Here are some topics I am interested:

Arithmetic of Quantum Circuits: This area is closely related to exact and approximate synthesis. Essentially the idea is to understand quantum circuits via the lens of number theoretic and more broadly algebraic methods. The first paper which showed a connection between exact synthesis and number theory is this one. Some beautiful work has been done by Peter Sarnak and his collaborators in this area, a letter he wrote about "Golden Gates" can be found here. Another related but distinct approach to synthesis was explored in some very interesting papers written by Vadym Kliuchnikov, Jon Yard and other collaborators and can be found here and here. In these papers synthesis is viewed nearly exclusively via the lens of quaternion algebras.

Quantum Error Correction: I am interested in constructing error correcting codes and understanding the relationship between different areas of mathematics and error correcting codes. Here is a book whose preface has a tantalizing table of connections between different topics in math and error correction. Lately I have become interested in constructing Quantum LDPC Codes and their connection with high dimensional expanders.

SIC-POVMs: Do SIC-POVMs exist in all dimensions? This turned out to be a very deep mathematical problem and has close connections with Stark conjectures and Hilbert's 12th problem. SIC-POVMs are also very interesting from the quantum information perspective because they show up all over the place for example in error correcting codes, contextuality and quantum state tomography. A extremely readable paper which summarizes the main results is this one, a nice talk on the number theoretic connection is this one.

Magic and Quantum Advantage: This is another interesting area and is closely related to the question of "What Supplies the Magic to Quantum Computation?". This area is central to answering the question of "What powers quantum computers?" and "What is the difference between classical and quantum computation? A comprehensive review written by some the leaders of the field can be found here.

Quantum Information Theory and Its Applications to Theoretical Physics: I am really interested in this area but have never got around to thinking about it for long durations, a paper I found very interesting is this one by Hayden and Preskill.

Graphical Calculi and its applications to Quantum Circuits: Symmetric Monoidal Categories provide an elegant framework to reason about quantum processes. ZX calculus and its extensions which have their roots in the theory of symmetric monoidal categories are some of the leading languages to model quantum circuits and understand error correcting codes. A couple of well written introductions can be found here and here.

I did my master's thesis from the Computer Science department at University of Calgary, my thesis can be found here. All my research papers can be found here.