ZX-calculus

The ZX-calculus is a rigorous graphical language for reasoning about linear maps between qubits, which are represented as "ZX-diagrams". A ZX-diagram consists of a set of generators called "spiders" that represent specific tensors. These are connected together to form a tensor network similar to Penrose graphical notation. Due to the symmetries of the spiders and the properties of the underlying category, topologically deforming a ZX-diagram (i.e. moving the generators without changing their connections) does not affect the linear map it represents. In addition to the equalities between ZX-diagrams that are generated by topological deformations, the ZX-calculus also has a set of graphical rewrite rules for transforming ZX-diagrams into one another. The ZX-calculus is "universal" in the sense that any linear map between qubits can be represented as a ZX-diagram, and different sets of graphical rewrite rules are complete for different families of linear maps. ZX-diagrams can be seen as a generalisation of quantum circuit notation.
History.
The ZX-calculus was first introduced by Bob Coecke and Ross Duncan in 2008 as an extension of the categorical quantum mechanics school of reasoning. They introduced the fundamental concepts of spiders, strong complementarity and most of the standard rewrite rules.
In 2009 Duncan and Perdrix found the additional Euler decomposition rule for the Hadamard gate, which was used by Backens in 2013 to establish the first completeness result for the ZX-calculus. Namely that there exists a set of rewrite rules that suffice to prove all equalities between stabilizer ZX-diagrams, where phases are multiples of formula_1, up to global scalars. This result was later refined to completeness including scalar factors.
In 2017, a completion of the ZX-calculus for the approximately universal formula_2 fragment was found, in addition to two different completeness results for the universal ZX-calculus (where phases are allowed to take any real value).
Also in 2017 the book "Picturing Quantum Processes" was released, that builds quantum theory from the ground up, using the ZX-calculus. See also the 2019 book "Categories for Quantum Theory".
Informal introduction.
ZX-diagrams consist of green and red nodes called "spiders", which are connected by "wires". Wires may curve and cross, arbitrarily many wires may connect to the same spider, and multiple wires can go between the same pair of nodes. There are also Hadamard nodes, usually denoted by a yellow box, which always connect to exactly two wires.
ZX-diagrams represent linear maps between qubits, similar to the way in which quantum circuits represent unitary maps between qubits. ZX-diagrams differ from quantum circuits in two main ways. The first is that ZX-diagrams do not have to conform to the rigid topological structure of circuits, and hence can be deformed arbitrarily. The second is that ZX-diagrams come equipped with a set of rewrite rules, collectively referred to as the "ZX-calculus". Using these rules, calculations can be performed in the graphical language itself.
Generators.
The building blocks or generators of the ZX-calculus are graphical representations of specific states, unitary operators, linear isometries, and projections in the computational basis formula_3 and the Hadamard-transformed basis formula_4.