SZX-Calculus: Scalable Graphical Quantum Reasoning

Carette, Titouan; Horsman, Dominic; Perdrix, Simon

doi:10.4230/lipics.mfcs.2019.55

Cited by 17 publications

(16 citation statements)

References 11 publications

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“…Remark 5.1 The two diagrams at the bottom of the table of generators are called dimension-binder and dimension-splitter respectively, in the qubit case they are similar to the divider and gatherer in the as introduced in [2].…”

Section: Qufinite Zx-calculusmentioning

confidence: 99%

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

Wang¹

2021

Preprint

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ZX-calculus is graphical language for quantum computing which usually focuses on qubits. In this paper, we generalise qubit ZX-calculus to qudit ZXcalculus 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 [12] within the framework of qufinite ZX-calculus.

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Section: Qufinite Zx-calculusmentioning

confidence: 99%

“…Proof: Denote by A the d m × d m row-addition elementary matrix in (2). Let {A k |0 ≤ k ≤ d m − 1} be the set of the columns of A. Then…”

mentioning

confidence: 99%

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

Wang¹

2021

Preprint

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“…Remark 2.10. There is a graphical calculus called scalable ZX (SZX from here on) [CHP19] that is notable here because one can view SZX as a graphical calculus that represents (potentially enormous) ZX diagrams. That is to say that one can view SZX as a graphical calculus with diagrams and diagrammatic rules as laid out in Ref.…”

Section: Definition 24 (Soundness [Sel11]) a Graphical Calculus Is So...mentioning

confidence: 99%

“…That is to say that one can view SZX as a graphical calculus with diagrams and diagrammatic rules as laid out in Ref. [CHP19], with an interpretation that turns a SZX diagram into a ZX diagram. We highlight this because all the other graphical calculi considered in this thesis have interpretations that turn diagrams into matrices, as indeed SZX does in its original presentation.…”

Section: Definition 24 (Soundness [Sel11]) a Graphical Calculus Is So...mentioning

confidence: 99%

Graphical Calculi and their Conjecture Synthesis

Miller-Bakewell¹

2020

Preprint

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Categorical Quantum Mechanics, and graphical calculi in particular, has proven to be an intuitive and powerful way to reason about quantum computing. This work continues the exploration of graphical calculi, inside and outside of the quantum computing setting, by investigating the algebraic structures with which we label diagrams. The initial aim for this was Conjecture Synthesis; the algorithmic process of creating theorems. To this process we introduce a generalisation step, which itself requires the ability to infer and then verify parameterised families of theorems. This thesis introduces such inference and verification frameworks, in doing so forging novel links between graphical calculi and fields such as Algebraic Geometry and Galois Theory. These frameworks inspired further research into the design of graphical calculi, and we introduce two important new calculi here. First is the calculus RING, which is initial among ring-based qubit graphical calculi, and in turn inspired the introduction and classification of phase homomorphism pairs also presented here. The second is the calculus ZQ, an edge-decorated calculus which naturally expresses arbitrary qubit rotations, eliminating the need for non-linear rules such as (EU) of ZX. It is expected that these results will be of use to those creating optimisation schemes and intermediate representations for quantum computing, to those creating new graphical calculi, and for those performing conjecture synthesis.

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“…This normal form leads to a proof of completeness, though full of nontrivial techniques. In addition, we gain an advantage that no scalable techniques are needed, as considered in [3].…”

Section: Introductionmentioning

confidence: 99%

Algebraic complete axiomatisation of ZX-calculus with a normal form via elementary matrix operations

Wang¹

2020

Preprint

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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 ZXcalculus 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.

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SZX-Calculus: Scalable Graphical Quantum Reasoning

Cited by 17 publications

References 11 publications

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

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

Graphical Calculi and their Conjecture Synthesis

Algebraic complete axiomatisation of ZX-calculus with a normal form via elementary matrix operations

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