We propose a quantum algorithm which uses the number of qubits in an optima
l way and efficiently simulates a physical model with rich and complex dyna
mics described by the quantum sawtooth map. The numerical study of the effe
ct 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 gener
ated by these imperfections are more significant than the errors of random
noise in gate operations.