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Vorlesung Quantum Computing SS ‘08
1
A scalable system with well characterized qubits
Long relevant decoherence times, much longer than the gate operation time
A qubit-specific measurement capability A
A „universal“ set of quantum gates U
The ability to initialize the state of the qubits to a simple fiducial state, e.g. |00...0>
„DiVincenzo “ criteria
DiVincenzo: Fortschr. Phys. 48 (2000) 9-11, pp. 771-783
Vorlesung Quantum Computing SS ‘08
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Quantum Computing with Ions in Traps
How to trap ions
State preparation
Qubit operations
CNOT
Deutsch – Jozsa Algorithm
advantages/drawbacks
Vorlesung Quantum Computing SS ‘08
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Paul Trap
Nobel Prize 1989
centre is field free
quadrupole field x and y motions not coupled!
Chemnitz University
Vorlesung Quantum Computing SS ‘08
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Linear Trap
x
y
z
U1
RUac
Uac(t) = Ur + V0 cos Tteffective potential:
eff = x2 x2 + y
2 y2 + z2 z2
x = y >> z
(averaged over one rf cycle)
U2
z0
M. Sasura and V. Buzek: quant-ph/0112041
Vorlesung Quantum Computing SS ‘08
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Ions in a Linear Trap
z = 2qU12
mz02
typical operation parameters:
V0 = 300 – 800 VT/2 = 16 – 18 MHz
U12 = 2000 V
z0 = 5 mm
R = 1.2 mm
z/2 = 500 - 700 kHz
x,y/2 = 1.4 – 2 MHz (40Ca+)
70 m
40Ca+
24Mg+
Seidelin et al: Phys. Rev. Lett. 96, 253003 (2006)
Nägerl et al: Phys. Rev. A 61, 023405 (2000)
Vorlesung Quantum Computing SS ‘08
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quantum computing with ions
H H-1
calculation
U
preparation
read-out
|A|
time
time
the ions are prepared to be in their ground state
Doppler cooling side band cooling
1st step 2nd step
kBT << ħz
Vorlesung Quantum Computing SS ‘08
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Doppler cooling
when absorbing a photon, also the momentum is transferred
the net momentum of the spontaneousemission is zero
E = ħp = ħk
E = 0p = 0
E = ħp = ħk
k
absorption
for ions moving toward the laser beam the lightappears blue shifted → use a red detuned laser
= 0 + k ∙ v
Ca
Vorlesung Quantum Computing SS ‘08
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side band cooling
Doppler cooling gets down to kBT ≈ ħ
internal electronic ground and excited state |g,|e
trapped ions movingin harmonic potentialstates |n, n= 0,1,2…
cooling:
|g,n → |e,n-1|e,n-1 → |g,n-1
Vorlesung Quantum Computing SS ‘08
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ions used as qubits
electronic states as qubits (“pseudo-spin”)(CNOT, Deutsch-Jozsa Algorithm, Quantum-Byte)
hyperfine states as qubits(CNOT, error correction, Grover Algorithm)
Vorlesung Quantum Computing SS ‘08
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40Ca+ as qubit
42S1/2
42P1/2
42P3/2
32D3/2
32D5/2
397 nm 729 nm
854 nm
866 nm
|0
|1
quad
rupo
letr
ansi
tion
used for Laser cooling
quadrupole transition with relatively long relaxation time
for cooling: 866 nm transition has to be irradiated as well, otherwise charge carriers will be trapped in 32D3/2 orbital
fluorescence detection for read-out
dete
ctio
n
Nägerl et al: Phys. Rev. A 61, 023405 (2000)D
5/2 o
ccup
atio
n P
D
D5
/2 o
ccup
atio
n P
D
red sideband blue sidebandafter
Doppler cooling
Vorlesung Quantum Computing SS ‘08
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9Be+ as qubit
electron spin S = 1/2, ms = 1/2nuclear spin I = 3/2, mI = 1/2, 3/2
F = I + S, mF
22P3/222P1/2 12 GHz
22S1/2
|F=2, mF=2
|F=1, mF=1
|
| 1.25 GHz
Dop
pler
coo
ling
hyperfine levels have long relaxation times
sideband cooling |2,2|n → |1,1|n-1; induced spontaneous Raman transition |1,1|n-1 → |2,2|n-1
sideband cooling
detection with fluorescence after + excitation +
det
ectio
n
Monroe et al: Phys. Rev. Lett. 75, 4011 (1995)
red
blueafter Doppler cooling
after sideband cooling
Vorlesung Quantum Computing SS ‘08
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quantum computing
H H-1
calculation
U
preparation
read-out
|A|
time
time
quantum-bit (qubit)
0 1
a10 + a21 =a1a2
Vorlesung Quantum Computing SS ‘08
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qubit operations
how does the system evolve with time?
U(t)e ħ- iHQCt
^^ HQC = Htrap + Hion + Hman
^ ^ ^ ^
Splitting of S = ½ in external magnetic field:
22S1/2
|F=2, mF=2
|F=1, mF=1
|
| s/2 = 1.25 GHz
Hion = -ħs
2
1
-10
0^
B0 = 0.18 mT
Hion = -S∙B = - SzB0 = -LSz^ ^ ^
Vorlesung Quantum Computing SS ‘08
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qubit coupling
coulomb repulsion couples motional degrees of freedom
Htrap = (x2 xi
2 + y2 yi
2 + z2 zi
2 + ) + M2
pi2
M2
e2
40 |ri - rj| i=1 i=1
N N
j>i
trap potential eff
Ekin
coulomb potential
positions at rest
1 mode
2 modes
A. Steane: quant-ph/9608011
^
Vorlesung Quantum Computing SS ‘08
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vibration modes as qubits (bus)
centre of mass motion used as qubit
A. Steane: quant-ph/9608011
i=1 i=1Htrap = ( z
2 zi2 + ) = ħi ai
†ai
pi2
M2
M2
N N
i
z = 2qU12
mz02
z
z 3
J.F. Poyatos et al., Fortschr. Phys. 48, 785
Vorlesung Quantum Computing SS ‘08
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9Be+: the two qubit system
22P1/2 50 GHz
22S1/2
|F=2, mF=2
|F=1, mF=1
|
| |1| |0
|1
| |0s/2 = 1.25 GHz
z/2 = 11.2 MHz
22P3/2
|F=3, mF=3
|0 |aux
|F=2, mF=0
vibrational state: control qubit
hyperfine state: target qubit
Ramantransition + detection
~ 313 nm
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 =00BzB =
B1 cos tB1 sin t
B0
magnetic field rotating in x,y-plane
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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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
B1 cos t00
2 =
Vorlesung Quantum Computing SS ‘08
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spin flip in rotating framehttp://www.wsi.tu-muenchen.de/E25/members/HansHuebl/animations.htm
Vorlesung Quantum Computing SS ‘08
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qubit manipulation: laser interaction
Hman = - ∙ B = mS B = B1x cos(kz-t+)^
Hman ≈ (S+ei + S-e-i)
= m B1/2ħ
ħ2
frame of reference: H0 = ħsSz + ħza†a
only spin state is changed
i (S+aei - S-a†e-i)ħ2
for = s - z “red” side band
i (S+a†ei - S-ae-i)ħ2
for = s + z “blue” side band
change of vibrational state always implies change of spin state
Lamb-Dicke parameter: ≡ 2d0/<<
for = s
Vorlesung Quantum Computing SS ‘08
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qubit rotation
22P1/2 50 GHz
22S1/2
|F=2, mF=2
|F=1, mF=1
|
| |1| |0
|1
| |0s/2
|0 |aux
|F=2, mF=0
Qubit rotation on target qubit U/2,ion
Raman transition with detuning s
Duration of laser pulse: /2 rotation
2e =
iħ
Sy cos /4
cos /4
sin /4
- sin /4
1
1-1 11
√2= = U/2
Vorlesung Quantum Computing SS ‘08
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/2- rotation matrix
-
-
1√2
U/2,ion =
base vectors of the two–qubit register:
Transformation matrix:
U/2,ion = () 1√2
U/2,ion = () 1√2
U/2,ion = () 1√2
U/2,ion = () 1√2
Vorlesung Quantum Computing SS ‘08
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CNOT operation
-
Uph = Uph transformation matrix:
Transformation sequence:
U/2,ion =U-/2,ion Uph
-
-
12
-
-
-
= = UCNOT
Monroe et al: Phys. Rev. Lett. 75, 4714 (1995)
Vorlesung Quantum Computing SS ‘08
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phase rotation
22P1/2 50 GHz
22S1/2
|F=2, mF=2
|F=1, mF=1
|
| |1| |0
|1
| |0s/2
|0 |aux
|F=2, mF=0
Phase rotation on control qubit Uph
Raman transition between and auxiliary state
Full rotation by 2
Uph
Vorlesung Quantum Computing SS ‘08
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quantum computing
H H-1
calculation
U
preparation
read-out
|A|
time
time
quantum-bit (qubit)
0 1
a10 + a21 =a1a2
Vorlesung Quantum Computing SS ‘08
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Be ions: read-out spin state
|F=2, mF=2
| |1
| |0
22P3/2
|F=3, mF=3
22S1/2
|F=1, mF=1
| |1| |0
+ detection
read-out spin state via fluorescence
prepare desired initial state usingRaman pulses
| on blue side band → |1 on internal state → |1|
perform CNOT
cool system to | |0
Vorlesung Quantum Computing SS ‘08
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Be ions: read out vibrational state
|F=2, mF=2
| |1
| |0
22P3/2
|F=3, mF=3
22S1/2
|F=1, mF=1
| |1| |0
+ detection
read-out spin state
prepare same initial state and do CNOT
convert vibrational into spin state on red side band for | on blue side band for |
read-out spin state via fluorescence