Graphical Reasoning in Compact Closed Categories for Quantum Computation

Compact closed categories provide a foundational formalism for a variety of important domains, including quantum computation. These categories have a natural visualisation as a form of graphs. We present a formalism for equational reasoning about such graphs and develop this into a generic proof system with a fixed logical kernel for equational reasoning about compact closed categories. Automating this reasoning process is motivated by the slow and error prone nature of manual graph manipulation. A salient feature of our system is that it provides a formal and declarative account of derived results that can include `ellipses'-style notation. We illustrate the framework by instantiating it for a graphical language of quantum computation and show how this can be used to perform symbolic computation.Comment: 21 pages, 9 figures. This is the journal version of the paper published at AIS

Encoding !-tensors as !-graphs with neighbourhood orders

Diagrammatic reasoning using string diagrams provides an intuitive language for reasoning about morphisms in a symmetric monoidal category. To allow working with infinite families of string diagrams, !-graphs were introduced as a method to mark repeated structure inside a diagram. This led to !-graphs being implemented in the diagrammatic proof assistant Quantomatic. Having a partially automated program for rewriting diagrams has proven very useful, but being based on !-graphs, only commutative theories are allowed. An enriched abstract tensor notation, called !-tensors, has been used to formalise the notion of !-boxes in non-commutative structures. This work-in-progress paper presents a method to encode !-tensors as !-graphs with some additional structure. This will allow us to leverage the existing code from Quantomatic and quickly provide various tools for non-commutative diagrammatic reasoning.Comment: In Proceedings QPL 2015, arXiv:1511.0118

Pattern matching and pattern discovery algorithms for protein topologies

We describe algorithms for pattern matching and pattern learning in TOPS diagrams (formal descriptions of protein topologies). These problems can be reduced to checking for subgraph isomorphism and finding maximal common subgraphs in a restricted class of ordered graphs. We have developed a subgraph isomorphism algorithm for ordered graphs, which performs well on the given set of data. The maximal common subgraph problem then is solved by repeated subgraph extension and checking for isomorphisms. Despite the apparent inefficiency such approach gives an algorithm with time complexity proportional to the number of graphs in the input set and is still practical on the given set of data. As a result we obtain fast methods which can be used for building a database of protein topological motifs, and for the comparison of a given protein of known secondary structure against a motif database

Tunneling and Quantum Noise in 1-D Luttinger Liquids

We study non-equilibrium noise in the transmission current through barriers in 1-D Luttinger liquids and in the tunneling current between edges of fractional quantum Hall liquids. The distribution of tunneling events through narrow barriers can be described by a Coulomb gas lying in the time axis along a Keldysh (or non-equilibrium) contour. The charges tend to reorganize as a dipole gas, which we use to describe the tunneling statistics. Intra-dipole correlations contribute to the high-frequency ``Josephson'' noise, which has an algebraic singularity at $\omega=e^*V/\hbar$, whereas inter-dipole correlations are responsible for the low-frequency noise. Inter-dipole interactions give a $1/t^2$ correlation between the tunneling events that results in a $|\omega|$ singularity in the noise spectrum. We present a diagrammatic technique to calculate the correlations in perturbation theory, and show that contributions from terms of order higher than the dipole-dipole interaction should only affect the strength of the $|\omega|$ singularity, but its form should remain $\sim |\omega|$ to all orders in perturbation theory.Comment: RevTex, 9 figures available upon request, cond-mat/yymmnn

Noncommutativity and Discrete Physics

The purpose of this paper is to present an introduction to a point of view for discrete foundations of physics. In taking a discrete stance, we find that the initial expression of physical theory must occur in a context of noncommutative algebra and noncommutative vector analysis. In this way the formalism of quantum mechanics occurs first, but not necessarily with the usual interpretations. The basis for this work is a non-commutative discrete calculus and the observation that it takes one tick of the discrete clock to measure momentum.Comment: LaTeX, 23 pages, no figure

Point patterns occurring on complex structures in space and space-time: An alternative network approach

This paper presents an alternative approach of analyzing possibly multitype point patterns in space and space-time that occur on network structures, and introduces several different graph-related intensity measures. The proposed formalism allows to control for processes on undirected, directional as well as partially directed network structures and is not restricted to linearity or circularity

How to derive Feynman diagrams for finite-dimensional integrals directly from the BV formalism

The Batalin-Vilkovisky formalism in quantum field theory was originally invented to address the difficult problem of finding diagrammatic descriptions of oscillating integrals with degenerate critical points. But since then, BV algebras have become interesting objects of study in their own right, and mathematicians sometimes have good understanding of the homological aspects of the story without any access to the diagrammatics. In this note we reverse the usual direction of argument: we begin by asking for an explicit calculation of the homology of a BV algebra, and from it derive Wick's Theorem and the other Feynman rules for finite-dimensional integrals.Comment: 11 pages. Final versio