Richard Feynman suggested that computers that use quantum logic for information processing can simulate some quantum systems efficiently, even when this is not possible to computers based on classical logic. To simulate the dynamics of a quantum system usually means to describe it in terms of qubits, and its dynamics by a succession of logical gates – which are unitary transformations involving at most two qubits at time. This is the paradigm known as the Quantum Circuit Model, in analogy with the logical circuits of classical computers. Nevertheless, this way of “interpreting” the system and its dynamics is very artificial and detached from the expected inner workings of Nature. A more natural way of describing quantum systems and their dynamics within a computational perspective is given by Quantum Walks (QWs) and Quantum Cellular Automata (QCA). These two models also describe the system as collection of finite dimensional systems, but the dynamics is autonomous and generated by local interactions.
The evolution of a qubit in a QCA or in a QW is thus fully determined by the surrounding qubits and the interactions between them. Besides being a more natural way to describe physical systems, the structure of QWs and QCA is also more suitable for nowadays implementations. As QWs and QCA have a discrete space-time, they are more amendable to the development of coarse graining procedures. The aim of such procedures is to find an effective description of the system, without resorting to all its degrees of freedom. This effective description might be highly advantageous when treating many-body quantum systems. Moreover, when taking the limit of the spacing between cells and the time-step to zero, a continuous dynamic emerges, as described by partial differential equations. This has already been done for several quantum walks, and the dynamics of relativistic quantum fields was obtained. Quantum cellular automata and different models of quantum walks will be employed in this project allowing for the simulation of other systems. At each level of coarse graining the continuous limit will be obtained, and an effective dynamic will emerge. In this project we will exploit discrete space-time quantum computational models–based descriptions and harvest them for questions of physical interest. For instance, we will address the following questions: what kinds of phenomenology can emerge from a continuous limit of different interactions and distributions of QCA and alternative models of QWs? Is it possible to describe some of these dynamics in an efficient way by a classical computer, or are there intrinsic quantum properties that render the description unavoidably quantum? In the latter case, is there a simplified level of description where a classical description becomes possible.