KPZ is everywhere

We construct a family of chaotic brickwork quantum circuits exhibiting superdiffusive transport with dynamical exponent z = 3/2 in the full Kardar-Parisi-Zhang (KPZ) universality class. This is enabled by the equilibrium current associated to the inhomogeneous conserved density having a nontrivial dependence on the chemical potential. Using nonlinear fluctuating hydrodynamics, we derive the propagation velocity, broadening scale, and universal scaling form of the charge-correlation function from microscopic equilibrium properties. Tensor-network simulations of a two-site qutrit circuit reproduce these parameter-free predictions, including the full stationary KPZ profile. We further construct a three-site qubit circuit, where transport varies from diffusive to superdiffusive with chemical potential, while long-lived coherent quasiparticles delay convergence of the profile. Our results establish that KPZ-like superdiffusion in quantum systems is not restricted to fine-tuned integrable systems.