Dr. Nadish de Silva (nadish_de_silva@sfu.ca)

Project: Simulating and correcting quantum computers via the stabiliser formalism

Small quantum computers are currently under construction.  How will we verify that they are working correctly?  We might try to simulate their operation on an existing conventional supercomputer.  This would be slow and difficult, however, as quantum computers, by design, perform tasks beyond the abilities of conventional computers.  Thus, clever schemes have been devised to classically simulate quantum computers as efficiently as possible within limited regimes.  Studying this question also gives us insight into the poorly understood mechanisms that drive quantum computational power.

In the longer term, how will we protect highly sensitive quantum data from errors while operating on them?  As quantum data cannot be copied or directly observed, more clever schemes have been devised to enable quantum computation in the face of environmental noise.

This project will centre on the mathematics used to answer both these questions: the stabiliser formalism.  The goal will be to better understand the stabiliser formalism and to utilise it towards facilitating quantum error correction and verifying near-term quantum computers.  

The precise question tackled in the project, and the balance between theory and numerical computations, will depend on the interests and skills of the student.

Requirements: Strong background in mathematics and computer science. Coding skills in Python could be an asset but are not strictly necessary.

Dr. Ailene MacPherson (ailenem@sfu.ca)

Project: Approximate Bayesian Computation Methods for Local Adaptation

Bayesian inference is a powerful method in population biology, allowing researchers to integrate complementary datasets to understand ecological, evolutionary, and epidemiological processes shaping the natural world (Beaumont et al. 2002). When the underlying biological processes become complex, for example disease transmission in a social network, analytical methods become cumbersome and computational approximations must be used instead.  In this project you will develop, implement, and test an Approximate Bayesian Computation (ABC) inference method for inferring the evolutionary processes underlying local adaptation. Local adaptation, where populations adapt to local environmental conditions, has widespread evolutionary and ecological consequences–it is central to generating and maintaining the biological diversity we enjoy every day.  Understanding how local adaptation arises is also essential for developing robust conservation programs (Letterhos 2024).

This highly interdisciplinary USRA project will combine elements of mathematics, statistics, computer science, and biology to address a scientific question of practical relevance to conservation. The student will be introduced to the fundamentals of Bayesian inference, mathematical statistics, and the population biology background.  They will design and implement an ABC approach in python, compare this method against an existing analytical sub-case, test the performance of their method more generally, and summarize and disseminate their methods and results.  The student will be encouraged to develop and present their work at one of two possible local scientific meetings in Fall 2025.

Up to 2 students are invited to work on this project.

Requirements: Previous experience with python programming and Math 348, 360, 468 or 469.

Dr. Paul Tupper (pft3@sfu.ca)

Project: Improving Clustering Algorithms with Diversities

Clustering is the task of assigning a large number of unlabeled data points to a small number of classes. Typically, the data are points in a vector space and a metric is defined on the vector space, leading to a distance matrix for the data. When clustering, the metric is used to break up the data into a number of groups. But using just the metric may throw out a lot of information about the placement of the points in the space.

Diversities are an extension of the idea of a metric space, where not just pairs of points but all finite subsets of points are assigned a value. The diameter of a set of points in a metric space is a diversity, but there are many other natural examples. In this project the student will investigate using the perspective of diversities to devise new clustering algorithms, investigating what works well computationally on large data sets.

Up to 2 students are invited to work on this project.

Requirements: Basic programming skills. Strong background in mathematics.