  A symmetry-twisted boundary condition of the path integral provides a
suitable framework for the semi-classical analysis of nonperturbative quantum
field theories (QFTs), and we reinterpret it from the viewpoint of the Hilbert
space. An appropriate twist with the unbroken symmetry can potentially produce
huge cancellations among excited states in the state-sum, without affecting the
ground states; we call this effect "quantum distillation". Quantum distillation
can provide the underlying mechanism for adiabatic continuity, by preventing a
phase transition under $S^1$ compactification. We revisit this point via the 't
Hooft anomaly matching condition when it constrains the vacuum structure of the
theory on $\mathbb{R}^d$ and upon compactification. We show that there is a
precise relation between the persistence of the anomaly upon compactification,
the Hilbert space quantum distillation, and the semi-classical analysis of the
corresponding symmetry-twisted path integrals. We motivate quantum distillation
in quantum mechanical examples, and then study its non-trivial action in QFT,
with the example of the 2D Grassmannian sigma model $\mathrm{Gr}(N,M)$. We also
discuss the connection of quantum distillation with large-$N$ volume
independence and flavor-momentum transmutation.
