Luke Schaeffer

Abstract A general quantum circuit can be simulated in exponential time on a classical computer. 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 quantum graph state corresponding to $G$. We provide a classical simulation of this circuit with the runtime stated above for planar graphs and otherwise $n t^{\omega-1}$ 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.

Abstract Recent work by Bravyi et al. constructs a relation problem that a noisy constant-depth quantum circuit (QNC^0) can solve with near certainty (probability 1 - o(1)), but that any bounded fan-in constant-depth classical circuit (NC^0) fails with some constant probability. We show that this robustness to noise can be achieved in the other low-depth quantum/classical circuit separations in this area. In particular, we show a general strategy for adding noise tolerance to the interactive protocols of Grier and Schaeffer. As a consequence, we obtain an unconditional separation between noisy QNC^0 circuits and AC^0[p] circuits for all primes p \geq 2, and a conditional separation between noisy QNC^0 circuits and log-space classical machines under a plausible complexity-theoretic conjecture. A key component of this reduction is showing average-case hardness for the classical simulation tasks---that is, showing that a classical simulation of the quantum interactive task is still powerful even if it is allowed to err on some constant fraction of inputs. We show that is possible even for quantum tasks which are parity-L-hard to simulate. To do this, we borrow techniques from randomized encodings used in cryptography.