Quantum computers are designed based on quantum mechanics principle, they are most suitable to solve the Schrodinger equation, and linear PDEs (and ODEs) evolved by unitary operators.  Nonlinear PDEs are difficult due to the no-cloning theorem.
In this talk we will show how to construct quantum algorithms for (nonlinear) Hamiton-Jacobi equations. Such equations develop singularities (caustics) in finite time even if the initial data are smooth. Beyond the caustics two possible solutions could be introduced: multivalued solutions which arise in geometric optics, high frequence or semiclassical limits of linear waves, etc, and viscosity solutions which arise in optimal control, level set methods for front propagations, etc.  We will show how quantum algorithms can be developed for both solutions. 
We will also introduce a quantum scientific computing platform—“UnitaryLab”, which is a software for quantum algorithms for scientific computing problems. The version 1.0 aims at solving ordinary and partial differential equations by quantum algorithms.