  The Kardar-Parisi-Zhang (KPZ) universality class describes the coarse-grained
behavior of a wealth of classical stochastic models. Surprisingly, it was
recently conjectured to also describe spin transport in the one-dimensional
quantum Heisenberg model. We test this conjecture by experimentally probing
transport in a cold-atom quantum simulator via the relaxation of domain walls
in spin chains of up to 50 spins. We find that domain-wall relaxation is indeed
governed by the KPZ dynamical exponent $z = 3/2$, and that the occurrence of
KPZ scaling requires both integrability and a non-abelian SU(2) symmetry.
Finally, we leverage the single-spin-sensitive detection enabled by the
quantum-gas microscope to measure a novel observable based on spin-transport
statistics, which yields a clear signature of the non-linearity that is a
hallmark of KPZ universality.
