From quantum cellular automata to quantum lattice gases

A natural architecture for nanoscale quantum computation is that of a quantum cellular automaton. Motivated by this observation, in this paper we begin an investigation of exactly unitary cellular automata. After proving that there can be no nontrivial, homogeneous, local, unitary, scalar cellular automaton in one dimension, we weaken the homogeneity condition and show that there are nontrivial, exactly unitary, partitioning cellular automata. We find a one parameter family of evolution rules which are best interpreted as those for a one particle quantum automaton. This model is naturally reformulated as a two component cellular automaton which we demonstrate to limit to the Dirac equation. We describe two generalizations of this automaton, the second of which, to multiple interacting particles, is the correct definition of a quantum lattice gas.Comment: 22 pages, plain TeX, 9 PostScript figures included with epsf.tex (ignore the under/overfull \vbox error messages); minor typographical corrections and journal reference adde

When is a quantum cellular automaton (QCA) a quantum lattice gas automaton (QLGA)?

Quantum cellular automata (QCA) are models of quantum computation of particular interest from the point of view of quantum simulation. Quantum lattice gas automata (QLGA - equivalently partitioned quantum cellular automata) represent an interesting subclass of QCA. QLGA have been more deeply analyzed than QCA, whereas general QCA are likely to capture a wider range of quantum behavior. Discriminating between QLGA and QCA is therefore an important question. In spite of much prior work, classifying which QCA are QLGA has remained an open problem. In the present paper we establish necessary and sufficient conditions for unbounded, finite Quantum Cellular Automata (QCA) (finitely many active cells in a quiescent background) to be Quantum Lattice Gas Automata (QLGA). We define a local condition that classifies those QCA that are QLGA, and we show that there are QCA that are not QLGA. We use a number of tools from functional analysis of separable Hilbert spaces and representation theory of associative algebras that enable us to treat QCA on finite but unbounded configurations in full detail.Comment: 37 pages, 7 figures, with changes to explanatory text and updated figures, J. Math. Phys. versio

Development of a collision table for three dimensional lattice gases

Bibliography: pages 92-95.A lattice gas is a species of cellular automaton used for numerically simulating fluid flows. TransGas [9], the lattice gas code currently in use at the CSIR, is based on the FHP-I model [5], and is used to perform various two-dimensional flow simulations. In order to broaden the scope of the applications in which lattice gases can be used locally, the development of a three-dimensional lattice gas capability is required. The first major task in setting up a three dimensional-lattice gas is the construction of an efficient collision rule generator which will determine collision outcomes. For suitability to local applications, the collision rules should be chosen in such a way as to maximise the Reynolds coefficient of the flow, while conserving quantities such as mass and momentum. Part of the task thus becomes an optimisation problem. When expanding from two to three dimensions, the number of possible collision rules increases from 64 to 16777216. If a complete collision rule table is used for determining collision outcomes, storage problems are encountered on the available hardware. Selection and optimisation of collision rules cannot be done by hand when there are so many rules to choose from. Selection of rules is thus non-trivial. The work outlined in this thesis provides the CSIR with a 3-D lattice gas collision table which is well suited to the available hardware capabilities. The necessary theoretical background is considered, and a survey of the literature is presented. Based on the findings of this literature study, various methods of collision outcome determination are implemented which are considered to be suitable to the local needs, while remaining within the constraints set by hardware availability. An isometric collision algorithm, and a reduced collision table are generated and tested. A measure of the overall efficiency of a lattice gas model is determined by two factors, namely the computational efficiency and the implementation efficiency. In testing a collision table, the first is characterised by the rate at which post-collision states can be determined, and depends on the hardware and programming techniques. The second factor can be expressed by means of a number called the Reynolds coefficient, which is defined and discussed in the following chapters. The higher the Reynolds coefficient of a model, the greater the scope of flow regimes which may be simulated using it. Another advantage of having a high Reynolds coefficient is that the simulation time required for a given flow regime decreases as the Reynolds coefficient of the model increases. The overall efficiency of the isometric model is too low to be of practical use, but a significant improvement is obtained by using the method of reduced tables. In the isometric case, the number of collision outcomes that can be determined per second is similar to that of the reduced table, but the Reynolds coefficient is very much lower. Simulation of a flow regime with a Reynolds number of about 100, on a lattice of size 128³, over 20 thousand timesteps, making use of the isometric model, would take of the order of a few years to complete on the currently available hardware. Since the simulation parameters mentioned above are typical of the local requirements for lattice gas simulations, this method is obviously unsatisfactory. The isometric method does however serve as a useful introduction to three-dimensional lattice gas collision rule methods. The reduced collision table has been constructed so that it maintains semi-detailed balance, and the Boltzmann Reynolds coefficient has been calculated. In the reduced collision table model, the efficiency is higher than the isometric case in respect of both the rate at which collision outcomes can be determined, and in terms of the Reynolds coefficient. As a result of these improvements, the simulation time for the exact case mentioned above would reduce to the order of days, on the same hardware. This simulation time is sufficiently low for immediate practical application in the local environment

On the development of slime mould morphological, intracellular and heterotic computing devices

The use of live biological substrates in the fabrication of unconventional computing (UC) devices is steadily transcending the barriers between science fiction and reality, but efforts in this direction are impeded by ethical considerations, the field’s restrictively broad multidisciplinarity and our incomplete knowledge of fundamental biological processes. As such, very few functional prototypes of biological UC devices have been produced to date. This thesis aims to demonstrate the computational polymorphism and polyfunctionality of a chosen biological substrate — slime mould Physarum polycephalum, an arguably ‘simple’ single-celled organism — and how these properties can be harnessed to create laboratory experimental prototypes of functionally-useful biological UC prototypes. Computing devices utilising live slime mould as their key constituent element can be developed into a) heterotic, or hybrid devices, which are based on electrical recognition of slime mould behaviour via machine-organism interfaces, b) whole-organism-scale morphological processors, whose output is the organism’s morphological adaptation to environmental stimuli (input) and c) intracellular processors wherein data are represented by energetic signalling events mediated by the cytoskeleton, a nano-scale protein network. It is demonstrated that each category of device is capable of implementing logic and furthermore, specific applications for each class may be engineered, such as image processing applications for morphological processors and biosensors in the case of heterotic devices. The results presented are supported by a range of computer modelling experiments using cellular automata and multi-agent modelling. We conclude that P. polycephalum is a polymorphic UC substrate insofar as it can process multimodal sensory input and polyfunctional in its demonstrable ability to undertake a variety of computing problems. Furthermore, our results are highly applicable to the study of other living UC substrates and will inform future work in UC, biosensing, and biomedicine