Quantum Computing

1
Quantum Computing
State-of-the-Art
Nuclear Magnetic Resonance
2
Quantum Computing with NMR
Nuclear magnetic resonance
State preparation in an ensemble
Quantum Fourier transform
finding prime factors Shors algorithm
solid state concepts
3
NMR quantum computer
4
the qubits in liquid NMR
Jones in http//arxiv.org/abs/quant-ph/0106067
magnetic moment of nucleus much smaller than of
electron (1/1000)
measuring magnetic moment of a single nucleus not
possible
for reasonable S/N 1018 spins
5
spins in a magnetic field
Eint -mzB0 - gNIzB0 -gNmIhB0
DmI ? 1
DE hn - gNhB0 300 MHz (B0 7 T, 1H)
population difference 510-5
6
spin dynamics
g (My(t)Bz - Mz(t)By)
g My(t)Bz
Mxcos(wLt) Mysin(wLt)
g (Mz(t)Bx - Mx(t)Bz)
-g Mx(t)Bz
Mycos(wLt) - Mxsin(wLt)
g (Mx(t)By - My(t)Bx)
0
7
spin-lattice relaxation T1
1?
nuclei T1 hours days electrons T1 ms
0?
8
spin-spin relaxation T2
1?
0?
9
spin manipulation
Bloch equations
B1ltltB0
10
spin flipping in lab frame
http//www.wsi.tu-muenchen.de/E25/members/HansHueb
l/animations.htm
11
NMR technique
Lieven Vandersypen, PhD thesis
http//arxiv.org/abs/quant-ph/0205193
B0 7-10 T
B1 cos wt
cos wt
B1
B1
Brf 2
0
-sin wt
0
0
12
pulsed magnetic resonance
zero filling
Hanning window
on resonance
13
FID spectrum
14
selective excitation
15
rotating frame
applied RF generates a circularly polarized RF
field, which is static in the rotating frame
16
chemical shift
Cory et al. Fortschr. Phys. 48 (2000) 9-11, 875
The 13C protons feel a different effective
magnetic field depending on the chemical
environment
17
coupling between nuclear spins
Cory et al. Fortschr. Phys. 48 (2000) 9-11, 875
Ecoup h Jij mI(i) mI(j)
18
state preparation
calculation
read-out
time
U
?
?
?
?
H
H-1
?YAY?
time
system cannot be cooled to pure ground state
a mixed ensemble is described by the density
matrix
r y? ?y
19
density matrix
y? a00? b01? g10? d11?
(a b g d)
r y? ?y
pure state only one state in diagonal occupied
with P1
? Tr(r2) 1
mixed state states yi? occupied with Pi
? Tr(r2) lt1
20
states in an ensemble
mI -1/2
energy
mI 1/2
magnetic field
level occupation follows Boltzmann statistics
eb
for
e-b
for
21
pseudo pure states
1
eb e-b
-mzB0/kBT
with e 1 -
reduced density matrix
22
qubit representation
Iz
r
identity is omitted
23
time development
calculation
read-out
time
U
?
?
?
?
H
H-1
?YAY?
time
Liouville von Neumann equation
24
time development
1018 copies of the same nuclear spin
r(0)
req
25
refocusing
if wL ? wr, e.g., due to inhomogeneous B0, the
spin picks up a phase

r(0)
Ix
2r(t) -1?
26
2 qubits
Cory et al. Physica D 120 (1998), 82
2,3-dibromo-thiophene
27
Simple CNOT
spin
a
b
?
CNOT operation
?
energy
?
?
spin levels individually addressable
28
coupling between nuclear spins
Cory et al. Fortschr. Phys. 48 (2000) 9-11, 875

29
CNOT with Alanine
Cory et al. Fortschr. Phys. 48 (2000) 9-11, 875
ai?
ao?
bi?
bo?
ci?
co?
NO operation
UNO
p
UCNOT