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?