The ZX-calculus is a graphical language that goes beyond circuit diagrams. It â€˜splits the atomâ€™ of well-known quantum logic gates to reveal the compositional structure inside. The calculus works by generalising the ideas of Z and X operations, allowing us to break out of the circuit model while maintaining soundness of reasoning. In doing so we can show properties of circuits, entanglement states, and protocols, in a visually succinct but logically complete manner.

The ZX-calculus is forging the next generation of quantum software. Using the calculus gives optimisation strategies that performs state-of-the-art T-count reduction (an important metric for fault-tolerant computing) and gate compilation. The generators of the calculus correspond closely to the basic operations of lattice surgery in the surface code, giving a visual design and verification language for these codes; and ZX has also been used to discover novel error correction procedures. It comes with a scalable notation capable of representing repeated structures at arbitrary qubit scales. The calculus also acts in the crucial role of an intermediate representation in a new commercial quantum compiler.

The example derivation requires javascript in order to be displayed.

For a longer introduction to the ZX-calculus see the tutorial page.

We have implemented an in-browser demonstration of the T-count optimising procedure from this paper.

Although there are plenty of papers on ZX-calculus related work, many of those are on the development and properties of the language rather than on its applications. Here are a few papers that show the calculus' uses and also its most modern presentation.

- Reducing T-count with the ZX-calculus.
- A Near-Optimal Axiomatisation of ZX-Calculus for Pure Qubit Quantum Mechanics

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