A technique for simulating quantum dynamics in phase space is discussed.It makes use of ensembles of classical trajectories to approximate the distribution functions and their derivatives by implementing Adaptive Kernel Density Estimation.It is found to improve the accuracy and stability of the simulations compared to more conventional particle methods.Formulation of the method in higher dimensions is straightforward.
A technique for simulating quantum dynamics in phase space is discussed.It makes use of ensembles of classical trajectories to approximate the distribution functions and their derivatives by implementing Adaptive Kernel Density Estimation.It is found to improve the accuracy and stability of the simulations compared to more conventional particle methods.Formulation of the method in higher dimensions is straightforward.
A technique for simulating quantum dynamics in phase space i discussed. It makes use of ensembles of classical trajectories to approximate the distribution functions and their derivatives by implementing Adaptive Kernel Density Estimation. It is found to improve the accuracy and stability of the simulations compared to more conventional particle methods. Formulation of the method in higher dimensions is straightforward. Complete analytical description of physical systems exist only for the most simple ideal cases. The vast majority of physical problems require the solution of approximate equations most fre- quently by using numerical calculations. To this end, much effort has been devoted to the devel- opment of suitable techniques to be used in computer simulations. The increasing computational power makes possible the description of more complicated, that is, less ideal systems allowing for more precise predictions of their properties as well as for a better understanding of the underlying phenomena. Among the wealth of numerical approaches to solve dynamical equations by means of discretization, particle methods have gained attention in the last few decades as a conceptu- ally appealing alternative. Conventional numerical solvers rely on a regular discretization of the independent variables to facilitate the evaluation of derivatives to a given order of accuracy. As a consequence, a mesh needs to be defined as the base of the simulations, parts of which remain empty at any given simulation time even with the most sophisticated mesh schemes. The cen- tral idea of particle methods consists of representing functions by sets of particles conveniently located and weighted such that the function is appropriately sampled by the particles. Evolution of the functions is then accomplished by an equivalent evolution of the particles. This meshless or Lagrangian type of discretization has some intrinsic computational efficiency advantages because there is no wasted space. The advantage, however, comes at the expense of having to evaluate derivatives of functions on unstructured arrays of sampling points. The issues about convergence and stability of the resulting algorithms have been the subject of active and fruitfull research. Currently there are some well established techniques to solve dynamical equations with particle methods. Our interest is to provide an alternative ingredient to further improve the performance of numerical solutions of dynamical problems of physical relevance. In the present work we discuss, in the context of particle methods, the implementation of a technique for representing functions and derivatives with adaptive parameters. Adaptiveness require the parameters to be automatically ad- justed to optimal values according to the location of the particles. As shown below, the proposed scheme is conceptually simple and has been tested in a number of different situations. In some simple test models the present work focuses on quantum dynamics, previously treated with more
We solve quantum dynamical equations of simple systems by propagating ensembles of interacting trajectories. A scheme is proposed which uses adaptive kernel density estimation for representing probability distribution functions and their derivatives. The formulation is carried on in the Husimi representation to ensure the positiveness of the distribution functions. By comparing to previous work, the effect of changing representations is studied as well as the advantage of using adaptive kernels for the estimation of probability distributions. We found significant improvement in the accuracy of the results.
Quantum phase fields are crucial for various quantum technologies, including quantum optics and quantum computing. Accurate reconstruction of these fields is essential for various applications. Current methods often suffer from computational inefficiency and limited accuracy, particularly when dealing with complex systems. This paper introduces a novel method for quantum phase field reconstruction that leverages adaptive kernel selection based on the system's evolution. The core mechanism involves a neural network dynamically selecting the most appropriate kernel function for reconstruction, optimizing both speed and accuracy. We demonstrate the efficacy of this approach through simulations and preliminary experimental results. The paper highlights the potential for significant improvements in both computational efficiency and reconstruction fidelity.
Density estimation is a fundamental task in statistics and machine learning applications. Kernel density estimation is a powerful tool for non-parametric density estimation in low dimensions; however, its performance is poor in higher dimensions. Moreover, its prediction complexity scale linearly with more training data points. This paper presents a method for neural density estimation that can be seen as a type of kernel density estimation, but without the high prediction computational complexity. The method is based on density matrices, a formalism used in quantum mechanics, and adaptive Fourier features. The method can be trained without optimization, but it could be also integrated with deep learning architectures and trained using gradient descent. Thus, it could be seen as a form of neural density estimation method. The method was evaluated in different synthetic and real datasets, and its performance compared against state-of-the-art neural density estimation methods, obtaining competitive results.
This paper proposes a novel adaptive quantum kernel learning framework designed to overcome the limitations of classical kernel methods when applied to quantum state analysis. The core challenge lies in effectively mapping the high-dimensional space of quantum states into a manageable, low-dimensional feature space for traditional kernel-based learning algorithms. Existing methods often fail to capture the intricate correlations inherent in quantum states, leading to suboptimal results. Our approach introduces a self-adaptive kernel learning algorithm that dynamically adjusts the kernel function based on the observed quantum state correlations. This adaptation is achieved through a quantum-inspired optimization process utilizing variational quantum circuits. The circuits evolve the kernel parameters, guided by quantum state tomography data, effectively learning a kernel function tailored to the specific quantum system being analyzed. The key innovation is the integration of quantum mechanics directly into the kernel learning process, offering a potentially significant improvement in feature extraction and classification accuracy for quantum data. This work addresses a critical bottleneck in quantum machine learning, paving the way for more powerful and accurate quantum algorithms.
This paper explores the application of adaptive quantum sampling to probability density estimation, a critical task in various fields including physics, chemistry, and finance. Classical methods struggle with high-dimensional probability distributions, often requiring computationally intensive sampling techniques. We introduce a novel quantum algorithm that dynamically adjusts its sampling strategy based on observed data, significantly enhancing the accuracy of probability density estimation compared to traditional methods. The core mechanism leverages quantum circuits to implement a probabilistic sampling process, ensuring a more robust and efficient estimation of distributions. We detail the algorithm's architecture, key quantum operations, and demonstrate its effectiveness through simulations and illustrative examples. The resulting quantum algorithm holds the potential to unlock deeper insights into complex probability distributions, offering a significant advancement in statistical analysis.