Quantum circuit complexity for linearly polarised light

TL;DR

This paper introduces a quantum circuit complexity framework for open systems, specifically modeling the linear polarisation of light using real 2x2 density matrices and analyzing the optimal number of gates with a power-law relationship.

Contribution

It extends quantum circuit complexity to open systems and models linear polarisation of light with a novel mathematical formalism involving GKLS-like processes.

Findings

Optimal number of gates follows a power-law relationship.

Model applies to the dynamics of linearly polarised light interacting with polarisers.

Framework bridges quantum complexity with classical light polarisation phenomena.

Abstract

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. Our approach involves the analysis of sequences of real $2\times2$ density matrices. This mathematical model is physically exemplified by the Stokes density matrices, which delineate the linear polarisation of a quasi-monochromatic light beam, and the gates, which are viewed as quantum polarisers, whose states are also real $2\times2$ density matrices. The interaction between polariser-linearly polarised 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 cost function or tolerance or accuracy, we unearth that the optimal number of gates follows a power-law relationship.