  A new model of nonlinear charged quantum relativistic fluids is presented.
This model can be discretized into Discrete Time Quantum Walks (DTQWs), and a
new hybrid (quantum-classical) algorithm for implementing these walks on NISQ
devices is proposed. High resolution (up to $N=2^{17}$ grid points) hybrid
numerical simulations of relativistic and non-relativistic hydrodynamical
shocks on current IBM NISQs are performed with this algorithm and shown to
reproduce equivalent simulations on classical computers. This work demonstrates
that nonlinear fluid dynamics can be simulated on NISQs, and opens the door to
simulating other, quantum and non-quantum fluids, including plasmas, with more
general quantum walks and quantum automata.
