Below are some interesting papers and books that I really like. None of these are mine! For my papers see the homepage or my Google Scholar page.
Number Theory and Quantum Circuits
This is a very surprising connection: a lot of very deep ideas from number theory can be used to study quantum circuits. See this paper, which first noticed the connection. Peter Sarnak wrote this letter on “Golden Gates”, which describes this connection from a number-theoretic point of view. There are other papers that extend this number-theoretic connection to many different gate sets; see this and this. For more recent work, see this paper and this one, which make use of Hermitian lattices and class numbers!
Error-Correcting Codes
I really like old papers and books on this topic. See this book by Sloane, Nebe and Rains. It has a tantalizing table of connections right at the start and talks about a lot of interesting mathematics! Weight enumerators are one of the central objects of study in this book. These have also found applications in quantum coding theory; see this paper and this one. These ideas also have connections to questions in quantum Shannon theory; see this paper.
Non-locality and Contextuality
This area tries to address the conceptual differences between classical and quantum information. There are some really cool and deep connections to sheaves and algebraic topology. These structures have a surprising way of turning up in the study of quantum advantage; see this paper and this one. There is also a very interesting connection between contextuality and magic state distillation.
Mathematics of Quantum Computation
Quantum theory has a lot of pretty mathematical structures. Some examples are SIC-POVMs, MUBs and the Clifford Hierarchy. There is this paper that talks about some discrete mathematical structures, and there is also a very nice book called Geometry of Quantum States written by Ingemar Bengtsson and Karol Życzkowski. I am also interested in categorical quantum mechanics, ZX-calculus and, more generally, the development of graphical calculi to study different aspects of quantum information. There is an interesting book called Picturing Quantum Processes on this topic by Aleks Kissinger and Bob Coecke.