  Symmetries (and their spontaneous rupturing) can be used to protect and
engender novel quantum phases and lead to interesting collective phenomena. In
Ref. 1, the authors described a general dynamical decoupling (polyfractal)
protocol that can be used to engineer multiple discrete symmetries in many-body
systems. The present work expands on the former by studying the effect of
quasiperiodic and random noise on such a dynamical scheme. We find generally
that relaxation of engineered symmetry generators proceeds by i) an initial
relaxation on microscopic timescales to a prethermal plateau whose height is
independent of noise, ii) a linear relaxation regime with a noise-dependent
rate, followed by iii) a slow logarithmic relaxation regime that is only
present for quasiperiodic noise. We glean the essential features of these
regimes via scaling collapses and show that they can be generally explained by
the spectral properties of the various noise waveforms considered. In
particular, the quasiperiodic noise is characterised by highly time dependent
spectrum with a noise floor that mimics white noise, and peaks that grow
sharper with time. We argue that both the noise floor and peaks contribute to
the initial linear-in-time relaxation while the logarithmic regime is initiated
when the peaks become sufficiently well resolved and cease to contribute to
further relaxation. We provide numerical evidence to justify these findings.
