Measure and Forget Dynamics in Random Circuits

Abstract

Unrecorded or ``forgetful'' measurements -- physically, a local dephasing channel -- can arise naturally as syndrome-measurement errors in fault-tolerant protocols and as uncontrolled decoherence in open systems. We numerically study random Clifford circuits in which recorded projective $Z$ measurements compete with unrecorded (forgotten) $Z$ measurements, focusing on the dynamical, finite-depth regime relevant to decoding and quantum memory. In the forget-only limit the global entropy density thermalizes at a rate independent of system size, and the forgetting rate at which thermalization sets in decays with circuit depth as a power law $p_f^{\ast}\sim d^{v}$, consistent with the critical-depth scaling inversely with the noise rate found for related noisy-circuit quantities. With recorded measurements present, the mutual information $I(A{:}B)$ -- which quantifies the recoverable correlation between the two halves -- develops a finite-depth peak whose height grows sub-extensively with $N$: its effective scaling exponent $\alpha(p_f)$ decreases monotonically, so that forgetting drives the recoverable correlation from near-volume-law toward area-law scaling. We map the recoverability landscape in the $(p_m,p_f)$ plane and show that forgetting destroys the measurement-induced purification transition. These results complement earlier stat-mech analyses of noisy monitored circuits and quantify the competition between measurement and forgetting relevant to quantum error correction.