Vorlesung Quantum Computing SS 08 1 quantum parallelism a 1 F |00> + a 2 F |01> + a 3 F |10> + a 4 F |11> a 1 |00> + a 2 |01> + a 3 |10> + a 4 |11> input

Vorlesung Quantum Computing SS ‘08

1

quantum parallelism

a1 F |00>+

a2 F |01>+

a3 F |10>+

a4 F |11>

}{a1 |00>

+a2 |01>

+a3 |10>

+a4 |11>

}{input

b1 |00>+

b2 |01>+

b3 |10>+

b4 |11>

}{=

output

F

Superposition for input created by Hadamard gates

Functions are represented by unitary operators

Quantum state tomography – but how to get a “real” result?


2

Deutsch algorithm: table of truthD. Deutsch, Proceedings of the Royal Society of London A 400 (1985), 97

simplest example: 1-bit-to-1-bit function f : x → {0,1}

case 1

0

0

case 2

1

1

case 3

1

0

case 4

1

0

constant balanced

f(1)

f(0)

real or

false?

false true

f(0)f(1) 0 0 1 1


3

strength of quantum parallelism

calculate a function f(x) with x ∈ {0,…,K} for all x at once

classical quantum

function balanced(f:function)a=f(0)b=f(1)return a b

function balanced(Uf:function)1 = H 0 = H |2 = Uf 1

3 = Hx 2 = f(f(|Rreturn f(f(

use quantum computing only if f(x) is “expensive” to calculate(time, memory), e. g., spin dynamics for huge structures

problem: superposition cannot be read-out: it will always collapse to an eigenstate


4

Deutsch algorithm: principle

input

eigen-state

read-out

eigen-state

super-position

transferto

eigen-state

or

test-function

??

??

one run only!

0 1 2 3


5

quantum circuit

Uf

x x

y yf(x)

data register

target register

addition modulo 2= f(x) for y=0

0

√2

f(0)f(1)

√2

H2 =√2

= 12√2

√2n

1 xf(x)x

evaluation of function f(x) with n data qubits x and 1 target qubit

BUT: How to extract information?

√2


6

Deutsch algorithm: quantum circuit

1

0Uf

x x

y yf(x)

H

H

H

0 1 2 3

0

1H2 =√2

= 12√2

2Uf 1


7

true

2Uf 1(|0|1) (|f(0) - |1 f(0)) 12

Deutsch algorithm: find Uf

case 1

0

0

case 2

1

1

case 3

1

0

case 4

1

0f(1)

f(0)

Uf

Uf x,yx,yf(x)

Uf: |0,0 → |0,0 , |0,1 → |0,1 , |1,0 → |1,1 , |1,1 → |1,0

Uf =

CNOT

Uf: |0,0 → |0,1 , |0,1 → |0,0 , |1,0 → |1,0 , |1,1 → |1,1

Uf =

Z-CNOT

Uf: |0,0 → |0,1 , |0,1 → |0,0 , |1,0 → |1,1 , |1,1 → |1,0

Uf =

NOT

Uf: |0,0 → |0,0 , |0,1 → |0,1 , |1,0 → |1,0 , |1,1 → |1,1

Uf =

ID

2Uf 1(|0, f(0) - |0,1 f(0) + |1, f(1) - |1,1 f(1)) 12

2Uf 1(|0 + |1) (|f(0) - |1 f(0)) 12

false

information encoded in phase of x-qubit


8

Deutsch algorithm: get the answer

3Hx 2

1√2

1√2

false : H2x =H =

= =

12

1

-11

11√2

1

1

12

f (0) f (1)

true : H2x = H =

= =

12

1

-11

11√2

1

-1

12

1√2

1√2

f (0) f (1)

f (0) f (1)| f (0) - |1 f (0)

√2

read-outread-out


9

Deutsch algorithm: summary

evaluates a global property of a function f(x) with a single run

Example: f(x) → f(0) = 1, f(1) = 1

1

0Uf

x x

y yf(x)

H

H

H

0 1 2 3

01H2 =√2

= 12√2

2

2

12

1√2

H2x =H =

=

12

1

-11

11√2

1

1

12

2

2 Uf 1

12

-

-

--

12


10

Deutsch algorithm: implementation

40Ca+

42S1/2

42P1/2

42P3/2

32D3/2

32D5/2

397 nm 729 nm

854 nm

866 nm

|0

|1

quad

rupo

letr

ansi

tion

used for Laser cooling

dete

ctio

n

D5

/2 o

ccup

atio

n P

D

D5

/2 o

ccup

atio

n P

D

red sideband blue sideband

Roos et al: Phys. Rev. Lett. 83, 4713 (1999)


11

Deutsch algorithm: implementationGulde et al: Nature 421, 48 (2003)

case 1: Uf = ID

case 2: Uf = NOT

case 3: Uf = CNOT

case 4: Uf = Z-CNOT


12

composite pulses

example: imprecise B1-field

rotate from z to -z:

1) – pulse at y (180y)

2) three pulse sequence: 90y180x90y

M. H. Levitt: Prog. Nucl. Magn. Reson. Spectrosc. 18 (1986)

for small deviations (180-2)y

for small deviations (90-)y(180-2)x(90-)y


13

Deutsch algorithm: FidelityGulde et al: Nature 421, 48 (2003)


14

more than 1 ionNägerl et al: Phys. Rev. A 60, 145(1999)

switching between ions within 14 s


15

CNOT with 2 ionsSchmidt-Kaler et al: Nature 422, 408(2003)

3m

n=0 n=1

n=0 n=1

Control Target0 0

R+(,0)1 1CNOT: spin state changed for n=0

R+(,)e

e


16

CNOT with 2 ionsSchmidt-Kaler et al: Nature 422, 408(2003)

3m

n=0 n=1

n=0 n=1

Control Target0 0

R+(,0)21

CNOT: spin state changed for n=0R+(,)e

e

R(,0) 1


17

error sourcesSchmitd-Kaler et al: Nature 422, 408 (2003)

CNOT with 2 40Ca+ ions as qubits and vibration mode as coupling

source contribution

laser frequency noise~100 Hz (phase coherence) ~10%

laser intensity fluctuations ~1%

laser detuning error ~2%

residual thermal excitation ~4%

addressing error ~3%

off-resonant excitations(for tgate = 600 s ) ~4%


18

Needle in a haystack

Grover algorithm: search in an unsorted database

“Quantum mechanics helps in searching for a needle in a haystack”

Phys. Rev. Lett. 79, 325 (1997)

Lov Grover,Bell labs


19

Search in a database

Example: search for a specific number in a phonebook

• N = 2n entries with index x = 0…N-1.

• A “detector” function f(x):

f(x) = 0- Entry x is no solution:

f(x) = 1- Entry x is a solution:

We need:

0 0 0 0 0 1 0 0


20

The oracle

Grover’s algorithm minimizes calls to “oracle”

0 0 0 0 0 1 0 0

Classical: on average N/2 calls to oracle.

Quantum: number of calls √N.

0 1 0 0

|x: adresses of data register

f(x)=1 if entrance is solution


21

1

0 Hn

H

Hn

H

n01

1

Quantum circuit

Target register: oracle qubit |q prepared in 1 = |1

Data register: superposition of n=2 qubits |x 1 = |0

1 2 3

H2 =√2

= 12√2

f(x) = 1

R iterations


22

Oracle operator

UO x q|x |q f(x) |q is flipped, if |x points to register with solution

|q0 2H =√2

UO |x |q0 = (-1)f(x) |x |q0

√2

|x √2

|x

√2

|x√2

= |x

UOf(x) = 0:

f(x) = 1:


23

Grover’s algorithm

H2 =√2

= 12√2

Oracle qubit does not change: Look at data register only.

2 =

3.1 = UO 2 = (|00 - |01 + |10 + |11) 12

3.2 = H23.1 = 14 12

= (|00 + |01 - |10 + |11)

3.3 = C3.2 = (-1)x-0)-1 3.2 = (|00 - |01 + |10 - |11)12

3.4 = H23.3 = |01


24

Grover’s algorithm

H

H

H

H

UO |x |q0 = (-1)f(x) |x |q0

1

-1 1 11

√2H = C

-

-

-

C


25

Geometrical analysis

C = -1 + 2 |00| with 2 = Hnand2 = Hn

G= HnCHnUO= Hn(2|00|-1)HnUO=(2|22|-1)UO

|=√N-M

1 (1-f(x))xx

√M1 f(x)x

x|= Superpostion of solutions

Superpostion of “no-solutions”

|2= | + | N

N-M√ N

M√

= cos | + sin |2

2

|

|

|22


26

Geometrical analysis

2

|

|

|2

UO 2 = cos | - sin |2

2

UO|2

|2

UO|2

(2|22|-1)=|22| - (1- |22|)

= P2 – P2┴

:relexion at |2

G|2

G 2 = cos | + sin |32

32

Gk 2 =cos | + sin |k+1

2k+12

G = cos

cos

- sin

sin


27

Iterations

Necessary number R of iterations is CI to

2

2

= =

2

R CI = ≤

2

2

arcsinNM√

4 M

N√

NM << 1with error probability p ≤ sin2 =

2 NM

• For more iterations than R, error increases

=> One needs to know number M of solutions.


28

ion trap quantum computing: a summary

• qubit representation– hyperfine states (9Be+, 43Ca+)– electronic states (40Ca+)– vibrational modes

• qubit manipulation: laser irradiation

• initial state preparation: – Doppler and sideband cooling

• read-out: fluoresence


29

relaxation and operation

• electronic states: – energy relaxation time T1 ~ 1s– phase relaxation time T2 ~ 10 ms– gate operation time Tgate ~ 200 s– T2/Tgate ~ 50

• hyperfine states:– phase relaxation time T2 ~ 10s– gate operation time Tgate ~ 10 s– T2/Tgate ~ 106

source: Homepage group A. Steane http://www.physics.ox.ac.uk/users/iontrap/news.html


30

scaling to more than 10 ionsKielpinski, Monroe, Wineland: Nature 417, 709 (2002)

C. Monroe group, Michigan ‘06

Documents

Vorlesung Quantum Computing SS 08 1 quantum parallelism a 1 F |00> + a 2 F |01> + a 3 F |10> + a 4 F |11> a 1 |00> + a 2 |01> + a 3 |10> + a 4 |11> input