  We consider a model of monitored quantum dynamics with quenched spatial
randomness: specifically, random quantum circuits with spatially varying
measurement rates. These circuits undergo a measurement-induced phase
transition (MIPT) in their entanglement structure, but the nature of the
critical point differs drastically from the case with constant measurement
rate. In particular, at the critical measurement rate, we find that the
entanglement of a subsystem of size $\ell$ scales as $S \sim \sqrt{\ell}$;
moreover, the dynamical critical exponent $z = \infty$. The MIPT is flanked by
Griffiths phases with continuously varying dynamical exponents. We argue for
this infinite-randomness scenario on general grounds and present numerical
evidence that it captures some features of the universal critical properties of
MIPT using large-scale simulations of Clifford circuits. These findings
demonstrate that the relevance and irrelevance of perturbations to the MIPT can
naturally be interpreted using a powerful heuristic known as the Harris
criterion.
