Axiomatic definitions of Quantum Cellular Automata (QCA) are foundational in discrete space-time models of physics. Templates of axiomatic QCA serve as primary points of investigations concerning properties, structural results, and classes of QCA that are potentially significant for quantum algorithms and simulations. We examine, from a representation theory standpoint, the relation between the templates corresponding to the Schr\{o}dinger and the Heisenberg pictures of evolution for QCA over finite, unbounded configurations.
Axiomatic definitions of Quantum Cellular Automata (QCA) are foundational in discrete space-time models of physics. Templates of axiomatic QCA serve as primary points of investigations concerning properties, structural results, and classes of QCA that are potentially significant for quantum algorithms and simulations. We examine, from a representation theory standpoint, the relation between the templates corresponding to the Schr{o}dinger and the Heisenberg pictures of evolution for QCA over finite, unbounded configurations.
Quantum cellular automata (QCA) are discrete models of space and time homogeneous quantum field theories (QFTs) and regarded as natural candidates for quantum simulation. Description of a QCA over the separable Hilbert space of finite, unbounded configurations (UFC Hilbert space) with unitary state evolution is the {\it Schrödinger template}, and over the incomplete infinite tensor product algebra (ITPA) with evolution by algebra automorphism is the {\it Heisenberg template}. Whether every Heisenberg template admits an equivalent Schrödinger template is a foundational question, and one that has persisted as an open problem. In the present paper we prove that for every Heisenberg template an equivalent Schrödinger template exists. We frame the question from a representation theory standpoint, using constructs and results from the representation theory of finite and countably infinite dimensional vector spaces and from category theory to answer it. With the previously known existence of a Heisenberg template for every Schrödinger template, our result establishes the equivalence of the templates.
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 informationwithin 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.
Discretizing spacetime is often a natural step towards modelling physical systems. For quantum systems, if we also demand a strict bound on the speed of information propagation, we get quantum cellular automata (QCAs). These originally arose as an alternative paradigm for quantum computation, though more recently they have found application in understanding topological phases of matter and have} been proposed as models of periodically driven (Floquet) quantum systems, where QCA methods were used to classify their phases. QCAs have also been used as a natural discretization of quantum field theory, and some interesting examples of QCAs have been introduced that become interacting quantum field theories in the continuum limit. This review discusses all of these applications, as well as some other interesting results on the structure of quantum cellular automata, including the tensor-network unitary approach, the index theory and higher dimensional classifications of QCAs.
This thesis presents a model of Quantum Cellular Automata (QCA). The presented formalism is a natural quantization of the classical Cellular Automata (CA). It is based on a lattice of qudits, and an update rule consisting of local unitary operators that commute with their own lattice translations. One purpose of this model is to act as a theoretical model of quantum computation, similar to the quantum circuit model. The main advantage that QCA have over quantum circuits is that QCA make considerably fewer demands on the underlying hardware. In particular, as opposed to direct implementations of quantum circuits, the global evolution of the lattice in the QCA model does not assume independent control over individual \emph{qudits}. Rather, all qudits are to be addressed collectively in parallel. The QCA model is also shown to be an appropriate abstraction for space-homogeneous quantum phenomena, such as quantum lattice gases, spin chains and others. Some results that show the benefits of basing the model on local unitary operators are shown: computational universality, strong connections to the circuit model, simple implementation on quantum hardware, and a series of applications. A detailed discussion will be given on one particular application of QCA that lies outside either computation or simulation: single-spin measurement. This algorithm uses the techniques developed in this thesis to achieve a result normally considered hard in physics. It serves well as an example of why QCA are interesting in their own right.
Over an arbitrary commutative ring R, we develop a theory of quantum cellular automata. We then use algebraic K-theory to construct a space Q(X) of quantum cellular automata (QCA) on a given metric space X. In most cases of interest, π0Q(X) classifies QCA up to quantum circuits and stabilization. Notably, the QCA spaces are related by homotopy equivalences Q(∗)≃ΩnQ(Zn) for all n, which shows that the classification of QCA on Euclidean lattices is given by an Ω-spectrum indexed by the dimension n. As a corollary, we also obtain a non-connective delooping of the K-theory of Azumaya R-algebras, which may be of independent interest. We also include a section leading to the Ω-spectrum for QCA over C∗-algebras with unitary circuits.