The ZX calculus: A complete graphical calculus for classical circuits using spiders
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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 units and counits to the category TOF generated by the Toffoli gate and ancillary bits, showing that this yields the strictification of spans of powers of the two element set; 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 algebra of a discrete inverse category is the same as computing the “environment structure” of the classical structures of the base discrete inverse category. We show that in this setting, the classical channels and the discrete Cartesian completion are the same constructions. Therefore, in the case of TOF, freely adding a counit, constructing the category of quantum channels, and computing the discrete Cartesian completion are all equivalent to partial functions between powers of the two element set. By glueing together the free counit completion and the free unit completion, this yields the strictification of spans between powers of the two element set.
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