vorlesung quantum computing ss 08 1 quantum computing

Vorlesung Quantum Computing SS ‘08

1

Quantum Computing


2

Quantum Computing with NMR

Nuclear magnetic resonance

State preparation in an ensemble

Quantum Fourier transform

finding prime factors –Shor’s algorithm

solid state concepts

A

AB12

AB02 B01


3

NMR quantum computer


4

the qubits in liquid NMRJones in http://arxiv.org/abs/quant-ph/0106067

magnetic moment of nucleus much smaller than of electron (1/1000)

for reasonable S/N 1018 spins

measuring magnetic moment of a single nucleus not possible

qubit: spin 1/2 nucleus


5

spins in a magnetic field

mI = -1/2

mI = 1/2

B0ener

gy

magnetic field

E = h = - NħB0 ~ 300 MHz (B0 = 7 T, 1H)

Eint = -zB0 = - NIzB0 = -NmIħB0

mI = 1

population difference ~ 5∙10-5


6

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 =00Bz


7

spin-lattice relaxation T1

nuclei: T1 ~ hours – dayselectrons: T1 ~ ms

spin system is in excited state

relaxation to ground state due to spin-phonon interaction

read-out within T1

dMz


Mz M0

T1


8

spin-spin relaxation T2

magnetization in x,y-plane(superposition)

superposition decays because of dephasing

T1 relaxation to ground state

dMx


dMy


Mx

T2

My

T2


9

spin manipulation

Bloch equations

dMz


Mz M0

T1

dMx


Mx

T2

dMy


My

T2

B =B1 cos tB1 sin t

B0

magnetic field rotating in x,y-plane

B1<<B0


10

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


11

NMR technique

x

y

z

B0 ~ 7-10 T

Lieven Vandersypen, PhD thesis: http://arxiv.org/abs/quant-ph/0205193

Brf = 2 = + B1 cos t

00

cos tsin t

0B1

cos t-sin t

0B1


12

pulsed magnetic resonance

Lorentz shaped resonance with HWHM = 1/T2*

precessing spin changes flux in coils inducing a voltage signal damped with 1/T2

*

on resonance

off resonance

Fast Fourier Transform (FFT)

Hanning window + zero filling


13

FID spectrum


14

selective excitation

Pulse shapes

Lieven Vandersypen, PhD thesis: http://arxiv.org/abs/quant-ph/0205193


15

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


16

chemical shift

The 13C protons feel a different effective magnetic fielddepending on the chemical environment

local electron currents shield the field

the Zeeman splitting changes andthus the resonance frequency

Eint = -ħB0N(i)mI(i) (1-i) i

Cory et al.: Fortschr. Phys. 48 (2000) 9-11, 875


17

coupling between nuclear spinsCory et al.: Fortschr. Phys. 48 (2000) 9-11, 875

Ecoup = ħ Jij mI(i) mI(j)

Eint = -ħB0N(i)mI(i) (1-i)i+ ħJij mI(i)mI(j)

i≠j


18

state preparation

H H-1

calculation

U

preparation

read-out

|A|

time

time

a mixed ensemble is described by the density matrix

=

system cannot be cooled to pure ground state


19

density matrix

= =

() =

pure state: only one state in diagonal occupied with P=1

mixed state: states i occupied with Pi → Tr(

→ Tr(


20

states in an ensemble

level occupation follows Boltzmann statistics

mI = -1/2

mI = 1/2

ener

gy

magnetic field

p ~ e =e =–E/kBT -zB0/kBT eb

e-b

for

for


21

pseudo pure states

= pp= =

p

p00 eb

e-b00

eb + e-b

1

-zB0/kBTwith e ≈ 1 zB0

kBT

= + 1

10

02n

1 b

-b0

02n

1

density matrix can be written = 2-n (1 + )

access population scales with 2-n (n: number of qubits)

reduced density matrix


22

qubit representation

= + 1

10

02n

1 b

-b0

02n

1 1

-10

0Iz =

21

Iz

1

00

021 1

10

021 1

-10

021

Iz

0

10

021 1

10

021 1

-10

021

identity is omitted

Iz

Iz


23

time development

H H-1

calculation

U

preparation

read-out

|A|

time

time

Liouville – von Neumann equation

H,^iħ t

(t) = (t=0) = U(t) (t=0) U†(t)ħ- i H t^

e ħ i H t

^

e ^ ^


24

time development

eq= + 1

10

021 b

-b0

021

1018 copies of the same nuclear spin

zB0

kBTb = = ħL

2kBT

B =00B0

rotate spin to x,y plane by applying RF pulse 2

(t) = ħ- i H t^

e ħ i H t

^

e

eq= + 21 L

kBT1 Iz

^

(0+)= + 21 L

kBT1 Ix

^

Iy^

21 L

kBT1 Ix

^ cos Lt + sin Lt+=

H = LIz^^

(0+)eq


25

refocusing

(0+)= + 21 L

kBT1 Ix

^

2(t) -1 ħ- i H t^

e ħ i H t

^

eIx^

= Ix^ Iy

^cos Lt + sin Lt

if L ≠ r, e.g., due to inhomogeneous B0, the spin picks

up a phase

applying second RF pulse x inverts y-component:

2(t+) -1 Iy^Ix

^ cos Lt sin LtL


26

2 qubits

2,3-dibromo-thiophene

Cory et al.: Physica D 120 (1998), 82

b

a

Jab


27

ener

gy

Simple CNOT

CNOT operation

spin levels individually addressable

11100100

10110100

spinba

pulse inverts spin population


28

coupling between nuclear spinsCory et al.: Fortschr. Phys. 48 (2000) 9-11, 875

chemical shift

qubit coupling(always on)

^H = (aIza + bIz

b + cIzc) ^ ^ ^

+ 2(JabIzaIz

b+JacIzaIz

c+JbcIzbIz

c)^ ^ ^ ^ ^ ^


29

CNOT with Alanine

Z:-90ai

bi

ci

Y:-90 Y: 90 X:-90

Y:180 -Y:180

ao

bo

co

Jab

NO operation

Cory et al.: Fortschr. Phys. 48 (2000) 9-11, 875

UNO =iħ

tIzaIz

cJe

iħ

tIzaIz

cJe

iħ

Iye

iħ

Iye

-i 4e

iħ

Ixb

2eiħ

Iza

2eiħ

IzaIx

b

e

iħ

Iy2

eiħ

Iy2

eiħ

J t Iza Iz

b

e

UCNOT =

vorlesung quantum computing ss 08 1 quantum computing

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