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The Gottesman-Knill theorem asserts that a quantum circuit composed ofClifford gates can be efficiently simulated on a classical computer. Here werevisit this theorem and extend it to quantum circuits composed of Clifford andT gates, where T is the single-qubit 45-degree phase shift. We assume that thecircuit outputs a bit string x obtained by measuring some subset of w qubits.Two simulation tasks are considered: (1) computing the probability of a givenoutput x, and (2) sampling x from the output probability distribution. It isshown that these tasks can be solved on a classical computer in time$poly(n,m)+2^{0.5 t} t^3$ and $poly(n,m)+2^{0.23 t} t^3 w^3$ respectively,where t is the number of T-gates, m is the total number of gates, and n is thenumber of qubits. The proposed simulation algorithms may serve as averification tool for medium-size quantum computations that are dominated byClifford gates. The main ingredient of both algorithms is a subroutine forapproximating the norm of an n-qubit state which is given as a linearcombination of $\chi$ stabilizer states. The subroutine runs in time $\chi n^3\epsilon^{-2}$, where $\epsilon$ is the relative error. We also developtechniques for approximating tensor products of "magic states" by linearcombinations of stabilizer states. To demonstrate the power of the newsimulation methods, we performed a classical simulation of a hidden shiftquantum algorithm with 40 qubits, a few hundred Clifford gates, and nearly 50T-gates.

Improved classical simulation of quantum circuits dominated by Clifford gates