We present a completely new approach to quantum circuit optimisation, based on the ZX-calculus. We first interpret quantum circuits as ZX-diagrams, which provide a flexible, lower-level language for describing quantum computations graphically. Then, using the rules of the ZX-calculus, we give a simplification strategy for ZX-diagrams based on the two graph transformations of local complementation and pivoting and show that the resulting reduced diagram can be transformed back into a quantum circuit. While little is known about extracting circuits from arbitrary ZX-diagrams, we show that the underlying graph of our simplified ZX-diagram always has a graph-theoretic property called generalised flow, which in turn yields a deterministic circuit extraction procedure. For Clifford circuits, this extraction procedure yields a new normal form that is both asymptotically optimal in size and gives a new, smaller upper bound on gate depth for nearest-neighbour architectures. For Clifford+T and more general circuits, our technique enables us to to `see around' gates that obstruct the Clifford structure and produce smaller circuits than naive 'cut-and-resynthesise' methods.

中文翻译：

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量子电路的图论简化与 ZX 演算

我们提出了一种基于 ZX 演算的全新量子电路优化方法。我们首先将量子电路解释为 ZX 图，它提供了一种灵活的低级语言来以图形方式描述量子计算。然后，使用 ZX-演算的规则，我们给出了基于局部互补和旋转两种图变换的 ZX-图的简化策略，并表明得到的简化图可以转换回量子电路。虽然对从任意 ZX 图中提取电路知之甚少，但我们表明，我们简化的 ZX 图中的底层图始终具有称为广义流的图论特性，这反过来又产生了确定性的电路提取过程。对于克利福德赛道，这个提取过程产生了一个新的范式，它在大小上是渐近最优的，并且为最近邻架构提供了一个新的、更小的门深度上限。对于 Clifford+T 和更一般的电路，我们的技术使我们能够“观察”阻碍 Clifford 结构的门，并产生比简单的“剪切再合成”方法更小的电路。