Various Quantum Transforms

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Various Quantum Transforms

• Zhaosheng Bao Liang Jiang Chenyang Wang Lisa
  Wang Zhipeng Zhang

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Contents

• 1. Quantum Fourier Transform
• 2. Wavelet Transform
• 3. Quantum Wavelet Transform
• 4. Ridgelet Transform
• 5. Quantum Ridgelet Transform(not done)

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Quantum Fourier Transform
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Continuous Fourier Transform
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Discrete Fourier Transform
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Inverse Descrete Fourier Transform
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Quantum Fourier Transform
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Hadamard Gate
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Rk gate
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Control-Rk gate
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Quantum Circuit for Quantum Fourier Transform
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Wavelet Transform

• Haar Transform

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General Transform

• Project a function f to a set of basis vi

• Different transforms use to different sets of
  basis.
• A special set of basis the Haar Basis to
  represent all the functions f(x), x is in 0,1

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Haar Basis
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Haar Transform
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Haar Transform Matrix

• We only work within a finite area 0,1
• The set of basis is discrete
• The sample values of the function
• Haar transform matrix

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Haar Transform Matrix
Example of n3, H8
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Haar Transform Matrix

• It is proved that Haar Transform Matrix can be
  decomposed in to the following form

Where I is just identity matrix, W is just 22
Hadmard matrix, and ? is the shuffle matrix we
will mentioned later.
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Efficient Quantum Gates
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Some Efficient Quantum Gates

• Control NOT gate
• 2 bits shuffle gate ?4
• Perfect shuffle gate
• Controlled-(n,i) shuffle gate
• Controlled-k Hadamard gate

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Shuffle gate for two bits ?4
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Implementation of ?4
Three controlled not gates build a ?4 gate
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Perfect shuffle gate
Example of n4.
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Controlled-(n,i) shuffle gate
Note Zero-Control!
Example of n4, i3.
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Controlled-k Hadamard Gate
Note Zero-Control!
Example of kn-1
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Implement Haar Wavelet Transform by Quantum Gates
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Haar Transform Matrix

• Haar Transform Matrix can be decomposed in to the
  following form

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Haar Transform Circuit
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Haar Circuit Complexity

• There are n controlled Hadmard gates O(n)
• Each controlled shuffle gate has
  complexity O(i)
• The n shuffle gates have complexity O(n2)
• The circuit complexity is O(n2), much more
  efficient than the classical complexity O(n2n)

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Ridgelet Transform
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The Ridge Function
Wavelet ? Scale, Point Position Ridgelet ? Scale,
Line Position
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Ridgelet Transform and Radon Space
Let,
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Transform from Radon Domain
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Discrete Radon Transform
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Intuitive Understanding

• Summation of image pixels over a set of lines.
• p(p1) lines and each line contains p points.
• Two distinct points belong to one line.
• For all k, p parallel lines cover the plane.

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Quantum Ridgelet

• Not Done.

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Acknowledgements

• CBSSS Program
• Discussions with Sam