This is for me an extremely pressing question, as the limits of analytic techniques often have little to do with the limits of nature. There are several ways in which computers have contributed to string theory already, but they are not the type of things I care about. For instance, computer algebra was used (in the 1970s!) to derive the precise form of the 11d SUGRA action, but this is just a computer working like a human physicist, working algebraically, it isn't really using the power of the machine to simulate analytically intractable situations.
The main obstacle to simulations in systems which represent string theory as it is understood today is that there is no universal good method to put arbitrary supersymmetric theories on a lattice for simulation. You can do it in artificial ways, by putting the theories on a lattice without any supersymmetry, and fine tuning parameters, but this is in practice hopeless, as the supersymmetric points are very finely tuned in this point of view, and you will never extract anything from the simulation. There exceptions are the cases where you know the Nicolai map in an explicit local way. Simon Catterall has been working very diligently to expand the domain of this limited set of examples as far as possible, and perhaps there has been some breakthrough from Syracuse recently, I haven't checked in a while. This group is simulating supersymmetric theories, and there are other groups in Japan.
The Nicolai map is a stochastic system whose Parisi-Sourlas Supersymmetry implies a normal relativistic supersymmetry of the solutions of the stochastic equation. There are really only two meaty examples, unfortunately. Fortunately, one of them includes all of flat-space M-theory!
The two examples are 1-d SUSY-QM, which is in reality the stochastic equation:
$$ \dot{x} - V' = \eta $$
Which, when you formulate as a path integral, produces a determinant which reproduces the SUSY-QM Fermionic content. This gives the small research fields of shape-invariance (or Schrodinger generalized raising and lowering operators). You can generalize this to arbitrary 1-d systems with enough supersymmetry, and the BFSS model has a ton of supersymmetry, so you're in luck. This means we can right now simulate M-theory efficiently on a computer, using stochastic equations for BFSS matrices.
This is not solving all the world's problems, because BFSS is the most notoriously hard to understand version of AdS/CFT, since the entire space is reconstructed from the dynamics of the branes in large N limit, and the limits are hard, and the physics is opaque. Still, people have made simulations of this in recent years, although what results they got, I don't know. Even the classical solutions to BFSS are not understood in any way, nor is it even obvious what the classical solutions even mean physically (at least not to me).
The other case where the stochastic formulation is know is 2-d N=2,2 Wess-Zumino model, which is defined by the stochastic equations:
$$\partial_x \phi_1 + \partial_y \phi_2 + V_r = \eta_1$$
$$\partial_y\phi_1 - \partial_x\phi_2 + V_i = \eta_2 $$
Where (V_r + iV_i) together are the real and imaginary components of a holomorphic function of $\phi_1 + i \phi_2$.
These two examples have been the only real examples for going on 30 years now. It is clear simply from the form of supersymmetric Lagrangians that it should be possible to this trick in general somehow, for every single supersymmetric field theory. Once you do it, the supersymmetric theory is actually easier to simulate than the non-supersymmetric one. As a warning, I should tell you that banging my head on this (self-imposed) problem probably cost me a PhD. But it only takes one idea for a breakthrough, and it is going to come eventually, although I hope not in 100 years.
Once we know how to do stochastic formulation of some kind for general SUSY theories, then one can simulate AdS/CFT cases which are higher dimensional, where the physical applications are more direct, like N=4 SUSY gauge theory. From this, it is easier to answer the questions one is interested in. The stochastic formulation is the sole obstacle.
There are potential applications of computers in the "mechanical physicist" sense, to compute properties of OPE's and such things, but I think you are interested in direct simulation, because that's what I was interested in.
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