Toward the nonequilibrium thermodynamic analog of complexity and the Jarzynski identity

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.

47 pages, 7 figures, to appear in JHEP

References in corpus (14)

• Complexity and Shock Wave Geometries
• Quantum Computation as Geometry
• Ensemble and Trajectory Thermodynamics: A Brief Introduction
• Quantum Complexity and Negative Curvature
• Jarzynski-like equality for the out-of-time-ordered correlator
• Optimal control, geometry, and quantum computing
• Switchbacks and the Bridge to Nowhere
• How to Build the Thermofield Double State
• Aspects of The First Law of Complexity
• Resource theory of quantum uncomplexity
• Complexity and Floquet dynamics: non-equilibrium Ising phase transitions
• The Jarzynski Identity and the AdS/CFT Duality
• The geometry of quantum computation
• The Schwinger Action Principle and the Feynman Path Integral for Quantum Mechanics in Curved Space

Cited by in corpus (4)

• Resource theory of quantum uncomplexity
• Primordial Gravitational Wave Circuit Complexity
• Circuit Complexity in ${\mathcal{Z}}_2$ $\mathcal{EEF}\mathcal{T}$
• Complexity-constrained quantum thermodynamics