Vorlesung Quantum Computing SS 08 1 A scalable system with well characterized qubits Long relevant decoherence times, much longer than the gate operation

Vorlesung Quantum Computing SS ‘08

1

A scalable system with well characterized qubits

Long relevant decoherence times, much longer than the gate operation time

A qubit-specific measurement capability A

A „universal“ set of quantum gates U

The ability to initialize the state of the qubits to a simple fiducial state, e.g. |00...0>

„DiVincenzo “ criteria

DiVincenzo: Fortschr. Phys. 48 (2000) 9-11, pp. 771-783


2

Quantum Computing with Ions in Traps

How to trap ions

State preparation

Qubit operations

CNOT

Deutsch – Jozsa Algorithm

advantages/drawbacks


3

Paul Trap

Nobel Prize 1989

centre is field free

quadrupole field x and y motions not coupled!

Chemnitz University


4

Linear Trap

x

y

z

U1

RUac

Uac(t) = Ur + V0 cos Tteffective potential:

eff = x2 x2 + y

2 y2 + z2 z2

x = y >> z

(averaged over one rf cycle)

U2

z0

M. Sasura and V. Buzek: quant-ph/0112041

cmeyer

whether ions are trapped depends on rod diameter vs R and Ur only! (v does not come in!)


5

Potential


6

Ions in a Linear Trap

z = 2qU12

mz02

typical operation parameters:

V0 = 300 – 800 VT/2 = 16 – 18 MHz

U12 = 2000 V

z0 = 5 mm

R = 1.2 mm

z/2 = 500 - 700 kHz

x,y/2 = 1.4 – 2 MHz (40Ca+)

70 m

40Ca+

24Mg+

Seidelin et al: Phys. Rev. Lett. 96, 253003 (2006)

Nägerl et al: Phys. Rev. A 61, 023405 (2000)


7

quantum computing with ions

H H-1

calculation

U

preparation

read-out

|A|

time

time

the ions are prepared to be in their ground state

Doppler cooling side band cooling

1st step 2nd step

kBT << ħz


8

Doppler cooling

when absorbing a photon, also the momentum is transferred

the net momentum of the spontaneousemission is zero

E = ħp = ħk

E = 0p = 0

E = ħp = ħk

k

absorption

for ions moving toward the laser beam the lightappears blue shifted → use a red detuned laser

= 0 + k ∙ v

Ca


9

side band cooling

Doppler cooling gets down to kBT ≈ ħ

internal electronic ground and excited state |g,|e

trapped ions movingin harmonic potentialstates |n, n= 0,1,2…

cooling:

|g,n → |e,n-1|e,n-1 → |g,n-1

cmeyer

natuerliche linienbreiteatomic transition


10

ions used as qubits

electronic states as qubits (“pseudo-spin”)(CNOT, Deutsch-Jozsa Algorithm, Quantum-Byte)

hyperfine states as qubits(CNOT, error correction, Grover Algorithm)

cmeyer

2003bite 2005

cmeyer

CNOT 1995Error corr 2004Grover 2005


11

40Ca+ as qubit

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

quadrupole transition with relatively long relaxation time

for cooling: 866 nm transition has to be irradiated as well, otherwise charge carriers will be trapped in 32D3/2 orbital

fluorescence detection for read-out

dete

ctio

n

Nägerl et al: Phys. Rev. A 61, 023405 (2000)D

5/2 o

ccup

atio

n P

D

D5

/2 o

ccup

atio

n P

D

red sideband blue sidebandafter

Doppler cooling

cmeyer

allowed transitions have relaxation times of ns (GHz)pulse duration on quadrupole trans ~150 mu without relaxation

cmeyer

J=L+S (I=0)


12

9Be+ as qubit

electron spin S = 1/2, ms = 1/2nuclear spin I = 3/2, mI = 1/2, 3/2

F = I + S, mF

22P3/222P1/2 12 GHz

22S1/2

|F=2, mF=2

|F=1, mF=1

|

| 1.25 GHz

Dop

pler

coo

ling

hyperfine levels have long relaxation times

sideband cooling |2,2|n → |1,1|n-1; induced spontaneous Raman transition |1,1|n-1 → |2,2|n-1

sideband cooling

detection with fluorescence after + excitation +

det

ectio

n

Monroe et al: Phys. Rev. Lett. 75, 4011 (1995)

red

blueafter Doppler cooling

after sideband cooling


13

quantum computing

H H-1

calculation

U

preparation

read-out

|A|

time

time

quantum-bit (qubit)

0 1

a10 + a21 =a1a2


14

qubit operations

how does the system evolve with time?

U(t)e ħ- iHQCt

^^ HQC = Htrap + Hion + Hman

^ ^ ^ ^

Splitting of S = ½ in external magnetic field:

22S1/2

|F=2, mF=2

|F=1, mF=1

|

| s/2 = 1.25 GHz

Hion = -ħs

2

1

-10

0^

B0 = 0.18 mT

Hion = -S∙B = - SzB0 = -LSz^ ^ ^

cmeyer

orient z-axis along static magnetic field


15

qubit coupling

coulomb repulsion couples motional degrees of freedom

Htrap = (x2 xi

2 + y2 yi

2 + z2 zi

2 + ) + M2

pi2

M2

e2

40 |ri - rj| i=1 i=1

N N

j>i

trap potential eff

Ekin

coulomb potential

positions at rest

1 mode

2 modes

A. Steane: quant-ph/9608011

^

cmeyer

Epot = qU

cmeyer

M : mass of each ion


16

vibration modes as qubits (bus)

centre of mass motion used as qubit

A. Steane: quant-ph/9608011

i=1 i=1Htrap = ( z

2 zi2 + ) = ħi ai

†ai

pi2

M2

M2

N N

i

z = 2qU12

mz02

z

z 3

J.F. Poyatos et al., Fortschr. Phys. 48, 785


17

9Be+: the two qubit system

22P1/2 50 GHz

22S1/2

|F=2, mF=2

|F=1, mF=1

|

| |1| |0

|1

| |0s/2 = 1.25 GHz

z/2 = 11.2 MHz

22P3/2

|F=3, mF=3

|0 |aux

|F=2, mF=0

vibrational state: control qubit

hyperfine state: target qubit

Ramantransition + detection

~ 313 nm


18

spin dynamics

dMx

dt= (My(t)Bz Mz(t)By)

dMy

dt= (Mz(t)Bx Mx(t)Bz)

dMz

dt= (Mx(t)By My(t)Bx)

= My(t)Bz

= - Mx(t)Bz

=

dMdt

= M(t) x B

= Mycos(Lt) - Mxsin(Lt)

= Mxcos(Lt) + Mysin(Lt)

B =00BzB =

B1 cos tB1 sin t

B0

magnetic field rotating in x,y-plane


19

spin flipping in lab framehttp://www.wsi.tu-muenchen.de/E25/members/HansHuebl/animations.htm


20

rotating frame

xyz

xyz

cos tcos tsin t

- sin t0 0 1

00

=

r z

y

xxr

yr

tt

cos tcos tsin t

- sin t0 0 1

00 cos t

sin t 0

B1

cos t-sin t

0B1+Brf =

r

cos 2t

0Brf =

r 100

B1 -sin 2tB1+

constant

counter-rotating at twice RF

applied RF generates a circularly polarized RF field, which is static in the rotating frame

B1 cos t00

2 =


21

spin flip in rotating framehttp://www.wsi.tu-muenchen.de/E25/members/HansHuebl/animations.htm


22

qubit manipulation: laser interaction

Hman = - ∙ B = mS B = B1x cos(kz-t+)^

Hman ≈ (S+ei + S-e-i)

= m B1/2ħ

ħ2

frame of reference: H0 = ħsSz + ħza†a

only spin state is changed

i (S+aei - S-a†e-i)ħ2

for = s - z “red” side band

i (S+a†ei - S-ae-i)ħ2

for = s + z “blue” side band

change of vibrational state always implies change of spin state

Lamb-Dicke parameter: ≡ 2d0/<<

for = s


23

qubit rotation

22P1/2 50 GHz

22S1/2

|F=2, mF=2

|F=1, mF=1

|

| |1| |0

|1

| |0s/2

|0 |aux

|F=2, mF=0

Qubit rotation on target qubit U/2,ion

Raman transition with detuning s

Duration of laser pulse: /2 rotation

2e =

iħ

Sy cos /4

cos /4

sin /4

- sin /4

1

1-1 11

√2= = U/2

cmeyer

Rx(a) = e-iaX/2 = cos a/2 *I - i sin a/2 XRy(a) = e-iaY/2 = cos a/2 *I - i sin a/2 Y


24

/2- rotation matrix

-

-

1√2

U/2,ion =

base vectors of the two–qubit register:

Transformation matrix:

U/2,ion = () 1√2

U/2,ion = () 1√2

U/2,ion = () 1√2

U/2,ion = () 1√2


25

CNOT operation

-

Uph = Uph transformation matrix:

Transformation sequence:

U/2,ion =U-/2,ion Uph

-

-

12

-

-

-

= = UCNOT

Monroe et al: Phys. Rev. Lett. 75, 4714 (1995)


26

phase rotation

22P1/2 50 GHz

22S1/2

|F=2, mF=2

|F=1, mF=1

|

| |1| |0

|1

| |0s/2

|0 |aux

|F=2, mF=0

Phase rotation on control qubit Uph

Raman transition between and auxiliary state

Full rotation by 2

Uph


27

quantum computing

H H-1

calculation

U

preparation

read-out

|A|

time

time

quantum-bit (qubit)

0 1

a10 + a21 =a1a2


28

Be ions: read-out spin state

|F=2, mF=2

| |1

| |0

22P3/2

|F=3, mF=3

22S1/2

|F=1, mF=1

| |1| |0

+ detection

read-out spin state via fluorescence

prepare desired initial state usingRaman pulses

| on blue side band → |1 on internal state → |1|

perform CNOT

cool system to | |0


29

Be ions: read out vibrational state

|F=2, mF=2

| |1

| |0

22P3/2

|F=3, mF=3

22S1/2

|F=1, mF=1

| |1| |0

+ detection

read-out spin state

prepare same initial state and do CNOT

convert vibrational into spin state on red side band for | on blue side band for |

read-out spin state via fluorescence

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Vorlesung Quantum Computing SS 08 1 A scalable system with well characterized qubits Long relevant decoherence times, much longer than the gate operation