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I am new to the concept of topological quantum computation (TQC).

Recently I have been thinking about simulating a quantum computer on a classical computer. I know that if I use merely the unitary matrices, the process will be very slow due to the $2^n$ factor, as is pointed out by many SE answers, such as this to name one.

My question is that what if I don't simulate quantum computers that way? What if I simulate a TQC, by (for example) introducing defects on a 2D topological insulators, and do some "simulated" manipulations on such defects? Will that be faster than the brute-force matrix-multiplication method?

I suspect that in this way, all the calculations will be reasonably small since the simulation should scale linearly with more "qubits", which is just an ab-initio calculation.

Part of me thinks that simulating a TQC will be better, while part of me thinks that this can't be right, TQC should be equivalent to QC.

Is there anything I am missing? Or simulating a TQC is indeed faster and better? And if so, why is quantum supremacy has not been achieved yet?

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  • $\begingroup$ Related: How does topological quantum computing differ from other models of quantum computing? $\endgroup$
    – Sanchayan Dutta
    Commented May 1, 2019 at 14:53
  • $\begingroup$ Simulating a system that supports fibonacci anyons requires tracking a hilbert space with dimension exponential in the number of anyons. This space is known as the fusion space of the anyons. $\endgroup$
    – Simon Burton
    Commented May 3, 2019 at 11:05
  • $\begingroup$ @ Simon Burton I am proposing the TQC not by tracking the Hilbert space, but by performing ab-initio calculations like in real world. I suppose in that way, I don't need to keep track of the Hilbert space? $\endgroup$
    – wooohooo
    Commented May 4, 2019 at 1:16
  • $\begingroup$ "the simulation should scale linearly with more [defects]" Why do you think that it scales linearly? $\endgroup$
    – Craig Gidney
    Commented Jun 29, 2021 at 21:38

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In the paper Simulation of topological field theories by quantum computers by Freedman, Larsen, and Wang they prove that "TQFTs cannot be used to define a model of computation stronger than the usual quantum model BQP". So to answer your question, no, anyonic topological quantum computation in general is not inherently more powerful than general quantum computing. You should think of TQC as a special way to cook up quantum gates (unitary matrices) by braiding or measuring anyons and/or defects.

Now it is entirely possible that certain algorithms might "compile faster" with respective to a specific gate set arising from a given TQC model, but that is a different question...

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