We present an algorithm for efficiently simulating a quantum circuit in the graph formalism. In the graph formalism, we represent states as a linear combination of graphs with Clifford operations on their vertices. We show how a $\mathcal{C}_3$ gate such as the Toffoli gate or $\frac\pi8$ gate acting on a stabilizer state splits it into two stabilizer states. We also describe conditions for merging two stabilizer states into one. We discuss applications of our algorithm to circuit identities and finding low stabilizer rank representations of magic states.

中文翻译：

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将图形式主义扩展到高阶门

我们提出了一种算法，用于在图形式主义中有效地模拟量子电路。在图形式主义中，我们将状态表示为图在其顶点上进行 Clifford 操作的线性组合。我们展示了 $\mathcal{C}_3$ 门（例如 Toffoli 门或 $\frac\pi8$ 门）如何作用于稳定器状态将其分成两个稳定器状态。我们还描述了将两个稳定器状态合并为一个的条件。我们讨论了我们的算法在电路标识中的应用，并找到了魔法状态的低稳定器等级表示。