Fast simulation of planar Clifford circuits

David Gosset1,2,3, Daniel Grier1,4,5, Alex Kerzner1,2, and Luke Schaeffer1,2,6

1Institute for Quantum Computing, University of Waterloo, Canada
2Department of Combinatorics and Optimization, University of Waterloo, Canada
3Perimeter Institute for Theoretical Physics, Waterloo, Canada
4Cheriton School of Computer Science, University of Waterloo, Canada
5Department of Computer Science and Engineering and Department of Mathematics, University of California, San Diego, US
6Joint Center for Quantum Information and Computer Science, College Park, Maryland, US

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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}$ $\lt$ $n^{1.19}$ 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.

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[2] Travis L. Scholten, Carl J. Williams, Dustin Moody, Michele Mosca, William Hurley, William J. Zeng, Matthias Troyer, and Jay M. Gambetta, "Assessing the Benefits and Risks of Quantum Computers", arXiv:2401.16317, (2024).

[3] Ryan L. Mann, "Simulating quantum computations with Tutte polynomials", npj Quantum Information 7, 141 (2021).

[4] Sahar Atallah, Michael Garn, Sania Jevtic, Yukuan Tao, and Shashank Virmani, "Efficient classical simulation of cluster state quantum circuits with alternative inputs", Quantum 8, 1243 (2024).

[5] Shihao Zhang, Jiacheng Bao, Yifan Sun, Lvzhou Li, Houjun Sun, and Xiangdong Zhang, "High-performance parallel classical scheme for simulating shallow quantum circuits", arXiv:2103.00693, (2021).

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