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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
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