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For practical validation and usability, an implementation is provided in discopy, enabling numerical verification of the constructions and supporting future ZX-calculus-based matrix synthesis.",{"@graph":14,"@context":72},[15,34,55],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & 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\nQuanlong Wang 1 Richie Yeung 1;2  \n1 Quantinuum, 17 Beaumont Street, Oxford OX1 2NA, United Kingdom  \n2University of Oxford  \nIn linear algebra applications, elementary matrices hold a significant role. This paper presents a diagrammatic representation of all 2m 􀀂 2n-sized elementary matrices in algebraic ZX-calculus, showcasing their properties on inverses and transpose through diagrammatic rewriting. Additionally, the paper uses this representation to depict the Jozsa-style matchgate in algebraic ZX-calculus. To further enhance practical use, we have implemented this representation in discopy. Overall, this work sets the groundwork for more applications of ZX-calculus such as synthesising controlled matrices  \n[19] in quantum computing.  \n1 Introduction  \nMatrices are used everywhere in modern science, like machine learning [17] or quantum computing [18], to name a few. Meanwhile, there is a graphical language called ZX-calculus that could also deal with matrix calculations such as matrix multiplication and tensor product [3, 5] . Then there naturally arisesa question: why are people bothering with using diagrams for matrix calculations given that matrix technology has been applied with great successes? There are a few reasons for doing so. First, there is a lot of redundancy in matrix calculations which could be avoided in graphical calculus. For example, to prove the cyclic property of matrices tr(AB) = tr(BA), all the elements of the two matrices will be involved, while in graphical language like ZX-calculus, the proof of the cyclic property is almost a tautology [4] . Second, matrix calculations always have all the elements of matrices involved, thus a“global” operation, while in ZX-calculus, the operations are just diagram rewriting where only a part of a diagram is replaced by another sub-diagram according to certain rewriting rule, thus essentially a“local” operation which makes things much easier. Finally, graphical calculus is much more intuitive than matrix calculation, therefore a pattern/structure is more probably to be recognised in a graphical formalism. In fact, as a graphical calculus for matrix calculation, ZX-calculus has achieved plenty of successes in the field of quantum computing and information [1, 6, 13, 20]  \nFor research realm beyond quantum, traditional ZX-calculus [3] is inconvenient as “it lacks a way to directly encode the complex numbers” [26] . To remedy this inconvenience while still preserving its power, we introduced an “extended” version called algebraic ZX-calculus with new generators added in (not a real extension in the sense that the new generators can be expressed by the original generators)  \n[23] . While [22] proves the completeness of algebraic ZX calculus using elementary matrix operations, the set of elementary operations given are a minimal set (multiplying the bottom row, adding the bottom row to other rows) and are unsuitable for practical matrix synthesis.  \nIn this paper, we first provide diagrammatic representations for adding, multiplying and switching the rows and columns of a 2m 􀀂 2n matrix, diagrammatically prove some basic properties about their inversesand transposes. Then we show how to represent an arbitrary matrix of size 2m 􀀂 2n in ZX diagrams. Finally we show how this representation method could be used for representing matchgates [10] . We  \n2 Representing and Implementing Matrices Using Algebraic ZX-calculus  \nalso include an implementation of this representation method in discopy (see [link]), which numerically verifies these examples. The results of this paper can be generalised from the field of complex numbers to arbitrary commutative semirings due to the completeness of algebraic ZX-calculus over semirings [24] .  \nRelated Work: There is an established and related body of research known as Graphical Linear Algebra. [2] The diagrams in this formal","cbCaifeGJRdRGIfs","https://ap.wps.com/l/cbCaifeGJRdRGIfs","pdf",566139,25,"English","# Introduction\n## Motivation for using diagrams in matrix calculations\n## Limitations of traditional ZX-calculus\n# Algebraic ZX-calculus\n## Generators, interpretations, and rewriting rules","[{\"question\":\"What does the paper contribute regarding elementary matrices in algebraic ZX-calculus?\",\"answer\":\"It presents a diagrammatic representation for all 2^m × 2^n-sized elementary matrices in algebraic ZX-calculus and studies how these diagrams behave under inverses and transposes using diagram rewriting.\"},{\"question\":\"How are properties like inverses and transposes handled in the proposed framework?\",\"answer\":\"The paper proves basic properties of the inverses and transposes diagrammatically by applying rewriting rules that transform parts of ZX diagrams into equivalent sub-diagrams.\"},{\"question\":\"How is the representation validated or made practical?\",\"answer\":\"The representation method is implemented in discopy, where numerical verification is used to confirm the examples discussed in the paper.\"}]","Representing and Implementing Matrices Using Algebraic ZX-calculus | PDF",1792978316,63]