Abstract: The quantum walk is a subset of classical random walks. Considerable research has lately been conducted on quantum walks in relation to quantum computing. The walk may be thought of as a type of quantum cellular automata. In a broader sense, there is a one-to-one link between them. Furthermore, we look at another type of quantum cellular automata that may be thought of as a quantum version of stochastic cellular automata. The quantum walk formalism is a widely used and quite practical paradigm for modelling quantum systems such as Dirac equation simulations, discrete dynamics in both the low and high energy regimes, and the development of a multitude of quantum algorithms. The walker in a QW moves in a quantum superposition of pathways, and the ensuing interference serves as the foundation for a broad range of quantum algorithms, including quantum search, graph isomorphism issues, and rating nodes in a network.
To generate a block representation of n-dimensional quantum cellular automata, we apply the following general result to them. As a result, we have demonstrated that their generic, axiomatic formulation nevertheless produces a unifying, operational representation of them.

Some non-trivial ramifications are investigated, such as the fact that bijective non-reversible CA are not physical as closed systems, or that quantum information can move faster than classical information — within some specified dynamics. Finally, we present n-dimensional quantum cellular automata that can simulate anything else. This means that the initial configuration and local transition rule of any one-dimensional QCA may be stored within the universal QCA's initial configuration. The simulated QCA will then correspond to many phases of the universal QCA. In the sense that each cell in the simulated QCA is represented as a group of nearby cells in the universal QCA, the simulation preserves the topology. Because the encoding is linear, it bears no cost of the calculation.

Keywords: Physics, Quantum simulation, QCA, Quantum Walks

| DOI: 10.17148/IARJSET.2021.81141