quantum computing … applications in informatics and physics p. shor, 1994: factorization of large...
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Quantum computing …
Applications in informatics and physics
P. Shor, 1994: factorization of large numbers is polynomial on a quantum computer, exponential on a classical computer
L. Grover, 1997: data base search N1/2 quantum queries, N classical
simulation of Schrödinger equations or any unitary evolution
quantum cryptography / repeaters / quantum links
improved atomic clocks
understanding the fundamentals of quantum mechanics / Gedanken-Experimente
Experiments with entangled matter
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The prototype
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Quantum gate proposal(s)
21121
Further gate proposals: • Cirac & Zoller• Mølmer & Sørensen, Milburn• Jonathan & Plenio & Knight• Geometric phases
0111
1101
1010
0000
control bitcontrol bit target bittarget bit
controlled NOTcontrolled NOT
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D5/2
729 nm
|1>
|0>
internal qubit
Qubits in a single 40Ca+ ion
S1/2
motional qubit
|0>|1>1
n=0
2
|S,n> |D,n> : carrier transition ()
|S,n> |D,n±1> : sideband transition ()
"computational subspace"
|S,0>
|D,0>|D,1>
|S,1>
COHERENT LASER MANIPULATION (Rabi oscillations)
First single-ion quantum gate: Monroe et al. (Wineland), PRL 75, 4714 (1995).
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2 ions + motion = 3 qubits
With several ions, the motional qubits are sharedWith several ions, the motional qubits are shared
vibrational modes computational subspace: 2 ions, 1 mode
|S,S,0>
|D,S,0>|D,S,1>
|S,S,1>
|S,D,0>|S,D,1>
|D,D,0>|D,D,1>
laser on ion 2
laser on ion 1laser on ion 2
laser on ion 1
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Details of C-Z CNOT gate operation (Phase gate)
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Experimental techniques
conditions vs. achievements
Experimental techniques
conditions vs. achievements
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D. P. DiVincenzo, Quant. Inf. Comp. 1 (Special), 1 (2001)
● Qubits store superposition information, ion string, but scalability?scalable physical system
● Ability to initialize the state of the qubits ground state cooling
● Long coherence times, hard work much longer than gate operation time
● Universal set of quantum gates: Coherent pulses on Single bit and two bit gates carrier and sidebands,
addressing
● Qubit-specific measurement capability Shelving, imaging
Some requirements ...
See Toni's lecture today
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Innsbruck linear ion trap
|1>
|0>
|1>
|0>
Two 2-level systems
5mm
MHz5radial
MHz27.0 axial
+HV +HV
RF
RF
GND
GND
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P3/2
854 nm
393 nm
S1/2
P1/2
D3/2
397 nm
866 nm
s1D5/2
729 nm
Level scheme of 40Ca+
S1/2
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Zeeman structure in non-zero magnetic field: :
(+ motional degrees of freedom ...)
S1/2
D5/2
5/23/2
-3/2-5/2
- /21
- /21/21
/21
2-level-system
1/2 5/2
Zeeman structure of the S1/2 – D5/2 transition
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P3/2
S1/2
P1/2
D3/2
s1D5/2
729 nm
Manipulation by laser pulses on 729 nm transition(~ 1 ms coherence time)
|1>
|0>
qubit
Level scheme of 40Ca+
S1/2
Superpositions of S1/2(m=1/2) and D5/2(m=5/2) form qubits
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P3/2
S1/2
P1/2
D3/2
397 nm
866 nm D5/2 |1>
|0>
qubit
Level scheme of 40Ca+
S1/2
State detection by photon scattering on S1/2 to P1/2 transition at 397 nm (> 99% in ~ 3 ms)
Detector
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P3/2
854 nm
S1/2
P1/2
D3/2
D5/2
729 nm
|1>
|0>
qubit
Level scheme of 40Ca+
S1/2
Motional state preparation by sideband
cooling on 729 nm transition (> 99.9%)
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S1/2
D5/2
|n> = |0> |1> |2>
coupled system & transitions
g
e
2-level-atom harmonic trap
spectroscopy: carrier and sidebands
Laser detuning
n = 1n = -1
n = 0
0n
12
Motional sidebands
Rabi frequencies
1 n
n
Carrier:
Red SB:
Blue SB:
k <0|x2|0>1/2 «
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Excitation spectrum of the S1/2 – D5/2 transition
ax = 1.0 MHzrad = 5.0 MHz
(only one Zeeman
component)
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Excitation spectrum of two ions
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Sideband cooling of two ions
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Laser pulses for coherent manipulation
coh >> gate
AOM = acousto-optical modulator, based on Bragg diffraction
"Ampl" includes switching on/off
AOM = acousto-optical modulator, based on Bragg diffraction
"Ampl" includes switching on/off
Ampl
Ampl
cw laser
RF
AOM I
t
I
t
to trap
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0
0.2
0.4
0.6
0.8
1
0 100 200 300 400 500 6000
0.2
0.4
0.6
0.8
1
Tim e (µs)
Po
pula
tion
of D
sta
te
see also experiments at NISTRoos et al., PRL 83, 4713 (1999)
S1/2
D5/2
0
1
1
2
Quantum state engineering
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Blue sideband
Blue sideband/2 /2
D-s
tate
po
pu
lati
on
0 50 100 150 200 250 3000
0.2
0.4
0.6
0.8
1
Pulse length (s)
Rabi-flops on blue sideband
Ramsey Interference
0 500
0.2
0.4
0.6
0.8
1
D-s
tate
po
pu
lati
on
100 150 200 250 300
Pulse length (s)
Qubit rotations
|S,0>
|D,0>|D,1>
|S,1>
|S,0>
|D,0>|D,1>
|S,1>
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Addressing of ions in a string
Well-focussed laser beam
● beam steering with electro-optical deflector
● addressing waist ~ 2.5 - 3.0 mm
● < 1/400 intensity on neighbouring ion
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Individual ion detectionon CCD camera
5µm
quantum state populations pSS,pSD,pDS,pDD
|SS>|DS>
|DD> |SD>
Two-ion histogram (1000 experiments)
region 1 region 2
|SS>
|SD>
|DS>
|DD>
Quantum state discrimination
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Cirac-Zoller Quantum CNOT Gatewith two trapped ions
Cirac-Zoller Quantum CNOT Gatewith two trapped ions
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Detection
ion 1
motion
ion 2
,S D
,S D
0 0
control qubit
target qubit
SWAP
1 2
Cirac-Zoller two-ion controlled-NOT gate
SWAP-1
Preparation |S> = bright|D> = dark
CNOT
"bus" qubit
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Result : schematic
S S S S
S D S D
D D DS
D D D Scontrolcontrol targettarget
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SS SS DS DD
SD SD DD DS
Result : full time evolution
Pre
pa
rati
on
De
tec
tio
n
every point = 100 single measurements, line = calculation (no fit)
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Details of time evolution
Prep
aration
SW
AP
SW
AP
-1CNOT betweenmotion and ion 2
Detectio
n
SS SS
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input
output
expideal >|2
Measured fidelity (truth table)
F. Schmidt-Kaler et al., Nature 422, 408 (2003)
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|SS> + ei|DD>
{ |SS>+|DD>, |SD>+|DS>}
controlcontrol targettarget
(|S>+|D>)|S>
Experimental sequence:
Ion 1Ion 1
Ion 2Ion 2
CNOT /2
/2
/2
Deterministic entanglement
"Super-Ramsey experiment"
Detection: Parity check ...
CNOTCNOT
|S>|S>
local /2 rotation local /2 rotation
local (/2,) rotationlocal (/2,) rotation
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outputpreparation gate detect
Gate coherence
Pro
ject
ion
CNOT
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Fidelity = 0.5 ( PSS + PDD + contrast) = 71(3)%
Oscillation with 2 entanglement !
Parity and fidelity
Ion 1Ion 1
Ion 2Ion 2
CNOT /2
/2
/2
Phase P
arit
y: P
SS+
PD
D-P
DS-P
SD
54% contrast
|SS>+|DD> ↔ |SD>+|DS>
"super-Ramsey experiment"
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Examples of experimental
problems & solutions
Examples of experimental
problems & solutions
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computational subspace
,0S,1S
,0D,1D out of CS !
2~Rabi
1~Rabi
2π
naive idea : -pulse on blue SB composite SWAP (from NMR)
computational subspace
,0S,1S
,0D,1D
4
Gate pulses (I) : SWAP
(works if initial state is not |S,1>)
Swap information from internal into motional qubit and back
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1
2
3
,0 ,1D Son
4 ,1 ,2D Son
1
3
I. Chuang et al., Innsbruck (2002)
3-step composite SWAP operation
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computational subspace
Phase factor -1 for all except |D,0 >
,0S,1S
Phase factor -1 for |S,1 >
Cirac & Zoller (1995) Composite phase gate
,0S,1S
,0D,1D
22
Gate pulses (II) : Phase gate
use auxiliarylevel
0,Aux1,Aux
M. H. Levitt, Prog. NMR Spectrosc., 1986I. L. Chuang, Innsbruck, 2002
Phase factor conditioned on state
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1 1 1 1( , ) , 2 2,0 , 2 2,0R R R R R
1
2
3
4
,0 ,1S D2on
Composite phase gate (2 rotation)
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1 1 1 1( , ) 2, 2 ,0 2, 2 ,0R R R R R
4
3
2
1
2 also on
2,1, DS
Action on |S,1> - |D,2>
no populationoutside CS !
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ion 1
motion
ion 2
,S DSWAP-1
,S D
0 0SWAP
Ion 1Ion 1
Ion 2Ion 2
pulse sequence
control bitcontrol bit
target bittarget bit
Cirac-Zoller two-ion controlled-NOT operation
blue0
blue
c0
blue
blue½
0
blue½
0
blue
c
CNOT
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SS → SS
Details of time evolution
Ion 1Ion 1
Ion 2Ion 2
blue0
blue
c0
blue
blue½
0
blue½
0
blue
c
Time (s)
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AC Stark shift & its compensation
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AC Stark shift & its compensation
1/2 → -5/2
1/2 → -1/2
1/2 → 3/2
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Why Bell states ?
entangled massive particles, distinguishable
resource for quantum cryptography / repeaters / quantum links
improved atomic clocks
understanding the fundamentals of quantum mechanics /
EPR paradox, Gedanken-Experimente
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Generation of Bell states with three pulses
atom 1 atom 2 atom 2
Carrier pulses:
Blue sideband pulses
2/
result
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z
x
y
/2 - pulse
Rotation of the Bloch sphere prior to state measurement
Principle of tomography (1 atom)
Measurement of spin components
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SSSD
DSDD SSSDDSDD
SSSD
DSDD SSSDDSDD
F=0.91
Bell state generation & tomography
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SSSD
DSDD SSSDDSDD
SSSD
DSDD SSSDDSDD
F=0.90
Bell state generation & tomography
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SSSD
DSDD SSSDDSDD
SSSD
DSDD SSSDDSDD
F=0.88
Bell state generation & tomography
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SSSD
DSDD SSSDDSDD
SSSD
DSDD SSSDDSDD
F=0.91
Bell state generation & tomography
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Fidelity : F = 0.91
Peres-Horodecki criterion :
Violation of a CHSH inequality: S(0°,90°,45°,135°)(exp) = 2.53(6) > 2
E(exp) = 0.79 (4)
Entanglement of formation for a pair of qubits (Wooters ’98) :
Entanglement characterization
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Cirac-Zoller quantum CNOT gate with two trapped ionsCirac-Zoller quantum CNOT gate with two trapped ions
F. Schmidt-Kaler, C. Becher, J. E., H. Häffner, C. Roos, W. Hänsel, G. Lancaster, S. Gulde, M. Riebe, T. Deuschle,
I.L. Chuang, R. Blatt
F. Schmidt-Kaler et al., Nature 422, 408 (2003)
The works and the workers
Bell States of Atoms with Ultralong Lifetimes and Their Tomographic State Analysis
Bell States of Atoms with Ultralong Lifetimes and Their Tomographic State Analysis
C. F. Roos et al., Phys. Rev. Lett. 92, 220402 (2004)
Deutsch-Jozsa quantum algorithm with a single trapped ionDeutsch-Jozsa quantum algorithm with a single trapped ion
S. Gulde et al., Nature 421, 48-50 (2003).
Precision measurement and compensation of optical Stark shifts for an ion-trap quantum processor
Precision measurement and compensation of optical Stark shifts for an ion-trap quantum processor
H. Häffner et al., Phys. Rev. Lett. 90, 143602 (2003).
http://heart-c704.uibk.ac.at/papers.html