  Recent advancements of intermediate-scale quantum processors have triggered
tremendous interest in the exploration of practical quantum advantage. The
simulation of fluid dynamics, a highly challenging problem in classical physics
but vital for practical applications, emerges as a good candidate for showing
quantum utility. Here, we report an experiment on the digital simulation of
unsteady flows, which consists of quantum encoding, evolution, and detection of
flow states, with a superconducting quantum processor. The quantum algorithm is
based on the Hamiltonian simulation using the hydrodynamic formulation of the
Schr\"odinger equation. With the median fidelities of 99.97% and 99.67% for
parallel single- and two-qubit gates respectively, we simulate the dynamics of
a two-dimensional (2D) compressible diverging flow and a 2D decaying vortex
with ten qubits. The experimental results well capture the temporal evolution
of averaged density and momentum profiles, and qualitatively reproduce spatial
flow fields with moderate noises. This work demonstrates the potential of
quantum computing in simulating more complex flows, such as turbulence, for
practical applications.
