Phases and phase transition in Grover's algorithm with systematic noise

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 δc,gap~ L⁻¹, there is a ergodicity breaking transition such that a finite-dimensional manifold remains non-ergodic for δ < δc,gap. Computational power is lost at a much smaller disorder, δc,comp ~ L-1/22-L/2. We comment on relevance to non-systematic noise in realistic quantum computers, including cold atom, trapped ion, and superconducting platforms.