  We construct quantum algorithms to compute physical observables of nonlinear
PDEs with M initial data. Based on an exact mapping between nonlinear and
linear PDEs using the level set method, these new quantum algorithms for
nonlinear Hamilton-Jacobi and scalar hyperbolic PDEs can be performed with a
computational cost that is independent of M, for arbitrary nonlinearity.
Depending on the details of the initial data, it can also display up to
exponential advantage in both the dimension of the PDE and the error in
computing its observables. For general nonlinear PDEs, quantum advantage with
respect to M is possible in the large M limit.
