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The ZX-Calculus : A graphical calculus for multipartite qubit systems

Mohammed Al-Jaff

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

In this thesis we will give a presentation of a graphical/diagrammatic calculus for quantum systems involving interacting quantum observables such as multi-partite systems of qubits, the ZX-Calculu ...

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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.
ZX-calculus is graphical language for quantum computing which usually focuses on qubits. In this paper, we generalise qubit ZX-calculus to qudit ZX-calculus in any finite dimension by introducing suitable generators, especially a carefully chosen triangle node. As a consequence we obtain a set of rewriting rules which can be seen as a direct generalisation of qubit rules, and a normal form for any qudit vectors. Based on the qudit ZX-calculi, we propose a graphical formalism called qufinite ZX-calculus as a unified framework for all qudit ZX-calculi, which is universal for finite quantum theory due to a normal form for matrix of any finite size. As a result, it would be interesting to give a fine-grained version of the diagrammatic reconstruction of finite quantum theory [Selby2021reconstructing] within the framework of qufinite ZX-calculus.
This paper has two tightly intertwined aims: (i) to introduce an intuitive and universal graphical calculus for multi-qubit systems, the ZX -calculus, which greatly simplifies derivations in the area of quantum computation and information. (ii) To axiomatize complementarity of quantum observables within a general framework for physical theories in terms of dagger symmetric monoidal categories. We also axiomatize phase shifts within this framework. Using the well-studied canonical correspondence between graphical calculi and dagger symmetric monoidal categories, our results provide a purely graphical formalisation of complementarity for quantum observables. Each individual observable, represented by a commutative special dagger Frobenius algebra, gives rise to an Abelian group of phase shifts, which we call the phase group. We also identify a strong form of complementarity, satisfied by the Z - and X -spin observables, which yields a scaled variant of a bialgebra.
The ZX calculus is a graphical language for reasoning about quantum processes. In this paper, we develop a minimal pure-qubit ZX calculus based on the work of Vilmart [arXiv:1812.09114], Backens, Perdrix, and Wang [arXiv:1709.08903], and Stoltz [arXiv:2606.12383]. This resolves a problem that has remained open for nearly a decade, since completeness was first proved. Specifically, we show that (Ir)(I_r)(Ir​) is derivable and establish the necessity of (B)(B)(B) and (Ig)(I_g)(Ig​), yielding two complete and minimal rulesets.
We introduce a graphical calculus for multi-qutrit systems (the qutrit ZX-calculus) based on the framework of dagger symmetric monoidal categories. This graphical calculus consists of generators for building diagrams and rules for transforming diagrams, which is obviously different from the qubit ZX-calculus. As an application of the qutrit ZX-calculus, we give a graphical description of a (2, 3) threshold quantum secret sharing scheme. In this way, we prove the correctness of the secret sharing scheme in a intuitively clear manner instead of complicated linear algebraic operations.