12,590 research outputs found
Quantum Causal Graph Dynamics
Consider a graph having quantum systems lying at each node. Suppose that the
whole thing evolves in discrete time steps, according to a global, unitary
causal operator. By causal we mean that information can only propagate at a
bounded speed, with respect to the distance given by the graph. Suppose,
moreover, that the graph itself is subject to the evolution, and may be driven
to be in a quantum superposition of graphs---in accordance to the superposition
principle. We show that these unitary causal operators must decompose as a
finite-depth circuit of local unitary gates. This unifies a result on Quantum
Cellular Automata with another on Reversible Causal Graph Dynamics. Along the
way we formalize a notion of causality which is valid in the context of quantum
superpositions of time-varying graphs, and has a number of good properties.
Keywords: Quantum Lattice Gas Automata, Block-representation,
Curtis-Hedlund-Lyndon, No-signalling, Localizability, Quantum Gravity, Quantum
Graphity, Causal Dynamical Triangulations, Spin Networks, Dynamical networks,
Graph Rewriting.Comment: 8 pages, 1 figur
Topological reversibility and causality in feed-forward networks
Systems whose organization displays causal asymmetry constraints, from
evolutionary trees to river basins or transport networks, can be often
described in terms of directed paths (causal flows) on a discrete state space.
Such a set of paths defines a feed-forward, acyclic network. A key problem
associated with these systems involves characterizing their intrinsic degree of
path reversibility: given an end node in the graph, what is the uncertainty of
recovering the process backwards until the origin? Here we propose a novel
concept, \textit{topological reversibility}, which rigorously weigths such
uncertainty in path dependency quantified as the minimum amount of information
required to successfully revert a causal path. Within the proposed framework we
also analytically characterize limit cases for both topologically reversible
and maximally entropic structures. The relevance of these measures within the
context of evolutionary dynamics is highlighted.Comment: 9 pages, 3 figure
Causal graph dynamics
We extend the theory of Cellular Automata to arbitrary, time-varying graphs.
In other words we formalize, and prove theorems about, the intuitive idea of a
labelled graph which evolves in time - but under the natural constraint that
information can only ever be transmitted at a bounded speed, with respect to
the distance given by the graph. The notion of translation-invariance is also
generalized. The definition we provide for these "causal graph dynamics" is
simple and axiomatic. The theorems we provide also show that it is robust. For
instance, causal graph dynamics are stable under composition and under
restriction to radius one. In the finite case some fundamental facts of
Cellular Automata theory carry through: causal graph dynamics admit a
characterization as continuous functions, and they are stable under inversion.
The provided examples suggest a wide range of applications of this mathematical
object, from complex systems science to theoretical physics. KEYWORDS:
Dynamical networks, Boolean networks, Generative networks automata, Cayley
cellular automata, Graph Automata, Graph rewriting automata, Parallel graph
transformations, Amalgamated graph transformations, Time-varying graphs, Regge
calculus, Local, No-signalling.Comment: 25 pages, 9 figures, LaTeX, v2: Minor presentation improvements, v3:
Typos corrected, figure adde
Energy-based Analysis of Biochemical Cycles using Bond Graphs
Thermodynamic aspects of chemical reactions have a long history in the
Physical Chemistry literature. In particular, biochemical cycles - the
building-blocks of biochemical systems - require a source of energy to
function. However, although fundamental, the role of chemical potential and
Gibb's free energy in the analysis of biochemical systems is often overlooked
leading to models which are physically impossible. The bond graph approach was
developed for modelling engineering systems where energy generation, storage
and transmission are fundamental. The method focuses on how power flows between
components and how energy is stored, transmitted or dissipated within
components. Based on early ideas of network thermodynamics, we have applied
this approach to biochemical systems to generate models which automatically
obey the laws of thermodynamics. We illustrate the method with examples of
biochemical cycles. We have found that thermodynamically compliant models of
simple biochemical cycles can easily be developed using this approach. In
particular, both stoichiometric information and simulation models can be
developed directly from the bond graph. Furthermore, model reduction and
approximation while retaining structural and thermodynamic properties is
facilitated. Because the bond graph approach is also modular and scaleable, we
believe that it provides a secure foundation for building thermodynamically
compliant models of large biochemical networks
Relationship Between Quantum Walk and Relativistic Quantum Mechanics
Quantum walk models have been used as an algorithmic tool for quantum
computation and to describe various physical processes. This paper revisits the
relationship between relativistic quantum mechanics and the quantum walks. We
show the similarities of the mathematical structure of the decoupled and
coupled form of the discrete-time quantum walk to that of the Klein-Gordon and
Dirac equations, respectively. In the latter case, the coin emerges as an
analog of the spinor degree of freedom. Discrete-time quantum walk as a coupled
form of the continuous-time quantum walk is also shown by transforming the
decoupled form of the discrete-time quantum walk to the Schrodinger form. By
showing the coin to be a means to make the walk reversible, and that the
Dirac-like structure is a consequence of the coin use, our work suggests that
the relativistic causal structure is a consequence of conservation of
information. However, decoherence (modelled by projective measurements on
position space) generates entropy that increases with time, making the walk
irreversible and thereby producing an arrow of time. Lieb-Robinson bound is
used to highlight the causal structure of the quantum walk to put in
perspective the relativistic structure of quantum walk, maximum speed of the
walk propagation and the earlier findings related to the finite spread of the
walk probability distribution. We also present a two-dimensional quantum walk
model on a two state system to which the study can be extended.Comment: 12 pages and 1 figure, Published versio
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