ZX-Rules for 2-Qubit Clifford+T Quantum Circuits

Coecke, Bob; Wang, Quanlong

doi:10.1007/978-3-319-99498-7_10

Cited by 22 publications

(11 citation statements)

References 35 publications

(41 reference statements)

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“…Related works. Two completeness results on diagrammatic languages have been established recently [4,10], independently of the present work. In [4], a new language, the ZH-calculus is introduced.…”

Section: Introductionmentioning

confidence: 66%

“…Like in the ZW-calculus, the entries of a complex matrix can be directly represented in a ZH-diagram while the representation of the scalars is the cornerstone -and the main technicality -of the normal forms in ZX-diagrams. In [10], the authors show that a simpler axiomatisation of the ZX-calculus is enough to prove the equivalence of 2-qubit Clifford+T circuits. Surprisingly, the proposed axiomatisation is based on the use of diagrams which are not in the π 4 -fragment whereas all 2-qubit Clifford+T circuits are in this fragment.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A Generic Normal Form for ZX-Diagrams and Application to the Rational Angle Completeness

Jeandel

Perdrix

Vilmart

2019

2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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Recent completeness results on the ZX-Calculus used a third-party language, namely the ZW-Calculus. As a consequence, these proofs are elegant, but sadly non-constructive. We address this issue in the following. To do so, we first describe a generic normal form for ZX-diagrams in any fragment that contains Clifford+T quantum mechanics. We give sufficient conditions for an axiomatisation to be complete, and an algorithm to reach the normal form. Finally, we apply these results to the Clifford+T fragment and the general ZX-Calculus -for which we already know the completeness-, but also for any fragment of rational angles: we show that the axiomatisation for Clifford+T is also complete for any fragment of dyadic angles, and that a simple new rule (called cancellation) is necessary and sufficient otherwise.

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Section: Introductionmentioning

confidence: 66%

Section: Introductionmentioning

confidence: 99%

A Generic Normal Form for ZX-Diagrams and Application to the Rational Angle Completeness

Jeandel

Perdrix

Vilmart

2019

2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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“…We say that both of these calculi are complete. This completeness caused quite some noise at the time, but it didn't come entirely out of the blue, some earlier work including [4,6,69,75], and there also are some further elaborations [47,110]. Definition 4.3.…”

Section: Complete Compositionalitymentioning

confidence: 99%

Compositionality as we see it, everywhere around us

Coecke¹

2021

Preprint

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There are different meanings of the term "compositionality" within science: what one researcher would call compositional, is not at all compositional for another researcher. The most established conception is usually attributed to Frege, and is characterised by a bottomup flow of meanings: the meaning of the whole can be derived from the meanings of the parts, and how these parts are structured together.Inspired by work on compositionality in quantum theory, and categorical quantum mechanics in particular, we propose the notions of Schrödinger, Whitehead, and complete compositionality. Accounting for recent important developments in quantum technology and artificial intelligence, these do not have the bottom-up meaning flow as part of their definitions.Schrödinger compositionality accommodates quantum theory, and also meaning-as-context. Complete compositionality further strengthens Schrödinger compositionality in order to single out theories like ZX-calculus, that are complete with regard to the intended model. All together, our new notions aim to capture the fact that compositionality is at its best when it is 'real', 'non-trivial', and even more when it also is 'complete'.At this point we only put forward the intuitive and/or restricted formal definitions, and leave a fully comprehensive definition to future collaborative work.

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“…All spiders are equipped with phases, and more abstractly, they can be defined as certain Frobenius algebras [20]. In more recent versions of the ZX-calculus, more general phases are also allowed [39,23], as these exist for equally general abstract reasons, and this then brings us to the general case of the phaser.…”

Section: The Phasermentioning

confidence: 99%

Meaning updating of density matrices

Coecke¹,

Meichanetzidis²

2020

Preprint

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The DisCoCat model of natural language meaning assigns meaning to a sentence given: (i) the meanings of its words, and, (ii) its grammatical structure. The recently introduced DisCoCirc model extends this to text consisting of multiple sentences. While in DisCoCat all meanings are fixed, in DisCoCirc each sentence updates meanings of words. In this paper we explore different update mechanisms for DisCoCirc, in the case where meaning is encoded in density matrices-which come with several advantages as compared to vectors.Our starting point are two non-commutative update mechanisms, borrowing one from quantum foundations research [34,35], and the other one from [9,36]. Unfortunately, neither of these satisfies any desirable algebraic properties, nor are internal to the meaning category. By passing to double density matrices [2, 51] we do get an elegant internal diagrammatic update mechanism.We also show that (commutative) spiders can be cast as an instance of the update mechanism of [34,35]. This result is of interest to quantum foundations, as it bridges the work in Categorical Quantum Mechanics (CQM) with that on conditional quantum states. Our work also underpins implementation of textlevel Natural Language Processing (NLP) on quantum hardware, for which exponential space-gain and quadratic speed-up have previously been identified.We thank Jon Barrett, Stefano Gogioso and Rob Spekkens for inspiring discussions preceding the results in this paper, and Matt Pusey, Sina Salek and Sam Staton for comments on a previous version of the paper. KM was supported by the EPSRC National Hub in Networked Quantum Information Technologies.

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ZX-Rules for 2-Qubit Clifford+T Quantum Circuits

Cited by 22 publications

References 35 publications

A Generic Normal Form for ZX-Diagrams and Application to the Rational Angle Completeness

A Generic Normal Form for ZX-Diagrams and Application to the Rational Angle Completeness

Compositionality as we see it, everywhere around us

Meaning updating of density matrices

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