  We propose a classical simulation method for quantum circuits based on
decomposing unitary gates into a sum of stabilizer projectors. By only
decomposing the non-Clifford gates, we take advantage of the Gottesman-Knill
theorem and build a bridge between stabilizer-based simulation and
Feynman-path-type simulation. We give two variants of this method:
stabilizer-based path-integral recursion (SPIR) and stabilizer projector
contraction (SPC). We also analyze further advantages and disadvantages of our
method compared to the Bravyi-Gosset algorithm and recursive Feynman
path-integral algorithms. We construct a parametrized circuit ensemble and
identify the parameter regime in this ensemble where our method offers superior
performance. We also estimate the time cost for simulating quantum supremacy
experiments with our method and motivate potential improvements of the method.
