No abstract

The theory of stochastic processes impacts both physical and social sciences. At the molecular scale, stochastic dynamics is ubiquitous because of thermal fluctuations. The Fokker-Plank-Smoluchowski equation models the time evolution of the probability density of selected degrees of freedom in the diffusive regime and it is a workhorse of physical chemistry. In this paper, we report the development and implementation of a Variational Quantum Eigensolver procedure to solve the Fokker-Planck-Smoluchowski eigenvalue problem. We show that such an algorithm, typically adopted to address quantum chemistry problems, can be applied effectively to classical systems paving the way to new applications of quantum computers. We compute the conformational transition rate in a linear chain of rotors experiencing nearest-neighbor interaction. We provide a method to encode on the quantum computer the probability distribution for a given conformation of the chain and assess its scalability in terms of operations. Performance analysis on noisy quantum emulators and quantum devices (IBMQ Santiago) is provided for a small chain showing results in good agreement with the classical benchmark without further addition of any error mitigation technique.

Many processes in chemistry, physics, and biology depend on thermallyactivated events in which the system changes its state by surmounting anactivation barrier. Examples range from chemical reactions, protein folding,and nucleation events. Parameterized forms of the mean-field potential areoften employed in the stochastic modeling of activated processes. In thiscontribution, we explore the alternative of employing parameterized forms ofthe equilibrium distribution by means of the symmetric linear combination oftwo gaussian functions. Such a procedure leads to flexible and convenientmodels for the landscape and the energy barrier whose features are controlledby the second moments of the gaussian functions. The rate constants areexamined through the solution of the corresponding diffusion problem, that isthe Fokker-Planck-Smoluchowski equation specified according to theparameterized equilibrium distribution. The numerical calculations clearly showthat the asymptotic limit of large barriers does not agree with the results ofthe Kramers theory. The underlying reason is that the linear scaling of thepotential, the procedure justifying the Kramers theory, cannot be applied whendealing with parameterized forms of the equilibrium distribution. A differentkind of asymptotic analysis is then required and we introduce the appropriatetheory when the equilibrium distribution is represented as a symmetric linearcombination of two gaussian functions, first in the one-dimensional case andafterward in the multi-dimensional diffusion model.

Many processes in chemistry, physics, and biology depend on thermally activated events in which the system changes its state by surmounting an activation barrier. Examples range from chemical reactions, protein folding, and nucleation events. Parameterized forms of the mean-field potential are often employed in the stochastic modeling of activated processes. In this contribution, we explore the alternative of employing parameterized forms of the equilibrium distribution by means of the symmetric linear combination of two gaussian functions. Such a procedure leads to flexible and convenient models for the landscape and the energy barrier whose features are controlled by the second moments of the gaussian functions. The rate constants are examined through the solution of the corresponding diffusion problem, that is the Fokker-Planck-Smoluchowski equation specified according to the parameterized equilibrium distribution. The numerical calculations clearly show that the asymptotic limit of large barriers does not agree with the results of the Kramers theory. The underlying reason is that the linear scaling of the potential, the procedure justifying the Kramers theory, cannot be applied when dealing with parameterized forms of the equilibrium distribution. A different kind of asymptotic analysis is then required and we introduce the appropriate theory when the equilibrium distribution is represented as a symmetric linear combination of two gaussian functions, first in the one-dimensional case and afterward in the multi-dimensional diffusion model.