8.82: The Discrete or Quantum Fourier Transform

The continuous-variable Fourier transforms involving position and momentum are well known. In Dirac notation (see chapter 6 in A Modern Approach to Quantum Mechanics by John S. Townsend) they are,

\[ \begin{matrix} \langle \ | \Psi \rangle = \int \langle \ |x \rangle \langle x | \Psi \rangle dx & \text{and} & \langle x | \Psi \rangle = \int \langle x | p \rangle \langle \ | \Psi \rangle dp \end{matrix} \nonumber \]

where

\[ \langle x | p \rangle = \langle p | x \rangle * = \frac{1}{ \sqrt{2 \pi \hbar}} \text{exp} \left( i \frac{2 \pi px}{h} \right) = \frac{1}{ \sqrt{2 \pi \hbar}} \text{exp} \left( i \frac{px}{ \hbar} \right) \nonumber \]

Using the coordinate and momentum completeness relations

\[ \begin{matrix} \int |x \rangle \langle x | dx = 1 & \text{and} & \int |p \rangle \langle p |dp = 1 \end{matrix} \nonumber \]

we can write the following generic Fourier transforms.

\[ \begin{matrix} \langle p | = \int \langle p | x \rangle \langle x | dx & \text{and} & \langle x | = \int \langle x | p \langle p | dp \end{matrix} \nonumber \]

By analogy a discrete Fourier transform between the k and j indices can be created.

\[ \langle k | = \sum_{j-0}^{N-1} \langle k | j \rangle \langle j | \nonumber \]

were, again, by analogy

\[ \langle k | j \rangle = \frac{1}{ \sqrt{N}} \text{exp} \left( i \frac{2 \pi}{N}kj \right) \nonumber \]

so that

\[ \langle k | = \frac{1}{ \sqrt{N}} \sum_{j=0}^{N-1} \text{exp} \left( i \frac{2 \pi}{N} k j \right) \langle j | \nonumber \]

Summing over the k index and projecting on to |Ψ> yields a system of linear equations.

\[ \sum_{k=0}^{N-1} \langle k | \Psi \rangle = \frac{1}{ \sqrt{N}} \sum_{k=0}^{N-1} \sum_{j=0}^{N-1} \text{exp} \left( i \frac{2 \pi}{N} kj \right) \langle j | \Psi \rangle \nonumber \]

Like all systems it is expressible in matrix form. For example, with N=2 and \( \begin{pmatrix} 1 \\ 0 \end{pmatrix}\) as the operand we have,

\[ \frac{1}{ \sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \frac{1}{ \sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} \nonumber \]

Here the matrix operator is the well-known Hadamard transform. In this case it transforms spin-up in the z-direction to spin-up in the x-direction, or horizontal polarization to diagonal polarization, etc. Naturally it transforms spin-up in the x-direction to spin-up in the z-direction.

\[ \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \frac{1}{ \sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} \frac{1}{ \sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix} \nonumber \]

This, of course, also occurs with the continuous-variable Fourier transform.

\[ \langle x | \Psi \rangle \xrightarrow{FT} \langle p | \Psi \rangle \xrightarrow \langle x | \Psi \rangle \nonumber \]

The Mathcad implementation of the discrete or quantum Fourier transform (QFT) is now demonstrated.

\[ \begin{matrix} N = 2 & m = 0 .. N-1 & n = 0 .. N - 1 & QFT_{m,~n} = \frac{1}{ \sqrt{N}} \text{exp} \left( i \frac{2 \pi m n}{N} \right) \end{matrix} \nonumber \]

\[ QFT = \begin{pmatrix} 0.707 & 0.707 \\ 0.707 & -0.707 \end{pmatrix} \nonumber \]

\[ \begin{matrix} QFT \begin{pmatrix} 1\\ 0 \end{pmatrix} = \begin{pmatrix} 0.707 \\ 0.707 \end{pmatrix} & QFT \frac{1}{ \sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \\ QFT \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 0.707 \\ -0.707 \end{pmatrix} & QFT \frac{1}{ \sqrt{2}} \begin{pmatrix} 1 \\ -1 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \end{matrix} \nonumber \]

These calculations demonstrate that the QFT is a unitary operator:

\[ \text{QFT QFT} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \nonumber \]