  We investigate the transition to quantum chaos, induced by static
imperfections, for an operating quantum computer that simulates efficiently a
dynamical quantum system, the sawtooth map. For the different dynamical regimes
of the map, we discuss the quantum chaos border induced by static imperfections
by analyzing the statistical properties of the quantum computer eigenvalues.
For small imperfection strengths the level spacing statistics is close to the
case of quasi-integrable systems while above the border it is described by the
random matrix theory. We have found that the border drops exponentially with
the number of qubits, both in the ergodic and quasi-integrable dynamical
regimes of the map characterized by a complex phase space structure. On the
contrary, the regime with integrable map dynamics remains more stable against
static imperfections since in this case the border drops only algebraically
with the number of qubits.
