Simple ZX and ZH calculi for arbitrary finite dimensions, via discrete integrals
2023
Online
report
The ZX calculus and the ZH calculus use diagrams to denote and to compute properties of quantum operations, and other multi-linear operators described by tensor networks. These calculi involve 'rewrite rules', which are algebraic manipulations of the tensor networks through transformations of diagrams. The way in which diagrams denote tensor networks is through a semantic map, which assigns a meaning to each diagram in a compositional way. Slightly different semantic maps, which may prove more convenient for one purpose or another (e.g., analysing unitary circuits versus analysing counting complexity), give rise to slightly different rewrite systems. Through a simple application of measure theory on discrete sets, we describe a semantic map for ZX and ZH diagrams for qudits of any dimension D>1, well-suited to represent unitary circuits, and admitting simple rewrite rules. In doing so, we reproduce the 'well-tempered' semantics of [arXiv:2006.02557] for ZX and ZH diagrams in the case D=2. We demonstrate rewrite rules for the 'stabiliser fragment' of the ZX calculus and a 'multicharacter fragment' of the ZH calculus; and demonstrate relationships which would allow the two calculi to be used interoperably as a single 'ZXH calculus'.
Comment: 17 pages of main text, 27 pages of appendices, many equations involving diagrams
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Simple ZX and ZH calculi for arbitrary finite dimensions, via discrete integrals
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Autor/in / Beteiligte Person: | de Beaudrap, Niel ; East, Richard D. P. |
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Veröffentlichung: | 2023 |
Medientyp: | report |
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