Complete ZX-Calculi for the Stabiliser Fragment in Odd Prime Dimensions

  title={Complete ZX-Calculi for the Stabiliser Fragment in Odd Prime Dimensions},
  author={Robert Ivan Booth and Titouan Carette},
We introduce a family of ZX-calculi which axiomatise the stabiliser fragment of quantum theory in odd prime dimensions. These calculi recover many of the nice features of the qubit ZX-calculus which were lost in previous proposals for higher-dimensional systems. We then prove that these calculi are complete, i.e. provide a set of rewrite rules which can be used to prove any equality of stabiliser quantum operations. Adding a discard construction, we obtain a calculus complete for mixed state… 

Figures from this paper

Simplification Strategies for the Qutrit ZX-Calculus

The main contribution of this work is the derivation of efficient rewrite strategies for the stabiliser fragment of the qutrit ZX-calculus, which constitutes a first non-trivial step towards the simplification ofqutrit quantum circuits.



Completeness of Graphical Languages for Mixed States Quantum Mechanics

A new construction, the discard construction, is introduced, which transforms any †-symmetric monoidal category into a symmetric Monoidal category equipped with a discard map, which provides an extension for several graphical languages that are proved to be complete for general quantum operations.

A Simplified Stabilizer ZX-calculus

It is proved that meta-rules like 'colour symmetry' and 'upside-down symmetry', which were considered as axioms in previous versions of the stabilizer ZX-calculus, can in fact be derived.

ZX-calculus for the working quantum computer scientist

This review discusses Clifford computation and graphically prove the Gottesman-Knill theorem, a recently introduced extension of the ZX-calculus that allows for convenient reasoning about Toffoli gates, and the recent completeness theorems that show that, in principle, all reasoning about quantum computation can be done using Zx-diagrams.

Completeness of the ZH-calculus

There are various gate sets used for describing quantum computation. A particularly popular one consists of Clifford gates and arbitrary single-qubit phase gates. Computations in this gate set can be

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

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.

Making the stabilizer ZX-calculus complete for scalars

This work replaces those scalar-free rewrite rules with correctly scaled ones and shows that, by adding one new diagram element and a new rewrite rule, the calculus can be made complete for pure qubit stabilizer quantum mechanics with scalars.

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

A new graphical calculus is presented that is sound and complete for universal quantum computation by demonstrating the reduction of any diagram to an easily describable normal form, which suggests that this calculus will be significantly more convenient for reasoning about the interplay between classical non-linear behaviour and purely quantum operations.

Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus

A simplification strategy for ZX-diagrams is given based on the two graph transformations of local complementation and pivoting and it is shown that the resulting reduced diagram can be transformed back into a quantum circuit.

Qufinite ZX-calculus: a unified framework of qudit ZX-calculi

This paper generalises qubit ZX-calculus to qudit ZX -calculus in any finite dimension by introducing suitable generators, especially a carefully chosen triangle node, and obtains a set of rewriting rules which can be seen as a direct generalisation of qubit rules, and a normal form for any qudit vectors.

The ZX-calculus is complete for stabilizer quantum mechanics

The ZX-calculus is a graphical calculus for reasoning about quantum systems and processes. It is known to be universal for pure state qubit quantum mechanics (QM), meaning any pure state, unitary