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

Abstract

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.