Categorical algebra and ZX-calculus

Abstract

The main goal of this work is to use dagger-symmetric monoidal categories to build a visually intuitive diagrammatic language that is capable of reflecting the special properties that quantum systems have. We will provide a minimal set of generators so that we can draw any quantum computing circuit and derive a set of rules that define equivalences between diagrams.

Notation: We will use the Dirac, or \emph{bra-ket}, notation for complex vector spaces and the linear operators on them. A \emph{ket} $\ket{v}$ denotes a vector from the vector space $V$. It represents the state of a quantum system. A \emph{bra} $\bra{v}$ represents the linear form $v : V \to \mathbb{C}$ that maps each vector $u \in V$ to the number $\braket{v|u} \in \mathbb{C}$, where $\braket{v|u}$ represents the inner product of the vectors $v$ and $u$.