arXiv subjects

Jonathan Gorard

Publications and source records attributed to Jonathan Gorard.

23 records · Page 2Linked to original sources

arXiv2021

Fast Automated Reasoning over String Diagrams using Multiway Causal Structure

We introduce an intuitive algorithmic methodology for enacting automated rewriting of string diagrams within a general double-pushout (DPO) framework, in which the sequence of rewrites is chosen in accordance with the causal structure of the underlying diagrammatic calculus. The combination of the rewriting structure and the causal structure may be elegantly formulated as a weak 2-category equipped with both total and partial monoidal bifunctors, thus providing a categorical semantics for the full multiway evolution causal graph of a generic Wolfram model hypergraph rewriting system. As an illustrative example, we show how a special case of this algorithm enables highly efficient automated simplification of quantum circuits, as represented in the ZX-calculus.

cs.LO↗

arXiv2021

ZX-Calculus and Extended Wolfram Model Systems II: Fast Diagrammatic Reasoning with an Application to Quantum Circuit Simplification

This article presents a novel algorithmic methodology for performing automated diagrammatic deductions over combinatorial structures, using a combination of modified equational theorem-proving techniques and the extended Wolfram model hypergraph rewriting formalism developed by the authors in previous work. We focus especially upon the application of this new algorithm to the problem of automated circuit simplification in quantum information theory, using Wolfram model multiway operator systems combined with the ZX-calculus formalism for enacting fast diagrammatic reasoning over linear transformations between qubits. We show how to construct a generalization of the deductive inference rules for Knuth-Bendix completion in which equation matches are selected on the basis of causal edge density in the associated multiway system, before proceeding to demonstrate how to embed the higher-order logic of the ZX-calculus rules within this first-order equational framework. After showing explicitly how the (hyper)graph rewritings of both Wolfram model systems and the ZX-calculus can be effectively realized within this formalism, we proceed to exhibit comparisons of time complexity vs. proof complexity for this new algorithmic approach when simplifying randomly-generated Clifford circuits down to pseudo-normal form, as well as when reducing the number of T-gates in randomly-generated non-Clifford circuits, with circuit sizes ranging up to 3000 gates, illustrating that the method performs favorably in comparison with existing circuit simplification frameworks, and also exhibiting the approximately quadratic speedup obtained by employing the causal edge density optimization. Finally, we present a worked example of an automated proof of correctness for a simple quantum teleportation protocol, in order to demonstrate more clearly the internal operations of the theorem-proving procedure.

cs.LO↗

arXiv2020

ZX-Calculus and Extended Hypergraph Rewriting Systems I: A Multiway Approach to Categorical Quantum Information Theory

Categorical quantum mechanics and the Wolfram model offer distinct but complementary approaches to studying the relationship between diagrammatic rewriting systems over combinatorial structures and the foundations of physics; the objective of the present article is to begin elucidating the formal correspondence between the two methodologies in the context of the ZX-calculus formalism of Coecke and Duncan for reasoning diagrammatically about linear maps between qubits. After briefly summarizing the relevant formalisms, and presenting a categorical formulation of the Wolfram model in terms of adhesive categories and double-pushout rewriting systems, we illustrate how the diagrammatic rewritings of the ZX-calculus can be embedded and realized within the broader context of Wolfram model multiway systems, and illustrate some of the capabilities of the software framework (ZXMultiwaySystem) that we have developed specifically for this purpose. Finally, we present a proof (along with an explicitly computed example) based on the methods of Dixon and Kissinger that the multiway evolution graphs and branchial graphs of the Wolfram model are naturally endowed with a monoidal structure based on rulial composition that is, furthermore, compatible with the monoidal product of ZX-diagrams.

cs.LO↗

arXiv2019

Quantum Cellular Automata, Black Hole Thermodynamics, and the Laws of Quantum Complexity

This paper introduces a new formalism for quantum cellular automata (QCAs), based on evolving tensor products of qubits using local unitary operators. It subsequently uses this formalism to analyze and validate several conjectures, stemming from a formal analogy between quantum computational complexity theory and classical thermodynamics, that have arisen recently in the context of black hole physics. In particular, the apparent resonance and thermalization effects present within such QCAs are investigated, and it is demonstrated that the expected exponential relationships between the quantum circuit complexity of the evolution operator, the classical entropy of the equilibrium QCA state, and the characteristic equilibration time of the QCA, all hold within this new model. Finally, a rigorous explanation for this empirical relationship is provided, as well as for the relationship with black hole thermodynamics, by drawing an explicit mathematical connection with the mean ergodic theorem, and the ergodicity of k-local quantum systems.

quant-ph↗

arXiv2016

Uniqueness Trees: A Possible Polynomial Approach to the Graph Isomorphism Problem

This paper presents the novel `uniqueness tree' algorithm, as one possible method for determining whether two finite, undirected graphs are isomorphic. We prove that the algorithm has polynomial time complexity in the worst case, and that it will always detect the presence of an isomorphism whenever one exists. We also propose that the algorithm will equivalently discern the lack of an isomorphism whenever one does not exist, and some initial justifications are given for this proposition, although it cannot yet be rigorously proven. Finally, we present experimental evidence for both the effectiveness and efficiency of the uniqueness tree method, using data gathered from a practical implementation of the algorithm. Some consequences and directions for further research are discussed.

cs.DM↗