Large Fourier transforms never exactly realized by braiding conformal blocks

TL;DR

This paper proves that large Fourier transforms cannot be exactly implemented using braiding conformal blocks in topological quantum computing, implying the necessity of approximation methods.

Contribution

It demonstrates the fundamental limitation of braiding conformal blocks in realizing exact large Fourier transforms in topological quantum computing.

Findings

Large Fourier transforms cannot be exactly realized by braiding conformal blocks.

Approximation is necessary for implementing Fourier transforms in this model.

The result applies to the Fibonacci (2+1)-topological quantum field theory.

Abstract

Fourier transform is an essential ingredient in Shor's factoring algorithm. In the standard quantum circuit model with the gate set $\{\U(2), \textrm{CNOT}\}$, the discrete Fourier transforms $F_N=(\omega^{ij})_{N\times N},i,j=0,1,..., N-1, \omega=e^{\frac{2\pi i}{N}}$, can be realized exactly by quantum circuits of size $O(n^2), n=\textrm{log}N$, and so can the discrete sine/cosine transforms. In topological quantum computing, the simplest universal topological quantum computer is based on the Fibonacci (2+1)-topological quantum field theory (TQFT), where the standard quantum circuits are replaced by unitary transformations realized by braiding conformal blocks. We report here that the large Fourier transforms $F_N$ and the discrete sine/cosine transforms can never be realized exactly by braiding conformal blocks for a fixed TQFT. It follows that approximation is unavoidable to implement the Fourier transforms by braiding conformal blocks.