ZXcalculus

ZX-calculus is a graphical language for reasoning about quantum computation and information, using diagrams to represent linear maps between qubits. Diagrams are built from two kinds of nodes, called Z-spiders and X-spiders, each labeled by an angle (a phase). Wires carry quantum systems; composition is sequential or parallel; the Hadamard gate is used to switch between the Z and X bases.

Z-spiders are usually drawn green and X-spiders red; a spider can have multiple legs and a phase angle that modifies the map. A Hadamard edge connects a Z-spider to an X-spider, implementing a basis change. The value of a diagram is the linear map obtained by composing and tensoring the mapped components.

Rewrite rules encode the algebraic properties used to manipulate diagrams. The spider fusion rule merges two same-color spiders connected by a wire, adding their phases. The color-change rule uses Hadamard edges to switch between Z and X spiders. Additional rules express the bialgebra and Hopf structure that governs interactions between the two bases, ensuring consistent composition and allowing simplifications to proceed.

Semantically, each diagram denotes a linear map between finite-dimensional Hilbert spaces; the rules are sound, meaning equivalent diagrams denote the same map. The ZX-calculus is universal for pure qubit quantum mechanics, and complete axiomatisations exist for important fragments (notably Clifford+T); ongoing work extends completeness to broader classes. It is widely used to reason about and simplify quantum circuits and protocols.

History and tools: introduced by Coecke and Duncan in the late 2000s, the calculus has become a standard in quantum diagrammatic reasoning. It underpins tools such as PyZX and Quantomatic, which automate diagrammatic rewrites for circuit optimization and verification. Applications include quantum circuit synthesis, teleportation and error-correction proofs, and visualization of entanglement structure.