# QCD and QED with unlimited computational power - how precise are they going to be?

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.

• I don't know enough Quantum Mechanics to give you a real answer, but as someone who lives with simulations every day, my comment would be: "All models are wrong; some are useful." – Cort Ammon Dec 25 '14 at 18:41
• It is widely known that "the radius of convergence of this power series is 0", but, in my humble opinion, it is a wrong statement. In fact, nobody uses these series directly because of the infrared divergence. IR divergence is avoided with summing up a part of the series into a finite function of $\alpha$, thus, the convergence of the remaining series has not been studied yet! This finite function (a sum of IR diagrams) is a function, which is not expanded in series, but used directly. It means using another initial approximation for, say, a scattering amplitude. (To be continued.) – Vladimir Kalitvianski Dec 25 '14 at 18:46
• Continued: The Dyson's argument does not apply since we are not obliged expand what we can have exactly in a compact and finite form. – Vladimir Kalitvianski Dec 25 '14 at 18:47
• At some point it becomes useless to improve the evaluate QED on its own because the contributions from the other sectors of the standard model will be larger than the improvement from the calculation. I think that's already the case for chiral contributions from the electroweak force. – CuriousOne Dec 25 '14 at 20:25
• Also, lattice QCD is a program to non-perturtabatively calculate out QCD phenomenon, without resorting to this series. – Jerry Schirmer Feb 12 '15 at 14:02

Part b) is a big mathematical physics topic in its own right. The divergent tail of an asymptotic series is not garbage, rather it contains a lot of information that together with some additional information can be used to compute non-perturbative effects. A general introduction to this topic is given here.

There are different approaches possible, some require that many terms are known. In practice this isn't very useful for field theory as only a few terms are usually known, but here the question is about unlimited computational power, and then these techniques are useful. E.g. One can the consider resumming the series using differential approximants. This has been used to yield accurate values for critical exponents, but it typically requires dozens of terms of a (divergent) series expansion.

In QCD, computational power is the principal limiting factor. For example, experimental measurements of the proton mass are about a million times more precise that the best available first principles calculation of the same thing from QCD.

If we had more computational power, we could use a variety of quite precise measurements of hadron properties to get a very precise determination of fundamental constants like the strong force coupling constant and quark masses, and these precise constant measurements, in turn, would make everything else more precise to levels comparable to QED.

Consider a problem that might take an hour or two to solve with a calculator in QED to exquisite precision (pretty much only limited for practical purposes by the available precision of our measurements of the physical constants that we plug into it). Doing calculations for a comparable problem in QCD with many more loops and literally years of calculating on reasonably powerful computers is still going to leave you with far less precision than the QED calculation.

In principle, the level of precision you could get in QCD with unlimited computational power would be about as precise as the precision of the most accurate proton and neutron mass measurements you could manage (currently about nine significant digits, rather than three of theoretical calculations today). You can get as precise as experimental data to calibrate your results can get as long as you have the computational power to do it. It would be a two step process, because right now, limited precision in QCD calculations implies imprecisely known physical constants which fundamentally limit the precision of any calculation. So, first you'd have to use your unlimited computational power to determine the relevant constants more precisely and then you'd have to do the calculations with the refined constants.

The difference, of course, has a lot to do with how quickly the loops converge. Five loops would be overkill for most applied QED calculations giving you more precision than you can measure. But, five loops in QCD will get you two or three significant digit accuracy. It would probably take dozens of QCD loops to get calculations a precise as a three or four loop QED calculation and might take decades with current computational resources and techniques to calculate in QCD.

Now lurking out there is the question of whether there is some profoundly more efficient way to calculate amplitudes which developments like the amplitudehedron (and lots of other suggestive evidence) tends to imply that there is if we could just figure out better ways to ignore mutually canceling terms and terms that do not contribute significantly to the final result. Cleverness can make up for a huge amount of computational brute force. But, so far, we haven't found any Holy Grails that make the calculation more tractable.

On the other hand, QCD calculations tend to be interchangeable parts that recur frequently. You only need to do calculations for any given problem once and often those calculations can have significant modularity (e.g. a particular decay branching fraction may come up half a dozen times in calculating all of the possible decay modes of an energetic jet, but it only has to be calculated once).