Quantum algorithms for QED computations related to the fine structure constants
The general belief seems to be that the expansion in $\alpha$ is an asymptotic series but not a convergent series. The handwaving estimate is that in $\sum_k c_k \alpha^k$, the scaling for the coefficients is roughly $c_k \sim k!$. So, since $\alpha \simeq 1/137$ the terms will start to get bigger rather than smaller for $k$ larger than around 137. (I assume there is serious literature on this topic, but I'm not too familiar with it. The above is what high energy physicists have told me in casual conversation.)

The fine structure constant itself is not something for which anyone knows a purely theoretical formula. So getting more digits of $\alpha$ is a fundamentally different problem from getting more digits of $\pi$. That being said, the challenges are both experimental and computational. Various experiments in particle accelerators and atomic physics laboratories are dedicated to making progressively more precise measurements of fundamental constants such as $\alpha$. Often, the high-precision theoretical calculation of the factors relating the experimentally observed quantities (such as scattering probabilities or spectral lines) to fundamental constants of interest such as $\alpha$ is very difficult and computationally heavy. The computational side can be just as much a limiting factor as the experimental side in these precision metrology problems. (Some of my coworkers at NIST specialize in this sort of thing.)

In the quantum algorithm that Keith and John and I developed, we do not use a perturbative expansion in powers of the coupling constant. The simulation algorithm is much more directly analogous to an actual experiment. However, one advantage over experiment is that in a simulation we are free to change $\alpha$ to whatever value we want. So, by computing scattering amplitudes at varying $\alpha$ it might be easier to determine the individual coefficients $c_k$ than it is in the real world. However, the study of quantum algorithms for simulating quantum field theories is in its infancy. The extraction of such coefficients is one of the numerous interesting questions that have not really been explored yet! Also, our algorithms don't yet tackle QED but rather some simplified models.

Today, we primarily have two classical algorithms for QFT: Feynman diagrams and lattice simulations. Feynman diagrams break down at strong coupling or high precision, as discussed above. Lattice calculations are mostly only good for computing static quantities such as binding energies (e.g. the mass of the proton), rather than dynamic quantities such as scattering amplitudes. This is because the lattice calculations use imaginary time. (Also, for certain condensed matter systems that are highly frustrated, even finding static quantities such as ground state energies is exponentially hard. It is not clear to me to what extent this phenomenon is relevant to high energy physics.) There is also a current research program on speeding up the computation of scattering amplitudes in supersymmetric quantum field theories. You may have heard about the "amplituhedron", which is part of this. This seems kind of exciting, but it's recent stuff and I don't know too much about it. 

So, there is room for exponential speedup by quantum computation in the case that you wish to compute dynamical quantities such as scattering amplitudes with high precision or in a strongly coupled quantum field theory. My papers with Keith and John work out polynomial-time quantum algorithms for computing scattering amplitudes in simple quantum field theories which can be strongly coupled. We would like to extend our algorithms to simulate more complete models such as QED and QCD but we are not there yet. Doing so involves nontrivial challenges, but my feeling is that quantum computers should be able to compute scattering amplitudes in quantum field theories in polynomial time quite generally.

So, that's the perspective based on known classical and quantum algorithms. There is also a perspective from complexity theory. For many classes of physical systems, the problem of computing transition amplitudes to polynomial precision is BQP-complete, and the problem of computing ground energies is QMA-complete. So, for worst case instances we expect quantum computers to compute transition amplitudes in polynomial time, whereas classical computers require exponential time. We expect both quantum and classical computers (as well as nature itself) to require exponential time to find ground states in the worst case. The question is whether the worst case instances of the computational problems look like any real physics. In the context of condensed matter physics, the answer is basically yes, I would say. In the context of high energy physics, one can construct BQP-hard instances of the scattering amplitude problem that correspond at least loosely to something a physicist might need to calculate. (We are currently working on a paper about this.) Whether one can construct QMA-hard instances of the problem of computing a vacuum state for a quantum field theory is something that I have not really thought about. However, I think this could be done if one is willing to allow non-translationally-invariant external fields.