My question is about quantum algorithms for QED (quantum electrodynamics) computations related to the fine structure constants. Such computations (as explained to me) amounts to computing Taylor-like series $$\sum c_k\alpha^k,$$ where $\alpha$ is the fine structure constant (around 1/137) and $c_k$ is the contribution of Feynman diagrams with $k$-loops.
This question was motivated by Peter Shor's comment (about QED and the fine structure constant) in a discussion regarding quantum computers on my blog. For some background here is a relevant Wikipedia article.
It is known that the first few terms of this computation gives very accurate estimations for relations between experimental outcomes which are with excellent agreement with experiments. (Perhaps even the best agreement between theory and experiments in the history of physics.)
However, the precision of these computations is limited by three important (and related) factors
a) Computational - the computations are very heavy and computing more terms is beyond our computational powers.
b) Mathematical-physical - at some point the computation will explode - in other words, the radius of convergence of this power series is 0.
c) Physical - The precision of the computation is limited because it does not take into account other forces and fields
My questions in short is: How much better results could we expect from these QED computations had we have an unlimited computation power.
In more details:
Question 1: With unlimited computational power what is the expected precision we can get taking only into the blow up of the coefficients. Namely, are there estimations for how many terms in the expansion before we witness explosion and what is the quality of the approximation we can expect when we use these terms.
Update: As Vladimir noted in a comment (and in fact also Steven wrote) it is believed that the radius of convergence is zero but there is no full argument that this is the case.
Steven Jordan in an answer to a related question (see below) mentioned a very rough heuristic explanation that $c_k$ behaves like $k!$ and that therefore the explosion of the coefficients will not occur until $k! 1/137^k$ starts increasing. This suggest that we can have 137 or so meaningful terms. (If $c_k$ accounts to $k!$ terms with cancellation perhaps we can replace $k!$ by its square root.)
A second related question is:
Question 2: With unlimited computational power what is the expected precision we can get when we take into account the effect of other fields not accounted for by the QED.
I am also interested to know if there are efficient quantum algorithms to compute this expansion . The paper: Stephen Jordan, Keith Lee, and John Preskill, Quantum Algorithms for Quantum Field Theories, may lead to efficient quantum algorithms for at least some versions of these computations. I asked this question on the sister theoretical computer science site to which Stephen Jordan gave an excellent answer.
The same question can be asked about QCD computations for properties of the proton or neutron. For examples, computation for the mass of the proton.
Question 3: Can we estimate for QCD computations for the mass of the proton what will be level of precision that can be achieved assuming we had unlimited computation power, and how it is compared to current precision.