Quantum Computing for Computer Scientists (9781139634120) by Yanofsky Noson S.; Mannucci Mirco A

Author:Yanofsky, Noson S.; Mannucci, Mirco A.
Language: eng
Format: epub
Publisher: Cambridge Univ Pr

Figure 6.7. The action of DFT†.

[P(ω0), P(ω1), P(ω2),…, P(ωk),…, P(ωM−1)]T is the vector of the values of the polynomial at the powers of the Mth root of unity.

Let us define the discrete Fourier transform, denoted DFT, as

Formally, DFT is defined as

It is easy to see that DFT is a unitary matrix: the adjoint of this matrix, DFT†, is formally defined as

To show that DFT is unitary, let us multiply

If k = j, i.e., if we are along the diagonal, this becomes

If k≠ j, i.e., if we are off the diagonal, then we get a geometric progression which sums to 0. And so DFT * DFT† = I.

What task does DFT† perform? Our text will not get into the nitty-gritty of this important operation, but we shall try to give an intuition of what is going on. Let us forget about the normalization for a moment and think about this intuitively. The matrix DFT acts on polynomials by evaluating them on different equally spaced points of the circle. The outcomes of those evaluations will necessarily have periodicity because the points go around and around the circle. So multiplying a column vector with DFT takes a sequence and outputs a periodic sequence. If we start with a periodic column vector, then the DFT will transform the periodicity. Similarly, the inverse of the Fourier transform, DFT†, will also change the periodicity. Suffice it to say that the DFT† does two tasks as shown in Figure 6.7:

It modifies the period from r to

It eliminates the offset.

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