  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.

Quantum Physics;Computational Complexity