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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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  • $\begingroup$ 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 May 3 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? $\endgroup$ – water liu May 4 at 1:16

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