We propose examples of a hybrid quantum-classical simulation where a classical computer assisted by a small quantum processor can efficiently simulate a larger quantum system. First, we consider sparse quantum circuits such that each qubit participates in O(1) two-qubit gates. It is shown that any sparse circuit on n + k qubits can be simulated by sparse circuits on n qubits and a classical processing that takes time 2O(k)poly(n). Second, we study Pauli-based computation (PBC), where allowed operations are nondestructive eigenvalue measurements of n-qubit Pauli operators. The computation begins by initializing each qubit in the so-called magic state. This model is known to be equivalent to the universal quantum computer. We show that any PBC on n + k qubits can be simulated by PBCs on n qubits and a classical processing that takes time 2O(k)poly(n). Finally, we propose a purely classical algorithm that can simulate a PBC on n qubits in a time 2αnpoly(n), where α≈ 0.94. This improves upon the brute-force simulation method, which takes time 2npoly(n). Our algorithm exploits the fact that n-fold tensor products of magic states admit a low-rank decomposition into n-qubit stabilizer states.