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arXiv:1911.06752 · Electronic Proceedings in Theoretical Computer Science

An Algebraic Axiomatisation of ZX-calculus

Quanlong Wang

Published Sep 6, 2021• 14 citations
arXivHTMLPDF

Abstract

ZX-calculus is a graphical language for quantum computing which is complete in the sense that calculation in matrices can be done in a purely diagrammatic way. However, all previous universally complete axiomatisations of ZX-calculus have included at least one rule involving trigonometric functions such as sin and cos which makes it difficult for application purpose. In this paper we give an algebraic complete axiomatisation of ZX-calculus instead such that there are only ring operations involved for phases. With this algebraic axiomatisation of ZX-calculus, we are able to establish for the first time a simple translation of diagrams from another graphical language called ZH-calculus and to derive all the ZX-translated rules of ZH-calculus. As a consequence, we have a great benefit that all techniques obtained in ZH-calculus can be transplanted to ZX-calculus, which can't be obtained by just using the completeness of ZX-calculus.

DOI
10.4204/eptcs.340.16

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ZX-calculus is a graphical language for quantum computing which is complete in the sense that calculation in matrices can be done in a purely diagrammatic way. However, all previous universally complete axiomatisations of ZX-calculus have included at least one rule involving trigonometric functions such as sin and cos which makes it difficult for application purpose. In this paper we give an algebraic complete axiomatisation of ZX-calculus instead such that there are only ring operations involved for phases. With this algebraic axiomatisation of ZX-calculus, we are able to establish for the first time a simple translation of diagrams from another graphical language called ZH-calculus and to derive all the ZX-translated rules of ZH-calculus. As a consequence, we have a great benefit that all techniques obtained in ZH-calculus can be transplanted to ZX-calculus, which can't be obtained by just using the completeness of ZX-calculus.
In this paper we give a complete axiomatisation of qubit ZX-calculus via elementary transformations which are basic operations in linear algebra. This formalism has two main advantages. First, all the operations of the phases are algebraic ones without trigonometry functions involved, thus paved the way for generalising complete axiomatisation of qubit ZX-calculus to qudit ZX-calculus and ZX-calculus over commutative semirings. Second, we characterise elementary transformations in terms of ZX diagrams, so a lot of linear algebra stuff can be done purely diagrammatically.
The ZX-Calculus is a graphical language for diagrammatic reasoning in quantum mechanics and quantum information theory. It comes equipped with an equational presentation. We focus here on a very important property of the language: completeness, which roughly ensures the equational theory captures all of quantum mechanics. We first improve on the known-to-be-complete presentation for the so-called Clifford fragment of the language - a restriction that is not universal - by adding some axioms. Thanks to a system of back-and-forth translation between the ZX-Calculus and a third-party complete graphical language, we prove that the provided axiomatisation is complete for the first approximately universal fragment of the language, namely Clifford+T. We then prove that the expressive power of this presentation, though aimed at achieving completeness for the aforementioned restriction, extends beyond Clifford+T, to a class of diagrams that we call linear with Clifford+T constants. We use another version of the third-party language - and an adapted system of back-and-forth translation - to complete the language for the ZX-Calculus as a whole, that is, with no restriction. We briefly discuss the added axioms, and finally, we provide a complete axiomatisation for an altered version of the language which involves an additional generator, making the presentation simpler.
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 elegantly described by the ZX-calculus, a graphical language for a class of string diagrams describing linear maps between qubits. The ZX-calculus has proven useful in a variety of areas of quantum information, but is less suitable for reasoning about operations outside its natural gate set such as multi-linear Boolean operations like the Toffoli gate. In this paper we study the ZH-calculus, an alternative graphical language of string diagrams that does allow straightforward encoding of Toffoli gates and other more complicated Boolean logic circuits. We find a set of simple rewrite rules for this calculus and show it is complete with respect to matrices over Z[12] , which correspond to the approximately universal Toffoli+Hadamard gateset. Furthermore, we construct an extended version of the ZH-calculus that is complete with respect to matrices over any ring R where 1+1 is not a zero-divisor.
The ZX-Calculus is a graphical language for diagrammatic reasoning in quantum mechanics and quantum information theory. An axiomatisation has recently been proven to be complete for an approximatively universal fragment of quantum mechanics, the so-called Clifford+T fragment. We focus here on the expressive power of this axiomatisation beyond Clifford+T Quantum mechanics. We consider the full pure qubit quantum mechanics, and mainly prove two results: (i) First, the axiomatisation for Clifford+T quantum mechanics is also complete for all equations involving some kind of linear diagrams. The linearity of the diagrams reflects the phase group structure, an essential feature of the ZX-calculus. In particular all the axioms of the ZX-calculus are involving linear diagrams. (ii) We also show that the axiomatisation for Clifford+T is not complete in general but can be completed by adding a single (non linear) axiom, providing a simpler axiomatisation of the ZX-calculus for pure quantum mechanics than the one recently introduced by Ng&Wang.
Recent developments in the ZX-Calculus have resulted in complete axiomatisations first for an approximately universal restriction of the language, and then for the whole language. The main drawbacks were that the axioms that were added to achieve completeness were numerous, tedious to manipulate and lacking a physical interpretation. We present in this paper two complete axiomatisations for the general ZX-Calculus, that we believe are optimal, in that all their equations are necessary and moreover have a nice physical interpretation. To do so, we introduce the singular-value decomposition of a ZX-diagram, and use it to show that all the rules of the former axiomatisation are provable with the new one.
We introduce here a new axiomatisation of the rational fragment of the ZX-calculus, a diagrammatic language for quantum mechanics. Compared to the previous axiomatisation introduced in [8], our axiomatisation does not use any metarule , but relies instead on a more natural rule, called the cyclotomic supplementarity rule, that was introduced previously in the literature. Our axiomatisation is only complete for diagrams using rational angles , and is not complete in the general case. Using results on diophantine geometry, we characterize precisely which diagram equality involving arbitrary angles are provable in our framework without any new axioms, and we show that our axiomatisation is continuous, in the sense that a diagram equality involving arbitrary angles is provable iff it is a limit of diagram equalities involving rational angles. We use this result to give a complete characterization of all Euler equations that are provable in this axiomatisation.