Variational theory of the Ricci curvature tensor dynamics

In this letter a new Lagrangian variational principle is proved to hold for the Einstein field equations, in which the independent variational tensor field is identified with the Ricci curvature tensor $$R^{\mu \nu }$$ R μ ν rather than the metric tensor $$g_{\mu \nu }$$ g μ ν . The corresponding Lagrangian function, denoted as $$L_{R}$$ L R , is realized by a polynomial expression of the Ricci 4-scalar $$R\equiv g_{\mu \nu }R^{\mu \nu }$$ R ≡ g μ ν R μ ν and of the quadratic curvature 4-scalar $$\rho \equiv R^{\mu \nu }R_{\mu \nu }$$ ρ ≡ R μ ν R μ ν . The Lagrangian variational principle applies both to vacuum and non-vacuum cases and for its validity it demands a non-vanishing, and actually also positive, cosmological constant $$\Lambda >0$$ Λ > 0 . Then, by implementing the deDonder–Weyl formalism, the physical conditions for the existence of a manifestly-covariant Hamiltonian structure associated with such a Lagrangian formulation are investigated. As a consequence, it is proved that the Ricci tensor can obey a Hamiltonian dynamics which is consistent with the solutions predicted by the Einstein field equations.

Damping Formation Mechanism and Damping Injection of Virtual Synchronous Generator Based on Generalized Hamiltonian Theory

Invertor as a virtual synchronous generator (VSG) to provide virtual inertia and damping can improve the stability of a microgrid, in which the damping is one of the fundamental problems in dynamics. From the view of the Hamiltonian dynamics, this paper researches the damping formation mechanism and damping injection control of VSG. First, based on the energy composition and dynamic characteristics of VSG, the differential equations system of VSG is established and is transformed into the generalized Hamiltonian system. Second, the effects of the three parameters of VSG, the damping coefficient D, active power droop coefficient, and time constant of excitation TE on damping characteristics are researched from a dynamic perspective, and simulation research is carried out with an isolated microgrid. Lastly, the control design method of Hamiltonian structure corrections used to add the damping factor and design the equivalent control inject damping to improve the stability of the isolated microgrid. Research shows that the voltage and frequency stability of the isolated microgrid can be effectively improved by selecting three key parameters of VSG and damping injection control. The innovations of this paper are 1. The Hamiltonian model of the inverter is deduced and established by taking the inverter as a virtual generator. 2. Based on the Hamiltonian model, damping characteristics of inverter in the microgrid are studied. 3. Hamiltonian structure correction method is applied to the inverter, and equivalent damping injection is designed to improve the stability of the microgrid.

From Geometry to Coherent Dissipative Dynamics in Quantum Mechanics

Starting from the geometric description of quantum systems, we propose a novel approach to time-independent dissipative quantum processes according to which energy is dissipated but the coherence of the states is preserved. Our proposal consists of extending the standard symplectic picture of quantum mechanics to a contact manifold and then obtaining dissipation by using appropriate contact Hamiltonian dynamics. We work out the case of finite-level systems for which it is shown, by means of the corresponding contact master equation, that the resulting dynamics constitute a viable alternative candidate for the description of this subclass of dissipative quantum systems. As a concrete application, motivated by recent experimental observations, we describe quantum decays in a 2-level system as coherent and continuous processes.

Quantum-Inspired Magnetic Hamiltonian Monte Carlo

Hamiltonian Monte Carlo (HMC) is a Markov Chain Monte Carlo algorithm that is able to generate distant proposals via the use of Hamiltonian dynamics, which are able to incorporate first-order gradient information about the target posterior. This has driven its rise in popularity in the machine learning community in recent times. It has been shown that making use of the energy-time uncertainty relation from quantum mechanics, one can devise an extension to HMC by allowing the mass matrix to be random with a probability distribution instead of a fixed mass. Furthermore, Magnetic Hamiltonian Monte Carlo (MHMC) has been recently proposed as an extension to HMC and adds a magnetic field to HMC which results in non-canonical dynamics associated with the movement of a particle under a magnetic field. In this work, we utilise the non-canonical dynamics of MHMC while allowing the mass matrix to be random to create the Quantum-Inspired Magnetic Hamiltonian Monte Carlo (QIMHMC) algorithm, which is shown to converge to the correct steady state distribution. Empirical results on a broad class of target posterior distributions show that the proposed method produces better sampling performance than HMC, MHMC and HMC with a random mass matrix.

Strong and uniform boundedness of groups

A group [Formula: see text] is called bounded if every conjugation-invariant norm on [Formula: see text] has finite diameter. We introduce various strengthenings of this property and investigate them in several classes of groups including semisimple Lie groups, arithmetic groups and linear algebraic groups. We provide applications to Hamiltonian dynamics.

Exploiting anticommutation in Hamiltonian simulation

Quantum computing can efficiently simulate Hamiltonian dynamics of many-body quantum physics, a task that is generally intractable with classical computers. The hardness lies at the ubiquitous anti-commutative relations of quantum operators, in corresponding with the notorious negative sign problem in classical simulation. Intuitively, Hamiltonians with more commutative terms are also easier to simulate on a quantum computer, and anti-commutative relations generally cause more errors, such as in the product formula method. Here, we theoretically explore the role of anti-commutative relation in Hamiltonian simulation. We find that, contrary to our intuition, anti-commutative relations could also reduce the hardness of Hamiltonian simulation. Specifically, Hamiltonians with mutually anti-commutative terms are easy to simulate, as what happens with ones consisting of mutually commutative terms. Such a property is further utilized to reduce the algorithmic error or the gate complexity in the truncated Taylor series quantum algorithm for general problems. Moreover, we propose two modified linear combinations of unitaries methods tailored for Hamiltonians with different degrees of anti-commutation. We numerically verify that the proposed methods exploiting anti-commutative relations could significantly improve the simulation accuracy of electronic Hamiltonians. Our work sheds light on the roles of commutative and anti-commutative relations in simulating quantum systems.