Joseph Collins

Computer and Information Sciences, University of Strathclyde

Tenure since 2017

Infinite Dimensional Categorical Quantum MechanicsJoseph Collins

Where a classical computer uses strings of bits which can take either the value 1 or 0 to encode information, a quantum computer uses quantum bits (or a qubit) which, under certain conditions, can be in superposition of 1 and 0. This means that the information encoded with qubits is governed by the laws of quantum mechanics. Thus, quantum computers are able to accomplish tasks which are provably impossible on classical computers, but this presents a great deal of challenges in developing the theory surrounding quantum computers.

In physics, objects with quantum mechanical properties are modelled by mathematical constructions called Hilbert spaces. This is a very effective tool for modelling a few of these objects, but as the number of objects increase, the calculations become exponentially more complicated. Since a quantum computer will potentially be dealing with thousands of qubits, it is important that we develop mathematical tools specifically for quantum computers. Categorical quantum mechanics (CQM) is a construction that models quantum mechanics and is specialised for quantum computers, and it avoids this problem by generalising the underlying structure of Hilbert spaces.

This project aims to extend CQM to infinite dimensions, as currently CQM can only model finite dimensional Hilbert spaces. This limits the universality and usefulness, as this means that there are some quantum mechanical systems and phenomena that CQM cannot yet model, and there are models for quantum computing that cannot yet be modelled by CQM.

Biography

I graduated with a first class degree from Heriot-Watt University in 2017 at the top of my class with an MMath, having been awarded the Roderick MacLeod Shearer Memorial Prize for excellence in Mathematics. In the summer of 2015, I researched the shuffle Hopf algebra and its applications to stochastic integrals, supervised by Dr SImon Malham, which was funded by the Carnegie Undergraduate Vacation Scholarship. This project involved knowledge of category theory, which is the field of mathematics integral to the PhD and is not taught in Heriot-Watt. This project introduced me to category theory and inspired me to research it further independently. Over the summer in 2016, I was supervised by Dr Mark Lawson as I studied the book “Groups and Representations” by Jonathan Alperin and Rowen Bell.