  The simulation of quantum dynamics calls for quantum algorithms working in
first quantized grid encodings. Here, we propose a variational quantum
algorithm for performing quantum dynamics in first quantization. In addition to
the usual reduction in circuit depth conferred by variational approaches, this
algorithm also enjoys several advantages compared to previously proposed ones.
For instance, variational approaches suffer from the need for a large number of
measurements. However, the grid encoding of first quantized Hamiltonians only
requires measuring in position and momentum bases, irrespective of the system
size. Their combination with variational approaches is therefore particularly
attractive. Moreover, heuristic variational forms can be employed to overcome
the limitation of the hard decomposition of Trotterized first quantized
Hamiltonians into quantum gates. We apply this quantum algorithm to the
dynamics of several systems in one and two dimensions. Our simulations exhibit
the previously observed numerical instabilities of variational time propagation
approaches. We show how they can be significantly attenuated through subspace
diagonalization at a cost of an additional $\mathcal{O}(MN^2)$ 2-qubit gates
where $M$ is the number of dimensions and $N^M$ is the total number of grid
points.
