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Hartmut HäffnerInstitute for Quantum Optics and Quantum InformationInnsbruck, Austria
Quantum computing with trapped ions
Berkeley, Nov 25th 2008
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• Basics of ion trap quantum computing
• Measuring a density matrix
• Quantum gates
• Deutsch Algorithm
000001010011
100011110011
Quantumprocessor
Cavity QED
Superconducting qubits
Trapped ions
Quantum dots
NMR
© A. Ekert
Which technology ?
000001010011
100011110011
Quantumprocessor
Cavity QED
Superconducting qubits
Trapped ions
Quantum dots
NMR
© A. Ekert
Which technology ?
Good things about ion traps:
Ions are excellent quantum memories; single qubit coherence times > 10 minutes have been demonstrated (Boulder 1991)
(
Ions can be controlled very well
Many ideas to scale ion traps
Bad things about ion traps:
Slow (~1 MHz)
Technically demanding
Why trapped ions ?
The hardware
S1/2
P1/2
D5/2
Qubit
Trapped ions form the quantum register
Innsbruck quantum processor
Trap electrodes
• Scalable physical system, well characterized qubits
• Ability to initialize the state of the qubits
• Long relevant coherence times, much longer than gate operation time
• “Universal” set of quantum gates
• Qubitspecific measurement capability
DiVincenzo criteria
1. Initialization in a pure quantum state P1/2 D5/2
=1s
S1/2
40Ca+
P1/2
S1/2
D5/2P1/2
S1/2
D5/2D5/2
S1/2
Experimental procedure
1. Initialization in a pure quantum state P1/2 D5/2
=1s
S1/2
40Ca+
P1/2
S1/2
D5/2
Dopplercooling Sideband
cooling
P1/2
S1/2
D5/2D5/2
S1/2
Experimental procedure
2. Quantum state manipulation on S1/2 – D5/2 transition
Quantum statemanipulation
1. Initialization in a pure quantum state P1/2
S1/2
D5/2D5/2
S1/2
Experimental procedure
P1/2 D5/2
=1s
S1/2
40Ca+
1. Initialization in a pure quantum state:
2. Quantum state manipulation on S1/2 – D5/2 transition
P1/2
S1/2
D5/2
Dopplercooling Sideband
cooling
P1/2
S1/2
D5/2
Quantum statemanipulation
P1/2
Fluorescencedetection
3. Quantum state measurement by fluorescence detection
S1/2
D5/2
Experimental procedure
P1/2 D5/2
=1s
S1/2
40Ca+
1. Initialization in a pure quantum state:
2. Quantum state manipulation on S1/2 – D5/2 transition
P1/2
S1/2
D5/2
Dopplercooling Sideband
cooling
P1/2
S1/2
D5/2
Quantum statemanipulation
P1/2
Fluorescencedetection
3. Quantum state measurement by fluorescence detection
S1/2
D5/2
Experimental procedure
5µm
50 experiments / s
Repeat experiments100200 times
Spatially resolveddetection withCCD camera
Two ions:
Ds
tate
pop
ulat
ion
Rabi oscillations
P1/2
S1/2
D5/2D5/2
Ds
tate
pop
ulat
ion
Rabi oscillations
P1/2
S1/2
D5/2D5/2
Ds
tate
pop
ulat
ion
Rabi oscillations
P1/2
S1/2
D5/2D5/2
The phase ...
+
++ =
+
++e =ωt
The phase ...
The phase ...
CCD
Paul trap
Fluorescencedetection
electrooptic deflector
coherentmanipulation of qubits
dichroicbeamsplitter
inter ion distance: ~ 4 µm
addressing waist: ~ 2 µm
< 0.1% intensity on neighbouring ions
10 8 6 4 2 0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Exc
itatio
nDeflector Voltage (V)
D
Addressing single qubits
Memory errors:
Bitflips
Dephasing
Operationial errors
technical imperfections …
Decoherence mechanisms
Ramsey Experiment
= 15.9 ms
π / 2 π / 2Ramsey Time
Dephasing of qubits
Zeeman shift
106 s
101 s
103 s
104 s
103 s
101 s
Single qubit gates
Two qubit gates (Geometric phase gates)
T
Single qubit coherence (magnetic field sensitive)
S
Coherence of the motion
105 s
102 s
100 s
102 s
Two qubit gates (CiracZoller approach)
T
Realized time scales
qubit
Raman transitions:
Excited state
Ground state
Long lived qubits
Raman transitions:
Excited state
Ground state
Long lived qubits
From: C. Langer et al., PRL 95, 060502 (2005), NIST
Level scheme of 9Be+:
Long lived qubits
From: C. Langer et al., PRL 95, 060502 (2005), NIST
Long lived qubits
106 s
101 s
103 s
104 s
103 s
101 s
Single qubit gates
Two qubit gates (Geometric phase gates)
T
Single qubit coherence (magnetic field sensitive)
S
Coherence of the motion
105 s
102 s
100 s
102 s
Two qubit gates (CiracZoller approach)
T
Realized time scales
Single qubit coherence (magnetic field insensitive)
S
Single qubit coherence (magnetic field insensitive + RF drive)
S
The common motionacts as the quantumbus.
Having the qubits interact
50 µm
The common motionacts as the quantumbus.
Having the qubits interact
harmonic trap
……
Ion motion
harmonic trap
…
2levelatom joint energy levels
…
Ion motion
carrier
carrier and sidebandRabi oscillationswith Rabi frequencies
LambDicke parameter
Coherent manipulation
• Introduction to quantum information
• Entangled states
• Teleportation
• Scaling of ion trap quantum computers
• Wiring up trapped ions
…
… …
…
Generation of Bell states
π
…
… …
…
Generation of Bell states
π/2
…
… …
…
Generation of Bell states
π…
… …
…
Generation of Bell states
π…
… …
…
Bell states with atoms
9Be+: NIST (fidelity: 97 %)
40Ca+: Oxford (83%)
111Cd+: Ann Arbor (79%)
25Mg+: Munich
40Ca+: Innsbruck (99%)
Generation of Bell states
Fluorescencedetection withCCD camera:
Coherent superposition or incoherent mixture ?
What is the relative phase of the superposition ?
Analysis of Bell states
Measurement of the density matrix:SSSDDS
DD SSSDDSDD
A measurement yields the zcomponent of the Bloch vector
=> Diagonal of the density matrix
Measuring a density matrix
A measurement yields the zcomponent of the Bloch vector
=> Diagonal of the density matrix
Rotation around the x or the yaxis prior tothe measurement yields the phase informationof the qubit.
Measuring a density matrix
A measurement yields the zcomponent of the Bloch vector
=> Diagonal of the density matrix
=> coherences of the density matrix
Rotation around the x or the yaxis prior tothe measurement yields the phase informationof the qubit.
Measuring a density matrix
SSSD
DSDD SSSDDSDD
SSSD
DSDD SSSDDSDD
SSSD
DSDD SSSDDSDD
Decoherence properties of qubits
Bellstate survives
Roos et al., Science 304, 1478 (2004)
Measurementof the center ion
A “real” thought experiment
Generalized Bell states
656100 measurements~ 10 h measurement time
Genuine 8particle entanglement
Häffner et al., Nature 438, 643 (2005)
Generalized Bell states
Quantum gates …
control target
...allows the realization of a universal quantum computer !
Having the qubits interact
control target
...allows the realization of a universal quantum computer !
Having the qubits interact
Most popular gates: CiracZoller gate (SchmidtKaler et al., Nature 422, 408 (2003)). Geometric phase gate (Leibfried et al., Nature 422, 412 (2003)). MølmerSørensen gate (Sackett et al., Nature 404, 256 (2000)).
Control bit
Target bit
Target
A controlledNOT operation
A controlledNOT operation
Ion 1:
Vibration:
Ion 2:
SWAP1SWAPControl qubit
Target qubit
A controlledNOT operation
Ion 1
Vibration
Ion 2
SWAP1SWAPControl qubit
Target qubit
Ion 1
Ion 2
Pulse sequence:Laser frequencyPulse lengthOptical phase
1
2
3
4
Composite phase gate (2π rotation)
4
3
2
1
Action on |S,1> |D,2>
0=ϕ 0=ϕ 2πϕ =
0 20 40 60 80 100 120 140 160 1800
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (µs)
D5/
2 e
xcita
tion
Single ion composite phase gate
Time (µs)
T
D5/
2 e
xcita
tion
0 50 100 150 200 250 3000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Phase gate
0.978 (5)
π/2π/2
Single ion composite CNOT gate
A controlledNOT operation
Ion 1
Vibration
Ion 2
SWAP1SWAPControl qubit
Target qubit
Ion 1
Ion 2
Pulse sequence:Laser frequencyPulse lengthOptical phase
OutputInput
Probability
Truth table of the CNOT
prepare CNOT detect
Using a CNOT to create a Bell state
output
Raman transitions between
Interaction of two ions via common motion.
n=0
n=1
MølmerSørensen gate
Raman transitions between
Interaction of two ions via common motion.
MølmerSørensen gate
Raman transitions between
Interaction of two ions via common motion.
MølmerSørensen gate
bicromatic beam applied to both ions
Technical realization
entangled ?
Entangling ions
J. Benhelm et al., Nature Physics 4, 463 (2008)
Theory: C. Roos, NJP 10, 013002 (2008)
gate duration
average fidelity: 99.3 (2) %
entangled
measure entanglementvia parity oscillations
Entangling ions
J. Benhelm et al., Nature Physics 4, 463 (2008)
Theory: C. Roos, NJP 10, 013002 (2008)
maximally entangled states
Gate concatenation
Gate performance
The MS gate operation is optimized by
• pulse shaping, minimizes offresonant excitation
• phase relation of amplitude modulated dualfrequency beam
C. Roos, NJP 10, 013002 (2008).
Optimization of the gate
• Scalable physical system, well characterized qubits / ?
• Ability to initialize the state of the qubits
• Long relevant coherence times, much longer than gate operation time
• “Universal” set of quantum gates
• Qubitspecific measurement capability
Often neglected:
• exceptional fidelity of operations
• low error rate also for large quantum systems
• all requirements have to met at the same time
DiVincenzo criteria
Cirac, Zoller, Kimble, Mabuchi Zoller, Tian, Blatt
Scaling of ion trap quantum computers
Its easy to have thousands of coherent qubits …but hard to control their interaction
Kielpinski, Monroe, Wineland
The Michigan T trap
An implementation of the Deutschalgoritm …
Deutsch‘s problem: Introduction
Decide which class the coin is: False (equal sides) or Fair
A single measurement does NOT give the right answer
Front
Back
Deutsch‘s problem: Mathematical formulation
4 possible coins are representend by 4 functions
false
fair0110f(1)
(
10 10f(0)
(
Case 4Case 3Case 2Case 1
BalancedConstant
false
fair
ZCNOTCNOTNOTIDz + f(x)
)
0110f(1)
(
10 10f(0)
(
Case 4Case 3Case 2Case 1
BalancedConstant
Deutsch‘s problem: Mathematical formulation
4 possible coins are representend by 4 functions
Physically reversible process realized by a unitary transformation+
Deutsch Jozsa quantum circuit
f1
f2
f3
f4
1000010000100001
1000010000010010
0100100000100001
0100100000010010
Ufn
x x
z f(x) + z
NOT
ID
CNOT
ZCNOT
Case Logic Quantum circuit Matrix
Quantum analysis gives the right answer after a single measurement!
•D. Deutsch, R. Josza, Proc. R. Soc. London A439, 553 (1992)
D
•M. Nielsen, I. Chuang, QC and QI, Cambridge (2000)
M
Uf
x
z
x
z + f(x)
z
|0>
|1>
Deutsch Jozsa quantum circuit
π/2
π/2
π/2
π/2
Uf
x
z
x
z + f(x)
z
|0>
|1>
No information in the second qubit
electronic qubit
motional qubit
π/2
π/2
π/2
π/2
D5/2
729 nm
|1>
|0>
internal qubit
Qubits in 40Ca+
S1/2
motional qubit
|0>|1>1
n=0
2
computational subspace
|S,0>
|D,0>|D,1>
|S,1>
Uf
x
z
x
z + f(x)
z
|0>
|1>
No information in the second qubit
electronic qubit
motional qubit
π/2
π/2
π/2
π/2
Deutsch Jozsa quantum circuit
f1
f2
f3
f4
1000010000100001
1000010000010010
0100100000100001
0100100000010010
Ufn
x x
z f(x) + z
NOT
ID
CNOT
ZCNOT
Case Logic Quantum circuit Matrix
Deutsch Jozsa: Realization
x x
z f(x) + z
Deutsch Jozsa: Realization
x x
z f(x) + z
Swap Swap
|S,0>
|D,0>|D,1>
|S,1>
3step composite SWAP operation
1
2
3
1
3
I. Chuang et al., Innsbruck (2002)
I
Deutsch Jozsa: Realization
x x
z f(x) + z
Swap Swap
Deutsch Jozsa: Realization
x x
z f(x) + z
Swap Swap
Phasegate
1
2
3
4
Composite phase gate (2π rotation)
4
3
2
1
Action on |S,1> |D,2>
Deutsch Jozsa: Realization
x x
z f(x) + z
Swap Swap
Phasegate
Deutsch Jozsa: Realization
x x
z f(x) + z
Swap Swap
Phasegate
Deutsch Jozsa: Realization
x x
z f(x) + z
Swap Swap
Phasegate
Phasegate
Deutsch Jozsa: Realization
Deutsch Jozsa: Realization
Time (µs)
T
Deutsch Jozsa: Result
0.986(4)
0
0.931(9)
0
0.90(1)
0
measured |<1|w>|2
1111expected |<1|w>|2
0.975(2)
0
0.975(4)
0
0.087(6)
0
0.019(6)
0
measured |<1|a>|2
1100expected |<1|a>|2
Case 4Case 3Case 2Case 1
BalancedConstant
S. Gulde et al., Nature 412, 48 (2003)
• Basics of ion trap quantum computing
• Measuring a density matrix
• Quantum gates
• Deutsch Algorithm
Conclusions
Berkeley, Nov 25th 2008
€$
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FWF SFB
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