  Entanglement asymmetry, which serves as a diagnostic tool for symmetry
breaking and a proxy for thermalization, has recently been proposed and studied
in the context of symmetry restoration for quantum many-body systems undergoing
a quench. In this Letter, we investigate symmetry restoration in various
symmetric random quantum circuits, particularly focusing on the U(1) symmetry
case. In contrast to non-symmetric random circuits where the U(1) symmetry of a
small subsystem can always be restored at late times, we reveal that symmetry
restoration can fail in U(1) symmetric circuits for certain small
symmetry-broken initial states in finite-size systems. In the early-time
dynamics, we observe an intriguing quantum Mpemba effect implying that symmetry
is restored faster when the initial state is more asymmetric. Furthermore, we
also investigate the entanglement asymmetry dynamics for SU(2) and $Z_{2}$
symmetric circuits and identify the presence and absence of the quantum Mpemba
effect for the corresponding symmetries, respectively. A unified understanding
of these results is provided through the lens of quantum thermalization with
conserved charges.
