论文标题
ZX＆-CALCULUS：使用蜘蛛的经典电路的完整图形微积分
The ZX&-calculus: A complete graphical calculus for classical circuits using spiders
论文作者
论文摘要
我们为Z和X蜘蛛生成的ZX-Calculus片段（对应于复制和添加）以及NOT GATE和GATE。 为了证明完整性，我们自由地将单位和符合添加到Toffoli Gate和Ancillary Bits生成的类别TOF中，这表明这产生了有限序的完整子类别，并具有两个对象功率的功能；然后在此类别和ZX＆之间执行两种翻译。 与ZX-Calculus的某些碎片相反，向TOF的某种延伸的翻译是自然的选择，因为Toffoli Gate的乘法性质。 为此，我们表明，在离散逆类别的半烟叶代数中自由添加库，与构建笛卡尔完成相同。特别是，对于离散的逆类别，经典渠道的类别，笛卡尔完成和添加Counit都会产生相同的类别。 因此，将这些构造应用于TOF会产生有限序的完整子类别，并具有两个对象幂的幂。 通过将免费的顾问完成和免费的单位完成粘合在一起，这产生了“量子的多重相关”。
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."
