  We propose a quantum algorithm which uses the number of qubits in an optimal
way and efficiently simulates a physical model with rich and complex dynamics
described by the quantum sawtooth map. The numerical study of the effect of
static imperfections in the quantum computer hardware shows that the main
elements of the phase space structures are accurately reproduced up to a time
scale which is polynomial in the number of qubits. The errors generated by
these imperfections are more dangerous than the errors of random noise in gate
operations.
