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In a lecture, recorded on Youtube, Gil Kalai presents a 'deduction' for why topological quantum computers will not work. The interesting part is that he claims this is a stronger argument than the argument against fault tolerant computing in general.

If I understand his argument correctly, he states that

  1. A (hypothetical) quantum computer without quantum error correction can simulate the system of anyons representing the qubit in a topological quantum computer.

  2. Therefore, any quantum computer based on these anyons must have at least as much noise as a quantum computer without quantum error correction. As we know that our noisy quantum computer is insufficient for universal quantum computation, topological quantum computers based on anyons cannot provide universal quantum computation either.

I think step 2 is sound, but I have some doubts on step 1 and why it implies 2. In particular:

  • Why can a quantum computer without error correction simulate the system of anyons?
  • If it can simulate the system of anyons, is it possible that it can only do so with low probability and hence cannot simulate the topological quantum computer with the same fault tolerance as the system of anyons?
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A topological quantum computer could be made by using an exotic phase of matter in which anyons arise as localized effects (such as quasiparticles or defects). In this case, errors typically cost energy, and so the probability is suppressed for small temperatures (though it will never be zero).

A topological quantum computer could also be made (or one could also say simulated) by a standard gate model quantum computer, such as one based on qubits.

In either case, we are using a noisy medium to engineer a system of anyons. And so we will get a noisy system of anyons. The effects of the noise will cause our anyons to wander around, as well as causing pair creations of additional anyons, etc. If these effects are not accounted for, it will cause errors in any topological quantum computation that we intend to do. So in this sense, his arguments are correct.

The important point to note, therefore, is that we must not fail to account for the errors. We must look at the system, keep track of where all anyons are, try to identify which ones we are using, and identify how to clear away the ones that have been created in error. This means that we must do error correction within the topological quantum computer.

The promise of TQC is mainly that there should be ways to engineer topological phases that will have less noise. They should therefore require less error correction. But they will definitely need some.

For a gate model quantum computer simulating a topological quantum computer, the benefits are that topological error correction is quite straightforward and has high thresholds. The surface codes are examples of this. But we don't usually think of this as a gate model QC simulating a topological QC. We just think of it as a good example of a quantum error correcting code.

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  • $\begingroup$ So, you mean that not all topological quantum computers (in particular, the "ways to engineer topological phases that will have less noise"?) can be simulated by noisy quantum computers? And that therefore the answer to my first question is 'it cannot always do so'? $\endgroup$
    – Discrete lizard
    Apr 16, 2018 at 11:55
  • $\begingroup$ @Discretelizard Any noisy quantum computer can simulate a TQC (assuming they aren't too noisy). But if the TQC implements error correction (as it should) we don't usually think of it as a 'simulation'. We usually just think of it as a particular kind of (topological) error correcting protocol that we can implement. I made some edits to make this a bit clearer. $\endgroup$
    – James Wootton
    Apr 16, 2018 at 12:09
  • $\begingroup$ Since we can consider the 'simulation' as a form of quantum error correction, this argument reduces to Kalai's arguments against fault tolerant computing in general. So, it seem that Kalai's claim that this argument is stronger than his general argument is false. $\endgroup$
    – Discrete lizard
    Apr 16, 2018 at 14:15
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    $\begingroup$ The idea that no error correction is required for TQC was a common misconception when this video was published. So there was a need for this argument to be made, and it was a very strong claim. But for fully implemented TQC, he'll have to rely on his other (less strong) arguments. $\endgroup$
    – James Wootton
    Apr 17, 2018 at 8:01

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