A graph theoretical approach to states and unitary operations

TL;DR

This paper develops a graph-theoretical framework to represent and implement local unitary operations on quantum states using signless Laplacian matrices, enabling graphical visualization of quantum transformations.

Contribution

It introduces the concept of local unitary equivalent graphs and demonstrates their use in visualizing quantum gates and state transformations graphically.

Findings

Graph switching can implement CNOT gate operations.

Graphical representation of Bell state generation is achieved.

Local unitary transformations are visualized through graph modifications.

Abstract

Building upon our previous work, on graphical representation of a quantum state by signless Laplacian matrix, we pose the following question. If a local unitary operation is applied to a quantum state, represented by a signless Laplacian matrix, what would be the corresponding graph and how does one implement local unitary transformations graphically? We answer this question by developing the notion of local unitary equivalent graphs. We illustrate our method by a few, well known, local unitary transformations implemented by single-qubit Pauli and Hadamard gates. We also show how graph switching can be used to implement the action of the CNOT gate, resulting in a graphical description of Bell state generation.