $|k\tau| \rightarrow 0$, the absolute values of the Bogoliubov coefficients $\alpha$ and $\beta$ converge. The vertical dashed line separates the super-Hubble (left) and sub-Hubble (right) regions.
• Figure 5: Log-log plot of the upper bounds on the complexity (computed geometrically and via a gate counting approach) of a scalar field mode in de Sitter spacetime. The vertical dashed line separates the super-Hubble (left) and sub-Hubble (right) regions. The vertical dotted line indicated the location of the peak at $|k\tau|=\frac{1}{\sqrt{2}}$.