Abstract: This talk introduces the ZX-calculus, a powerful graphical language for reasoning about quantum computations. I will start with an overview of process theories, a general framework for describing how processes act upon different types of information. I then focus on the process theory of quantum circuits, where each function (or gate) is a unitary linear transformation acting upon qubits. The ZX-calculus simplifies the set of available gates in terms of two atomic operations: Z and X spiders, which generalize rotations around the Z and X axes of the Bloch sphere. I demonstrate how to translate quantum circuits into ZX-diagrams and how to simplify ZX diagrams using a set of seven equivalences. Through examples and illustrations, I hope to convey that the ZX-calculus provides an intuitive and powerful tool for reasoning about quantum computations, allowing for the derivation of equivalences between circuits. By the end of the talk listeners should be able to understand equations written in the ZX-calculus and potentially use them in their own work.

Six-functor formalisms are ubiquitous in mathematics, and I will start this talk by giving a quick introduction to them. A three-functor formalism is, as the name suggests, (the better) half of a six-functor formalism. I will discuss what it means for such a three-functor formalism to be unitary, and why commutative Von Neumann algebras (and hence, by the Gelfand-Naimark theorem, measure spaces) admit a unitary three-functor formalism that can be viewed as mixing sheaf theory with functional analysis. Based on joint work with André Henriques.