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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
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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=$UU
2x2OK (NOT)
Sorry, not dagger but dollar! Uti = )exp( H
-
3
(2)
=
=
0001
01
0
01
10000 BA
=
0010
01
=
0100
10
=
1000
11
n-qubit 2n2-qubit
=
0100100000100001
ABC
2-qubit
2nx2nOK ()
ab
baXOR
1011111001010000
AB
A
B
A
BA
Physical Realization
Cavity QED
Magnetic resonance
Ion trap
Superconductor
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4
DiVincenzos Criteria1. Well defined extensible qubit array2.
Preparable in the 000 state3. Long decoherence time4. Universal set
of gate operations5. Single quantum measurements
NMR
1. 1/2qubit2. 3. 4. 1-qubitJ-2-qubit5.
qubit
C
C C
C
Br
BrS
H
H
2,3-dibromo-thiophene
2-qubit !!
0
1
1/2,0,1
(~ 10T)
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5
Bulk ensemble
>
610 deviation
+
=+=
1001
41001
21)
2(
21 zeq 1
1-qubit
0
1
]]1001[[
1000000000000001
21
21
=+=+= zzBz
Az II 11
]]31111113[[ =++=Cz
Bz
Az III
2-qubit
3-qubit
B0
z
xy
(free induction decay)
pulse2/
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8
Free Induction Decay
FFT BA
ABJ2 ABJ2
B
ABB J
ABB J +
ABA J +
ABA J
00
11
0110
]]1001[[ =00 01 10 11
]]31111113[[ =
]]31111131[[$ =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=
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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
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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
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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/ =
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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
=
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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
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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
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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 ?
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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
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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
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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??
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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
