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Variational algorithms use parameterized circuits to encode the potential solutions. This brings a whole new world of issues on the table.

Most of the quantum PDE solvers encode the solution in the quantum state amplitudes. This encoding poses a serious limitation on what data can be retrieved from the quantum computer.

For time-dependent PDEs, another boundary condition has to be provided: the initial condition. This leads to another data input problem, that might be hard to solve.

The domain of definition of a PDE has bounds. A value need to be attached to each point on the boundary. This can only be done via data encoding, which might be problematic for a quantum computer.

Partial Differential Equations (PDEs) are solved on a specific domain, called the domain of definition. But transmitting the limits of this domain to the quantum computers might be a limiting step in solving PDEs.