Various Quantum Transforms Zhaosheng Bao; Liang Jiang; Chenyang Wang; Lisa Wang; Zhipeng Zhang;

Various Quantum Transforms

Zhaosheng Bao; Liang Jiang; Chenyang Wang; Lisa Wang; Zhipeng Zhang;

Contents

1. Quantum Fourier Transform2. Wavelet Transform3. Quantum Wavelet Transform4. Ridgelet Transform5. Quantum Ridgelet Transform(not done)

Quantum Fourier Transform

Continuous Fourier Transform

ft FF

2tftt

then

ft 12

2tF

Discrete Fourier Transform

ftwheret 0, 1, 2, ..., N 1

Inverse Descrete Fourier Transform

Quantum Fourier Transform

N 2n

Hadamard Gate

)10(2

10 H

)10(2

11 H

12 1 11 1

Rk gate

1 0

0 22k00 kR

11 2/2 kike

R

Control-Rk gate

1 0 0 00 1 0 00 0 1 0

0 0 0 22k

0000 kRc0101 kRc

1010 kRc 1111 2/2 kike

Rc

Quantum Circuit for Quantum Fourier Transform

On2

Wavelet Transform

— Haar Transform

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]

Haar Basis

00

00

10 1

1

20

21

22

23

Haar Transform

Haar Transform Matrix

We only work within a finite area [0,1]The set of basis is discreteThe sample values of the functionHaar transform matrix

{ ( )}if x { , }jf v n2

Transform Matrix H

Haar Transform Matrix

Example of n=3, H8

Haar Transform Matrix

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

1

42 2 2 2 2 2 4

2 2 2 2 2 2

( ) ( ) ( )

( ) ( ( )) ( )

n n i n i n

n n i i n

H I I

W I I W I I W

Where I is just identity matrix, W is just 2*2 Hadmard matrix, and Π is the shuffle matrix we will mentioned later.

Efficient Quantum Gates

Some Efficient Quantum Gates

Control NOT gate2 bits shuffle gate Π4 Perfect shuffle gateControlled-(n,i) shuffle gateControlled-k Hadamard gate

2n

k-12 2W I

i2 2 2n iI

Shuffle gate for two bits Π4

Implementation of Π4

Three controlled not gates build a Π4 gate

Perfect shuffle gate

2n

Example of n=4.

Controlled-(n,i) shuffle gate

i2 2 2n iI

Example of n=4, i=3.

Note: Zero-Control!

Controlled-k Hadamard Gate

k-12 2W I

Example of k=n-1

Note: Zero-Control!

Implement Haar Wavelet Transform by Quantum Gates

Haar Transform Matrix

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

1

42 2 2 2 2 2 4

2 2 2 2 2 2

( ) ( ) ( )

( ) ( ( )) ( )

n n i n i n

n n i i n

H I I

W I I W I I W

Haar Transform Circuit

1

42 2 2 2 2 2 4

2 2 2 2 2 2

( ) ( ) ( )

( ) ( ( )) ( )

n n i n i n

n n i i n

H I I

W I I W I I W

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(n*2n)

i2 2 2n iI

Ridgelet Transform

The Ridge Function

a

bxx

axba

sincos1

)( 21,,

Wavelet Scale, Point PositionRidgelet Scale, Line Position

Ridgelet Transform and Radon Space

2

)()(),,( ,, xdxfxbaRIT baf

2

)(sincos1 21 xdxf

a

bxx

a

Let,

xdxfxxR f )()sincos(),(2

21

Transform from Radon Domain

dtReF fit

f

),()sin,cos(

dRbaRIT baff )(),(),,( ,

Discrete Radon Transform

plp

plk

Ljik

p

p

ZjjlL

ZiplkijjiL

jifp

lr

pZ

Zjif

lk

:),(

),(mod:),(

],[1

][

1,,1,0

grid finite aon ],[

,

,

),(

2

,

Intuitive Understanding Summation of image pixels over a set of lines. p(p+1) lines and each line contains p points. Two distinct points belong to one line. For all k, p parallel lines cover the plane.

Quantum Ridgelet

Not Done.

Acknowledgements

CBSSS ProgramDiscussions with Sam