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IEEE International Conference on Quantum Computing and Engineering (QCE)

Estimation of the probabilistic description of quantum physical systems

Abdessamad Belfakir, Randa Herzallah

Published Sep 1, 2022• 0 citations
Publisher

Abstract

Finding the analytic solution of the Liouville-von Neumann equation is mathematically very challenging. Thus, in this paper, we propose an alternative method that estimates and predicts the dynamics of quantum systems through the estimation of the probability distribution describing the vectorisation of the associated density operator involved in the Liouville-von Neumann equation. In particular, we show that the dynamics of the density operator and measurement states of a quantum system can be characterised by complex Gaussian probability density functions. Unlike conventional methods that provide a deterministic estimate of the density operator, our probabilistic framework provide a general estimation method that also characterises the uncertainty and imperfections in quantum systems due to noises and other sources. Comparison of exact and estimated solutions for a real molecule interacting with an external electric field demonstrates the effectiveness of the proposed estimation method.

DOI
10.1109/qce53715.2022.00034

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Finding the analytic solution of the Liouville-von Neumann equation is mathematically very challenging. Thus, in this paper, we propose an alternative method that estimates and predicts the dynamics of quantum systems through the estimation of the probability distribution describing the vectorisation of the associated density operator involved in the Liouville-von Neumann equation. In particular, we show that the dynamics of the density operator and measurement states of a quantum system can be characterised by complex Gaussian probability density functions. Unlike conventional methods that provide a deterministic estimate of the density operator, our probabilistic framework provide a general estimation method that also characterises the uncertainty and imperfections in quantum systems due to noises and other sources. Comparison of exact and estimated solutions for a real molecule interacting with an external electric field demonstrates the effectiveness of the proposed estimation method.
A new control method that considers all sources of uncertainty and noises that might affect the time evolutions of quantum physical systems is introduced. Under the proposed approach, the dynamics of quantum systems are characterised by probability density functions (pdfs), thus providing a complete description of their time evolution. Using this probabilistic description, the proposed method suggests the minimisation of the distance between the actual pdf that describes the joint distribution of the time evolution of the quantum system and the external electric field, and the desired pdf that describes the system target outcome. We start by providing the control solution for quantum systems that are characterised by arbitrary pdfs. The obtained solution is then applied to quantum physical systems characterised by Gaussian pdfs and the form of the optimised controller is elaborated. Finally, the proposed approach is demonstrated on a real molecular system and two spin systems showing the effectiveness and simplicity of the method.
A new control method that considers all sources of uncertainty and noises that might affect the time evolutions of quantum physical systems is introduced. Under the proposed approach, the dynamics of quantum systems are characterised by probability density functions (pdfs), thus providing a complete description of their time evolution. Using this probabilistic description, the proposed method suggests the minimisation of the distance between the actual pdf that describes the joint distribution of the time evolution of the quantum system and the external electric field, and the desired pdf that describes the system target outcome. We start by providing the control solution for quantum systems that are characterised by arbitrary pdfs. The obtained solution is then applied to quantum physical systems characterised by Gaussian pdfs and the form of the optimised controller is elaborated. Finally, the proposed approach is demonstrated on a real molecular system and two spin systems showing the effectiveness and simplicity of the method.
A new control method that considers all sources of uncertainty and noises that might affect the time evolutions of quantum physical systems is introduced. Under the proposed approach, the dynamics of quantum systems are characterised by probability density functions (pdfs), thus providing a complete description of their time evolution. Using this probabilistic description, the proposed method suggests the minimisation of the distance between the actual pdf that describes the joint distribution of the time evolution of the quantum system and the external electric field, and the desired pdf that describes the system target outcome. We start by providing the control solution for quantum systems that are characterised by arbitrary pdfs. The obtained solution is then applied to quantum physical systems characterised by Gaussian pdfs and the form of the optimised controller is elaborated. Finally, the proposed approach is demonstrated on a real molecular system and two spin systems showing the effectiveness and simplicity of the method.
The starting point of the classical description of motion is the Newton equations that yield a phase space trajectory (rN (t), pN (t)) for a given initial condition (rN (0), pN (0)). Alternatively one may describe classical motion in the framework of the Liouville equation (Section (1.2.2)) that describes the time evolution of the phase space probability density f (rN , pN ; t). For a closed system fully described in terms of a well specified initial condition, the two descriptions are completely equivalent. Probabilistic treatment becomes essential in reduced descriptions that focus on parts of an overall system, as was demonstrated in Sections 5.1–5.3 for equilibrium systems, and in Chapters 7 and 8 that focus on the time evolution of classical systems that interact with their thermal environments. This chapter deals with the analogous quantum mechanical problem. Within the limitations imposed by its nature as expressed, for example, by Heisenbergtype uncertainty principles, the Schrödinger equation is deterministic. Obviously it describes a deterministic evolution of the quantum mechanical wavefunction. The analog of the phase space probability density f (rN , pN ; t) is now the quantum mechanical density operator (often referred to as the “density matrix”), whose time evolution is determined by the quantum Liouville equation. Again, when the system is fully described in terms of a well specified initial wavefunction, the two descriptions are equivalent. The density operator formalism can, however, be carried over to situations where the initial state of the system is not well characterized and/or a reduced description of part of the overall system is desired. Such situations are considered later in this chapter.
Abstract The review of a new probability representation of quantum states is presented, where the states are described by conventional probability distribution functions. The invertible map of the probability distribution onto density operators in the Hilbert space is found using the introduced operators called a quantizer–dequantizer, which specify the invertible map of operators of quantum observables onto functions and a product of the operators onto an associative product (star product) of the functions. Examples of a quantum oscillator and a spin-1/2 particle are considered. The kinetic equations for probabilities, specifying the evolution of the states of a quantum system, which are equivalent to Schrödinger and von Neumann equations, are derived explicitly.
Abstract This study introduces an innovative control methodology designed specifically for quantum physical systems. It bridges a critical gap by accounting for the full spectrum of uncertainties and noise that could affect the time evolution of these systems. In contrast to conventional methods, our pioneering approach utilizes probability density functions (pdfs) to characterize quantum dynamics, providing a more detailed and accurate description of their temporal behavior. We propose a strategy that seeks to minimize the discrepancy between the actual pdf, which encapsulates the combined dynamics of the quantum system and an external electric field, and a desired pdf that aligns with the system’s intended outcomes. This strategy marks a significant shift from traditional quantum control techniques. Initially, we present a solution for controlling quantum systems defined by general pdfs. This solution is then demonstrated on quantum systems described by Gaussian pdfs, with an in-depth account of the resulting optimized controller’s structure. The study culminates with practical demonstrations, showcasing the approach’s efficacy and practicality, thus endorsing its potential as a formidable instrument in quantum control. Graphical abstract