# Dirac Equation computational complexity

## Main Question or Discussion Point

How fast does the computational complexity of the Dirac equation, with regards to full* solution, grow with number of particles N? can we specify the order of time t(N) for this solution in terms of t(N=1)?

(I assume that number of protons, neutrons and electrons combined is N - i.e. that complexity grows similarly with particles of every kind.)

*Full solution meaning a numerical solution reached without making any approximations or neglecting any terms (beyond those neglected even for cases we can solve "exactly" like that of a hydrogen atom)

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An exact solution (or, at least, one that can be computed to arbitrarily good precision) means, in effect, simulating the time evolution of the system. With the best known algorithms for classical simulations of quantum systems, simulation complexity of arbitrary composite systems grows exponentially in the number of subsystems. It's not possible to do better than that in full generality. That's pretty much the main impetus behind quantum computing. A quantum computer can simulate other quantum systems (including, it seems, particles in quantum field theory—this a current research project by Preskill and others) in polynomial time.

Certain special cases can have efficient classical solutions, of course. In this case, it depends on what you mean by solutions to the Dirac equation. If you mean the equation for free fields, then, well, the solution is basically constant time. Once you've solved it, you can apply it to as many fermions as you like. But if you mean an interacting Dirac equation—mediated by, say, a Yukawa interaction or the EM field—then you're most likely looking at exponential classical complexity. If you can do better, you can probably become quite famous...

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An exact solution (or, at least, one that can be computed to arbitrarily good precision) means, in effect, simulating the time evolution of the system. With the best known algorithms for classical simulations of quantum systems, simulation complexity of arbitrary composite systems grows exponentially in the number of subsystems. It's not possible to do better than that in full generality.
Ah, so t(N) = t(N=1)*e^(kN) in full generality for some constant k, is the best we can guarantee?

Let us take the case of atomic/molecular systems, can we predict k for that case?

But if you mean an interacting Dirac equation—mediated by, say, a Yukawa interaction or the EM field—then you're most likely looking at exponential classical complexity. If you can do better, you can probably become quite famous...
Yes. We could consider simple chemical systems for now.