  The classical simulation of quantum circuits is of central importance for
benchmarking near-term quantum devices. The fact that gates belonging to the
Clifford group can be simulated efficiently on classical computers has
motivated a range of methods that scale exponentially only in the number of
non-Clifford gates. Here, we consider the expectation value problem for
circuits composed of Clifford gates and non-Clifford Pauli rotations, and
introduce a heuristic perturbative approach based on the truncation of the
exponentially growing sum of Pauli terms in the Heisenberg picture. Numerical
results are shown on a Quantum Approximate Optimization Algorithm (QAOA)
benchmark for the E3LIN2 problem and we also demonstrate how this method can be
used to quantify coherent and incoherent errors of local observables in
Clifford circuits. Our results indicate that this systematically improvable
perturbative method offers a viable alternative to exact methods for
approximating expectation values of large near-Clifford circuits.
