Fast simulation of planar Clifford circuits

TL;DR

The paper advances classical simulation of restricted quantum computations by exploiting planarity and treewidth in Clifford circuits. It introduces a framework that maps graph-state measurement tasks to tree-decomposition-guided Clifford circuits and uses two subroutines—sampling and a correction process—to produce correct output samples efficiently. The main technical results show a planar-graph sampling algorithm running in $\tilde{O}(n^{\omega/2})$ time (with $\omega$ the matrix-multiplication exponent) and extend this to planar constant-depth Clifford circuits, aided by a planar linear-system solver and a new affine-stabilizer framework. These methods yield substantial improvements over previous cubic-time algorithms and illuminate the boundary between quantum and classical capabilities for restricted circuit families, with potential applications to Clifford tensor networks and planar Clifford circuits.

Abstract

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 $n^{ω/2}<n^{1.19}$ which samples from the output distribution obtained by measuring all $n$ qubits of a planar graph state in given Pauli bases. Here $ω$ 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 $nt^{ω-1}$ 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.

Figures (15)

• Figure 1: Simulation time in seconds vs. grid length $\ell$ of an $\ell \times \ell$ grid. The data is taken as an average over 7 random trials from $\ell = 2$ to $\ell = 1000$ by $2$'s. We cut off the naïve algorithm at $\ell=408$ due to memory limitations. The simulations were conducted on a desktop computer using a modified version of the CHP Clifford simulator aaronson+gottesman:2004gridCHPpp. Our simulations do not use fast matrix multiplication. See \ref{['sec:software_implementation']} for the implementation details.
• Figure 2: Depiction of the recursive algorithm for graph state simulation on a 2D grid. Blue vertices are measured and red are unmeasured. (a) Quantum graph states corresponding to the 5 connected components of this graph are prepared and the qubits at the blue vertices are measured. (b) $\operatorname{CZ}$ gates are applied to connect them together. (c) Single-qubit measurements are performed on all qubits corresponding to red vertices not on the perimeter.
• Figure 3: A 5-coarse-graining from a nonplanar graph to a planar graph.
• Figure 4: (a) Planar graph on 8 vertices. (b) A tree decomposition of width 2 treed.
• Figure 5: Depiction of the transformation used in \ref{['lem:td2']} which maps a tree decomposition to a nice tree decomposition.

Theorems & Definitions (76)

• Theorem 1
• Theorem 2: Special case of \ref{['thm:planargss']}
• Theorem 3
• Theorem 4: Special case of \ref{['thm:linearsolver']}
• Theorem 5
• Theorem 6: informal
• Theorem 7: informal
• Theorem 8: Extension of Theorem \ref{['thm:planargss']}
• Claim 9
• proof