Non-Unitary Quantum Cellular Automata

The fundamental physical principles of causality and conservation of information impose stringent limitations on the time evolution of physical systems. Within the most basic framework where both space and time are discretized and causality is conserved, many-body systems can be characterized using cellular automata (CAs). These are systems with discrete variables evolving according to update rules dependent solely on local neighborhood interactions (which include for instance the shift map). Quantum cellular automata (QCAs) are analogous models incorporating quantum updates. Although their approximations may appear simplistic when compared to the complexities of realistic many-body dynamics, QCAs offer valuable models for investigating various facets of quantum dynamics. In particular, comparatively little research has been done on non-unitary QCAs even though some of the most useful classical CAs are non-invertible. This thesis examines non-unitary QCAs to model classical and quantum stochastic physics via both discrete- and continuous-time dynamics in one dimension. First, non-equilibrium phase transitions are demonstrated by a novel non-unitary QCA model. The model is based on a well-studied classical model in the directed percolation universality class, which provides a straightforward model for studying associate quantum dynamics due to its partitioning scheme. Corresponding transitions between its absorbing and percolating phases could be observed for both discrete-time and continuous-time dynamics, where the addition of a Hamiltonian is shown to generate coherences at the fixed point while preserving the properties of the dynamical phase [1]. Second, non-unitary QCAs are used to study the density classification (DC) task. This is a computation which maps global density information to a local density, where two approaches are considered: one that preserves the number density and one that performs majority voting. For the DC, two QCAs are introduced that reach the fixed point solution in a time scaling quadratically with the system size. One of the QCAs is based on a known classical probabilistic CA which has been studied in the context of DC [2]. The second QCA for DC is a new quantum model that is designed to demonstrate additional quantum features and is restricted to only two-body interactions. Both can be generated by continuous-time Lindblad dynamics. A third QCA is a hybrid rule defined by discrete-time three-body interactions that is shown to solve the majority voting problem within a time that scales linearly with the system size [3]. Third, a more general classification of non-unitary QCAs is studied based on the index theory for unitary QCAs [4]. The index represents a measure for the net flow of quantum information across a boundary that is invariant under finite-depth local unitary circuits, but is not defined for open quantum systems. A new measure of information flow is proposed for non-unitary QCAs called the information current. While the new measure is not rigid, it can be computed locally based on the matrix-product operator representation of the map and captures the fact that an environment can assist in the transport of information. This is demonstrated by several exemplary dynamics ranging from dissipative local maps to integrable models in one dimension [5].