  Quantum state designs, by enabling an efficient sampling of random quantum
states, play a quintessential role in devising and benchmarking various quantum
protocols with broad applications ranging from circuit designs to black hole
physics. Symmetries, on the other hand, are expected to reduce the randomness
of a state. Despite being ubiquitous, the effects of symmetry on quantum state
designs remain an outstanding question. The recently introduced projected
ensemble framework generates efficient approximate state t-designs by hinging
on projective measurements and many-body quantum chaos. In this work, we
examine the emergence of state designs from the random generator states
exhibiting symmetries. Leveraging on translation symmetry, we analytically
establish a sufficient condition for the measurement basis leading to the state
t-designs. Then, by making use of the trace distance measure, we numerically
investigate the convergence to the designs. Subsequently, we inspect the
violation of the sufficient condition to identify bases that fail to converge.
We further demonstrate the emergence of state designs in a physical system by
studying the dynamics of a chaotic tilted field Ising chain with periodic
boundary conditions. We find faster convergence of the trace distance in the
initial time, however, it saturates to a finite value deviating from random
matrix prediction at a late time, in contrast to the case with open boundary
condition. To delineate the general applicability of our results, we extend our
analysis to other symmetries. We expect our findings to pave the way for
further exploration of deep thermalization and equilibration of closed and open
quantum many-body systems.
