In this article, we investigate various physical implications of quantum circuit complexity using squeezed state formalism of Primordial Gravitational Waves (PGW). Recently quantum information theoretic concepts such as entanglement entropy, and complexity are becoming pivotal role to understand the dynamics of quantum system even in the diverse fields such as high energy physics and cosmology. This paper is devoted in studying quantum circuit complexity of PGW for various cosmological models such as De Sitter, Inflation, Radiation, Reheating, Matter, Bouncing, Cyclic and Black hole gas model etc. We compute complexity measure using both Covariance and Nielsens wave function method for three different choices of vacua: Motta Allen, $alpha$ and Bunch Davies. Besides computing circuit complexity, we have also computed Von-Neumann entanglement entropy. By making the comparison of complexity with entanglement entropy, we are able to probe various features regarding dynamics of evolution for different cosmological models. Because entanglement entropy is independent of the squeezing angle, we are able to understand more details of the system using Nielsens measure of complexity which is dependent on both squeezing parameter and angle. This implies that quantum complexity could indeed be a useful probe to study quantum features in cosmological scale. Quantum complexity is also becoming a powerful technique to understand the chaotic behavior and random fluctuations of quantum fields. Using the growth of complexity, we are able to compute quantum Lyapunov exponent for various cosmological models and give comment on its chaotic nature.
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