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Towards the non-equilibrium thermodynamics of the complexity and the Jarzynski identity

Chen Bai, Wenhao Li, Xian-Hui Ge

Published Jul 20, 2021• 5 citations
OpenAlex

Abstract

We discuss, from a non-equilibrium thermodynamics perspective, the proposals that quantum computational complexity can be described 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 the 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 for describing quantum complexity. In this paper, we mainly utilize this framework to prove the second law of complexity and obtain a fluctuation-dissipation theorem based on the complexity version of the Jarzynski identity. Besides, we choose the transverse field Ising model as an example to test our results numerically.

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We discuss, from a non-equilibrium thermodynamics perspective, the proposals that quantum computational complexity can be described 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 the 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 for describing quantum complexity. In this paper, we mainly utilize this framework to prove the second law of complexity and obtain a fluctuation-dissipation theorem based on the complexity version of the Jarzynski identity. Besides, we choose the transverse field Ising model as an example to test our results numerically.
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.
Fluctuation relations allow for the computation of equilibrium properties, like free energy, from an ensemble of non-equilibrium dynamics simulations. Computing them for quantum systems, however, can be difficult, as performing dynamic simulations of such systems is exponentially hard on classical computers. Quantum computers can alleviate this hurdle, as they can efficiently simulate quantum systems. Here, we present an algorithm utilizing a fluctuation relation known as the Jarzynski equality to approximate free energy differences of quantum systems on a quantum computer. We discuss under which conditions our approximation becomes exact, and under which conditions it serves as a strict upper bound. Furthermore, we successfully demonstrate a proof-of-concept of our algorithm using the transverse field Ising model on a real quantum processor. The free energy is a central thermodynamic property that allows one to compute virtually any equilibrium property of a physical system. Thus, as quantum hardware continues to improve, our algorithm may serve as a valuable tool in a wide range of applications including the construction of phase diagrams, prediction of transport properties and reaction constants, and computer-aided drug design in the future.
Modern theoretical physics rests on the interplay between quantum dynamics, thermodynamics, and spacetime geometry.Quantum mechanics describes microscopic evolution through unitary time dynamics, while thermodynamic systems exhibit irreversible entropy production. Meanwhile, geometric formulations of quantum mechanics and statistical physics suggest that physical laws may possess an underlying geometric structure. In this work we introduce a unified framework in which quantum evolution occurs on a complex-time manifold that incorporates both dynamical time and thermodynamic flow. We construct a geometric variational principle on this extended temporal manifold and derive a generalized quantum evolution equation that contains both Hamiltonian dynamics and geometric diffusion terms. The resulting theory provides a natural bridge between quantum mechanics, statistical mechanics, and informationgeometry. We explore the implications of the framework for quantum statistical systems, quantum field theory, and information flow in gravitational systems.
In this paper, we give a pedagogical introduction to the ideas of quantum thermodynamics and work fluctuations, using only basic concepts from quantum and statistical mechanics. After reviewing the concept of work as usually taught in thermodynamics and statistical mechanics, we discuss the framework of non-equilibrium processes in quantum systems together with some modern developments, such as the Jarzynski equality and its connection to the second law of thermodynamics. We then apply these results to the problem of magnetic resonance, where all calculations can be done exactly. It is shown in detail how to build the statistics of the work, both for a single particle and for a collection of non-interacting particles. We hope that this paper will serve as a tool to bring the new student up to date on the recent developments in non-equilibrium thermodynamics of quantum systems.
The Jarzynski equality, which relates equilibrium free-energy difference to an average of non-equilibrium work, plays a central role in modern non-equilibrium statistical thermodynamics. In this paper, we study a weaker consequence of this relation, known as Jarzynski's inequality, which can be formally obtained from the Jarzynski equality via Jensen's inequality. We identify and analyze several extensions of Jarzynski's inequality that go beyond its direct derivation from the Jarzynski equality. In particular, we consider chemical systems both in the linear-response regime and away from linear thermodynamics. Furthermore, by employing functional-integral techniques, we extend Jarzynski's inequality to many-body statistical systems described by quantum field theory. Salient issues, such as connections of the Jarzynski inequality with the maximum work theorem and the Landau--Lifshitz theory of fluctuations, are also discussed.
Abstract It is conjectured that in the geometric formulation of quantum computing, one can study quantum complexity through classical entropy of statistical ensembles established non-relativistically in the group manifold of unitary operators. The kinetic and positional decompositions of statistical entropy are conjectured to correspond to the Kolmogorov complexity and computational complexity, respectively, of corresponding quantum circuits. In this paper, we claim that by applying the virial theorem to the group manifold, one can derive a generic relation between Kolmogorov complexity and computational complexity in the thermal equilibrium.