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Register a new userAlgebraic complete axiomatisation of ZX-calculus with a normal form via elementary matrix operations
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Quanlong Wang
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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 ZX-calculus 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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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 cant be obtained by just using the completeness of ZX-calculus.
ZX-calculus is a strict mathematical formalism for graphical quantum computing which is based on the field of complex numbers. In this paper, we extend its power by generalising ZX-calculus to such an extent that it is universal both in an arbitrary commutative ring and in an arbitrary commutative semiring. Furthermore, we follow the framework of arXiv:2007.13739 to prove respectively that the proposed ZX-calculus over an arbitrary commutative ring (semiring) is complete for matrices over the same ring (semiring), via a normal form inspired from matrix elementary operations such as row addition and row multiplication. This work could lead to various applications including doing elementary number theory in string diagrams.
We give a complete presentation for the fragment, ZX&, of the ZX-calculus generated by the Z and X spiders (corresponding to copying and addition) along with the not gate and the and gate. To prove completeness, we freely add a unit and counit to the category TOF generated by the Toffoli gate and ancillary bits, showing that this yields the full subcategory of finite ordinals and functions with objects powers of two; and then perform a two way translation between this category and ZX&. A translation to some extension of TOF, as opposed to some fragment of the ZX-calculus, is a natural choice because of the multiplicative nature of the Toffoli gate. To this end, we show that freely adding counits to the semi-Frobenius algebras of a discrete inverse category is the same as constructing the Cartesian completion. In particular, for a discrete inverse category, the category of classical channels, the Cartesian completion and adding counits all produce the same category. Therefore, applying these constructions to TOF produces the full subcategory of finite ordinals and partial maps with objects powers of two. By glueing together the free counit completion and the free unit completion, this yields qubit multirelations.
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.
We introduce here a new axiomatisation of the rational fragment of the ZX-calculus, a diagrammatic language for quantum mechanics. Compared to the previous axiomatisation introduced in [8], our axiomatisation does not use any metarule , but relies instead on a more natural rule, called the cyclotomic supplementarity rule, that was introduced previously in the literature. Our axiomatisation is only complete for diagrams using rational angles , and is not complete in the general case. Using results on diophantine geometry, we characterize precisely which diagram equality involving arbitrary angles are provable in our framework without any new axioms, and we show that our axiomatisation is continuous, in the sense that a diagram equality involving arbitrary angles is provable iff it is a limit of diagram equalities involving rational angles. We use this result to give a complete characterization of all Euler equations that are provable in this axiomatisation.
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