This study simulates random movement and aggregation of particles in two-dimensional space based upon both quantum and classical mechanics. Using an original computer program to perform the calculations, the objective is to compare how quantum effects influence the random movement of a particle in comparison to the classical random movement. These effects are further studied by analyzing how the amassing of particles around a "seed" is affected by the differences in the random movement. Using the classical models that were generated as the basis of comparison, the initial results show that the quantum model aggregate grows at a slower ratc than the classical case. Also, the quantum model grows in a more amorphous manner than the clear branching of the classical example. In an effort to more accurately simulate the behavior of the probability function as it encounters other particles, both the quantum and classical models were adjusted. This yielded a quantum aggregation that developed more similarly than the classical model. The primary difference in the quantum model was a noticeable lack of symmetry as the particles amassed around the seeded particles. It is possible that this iteration of the quantum model develops more rapidly then the classical model, though more simulations are needed to further test this. The effect other particles have on the development of the probability function also needs to be further examined lo ensure that it is being modeled as accurately as possible.
This study simulates random movement and aggregation of particles in two-dimensional space based upon both quantum and classical mechanics. Using an original computer program to perform the calculations, the objective is to compare how quantum effects influence the random movement of a particle in comparison to the classical random movement. These effects are further studied by analyzing how the amassing of particles around a "seed" is affected by the differences in the random movement. Using the classical models that were generated as the basis of comparison, the initial results show that the quantum model aggregate grows at a slower ratc than the classical case. Also, the quantum model grows in a more amorphous manner than the clear branching of the classical example. In an effort to more accurately simulate the behavior of the probability function as it encounters other particles, both the quantum and classical models were adjusted. This yielded a quantum aggregation that developed more similarly than the classical model. The primary difference in the quantum model was a noticeable lack of symmetry as the particles amassed around the seeded particles. It is possible that this iteration of the quantum model develops more rapidly then the classical model, though more simulations are needed to further test this. The effect other particles have on the development of the probability function also needs to be further examined lo ensure that it is being modeled as accurately as possible.
Though classical random walks have been studied for many years, research concerning their quantum analogues, quantum random walks, has only come about recently. Numerous simulations of both types of walks have been run and analyzed, and are generally well-understood. Research pertaining to one of the more important properties of classical random walks, namely, their ability to build fractal structures in diffusion-limited aggregation, has been particularly noteworthy. However, only now has research begun in this area in regards to quantum random motion. The study of random walks and the structures they build has various applications in materials science. Since all processes are quantum in nature, it is important to consider the quantum variant of diffusion-limited aggregation. Recognizing that Schrödinger equation and a classical random walk are both diffusion equations, it is possible to connect and compare them. Using similar parameters for both equations, we ran various simulations aggregating particles. Our results show that particles moving according to Schrödinger equation can create fractal structures, much like the classical random walk. Furthermore, the fractal dimensions of these quantum diffusion-limited aggregates vary between 1.43 and 2, depending on the size of the initial wave packet.
In the world of computational probing, random walks and quantum walks are two separate but tightly linked paradigms. Classical stochastic processes like random walks are based on the random movement of particles in mathematical space and have been extensively used in data mining, network analysis, computer vision, etc. Quantum walks, by contrast generalize random walks into the quantum realm, taking the benefit of the quantum superposition and interference in order to investigate multiple paths at the same time. Quantum walks provide significant potential for algorithmic development: in quantum computing and communication, and in Scientific modelling and simulation. This paper provides an in-depth survey of random walks and quantum walks, outlining historical context, fundamental principles, applications, challenges, and future prospects. In this paper, motivated by the abstractness of both random walks and quantum walks, we survey the field of computational exploration at large scales, coalescing the myriad notions and difficult challenges addressed by existing research efforts on random and quantum walks, inferring a nuanced and multifaceted view on how both random walks and quantum walks are involved in paving the way to future development in many fields.
Quantum computers have sparked significant interest in recent years due to their potential to revolutionize various fields. One area where quantum computing shows great promise is in the study of quantum walks, a quantum counterpart of the classical random walk algorithm that has been foundational in scientific research. While both quantum and classical random walks involve a "walker" moving through a space or graph, quantum walks differ fundamentally due to quantum principles such as superposition, leading to unique behaviors like linear spreading and localization. This thesis investigates the quantum walk simulation, with a particular focus on the Quantum-Walk-Replicating-Random-Walk (QWRW) model. Unlike traditional quantum walks, which sum over all possible paths with complex interference effects, the QWRW approach models the walk as a series of distinct, classical-like steps. This trajectory-based perspective offers a novel way to analyze the walker's position and movement, avoiding the complexities of quantum interference. The QWRW model is particularly valuable in understanding key phenomena of quantum walks, such as linear spreading and localization, by providing insights into the directional properties of quantum walkers. By defining transition probabilities in both space and time, the QWRW model offers a detailed framework for examining the spatial and temporal characteristics of quantum walks, enhancing our understanding of their behavior and potential applications. This study aims to bridge the gap between classical and quantum walks, contributing to the broader field of quantum computing and its practical implications.
Random walks are fundamental models of stochastic processes with applications in various fields, including physics, biology, and computer science. We study classical and quantum random walks under the influence of stochastic resetting on arbitrary networks. Based on the mathematical formalism of quantum stochastic walks, we provide a framework of classical and quantum walks whose evolution is determined by graph Laplacians. We study the influence of quantum effects on the stationary and long-time average probability distribution by interpolating between the classical and quantum regime. We compare our analytical results on stationary and long-time average probability distributions with numerical simulations on different networks, revealing differences in the way resets affect the sampling properties of classical and quantum walks.
Quantum walks and random walks bear similarities and divergences. One of the most remarkable disparities affects the probability of finding the particle at a given location: typically, almost a flat function in the first case and a bell-shaped one in the second case. Here I show how one can impose any desired stochastic behavior (compatible with the continuity equation for the probability function) on both systems by the appropriate choice of time- and site-dependent coins. This implies, in particular, that one can devise quantum walks that show diffusive spreading without losing coherence as well as random walks that exhibit the characteristic fast propagation of a quantum particle driven by a Hadamard coin.
This paper investigates the application of quantum-inspired concepts to enhance the performance of random walk algorithms, specifically within the domain of global optimization. Classical random walk algorithms frequently struggle with convergence issues, frequently becoming trapped within local optima, a significant limitation when dealing with complex, high-dimensional search spaces. This research proposes a novel approach where random walk behavior is modeled based on principles derived from quantum mechanics, incorporating superposition and interference effects to facilitate more extensive and efficient exploration of the solution landscape. The core idea is to replace the deterministic step size in traditional random walks with a probabilistic movement governed by a quantum-like probability distribution. We explore the mathematical formulation of this process, detailing the probabilities of transition between states and the impact of interference. The resulting algorithms demonstrate improved convergence rates and a reduced tendency to get stuck in local optima compared to standard random walk methods. The paper concludes by outlining potential future research directions, including adaptive parameter tuning and integration with other optimization techniques.