# ZH: A Complete Graphical Calculus for Quantum Computations Involving Classical Non-linearity

@article{Backens2019ZHAC, title={ZH: A Complete Graphical Calculus for Quantum Computations Involving Classical Non-linearity}, author={Miriam Backens and Aleks Kissinger}, journal={Electronic Proceedings in Theoretical Computer Science}, year={2019} }

We present a new graphical calculus that is sound and complete for a universal family of quantum circuits, which can be seen as the natural string-diagrammatic extension of the approximately (real-valued) universal family of Hadamard+CCZ circuits. The diagrammatic language is generated by two kinds of nodes: the so-called `spider' associated with the computational basis, as well as a new arity-$N$ generalisation of the Hadamard gate, which satisfies a variation of the spider fusion law. Unlike…

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## References

SHOWING 1-10 OF 26 REFERENCES

### A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics

- MathematicsLICS
- 2018

The ZX-Calculus is made complete for the so-called Clifford+T quantum mechanics by adding two new axioms to the language, and it is proved that the π/4-fragment of the ZX -Calculus represents exactly all the matrices over some finite dimensional extension of the ring of dyadic rationals.

### Interacting quantum observables: categorical algebra and diagrammatics

- MathematicsArXiv
- 2009

The ZX-calculus is introduced, an intuitive and universal graphical calculus for multi-qubit systems, which greatly simplifies derivations in the area of quantum computation and information and axiomatize phase shifts within this framework.

### Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning

- Physics
- 2017

This entirely diagrammatic presentation of quantum theory represents the culmination of ten years of research, uniting classical techniques in linear algebra and Hilbert spaces with cutting-edge developments in quantum computation and foundations.

### Both Toffoli and controlled-NOT need little help to do universal quantum computing

- PhysicsQuantum Inf. Comput.
- 2003

It is proved that Controlled-NOT plus any single-qubit real gate that does not preserve the computational basis and is not Hadamard (or its alike) are universal for quantum computing.

### The algebra of entanglement and the geometry of composition

- MathematicsArXiv
- 2017

The ZW calculus is presented, the first complete diagrammatic axiomatisation of the theory of qubits, and a notion of regular polygraph is proposed, barring cell boundaries that are not homeomorphic to a disk of the appropriate dimension, and the existence of weak units is equivalent to a representability property.

### A graphical approach to measurement-based quantum computing

- Computer Science, PhysicsQuantum Physics and Linguistics
- 2013

This work demonstrates the use of the ZX-calculus in reasoning about measurement-based quantum computing, where the graphical syntax directly captures the structure of the entangled states used to represent computations, and shows that the notion of information flow within the entangledStates gives rise to rewriting strategies for proving the correctness of quantum programs.

### A Simple Proof that Toffoli and Hadamard are Quantum Universal

- Physics, Philosophy
- 2003

Recently Shi proved that Toffoli and Hadamard are universal for quantum computation. This is perhaps the simplest universal set of gates that one can hope for, conceptually; It shows that one only…

### Y-Calculus: A language for real Matrices derived from the ZX-Calculus

- Mathematics, Computer ScienceQPL
- 2017

A ZX-like diagrammatic language devoted to manipulating real matrices - and rebits -, with its own set of axioms, is introduced, and it is proved that some restriction of the language is complete.

### Hierarchy of universal entanglement in 2D measurement-based quantum computation

- Physics, Computer Science
- 2015

This work utilizes recent advances in the subject of symmetry-protected topological order (SPTO) to introduce a novel MQC resource state, whose physical and computational behavior differs fundamentally from the cluster state.

### Quantomatic: A proof assistant for diagrammatic reasoning

- Computer ScienceCADE
- 2015

This work briefly outlines the theoretical basis of Quantomatic's rewriting engine, then gives an overview of the core features and architecture and gives a simple example project that computes normal forms for commutative bialgebras.