We study the quantum evolution of a non-Hermitian qubit realized as a submanifold of a dissipative superconducting transmon circuit. Real-time tuning of the system parameters to encircle an exceptional point results in nonreciprocal quantum state transfer. We further observe chiral geometric phases accumulated under state transport, verifying the quantum coherent nature of the evolution in the complex energy landscape and distinguishing between coherent and incoherent effects associated with exceptional point encircling. Our work demonstrates an entirely new method for control over quantum state vectors, highlighting new facets of quantum bath engineering enabled through dynamical non-Hermitian control.