  We show that a K\"ahler-Ricci soliton on a Fano manifold can always be
smoothly approximated by a sequence of relative anticanonically balanced
metrics, also called quantized K\"ahler-Ricci solitons. The proof uses a
semiclassical estimate on the spectral gap of an equivariant Berezin transform
to extend a strategy due to Donaldson, and can be seen as the quantization of a
method due to Tian and Zhu, using quantized Futaki invariants as obstructions
for quantized K\"ahler-Ricci solitons. As corollaries, we recover the
uniqueness of K\"ahler-Ricci solitons up to automorphisms, and show how our
result also applies to K\"ahler-Einstein Fano manifolds with general
automorphism group.
