Quantum circuit complexity for linearly polarized light

In this study, we explore a form of quantum circuit complexity that extends to open systems. To illustrate our methodology, we focus on a basic model where the projective Hilbert space of states is depicted by the set of orientations in the Euclidean plane. Specifically, we investigate the dynamics of mixed quantum states as they undergo interactions with a sequence of gates. The latter aim to accurately adjust the path from referent to target, aligning it as closely as possible with the path we have chosen. Our approach involves the analysis of sequences of real $2\ifmmode\times\else\texttimes\fi{}2$ density matrices. This mathematical model is physically exemplified by the Stokes density matrices, which delineate the linear polarization of a quasimonochromatic light beam, and the gates, which are viewed as quantum polarizers, whose states are also real $2\ifmmode\times\else\texttimes\fi{}2$ density matrices. The interaction between polarizer-linearly polarized light is construed within the context of this quantum formalism. Each density matrix for the light evolves in an approach analogous to a Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) process during the time interval between consecutive gates. Notably, when considering an upper limit for the tolerance or accuracy, we unearth that the number of gates follows a power-law relationship which gives an upper bound of the complexity.