Toward the nonequilibrium thermodynamic analog of complexity and the Jarzynski identity

TL;DR

This paper develops a novel nonequilibrium complexity framework by deriving a complexity analog of the Jarzynski identity, linking quantum complexity geometry with thermodynamic fluctuation principles, and exploring implications for holographic fluctuations.

Contribution

It introduces a complexity version of the Jarzynski identity using complexity geometry, bridging quantum computational complexity with nonequilibrium thermodynamics.

Findings

Derived a complexity Jarzynski identity in the space of unitaries

Strengthened evidence for a resource theory of uncomplexity

Proposed a fluctuation-dissipation theorem for complexity

Abstract

The Jarzynski identity can describe small-scale nonequilibrium systems through stochastic thermodynamics. The identity considers fluctuating trajectories in a phase space. The complexity geometry frames the discussions on quantum computational complexity using the method of Riemannian geometry, which builds a bridge between optimal quantum circuits and classical geodesics in the space of unitary operators. Complexity geometry enables the application of the methods of classical physics to deal with pure quantum problems. By combining the two frameworks, i.e., the Jarzynski identity and complexity geometry, we derived a complexity analog of the Jarzynski identity using the complexity geometry. We considered a set of geodesics in the space of unitary operators instead of the trajectories in a phase space. The obtained complexity version of the Jarzynski identity strengthened the evidence for the existence of a well-defined resource theory of uncomplexity and presented an extensive discussion on the second law of complexity. Furthermore, analogous to the thermodynamic fluctuation-dissipation theorem, we proposed a version of the fluctuation-dissipation theorem for the complexity. Although this study does not focus on holographic fluctuations, we found that the results are surprisingly suitable for capturing their information. The results obtained using nonequilibrium methods may contribute to understand the nature of the complexity and study the features of the holographic fluctuations.