# Quantum Graph Theory

Name of applicant

David Earl Roberson

Institution

Technical University of Denmark

Amount

DKK 4,997,736

Year

2021

Type of grant

Semper Ardens: Accelerate

## What?

The phenomenon of quantum entanglement gives rise to such counterintuitive physical behaviors that even Einstein declared it "spooky action at-a-distance". By exploiting this phenomenon we can perform tasks that would be impossible under the laws of classical physics. This project focuses on one such task where two parties employ quantum entanglement to convince a third that two given networks, also known as "graphs", have the same structure. Whenever this is possible the graphs are said to be "quantum isomorphic". This project will use quantum isomorphism to develop deep connections between quantum information theory, discrete mathematics, and noncommutative mathematics. This work will advance the nascent field of quantum graph theory into a major area of research.

## Why?

Quantum entanglement is a fundamental resource in the rapidly developing fields of quantum information and computation. However, we still do not have a complete understanding of this phenomenon. The connections to discrete mathematics and noncommutative mathematics developed in this project will shed new light on this phenomenon and provide new tools for understanding and making use of entanglement. Conversely, these connections will provide novel perspectives on problems in discrete mathematics and noncommutative mathematics, and will allow us to transfer techniques between these fields. This will lead to new breakthroughs in areas where progress has slowed or halted.

## How?

Our starting point is our recent discovery that two graphs are quantum isomorphic if and only if they have the same number of any given planar substructure. This result and its proof weaves together ideas from quantum information, discrete mathematics, and noncommutative mathematics. From here we will move out in three complementary directions. We will 1) develop combinatorial models of objects from noncommutative mathematics, 2) use tools from noncommutative mathematics to investigate variants of quantum isomorphism arising from different physical theories, and 3) establish a cohesive theory of relations arising from counting various types of substructures. The Young Researcher Fellowship will allow me to assemble a team to tackle the challenges of all three lines of research.