Title: Towards the Non-equilibrium thermodynamics of Complexity and Jarzynski Identity

Abstract: We discuss, from a non-equilibrium thermodynamics perspective, the proposals that quantum computational complexity can be described \textit{via} geometric approaches, i.e. complexity geometry. We review the basic concepts of complexity geometry and the principle of minimum complexity. We describe how to construct a partition function on special unitary group manifold, which may be extended to derive the Jarynzki identity from a Fokker-Planck equation with a sink term. In this way, we obtain, by using Hamilton-Jacobi equation, the Jarzynski identity that directly relates to quantum computational complexity for what might be carried out to provide a non-equilibrium framework describing complexity. Besides, we choose the transverse field Ising model as an example to test our results numerically.