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Introduction to NMR Quantum Information Processing Eisuke Abe (Keio University) December 18, 2002 Towards scalable quantum computationat RIEC, Tohoku University 参考文献 量子計算全般 Quantum Computation and Quantum Information, M. A. Nielsen and I. L. Chuang, Cambridge University Press (2000) NMR量子計算 (http://arxiv.org/) quant-ph/0205193, Lieven M. K. Vandersypen Ph.D Thesis (224 pages), Stanford University quant-ph/0207172, R. Laflamme et. al. 44 pages

# Introduction to NMR Quantum Information Processing Introduction to NMR Quantum Information Processing Eisuke Abe (Keio University) December 18, 2002 “Towards scalable quantum computation”

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• 1

Introduction to NMR Quantum Information Processing

Eisuke Abe (Keio University)December 18, 2002

Towards scalable quantum computationat RIEC, Tohoku University

Quantum Computation and Quantum Information, M. A. Nielsen and I. L. Chuang, Cambridge University Press (2000)

NMR (http://arxiv.org/)quant-ph/0205193, Lieven M. K. Vandersypen

Ph.D Thesis (224 pages), Stanford University

quant-ph/0207172, R. Laflamme et. al.44 pages

• 2

NMR

1-qubitNOT

Deutsch-Jozsa

(1)

01?

10? 01

10

NOT

12

102

11

12

102

10

+

H

=

01

0

1-qubit 21-qubit

=

10

1 ),(10 C+=

=

=

0110

x

1-qubit

=11

112

1H

1=\$UU

2x2OK (NOT)

Sorry, not dagger but dollar! Uti = )exp( H

• 3

(2)

=

=

0001

01

0

01

10000 BA

=

0010

01

=

0100

10

=

1000

11

n-qubit 2n2-qubit

=

0100100000100001

ABC

2-qubit

2nx2nOK ()

ab

baXOR

1011111001010000

AB

A

B

A

BA

Physical Realization

Cavity QED

Magnetic resonance

Ion trap

Superconductor

• 4

DiVincenzos Criteria1. Well defined extensible qubit array2. Preparable in the 000 state3. Long decoherence time4. Universal set of gate operations5. Single quantum measurements

NMR

1. 1/2qubit2. 3. 4. 1-qubitJ-2-qubit5.

qubit

C

C C

C

Br

BrS

H

H

2,3-dibromo-thiophene

2-qubit !!

0

1

1/2,0,1

(~ 10T)

• 5

Bulk ensemble

>

610 deviation

+

=+=

1001

41001

21)

2(

21 zeq 1

1-qubit

0

1

]]1001[[

1000000000000001

21

21

=+=+= zzBz

Az II 11

]]31111113[[ =++=Cz

Bz

Az III

2-qubit

3-qubit

B0

z

xy

(free induction decay)

pulse2/

• 8

Free Induction Decay

FFT BA

ABJ2 ABJ2

B

ABB J

ABB J +

ABA J +

ABA J

00

11

0110

]]1001[[ =00 01 10 11

]]31111113[[ =

]]31111131[[\$ =UU

000 001 010 011 100 101 110 111ABC

U

)31(:)11(:)13(:)11( ++++

00 01 10 11A,B,C4 (::)

2:1:2:1=)31(:)11(:)13(:)11( ++

1:0:1:0=)31(:)11(:)11(:)31( ++

1:0:0:1=

• 9

(Temporal averaging) (Logical labeling) (Spatial averaging)

2-qubit

[ ]

=

==

0000000000000001

0001

0001

0000pure

effpureeq orU

00002

1 += 1neff

)(Tr)(Tr FF pureeff =Q

??

==

dc

ba

eq1

==

cb

da

UU eq\$

2

==

bd

ca

UU eq2\$2

3

0000)14()1(

11

13

321

+

=

=

= ++

aa

aa

aa

eff

1

1=+++ dcbaQ

==

0100001010000001

BAABCCU U

• 10

]]0101[[3 =

2

A

]]1001[[1 =

1000 1101

]]0011[[2 =00 01 10 11

B2 3

4

00

12335251544122553

5234331425133241131512

2533114344552213244551

NOTNOTNOTNOTNOTNOTNOT

CCCCCCCCCCCCCCCCCCCCCCCCCCC

1 (::)

5-qubit

• 11

]]31111113[[\$ =UU

110010

101001 ABC

A=0

+

00

01

4

11

11

]]31111113[[ =000 001 010 011 100 101 110 111

=

1000000000000100000000100001000000001000010000000010000000000001

U

000

001 010

011

100

101 110

111

00

11

01 10

• 12

BC1001ANOT

010110001101110010101001

UABC

U ?

111011111011111011

BAC CAC

B1ANOTC1ANOT

(BC11A2NOT)

B or CANOT

BACACABA CCCCU ==

A

B

C

Spin Hamiltonian

+

+

+

1000010000100001

2000

000000000

21 AB

BA

BA

BA

BA

J

Bz

AzAB

BzB

AzA IIJII 2 +=H

=

2

ZeemanLarmor J-

0B

100.0040.0519F

1.0710.7113C

99.9942.581H

(%)

(MHz/T)

z

??

x

B0

x

)sincos( ttBrf = yxBLrr

xy

x

z

B0

Brf

zxBeff )( 0

+= BBrf

=B0

B0

B0

• 14

1-qubit

=B0

pulse2/

xy-x-y

z

pulse

Brfx

L++++=

= !3!2!

32

0

AAAnAe

n

nA 1

AxixixA sincos)exp( += 1A

=

=

=

)2

exp(0

0)2

exp()(

2cos

2sin

2sin

2cos

)(

2cos

2sin

2sin

2cos

)(

i

iRR

i

iR zyx

)2/exp(

1lim)(

=

=

n

n nIi

R

(x,y,z) (!!)

===

==

==

iiXXYRZRY

ii

RX zyx 001

)2/(1111

21)2/(

11

21)2/(

XYXRZRYRX zyx ==== )2/()2/()2/(

2/ =

• 15

J-Bz

AzAB

AzAA IIJI 2 +=H

A

BBBzI 02

10 =

AzABAA IJ )( =H

BA

BABA

BA

AB

00

11

0110

ABB J

ABB J +ABA J +

ABA J 00

11

0110

J-

==

)2/exp(0000)2/exp(0000)2/exp(0000)2/exp(

)2exp()(

JtiJti

JtiJti

IIJtitU BzAzJ

J-

Z

Z zJ =

J

==

10000000000001

)2/1(i

iiJUJ J

Jt

21

=

• 16

NOT

=

=

0100100000100001

i

JYXZZC BBBAAB

==

==

==

==

ii

iZZ

ii

ii

XX

i

iiZZYY

AB

BB

00000000100001

100100

001001

21

00001000000001

11001100

00110011

21

11

11

JZ

Z

Y X=

A

B

==

0100000

0000001

~i

iiJYXC BBAB

]][[~]][[~ \$ cdbaCdcbaC ABAB =

NOT

NOT

NOT A

B

0

1

0

0

0

1

1

1

0

1

1

0

BXBY Jx

y

B

xy

• 17

Refocusing1== )()()()( 2222 JBJBJAJA UYUYUYUY

2AY

xy

2AYA

2BY

xy

2BY

A

-

B

pulse J

NMR

CC C

F Br

Cl

H F F

Deutsch-Jozsa

chloroform

qubit AMHz5002/ A

qubit BMHz1252/ B

Hz215ABJ

bromotrifluoroethylene

MHz470F19

T7.110 =B

122.1Hz=ABJ

Hz8.53=BCJ

Hz0.75=ACJ

qubit A

qubit B

qubit C

kHz2.132/)( = BA

kHz5.92/)( = AC

Cl

Cl

• 18

Deutsch-Jozsa

fY

Y

Y

Y

x x

y )(xfy

01101

10100

f4f3f2f1

balancedconstantx

2 bit f (x)

2 f (x) constantbalanced

0

0

output

constant 0balanced 1

input qubit A 10 +

10

work qubit B

1111

001010

0101

100000

BCAB

NMR00

work bit1

D-Jx x

y )(xfy

f

BBBAAA

BBBBBAAA

JYYXYXYJYXXYXYXYf

== ))((3xyxfy = )(3

)4/1()4/1( 212

2 JUXJUfXf JBJB ==yyxfy == 1)(2

)4/1()4/1( 221 JUXJUXf JBJB=yyxfy == 0)(1

BBBAAAB JYYXYXYfXf == 32

4xyxfy = )(4

f ?

• 19

D-JA

constantbalanced

]]1001[[1 =

]]1100[[\$1 =UU

1000 1101

balanced

]]0011[[2 =

]]0110[[\$2 =UU

]]0101[[3 =

]]0101[[\$3 =UU

2

1111

001010

0101

100000

BCAB

D-JA (:)

49.9755160MHz

constant

balanced

• 20

=A

B

C

ABJ2/1

ACJ2/1

Y Y2X 2X

(::)]]31111113[[ ]]31111113[[

Cz

BzBC

Cz

AzAC

Bz

AzAB

CzC

BzB

AzA IIJIIJIIJIII 222 +++=H

==

Hz0.75Hz1.122

AC

AB

JJ

Refocusing

ACABA JJJ ==

X

Cz

AzA

Bz

AzA

AzAA IIJIIJI 22 +=H

A

Case.1: BC

Case.2:BC AzAAzAA IJI )2( =H

AzAA I=H

AY AY2Jx

x

y

y

U BC1001ANOTCase.1

Case.2

2

• 21

Nature

Phys. Rev. Lett.

Phys. Rev. Lett.

Nature

Phys. Rev. Lett.

J. Chem. Phys.

Nature

Nature

Phys. Rev. Lett.

Journal

2001IBM (Vandersypen)Shor (Factoring 15)7

2000IBM (Vandersypen)Order Finding5

1999IBM (Vandersypen)Logical Labeling

1998LANL (Neilsen)Teleportation

1998MIT (Cory)Error Correction

3

1998Oxford (Jones)Deutsch-Jozsa

1998Oxford (Jones)Grover

1998IBM (Chuang)Deutsch-Jozsa

1998IBM (Chuang)Grover

2

YearGroup (First Author)AlgorithmsQubits

Scalable ??

)(21

+= 1n

W.S.Warren, Science 277, 1688(1997)

10-qubit??

• 22

B. E. Kane, Nature (1998)

T. D. Ladd et. al., PRL (2002)

E. Knill et. al. (KLM), Nature (2001)

J. Q. You et. al., PRL (2002)

Nuclear Spin Linear Optics

Josephson junction

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