Quantum AlgorithmsPreliminaria

Quantum AlgorithmsPreliminaria

Artur Ekert

Computation

INPUT OUTPUT

010101

11

0010

Physics Inside(and outside)

THIS IS WHAT OUR LECTURES WILL BE ABOUT

Classical deterministic computation

Initial configuration(input)

Final configuration(output)

Configuration = complete specification of the state of the computer and data

Physically allowed operations, computational steps

Intermediate configurations

Classical deterministic computation

0

10 1

0

0

0 0

0

1 1

1

000 001 101 110

Computational steps – moves from one configuration to another – are performed by elementary operations on bits

Boolean Networks

0

10 1

0

0

0 0

0

1 1

1

NOT

OR AND

OR

0

0

0

1

1

0

Basic operations = logic gates

01

01

01

10

AND1

0OR

Wire, identity

NOT

11

Logical AND

00

Logical OR

Output 0 apart from the (1,1) input

Output 1 apart from the (0,0) input

Fan out

X

X

X

Classical probabilistic computation

1P 2P

3P 4P 1 2 3 4P PP PP

Input Possible outputs

Quantum computation

1A 2A

4A

1 2 3 4A A A A A

* *1 2

2

1 2 3 4

2 2

3

1 3 4

4

2

Re2

P A A A A

A A A

A A A

A

A

3A

Constructive or destructive interference: enhance correct outputssuppress wrong outputs

GOOD SIDE: extra computational powerBAD SIDE: sensitive to decoherence

Quantum computation

0

0

0 10 1

2

0

0

10 1

2

0

Initial configurationof the three qubits

000

000

001

1

2

1

2

000

100

001

101

1

2

1

2

0

1

2

1

2

Bits and Qubits

1 0 1 0 10

BIT QUBIT

Quantum Boolean Networks

H0

0

H0

10 1

2

0

0

0

0 10 1

2

0

0

10 1

2

0

0

H 0

000 1000 001

2 1

000 1002

Quantum operations

H0

0

H0

10 1

2

0

H 0

Single qubit gates

0 0

1 1ie

1 0

0 iRe

H

10 0 1

21

1 0 12

1 11

1 12H

Hadamard

Continuous set of phase gates

kR 2

2

0 0

1 1k

i

e

2 / 2

1 0

0kk i

Re

Discrete set of phase gates

Single qubit interference

0

10

1

2 1 00 cos

2P

2 1 01 sin

2P

0

1

H H

1 0 1

2

1

0

0P

0

Any single qubit interference

H H

0 10 1

2 1

0 12

ie cos 0 sin 12 2

i

INPUT OUTPUT

cos sin1 1 1 0 1 11 1 2 21 1 0 1 12 2 sin cos

2 2

ii

ie

ei

in the matrix form

Any unitary operation on a qubit

H H

0 cos 0 sin 12 2

iie

INPUT OUTPUT

cos sin1 0 1 1 1 0 1 11 1 2 20 1 1 0 1 12 2 sin cos

2 2

ii i

i i

ie

e eie e

in the matrix form – the most general SU(2) operation on a single qubit

Possible implementations

© ENS Paris

Two and more qubits

1 0 1 101 5

Notation

1

0

1

2 1

00,1

d

d xx

x x

Operations on two qubits

1 0 0 0

0 1 0 0

0 0 0 1

0 0 1 0

Controlled-NOT

Controlled-U

U

0 0 0 0

0 1 0 1

1 0 1 1

1 1 1 0

0 0 0 0

0 1 0 1

1 0 1 0

1 1 1 1

U

U

1 0 0 0

0 1 0 0

0 0 0 0

0 0 0 0

U

Quantum interferometry revisited

H H

H H

Uu u

iU u e u

REMEMBER THIS TRICK !

Phases in a new way

H H

Uu

0

u

cos 0 sin 12 2

i

10 1

21 10 1

2 2

u

u u

1 10 1

2 21

0 12

i

u U u

e u

Entangled states

H

0

0

10 0 1 1

2

10 0 1 1

2

1 10 0 0 1 0 0 1

2 2

entangled

separable

Bell & GHZ states

H

100 0 0 1 1

21

01 0 1 1 021

10 0 0 1 121

11 0 1 1 02

H

1000 0 0 0 1 1 1

21

001 0 0 1 1 1 02

...ETC

Useful decomposition of any U in

SU(2)

1 1x xU B BA A

For any U in SU(2)

A A-1 B B-1

1xA A

Rotation by around some axis a

1xB B

Rotation by around some axis b

Recall that x represents rotation by around axis x

Rotation by twice the angle between axis a and b around

the axis perpendicular to a and b

1 1x xB BA A

Building controlled-U operations

A A-1 B B-1 U

=

A, A-1, B and B-1 are single qubit operations and can be constructed from the Hadamard and phase gates.

Controlled-U can be constructed from single qubit operations and the controlled-NOT gates.

Hence any controlled-U gate can be constructed from the Hadamard, the controlled-NOT and phase gates.

Toffoli Gate

=

2RH †2R 2R H

2R2

2

2

0 0

1 1 1i

e i

2

2

1 0

0k

ikRe

Controlled-controlled NOT

1x

2x

y

1x

2x

1 2x x y

Computes logical AND

1x

2x

y

1x

1 2SUM x x

1 2CARRY x x

Quantum adder

Quantum Networks

2RH †2R 2R H

Quantum adder

H

H

H

H

Quantum Hadamard transform

Quantum Hadamard Transform

H0

H0

0 1

0 1

20,1

1 10 0 0 0 0 1 1 0 1 1

4 4 xx

H0

H1

0 1

0 1

0

20,1

1 10 1 0 0 0 1 1 0 1 1 1

4 4x

x

x

Quantum Hadamard Transform

H0

H0

H0

H0

0 1

0 1

0 1

0 1

/ 2

0,1

10

2 nn

x

x

H0

H1

H1

H1

0 1

0 1

0 1

0 1

/ 20,1

11

2 n

z x

nx

z x

1 2 1 0...n nz z z z z

1 2 1 0...n nx x x x x 0 0 1 1 2 2 1 1... n nx z x z x z x z z x

Quantum Hadamard Transform

/ 20,1

11

2 n

z x

nx

z x

Is also known as the quantum Fourier transform on group 2n

Z

2Z group 0,10= the set with operation (addition mod 2)

2n

Z group 0,1n

0 = the set with operation (addition mod 2 bit by bit)

010101110010011

110101101011101

100000011001110

example for n=15

Quantum Fourier Transform

/ 20,1

11

2 n

z x

nx

z x

Quantum Fourier transform on group 2n

Z

21

/ 20

1

2

iN zxN

nx

z e x

Quantum Fourier transform on group NZ

Recall

H

10 0 1

21

1 0 12

1 11

1 12H

Hadamard

kR 2

2

0 0

1 1k

i

e

2 / 2

1 0

0kk i

Re

Discrete set of phase gates

Quantum Fourier Transform

H0x 02 0.0 1i xe

H

H0x

1x 2R

02 0.0 1i xe

1 02 0.0 1i x xe

H

H0x

1x 2R

02 0.0 1i xe

1 02 0.0 1i x xe

H2x 2 1 02 0.0 1i x x xe 2R 3R

F1

F2

F3

Quantum Fourier Transform

H

H0x

1x 2R

H2x 2R 3R

2R 3R 4RH3x

3y

2y

1y

0y

2

21

2

n

ixy

ny

x e y

Uniform family of networks

n Hadamard gates and n(n-1)/2 phase shifts, the size of the network = n(n+1)/2

Quantum Fourier Transform

H

H0x

1x 2R

H2x 2R 3R

2R 3R 4RH3x

3y

2y

1y

0y

1 2 0

22 0. ...2

1 1

2 2

n n n

ixy i x x x y

n ny y

x e y e y

Quantum Fourier Transform

H

H0x

1x 2R

02 0.0 1i xe

1 02 0.0 1i x xe

H2x 2 1 02 0.0 1i x x xe 2R 3R

F3

H

H 0x

1x2R

H 2x2R3R

Fy3

02 0.0 1i xe

1 02 0.0 1i x xe

2 1 02 0.0 1i x x xe

Quantum function evaluation

fy

x

: 0,1 0,1n

f Boolean function

0x1x2x3x4x

0x1x2x3x4x

x

( )y f x

( )x y x y f x

Quantum function evaluation

1my

x

: 0,1 0,1n m

f can be viewed as m Boolean functions

0x1x2x3x

0x1x2x3x

x

1 1( )m m

y f x fm-1

fm-22my

0y f0

…

……

…

……

……

2 2( )m m

y f x

0 0( )y f x

Quantum function evaluation :f X Y

Group X

Group (Y, )

x y x y f x

x y x y f x bit by bit addition – group 2Zm

mod2mx y x y f x modular addition – group m2Z

m

2

2

Z

Z mod2

m

m