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As a non specialist, for a single particle system it's easy to appreciate the concept of a pilot wave extending through all Euclidean space, guiding a particle which ends up at a location determined by the pilot wave and its initial location.

For multiple particles however the wave would presumably need more dimensions to reflect the configuration space of the system.

Is this correct, and if so where does the pilot wave reside?

A related question may be, if quantum computers give an exponential speedup for factorization, then according to pilot wave theory where does the computation take place?

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The "pilot wave" is just the usual wave function of quantum mechanics. If you have $N$ spinless particles, it is a map: $\psi: \mathbb{R}^{3N} \to \mathbb{C}$. This means it lives in the 3N-dimensional configuration space of the particles. I should emphasize that this is not a specialty of pilot wave theory but just the usual framework of quantum mechanics.

The special thing in pilot wave theory are the additional actual particles that actually have positions in $\mathbb{R}^3$ and thus link the abstract object $\psi$ with objects that can be thought of as moving in our physical world, as the objects that tables, chairs and people are made of.

I don't totally understand the intent of your last question, but let's try: The obvious answer is that if a computer computes something, this computation takes place inside of the computer. You know, configuration space is not really a "place" in any reasonable sense. It's an abstract mathematical way of describing the things that happen when, e.g. in a quantum computer, particles move back and forth.

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  • $\begingroup$ To elaborate on the last bit. A proponent of MWI would argue that a quantum computation takes place in other universes (which occasionally interfere with ours, allowing us to gain some knowledge of results). So presumably pilot wave proponents say the computation takes place in the pilot wave (which is exponentially faster for certain tasks as it lives in $\mathbb{R}^{3N}$). Somehow the latter sounds less plausible as an answer for where the computation took place, as it doesn't have a physical location. $\endgroup$
    – Sideshow Bob
    Commented Jan 11, 2018 at 15:11
  • $\begingroup$ @SideshowBob: Since I am a proponent of the pilot wave theory, I speak for myself when I say that it doesn't make sense to ask "where" with respect to the wave function. (The wave function is more of "how" because it determines how the particles in our 3d space move). One should try to avoid saying too weird things only because it's about quantum mechanics. $\endgroup$
    – Luke
    Commented Jan 11, 2018 at 17:27

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