A general quantum circuit can be simulated classically in exponential time. If it has a planar layout, then a tensor-network contraction algorithm due to Markov and Shi has a runtime exponential in the square root of its size, or more generally exponential in the treewidth of the underlying graph. Separately, Gottesman and Knill showed that if all gates are restricted to be Clifford, then there is a polynomial time simulation. We combine these two ideas and show that treewidth and planarity can be exploited to improve Clifford circuit simulation. Our main result is a classical algorithm with runtime scaling asymptotically as $n^{\omega/2}<n^{1.19}$ which samples from the output distribution obtained by measuring all $n$ qubits of a planar graph state in given Pauli bases. Here $\omega$ is the matrix multiplication exponent. We also provide a classical algorithm with the same asymptotic runtime which samples from the output distribution of any constant-depth Clifford circuit in a planar geometry. Our work improves known classical algorithms with cubic runtime. A key ingredient is a mapping which, given a tree decomposition of some graph $G$, produces a Clifford circuit with a structure that mirrors the tree decomposition and which emulates measurement of the corresponding graph state. We provide a classical simulation of this circuit with the runtime stated above for planar graphs and otherwise $nt^{\omega-1}$ where $t$ is the width of the tree decomposition. Our algorithm incorporates two subroutines which may be of independent interest. The first is a matrix-multiplication-time version of the Gottesman-Knill simulation of multi-qubit measurement on stabilizer states. The second is a new classical algorithm for solving symmetric linear systems over $\mathbb{F}_2$ in a planar geometry, extending previous works which only applied to non-singular linear systems in the analogous setting.

翻译：通用量子电路可以在指数时间内经典模拟。若电路具有平面布局，则马尔可夫和施提出的张量网络收缩算法运行时间与电路规模的平方根呈指数关系，更一般地，与底层图的树宽呈指数关系。另一方面，戈特斯曼和尼尔证明，若所有门限制为克利福德门，则可进行多项式时间模拟。我们结合这两个思路，表明树宽和平面性可用于改进克利福德电路模拟。我们的主要结果是一个经典算法，其渐近运行时间缩放为 $n^{\omega/2}<n^{1.19}$，该算法从通过测量平面图态的所有 $n$ 个量子比特（在给定泡利基下）得到的输出分布中采样。其中 $\omega$ 是矩阵乘法指数。我们还提供了一个具有相同渐近运行时间的经典算法，用于从平面几何中任意恒定深度克利福德电路的输出分布中采样。我们的工作改进了已知运行时间为三次方的经典算法。一个关键要素是一种映射，给定某个图 $G$ 的树分解，该映射产生一个结构镜像树分解的克利福德电路，并模拟相应图态的测量。我们对该电路进行经典模拟，对于平面图运行时间如前述，否则为 $nt^{\omega-1}$，其中 $t$ 是树分解的宽度。我们的算法包含两个可能具有独立价值的子程序。第一个是矩阵乘法时间版本的戈特斯曼-尼尔模拟，用于稳定子态的多量子比特测量。第二个是新的经典算法，用于在平面几何中求解 $\mathbb{F}_2$ 上的对称线性系统，扩展了先前仅适用于类似设定中非奇异线性系统的工作。