  While limitations on quantum computation by Markovian environmental noise are
well-understood in generality, their behavior for different quantum circuits
and noise realizations can be less universal. Here we consider a canonical
quantum algorithm - Grover's algorithm for unordered search on $L$ qubits - in
the presence of systematic noise. This allows us to write the behavior as a
random Floquet unitary, which we show is well-characterized by random matrix
theory (RMT). The RMT analysis enables analytical predictions for phases and
phase transitions of the many-body dynamics. We find two separate transitions.
At moderate disorder $\delta_{c,\mathrm{gap}}\sim L^{-1}$, there is a
ergodicity breaking transition such that a finite-dimensional manifold remains
non-ergodic for $\delta < \delta_{c,\mathrm{gap}}$. Computational power is lost
at a much smaller disorder, $\delta_{c,\mathrm{comp}} \sim L^{-1/2}2^{-L/2}$.
We comment on relevance to non-systematic noise in realistic quantum computers,
including cold atom, trapped ion, and superconducting platforms.
