  The $3D$ Ising transition, the most celebrated and unsolved critical
phenomenon in nature, has long been conjectured to have emergent conformal
symmetry, similar to the case of the $2D$ Ising transition. Yet, the emergence
of conformal invariance in the $3D$ Ising transition has rarely been explored
directly, mainly due to unavoidable mathematical or conceptual obstructions.
Here, we design an innovative way to study the quantum version of the $3D$
Ising phase transition on spherical geometry, using the "fuzzy
(non-commutative) sphere" regularization. We accurately calculate and analyze
the energy spectra at the transition, and explicitly demonstrate the
state-operator correspondence (i.e. radial quantization), a fingerprint of
conformal field theory. In particular, we have identified 13 parity-even
primary operators within a high accuracy and 2 parity-odd operators that were
not known before. Our result directly elucidates the emergent conformal
symmetry of the $3D$ Ising transition, a conjecture made by Polyakov half a
century ago. More importantly, our approach opens a new avenue for studying
$3D$ CFTs by making use of the state-operator correspondence and spherical
geometry.
