  Approximation based on perturbation theory is the foundation for most of the
quantitative predictions of quantum mechanics, whether in quantum many-body
physics, chemistry, quantum field theory or other domains. Quantum computing
provides an alternative to the perturbation paradigm, yet state-of-the-art
quantum processors with tens of noisy qubits are of limited practical utility.
Here, we introduce perturbative quantum simulation, which combines the
complementary strengths of the two approaches, enabling the solution of large
practical quantum problems using limited noisy intermediate-scale quantum
hardware. The use of a quantum processor eliminates the need to identify a
solvable unperturbed Hamiltonian, while the introduction of perturbative
coupling permits the quantum processor to simulate systems larger than the
available number of physical qubits. We present an explicit perturbative
expansion that mimics the Dyson series expansion and involves only local
unitary operations, and show its optimality over other expansions under certain
conditions. We numerically benchmark the method for interacting bosons,
fermions, and quantum spins in different topologies, and study different
physical phenomena, such as information propagation, charge-spin separation,
and magnetism, on systems of up to $48$ qubits only using an $8+1$ qubit
quantum hardware. We experimentally demonstrate our scheme on the IBM quantum
cloud, verifying its noise robustness and illustrating its potential for
benchmarking large quantum processors with smaller ones.
