  The finite element method is used to approximately solve boundary value
problems for differential equations. The method discretises the parameter space
and finds an approximate solution by solving a large system of linear
equations. Here we investigate the extent to which the finite element method
can be accelerated using an efficient quantum algorithm for solving linear
equations. We consider the representative general question of approximately
computing a linear functional of the solution to a boundary value problem, and
compare the quantum algorithm's theoretical performance with that of a standard
classical algorithm -- the conjugate gradient method. Prior work had claimed
that the quantum algorithm could be exponentially faster, but did not determine
the overall classical and quantum runtimes required to achieve a predetermined
solution accuracy. Taking this into account, we find that the quantum algorithm
can achieve a polynomial speedup, the extent of which grows with the dimension
of the partial differential equation. In addition, we give evidence that no
improvement of the quantum algorithm could lead to a super-polynomial speedup
when the dimension is fixed and the solution satisfies certain smoothness
properties.
