vorlesung quantum computing ss 08 1 quantum computing
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Vorlesung Quantum Computing SS ‘08
1
Quantum Computing
Vorlesung Quantum Computing SS ‘08
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
Vorlesung Quantum Computing SS ‘08
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NMR quantum computer
Vorlesung Quantum Computing SS ‘08
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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
Vorlesung Quantum Computing SS ‘08
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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
Vorlesung Quantum Computing SS ‘08
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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
Vorlesung Quantum Computing SS ‘08
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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
dt= (Mx(t)By My(t)Bx)
Mz M0
T1
Vorlesung Quantum Computing SS ‘08
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spin-spin relaxation T2
magnetization in x,y-plane(superposition)
superposition decays because of dephasing
T1 relaxation to ground state
dMx
dt= (My(t)Bz Mz(t)By)
dMy
dt= (Mz(t)Bx Mx(t)Bz)
Mx
T2
My
T2
Vorlesung Quantum Computing SS ‘08
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spin manipulation
Bloch equations
dMz
dt= (Mx(t)By My(t)Bx)
Mz M0
T1
dMx
dt= (My(t)Bz Mz(t)By)
Mx
T2
dMy
dt= (Mz(t)Bx Mx(t)Bz)
My
T2
B =B1 cos tB1 sin t
B0
magnetic field rotating in x,y-plane
B1<<B0
Vorlesung Quantum Computing SS ‘08
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spin flipping in lab framehttp://www.wsi.tu-muenchen.de/E25/members/HansHuebl/animations.htm
Vorlesung Quantum Computing SS ‘08
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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
Vorlesung Quantum Computing SS ‘08
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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
Vorlesung Quantum Computing SS ‘08
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FID spectrum
Vorlesung Quantum Computing SS ‘08
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selective excitation
Pulse shapes
Lieven Vandersypen, PhD thesis: http://arxiv.org/abs/quant-ph/0205193
Vorlesung Quantum Computing SS ‘08
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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
Vorlesung Quantum Computing SS ‘08
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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
Vorlesung Quantum Computing SS ‘08
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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
Vorlesung Quantum Computing SS ‘08
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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
Vorlesung Quantum Computing SS ‘08
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density matrix
= =
() =
pure state: only one state in diagonal occupied with P=1
mixed state: states i occupied with Pi → Tr(
→ Tr(
Vorlesung Quantum Computing SS ‘08
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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
Vorlesung Quantum Computing SS ‘08
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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
Vorlesung Quantum Computing SS ‘08
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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
Vorlesung Quantum Computing SS ‘08
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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 ^ ^
Vorlesung Quantum Computing SS ‘08
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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
Vorlesung Quantum Computing SS ‘08
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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
Vorlesung Quantum Computing SS ‘08
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2 qubits
2,3-dibromo-thiophene
Cory et al.: Physica D 120 (1998), 82
b
a
Jab
Vorlesung Quantum Computing SS ‘08
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ener
gy
Simple CNOT
CNOT operation
spin levels individually addressable
11100100
10110100
spinba
pulse inverts spin population
Vorlesung Quantum Computing SS ‘08
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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)^ ^ ^ ^ ^ ^
Vorlesung Quantum Computing SS ‘08
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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 =
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