Infinitely fast critical dynamics: Teleportation through temporal rare regions in monitored quantum circuits

We consider measurement-induced phase transitions in monitored quantum circuits with a measurement rate that fluctuates in time, remaining spatially uniform at each time. The spatially correlated fluctuations in the measurement rate disrupt the volume-law phase for low measurement rates; at a critical measurement rate, they give rise to an entanglement phase transition with ``ultrafast'' dynamics, i.e., space time ($x,t$) scaling $logx\ensuremath{\sim}{t}^{{\ensuremath{\psi}}_{\ensuremath{\tau}}}$. The ultrafast dynamics at the critical point can be viewed as a space-time-rotated version of an infinite-randomness critical point; despite the spatial locality of the dynamics, ultrafast information propagation is possible because of measurement-induced quantum teleportation. We identify temporal Griffiths phases on either side of this critical point. We provide a physical interpretation of these phases, and support it with extensive numerical simulations of information propagation and entanglement dynamics in stabilizer circuits. The implications of our results on the general stability of phase transitions and ordered phases to such temporal randomness are discussed.