Imperfect Graphs from Unitary Matrices -- I

TL;DR

This paper introduces a graph-theoretic framework called Topological Structure of Superpositions (TSS) that maps unitary matrices to directed graphs to analyze quantum operators' structural properties, aiding in the development of quantum algorithms.

Contribution

It presents a novel method to represent quantum operators as directed graphs, focusing on connectivity and reachability, which simplifies the analysis of quantum circuits without phase or amplitude details.

Findings

TSS effectively describes quantum gates like Hadamard and Pauli gates.

The framework offers a new perspective by viewing quantum circuits as discrete dynamical systems.

It isolates structural properties of quantum operators by discarding amplitude and phase information.

Abstract

Matrix representations of quantum operators are computationally complete but often obscure the structural topology of information flow within a quantum circuit \cite{nielsen2000}. In this paper, we introduce a generalized graph-theoretic framework for analyzing quantum operators by mapping unitary matrices to directed graphs; we term these structures \emph{Imperfect Graphs} or more formally as \emph{Topological Structure of Superpositions}(TSS) as a tool to devise better Quantum Algorithms. In this framework, we represent computational basis states as vertices. A directed edge exists between two vertices if and only if there is a non-zero amplitude transition between them, effectively mapping the support of the unitary operator. In this paper we deliberately discard probability amplitudes and phase information to isolate the connectivity and reachability properties of the operator. We demonstrate how TSS intuitively helps describe gates such as the Hadamard, Pauli-(X,Y,Z) gates, etc \cite{nielsen2000}. This framework provides a novel perspective for viewing quantum circuits as discrete dynamical systems \cite{childs2009,aharonov2001} Keywords: Quantum Algorithms, Unitary Matrix Approach, Topological Structure of Superpositions (TSS), Graph Theory