Completeness for arbitrary finite dimensions of ZXW-calculus, a unifying calculus

Poór, Boldizsár; Wang, Quanlong; Shaikh, Razin A.; Yeh, Lia; Yeung, Richie; Coecke, Bob

doi:10.1109/lics56636.2023.10175672

Cited by 7 publications

(6 citation statements)

References 48 publications

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“…The ZXW calculus is a complete diagrammatic language for qudit quantum computing [43]. Formally, let Vect d be the symmetric monoidal category with objects tensor products of qudits C d and morphisms given by linear maps between them.…”

Section: Zxw Calculusmentioning

confidence: 99%

“…. Note that here we made a slight change for the presentation of the qudit ZXW calculus in comparison to the version of [43]: we use the X spider instead of the Hadamard node as a generator now, so the definition of the former becomes a rule and the rule for the latter turns into a definition. Also we removed the (H1) rule of [43] from the rule set since we find it can be derived from other rules now.…”

Section: Zxw Calculusmentioning

confidence: 99%

“…The rewriting power and the ability of representing sums using W nodes motivated the development of the ZXW calculus, allowing the first diagrammatic treatment of Hamiltonians and exponentiation [46], as well as integration and differentiation [54]. A natural generalisation of ZXW to higher dimensions resulted in the first complete axiomatisation for qudit quantum computing [43].…”

Section: Introductionmentioning

confidence: 99%

“…the category of linear maps between qudits [43]. In Section 3 and Section 4 we present our main contributions: (1) we introduce a more expressive calculus ZW ∞ for the category of linear operators on the Fock space Vect N , (2) we give a truncated interpretation T d : ZW ∞ → ZXW d and prove Theorem 4.2 which allows to derive equalities in ZW ∞ by lifting them from ZXW d for 'big enough' d, (3) we give an axiomatisation of ZW ∞ for which we prove soundness via the lifting theorem.…”

Section: Introductionmentioning

confidence: 99%

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Light-Matter Interaction in the ZXW Calculus

de Felice,

Shaikh,

Poór

et al. 2023

Electron. Proc. Theor. Comput. Sci.

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In this paper we develop a graphical calculus to rewrite photonic circuits involving lightmatter interactions and non-linear optical effects. We introduce the infinite ZW calculus, a graphical language for linear operators on the bosonic Fock space which captures both linear and non-linear photonic circuits. This calculus is obtained by combining the QPath calculus, a diagrammatic language for linear optics, and the recently developed qudit ZXW calculus, a complete axiomatisation of linear maps between qudits. It comes with a "lifting" theorem allowing to prove equalities between infinite operators by rewriting in the ZXW calculus. We give a method for representing bosonic and fermionic Hamiltonians in the infinite ZW calculus. This allows us to derive their exponentials by diagrammatic reasoning. Examples include phase shifts and beam splitters, as well as non-linear Kerr media and Jaynes-Cummings light-matter interaction.

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Section: Zxw Calculusmentioning

confidence: 99%

Section: Zxw Calculusmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Light-Matter Interaction in the ZXW Calculus

de Felice,

Shaikh,

Poór

et al. 2023

Electron. Proc. Theor. Comput. Sci.

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“…[13] mentioned below. Some proposals capture all finite or infinite dimensions [59,66,57,30], but lack many of the nicer features of the qubit calculus. Of particular importance to our paper is Ref.…”

Section: Introductionmentioning

confidence: 99%

The Qupit Stabiliser ZX-travaganza: Simplified Axioms, Normal Forms and Graph-Theoretic Simplification

Poór,

Booth,

Carette

et al. 2023

Electron. Proc. Theor. Comput. Sci.

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We present a smorgasbord of results on the stabiliser ZX-calculus for odd prime-dimensional qudits (i.e. qupits). We derive a simplified rule set that closely resembles the original rules of qubit ZX-calculus. Using these rules, we demonstrate analogues of the spider-removing local complementation and pivoting rules. This allows for efficient reduction of diagrams to the affine with phases normal form. We also demonstrate a reduction to a unique form, providing an alternative and simpler proof of completeness. Furthermore, we introduce a different reduction to the graph state with local Cliffords normal form, which leads to a novel layered decomposition for qupit Clifford unitaries. Additionally, we propose a new approach to handle scalars formally, closely reflecting their practical usage. Finally, we have implemented many of these findings in DiZX, a new open-source Python library for qudit ZX-diagrammatic reasoning.

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