Numerical solution methods for quantum stochastic processes

Quantum information-processing techniques enable work extraction from a system's inherently quantum features, in addition to the classical free energy it contains. Meanwhile, the science of computational mechanics affords tools for the predictive modeling of non-Markovian classical and quantum stochastic processes. We combine tools from these two sciences to develop a technique for predictive work extraction from non-Markovian stochastic processes with quantum outputs. We demonstrate that this technique can extract more work than non-predictive quantum work extraction protocols, on the one hand, and predictive work extraction without quantum information processing, on the other. We discover a phase transition in the efficacy of memory for work extraction from quantum processes, which is without classical precedent. Our work opens up the prospect of machines that harness environmental free energy in an essentially quantum, essentially time-varying form.

Within a differential expression of the Heisenberg operator, the forward and backward evolution can be joined together along a closed time contour. This manipulation leads to a dramatic cancellation of oscillations due to the two individual propagators in the Heisenberg operator and the resulting forward-backward propagator is more tractable to semiclassical approximations. This article gives a detailed description of the forward-backward semiclassical dynamics (FBSD) formalism. The semiclassical propagators, especially those of the initial value representations (IVRs), are briefly discussed. The derivation of the FBSD based on the Herman–Kluk propagator is reviewed. Different FBSD formulations with other semiclassical IVRs are worked out and numerical calculations show that they are also capable of describing quantum dynamics semiquantitatively and all display accuracy similar to the classical Wigner method.

An efficient time-dependent (TD) Monte Carlo (MC) importance sampling method has recently been developed [G. Tao and W. H. Miller, J. Chem. Phys. 135, 024104 (2011)] for the evaluation of time correlation functions using the semiclassical (SC) initial value representation (IVR) methodology. In this TD-SC-IVR method, the MC sampling uses information from both time-evolved phase points as well as their initial values, and only the "important" trajectories are sampled frequently. Even though the TD-SC-IVR was shown in some benchmark examples to be much more efficient than the traditional time-independent sampling method (which uses only initial conditions), the calculation of the SC prefactor-which is computationally expensive, especially for large systems-is still required for accepted trajectories. In the present work, we present an approximate implementation of the TD-SC-IVR method that is completely prefactor-free; it gives the time correlation function as a classical-like magnitude function multiplied by a phase function. Application of this approach to flux-flux correlation functions (which yield reaction rate constants) for the benchmark H + H(2) system shows very good agreement with exact quantum results. Limitations of the approximate approach are also discussed.

Performance of planned activities not always leads to the achievement of planned results in innovation projects with a high degree of uncertainty. Management of such specific projects requires the usage of specific methods and processes. It is necessary to develop own stochastic process methods rather than use classical methods of deterministic management. The main peculiarity of stochastic processes of projects consists in the inability to carry out more than one experiment. Implementation of a project is a stochastic process, which is carried out only once. For the purposes of the research, stochastic project management is the process of organisation, planning and control over projects, in which the decisions and actions of managers are described by stochastic functions. If the project realisation is described by stochastic processes, the project management process is indeed described by stochastic processes. This research suggests a project management approach based on the focus on the project product and project management as stochastic processes based on stochastic functions. It is demonstrated that stochastic function values, which underlie the stochastic processes, are formed by the intellectual instruments of those involved in the project and are developed on the basis of their knowledge and skills. The latter, in turn, are the result of studying and practical work on the project. Therefore, it is difficult to predict decisions and actions of managers and contractors, even if internal and external influences on them are specified. Then, any decisions or actions of such persons with regard to the external observer are described by stochastic functions. The purpose of this study is to describe the processes of stochastic project management. The purpose of the study is the project management process. The subject of research is the methods of stochastic project management. Within the framework of the study, all groups of processes of stochastic control will be mathematically described and the conditions of their impact determined. The processes of stochastic project management are outlined and formally presented. These are the processes of project management organisation, the processes of project management support, processes of project content definition, processes of defining the resources, required for the project works, processes of project planning, processes of influence on contractors of actions (works).

An efficient method, the self-consistent hybrid method, is proposed for accurately simulating time-dependent quantum dynamics in complex systems. The method is based on an iterative convergence procedure for a dynamical hybrid approach. In this approach, the overall system is first partitioned into a “core” and a “reservoir” (an initial guess). The former is treated via an accurate quantum mechanical method, namely, the time-dependent multiconfiguration self-consistent field or multiconfiguration time-dependent Hartree approach, and the latter is treated via a more approximate method, e.g., classical mechanics, semiclassical initial value representations, quantum perturbation theories, etc. Next, the number of “core” degrees of freedom, as well as other variational parameters, is systematically increased to achieve numerical convergence for the overall quantum dynamics. The method is applied to two examples of quantum dissipative dynamics in the condensed phase: the spin-boson problem and the electronic resonance decay in the presence of a vibrational bath. It is demonstrated that the method provides a practical way of obtaining accurate quantum dynamical results for complex systems.

A coherent state phase space representation of operators, based on the Husimi distribution, is used to derive an exact expression for the symmetrized version of thermal correlation functions. In addition to the time and temperature independent phase space representation of the two operators whose correlation function is of interest, the integrand includes a non-negative distribution function where only one imaginary time and one real time propagation are needed to compute it. The methodology is exemplified for the flux side correlation function used in rate theory. The coherent state representation necessitates the use of a smeared Gaussian flux operator whose coherent state phase space representation is identical to the classical flux expression. The resulting coherent state expression for the flux side correlation function has a number of advantages as compared to previous formulations. Since only one time propagation is needed, it is much easier to converge it with a semiclassical initial value representation. There is no need for forward-backward approximations, and in principle, the computation may be implemented on the fly. It also provides a route for analytic semiclassical approximations for the thermal rate, as exemplified by a computation of the transmission factor through symmetric and asymmetric Eckart barriers using a thawed Gaussian approximation for both imaginary and real time propagations. As a by-product, this example shows that one may obtain "good" tunneling rates using only above barrier classical trajectories even in the deep tunneling regime.

A semiclassical initial value representation approach for molecular systems in Cartesian coordinates is combined with a recently proposed time averaging technique [J. Chem. Phys. 118, 7174 (2003)]. It is shown that a single trajectory can yield the zero-point energy of the water dimer with good accuracy for the model chosen when compared to fully constrained Cartesian semiclassical calculations. The convergence with respect to the number of averaging time origins is discussed.

Semiclassical approximation of real-time quantum effects is analyzed with the aid of the semiclassical initial value representation (SC-IVR) and Wigner distribution functions. We utilize these two ingredients to propose a new version of the semiclassical correlation function that contains, in principle, all quantum-mechanical effects. The advantage of this formulation is that it allows for a stepwise approximation specifically for real-time quantum effects based on a gradual inclusion of more degrees of freedom into the integral responsible for interference. From numerical calculations, this procedure does not seem to depend significantly on the choice coordinates if all degrees of freedom are coupled. This freedom from the coordinate choice removes possible ambiguities in applying this method. Several example cases are presented to demonstrate the usefulness of this approach.

Internal conversion is an inherently quantum mechanical process. To date, "ab initio" computation of internal conversion rates was limited to harmonic based approximations. These are questionable since the typical transition to the ground electronic state occurs at energies which are far from the harmonic limit. It is thus of interest to study the applicability of the Semiclassical Initial Value Representation (SCIVR) approach which is in principle amenable to "on the fly" studies even with "many" degrees of freedom. In this work we apply the Herman-Kluk-SCIVR methodology to compute the internal conversion rates for formaldehyde for a variety of initial vibronic states. The SCIVR computation gives reasonable agreement with experiment, while the harmonic approximation typically gives rates that are too high.

The mathematical model describing a light beam prepared in an arbitrary quantum optical state is a quasifree quantum stochastic process on the C* algebra of the canonical commutatation relations. For such quantum stochastic processes the concept of stochastic states is introduced. Stochastic quantum states have a classical analog in the following sense: If the light beam is prepared in a stochastic state, one can construct a generalized classical stochastic process, such that the distributions of the quantum observables and the classical random variables coincide. A sufficient algebraic condition for the stochasticity of a quantum state is formulated. The introduced formalism generalizes the Wigner representation from a single field mode to a continuum of modes. For the special case of a single field mode the stochasticity condition provides a new criterion for the positivity of the Wigner function related to the given state. As an example the quantized eletromagnetic field in empty space at temperature T=0 is discussed. It turns out that the corresponding classical stochastic process is not a white noise but a colored noise with a linearly increasing spectrum.

In work the approach to modelling of molecular-structural characteristics of polymers during process thermooxidizing destruction is considered, for a basis imitating modelling of stochastic processes of destruction of macromolecules of polymer is accepted.

The semiclassical (SC) initial value representation (IVR) for the time evolution operator e−iĤt/ℏ involves a phase space integral over the initial conditions of classical trajectories. It is shown in this paper how an IVR for the two time evolution operators in a typical quantum mechanical time correlation function, CAB(t)≡tr[ÂeiĤt/ℏB̂e−iĤt/ℏ], can be combined into one such phase space integral; i.e., time evolution from 0 to t and from t to 0 is combined into one overall SC-IVR propagation. This not only reduces the dimensionality of the phase space average, but the forward–backward (FB) nature of the net trajectory has a partial self-cancellation that reduces the oscillatory behavior of the integrand. Several applications of this FB-IVR to reactive flux correlation functions are presented to illustrate its possibilities.

Quantum stochastic processes are operator processes in Fock space adapted in a natural way with respect to the tensor product factorization of the space. One can describe the usual classical stochastic processes (e.g., Brownian, Poisson) as well as new variety of non-commuting processes in this language. In this framework, quantum stochastic calculus (like Ito calculus in the classical case) has been developed by Hudson and Parthasarathy. Using this calculus, one can rewrite the simple Wigner-Weisskopf model of a decaying atom. In this model, the system (atom) is coupled linearly to the environment (a bath of infinite degrees of freedom represented by a reducible bosonic quantum stochastic process at non-zero temperature). The quantum stochastic differential equation of evolution is solved, and the bath variables are averaged out by taking vacuum expectation values leading to a law of relaxation of the state of the system.

This paper studies the space-complexity of predicting the long-term behavior of a class of stochastic processes based on evolutions and measurements of quantum mechanical systems. These processes generalize a wide range of both quantum and classical space-bounded computations, including unbounded error computations given by machines having algebraic number transition amplitudes or probabilities. It is proved that any space s quantum stochastic process from this class can be simulated probabilistically with unbounded error in space O(s), and therefore deterministically in space O(s2).

The strong consistency of a general class of prediction error identification methods for stationary stochastic processes is demonstrated. In particular, the strong consistency of the maximum likelihood method for stationary Gaussian processes [4], [5] and of the quadratic loss prediction error method for stationary stochastic processes [1]-[3] follow as special cases of the general result.