8.82: The Discrete or Quantum Fourier Transform
- Page ID
- 149085
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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:
\[ \text{QFT QFT} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \nonumber \]