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Communications Physics

Mixed quantum-classical dynamics for near term quantum computers

Daniel Bultrini, Oriol Vendrell

Published Nov 15, 2023• 8 citations
Publisher

Abstract

Abstract Mixed quantum-classical dynamics is a set of methods often used to understand systems too complex to treat fully quantum mechanically. Many techniques exist for full quantum mechanical evolution on quantum computers, but mixed quantum-classical dynamics are less explored. We present a modular algorithm for general mixed quantum-classical dynamics where the quantum subsystem is coupled with the classical subsystem. We test it on a modified Shin-Metiu model in the first quantization through Ehrenfest propagation. We find that the Time-Dependent Variational Time Propagation algorithm performs well for short-time evolutions and retains qualitative results for longer-time evolutions.

DOI
10.1038/s42005-023-01451-2

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Abstract Mixed quantum-classical dynamics is a set of methods often used to understand systems too complex to treat fully quantum mechanically. Many techniques exist for full quantum mechanical evolution on quantum computers, but mixed quantum-classical dynamics are less explored. We present a modular algorithm for general mixed quantum-classical dynamics where the quantum subsystem is coupled with the classical subsystem. We test it on a modified Shin-Metiu model in the first quantization through Ehrenfest propagation. We find that the Time-Dependent Variational Time Propagation algorithm performs well for short-time evolutions and retains qualitative results for longer-time evolutions.
Mixed quantum-classical dynamics is a set of methods often used to understand systems too complex to treat fully quantum mechanically. Many techniques exist for full quantum mechanical evolution on quantum computers, but mixed quantum-classical dynamics are less explored. We present a modular algorithm for general mixed quantum-classical dynamics where the quantum subsystem is coupled with the classical subsystem. We test it on a modified Shin-Metiu model in the first quantization through Ehrenfest propagation. We find that the Time-Dependent Variational Time Propagation algorithm performs well for short-time evolutions and retains qualitative results for longer-time evolutions.
An algorithm for the simulation of quantum-classical dynamics is presented. Quantum-classical evolution is effected by a propagator exp(i (L) over capt) involving the quantum classical Liouville operator (L) over cap that describes the evolution of a quantum subsystem coupled to a classical bath. Such a mixed description provides a means to study the dynamics of complex many-body systems where certain degrees of freedom are treated quantum mechanically. The algorithm is constructed by decomposing the time interval Deltat into small segments of length At and successively applying the propagator in the short time segments to obtain the evolution for long times. The algorithm is shown to be a discretization of the iterated Dyson form of the propagator whose direct solution is vexatious. The sequential short-time propagation algorithm is applied to the spin-boson model for a range of values of the Kondo parameter and shown to be effective.
Quantum mechanics is not logically closed with respect to the classical world.Its formalism unfolds as the quantization of a sub-set of classical Hamiltonians.The interpretation of quantum theory in terms of the measurement process inevitably requires to deal with systems composed by a mixture of both classical and quantum degrees of freedom.Moreover, when energy can flow between the quantum and classical degrees of freedom (i.e., in the case of nonadiabatic dynamics), there are more theoretical difficulties in order to obtain a fully consistent quantum-classical formalism.In order to perform calculations, one can renounce to the usual Lie algebraic structure of well-established physical theories, adopt non-Hamiltonian brackets, and obtain a formalism for the dynamics and statistics of quantum-classical systems that has an affordable computational complexity.Recent progress in the algorithms for the sampling of nonadiabatic dynamics of quantum-classical systems at long time is reviewed here.
Mixed quantum-classical equations of motion are derived for a quantum subsystem of light (mass m) particles coupled to a classical bath of massive (mass M) particles. The equation of motion follows from a partial Wigner transform over the bath degrees of freedom of the Liouville equation for the full quantum system, followed by an expansion in the small parameter μ=(m/M)1/2 in analogy with the theory of Brownian motion. The resulting mixed quantum-classical Liouville equation accounts for the coupled evolution of the subsystem and bath. The quantum subsystem is represented in an adiabatic (or other) basis and the series solution of the Liouville equation leads to a representation of the dynamics in an ensemble of surface-hopping trajectories. A generalized Pauli master equation for the evolution of the diagonal elements of the density matrix is derived by projection operator methods and its structure is analyzed in terms of surface-hopping trajectories.
A general dynamical system composed of two coupled sectors is considered. The initial-time configuration of one of these sectors is described by a set of classical data while the other is described by standard quantum data. These dynamical systems will be named half quantum and the aim of this paper is to derive their dynamical evolution. The standard approach would be to use quantum mechanics to make predictions for the time evolution of the half quantum initial data. The main problem is how quantum mechanics can be applied to a dynamical system whose initial time configuration is not described by a set of fully quantum data. A solution to this problem is presented and used as a guideline to obtain a general formulation of coupled classical-quantum dynamics. Finally, a quantization prescription mapping a given classical theory to the corresponding half quantum one is presented.
Abstract The quantum speed limit (QSL) provides a fundamental upper bound on the speed of quantum evolution, but its evaluation in generic open quantum systems still presents a formidable computational challenge. Herein, we introduce a hybrid quantum-classical method for computing QSL times in multi-level open quantum systems. The method is based on a mixed Wigner–Heisenberg representation of the composite quantum dynamics, in which the open subsystem of interest is treated quantum mechanically and the bath is treated in a classical-like fashion. By solving a set of coupled first-order deterministic differential equations for the quantum and classical degrees of freedom, one can compute the QSL time. To demonstrate the utility of the method, we study the unbiased spin-boson model and provide a detailed analysis of the effect of the subsystem-bath coupling strength and bath temperature on the QSL time. In particular, we find a turnover of the QSL time in the strong coupling regime, which is indicative of a speed-up in the quantum evolution. We also apply the method to the Fenna–Matthews–Olson complex model and identify a potential connection between the QSL time and the efficiency of the excitation energy transfer at different temperatures.