quantum computer: basics, gates, algorithms - ulm · quantum computer: basics, gates, algorithms...
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![Page 1: Quantum computer: basics, gates, algorithms - Ulm · Quantum computer: basics, gates, algorithms Ulm, Germany: 40Ca+ • single qubit gate • various two qubit gates … baby-steps](https://reader030.vdocument.in/reader030/viewer/2022021715/5c747fb509d3f287228bf696/html5/thumbnails/1.jpg)
Quantum computer: basics, gates, algorithms
Ulm, Germany: 40Ca+
• single qubit gate• various two qubit gates… baby-steps shown
so far with ion quantum processors
• and how to reach a scalable device in future
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2-level-atom harmonic trap
Laser coupling
dressed system
„molecularFranck Condon“picture
......
Sn ,1−
Dn ,1−Dn,
Dn ,1+
Sn ,1+Sn,
„energyladder“picture
S
D
D
S
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2-level-atom harmonic trap
Laser coupling
dressed system
„molecularFranck Condon“picture
„energyladder“picture
S
D
D
S
......
Sn ,1−
Dn ,1−Dn,
Dn ,1+
Sn ,1+Sn,
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0,S
0,DP
rob.
forD
electronicexcitation
laser pulse length in µs
Coherent qubit rotation
Carrier flops
electronicexcitation
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0,S
0,D1,D
1,S
internalelectronicstate
Vibrational quanta
laser pulse length in µs
Coherent qubit rotation
Carrier flops
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Temporal sequence of quantum logic operationsINPUT OUTPUT
Basics of a quantum computer
Single qubit gate
two-qubitgatetime
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applications in physics and informatics
P. Shor, 1994: factorization of large numbers, L digits, is much moreefficient on a quantum computer than with a classical computer:
classical computer: ~exp(L1/3), quantum computer: ~ L2
L. Grover, 1997: search data base - quantum computer: ~ L
simulation of Schrödinger equations or any unitary evolutionspin interactions, quantum phase transitions
quantum cryptography / repeaters / quantum links
improved atomic clocksunderstanding the fundamentals of quantum mechanics / Gedanken-Experimente
Experiments with entangled matter
Why?
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The requirements for experimental qc
• Qubits store superposition information, scalable physical system
• Ability to initialize the state of the qubits
• Universal set of quantum gates: Single bit and two bit gates
• Long coherence times, much longer than gate operation time
• Qubit-specific measurement capability
D. P. DiVincenzo, Quant. Inf. Comp. 1 (Special), 1 (2001)
Qubit Transformation
10 βαψ +=
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Scalable device?
Experimental status
Quantum Information Roadmapshttp://qist.ect.it/ http://qist.lanl.gov/
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Quantum gate proposal
control bitcontrol bit target bittarget bit
• single bit rotations and quantum gates
• small decoherence• unity detection efficiency• scalable
J. I. Cirac P. ZollerW. Paul
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• single bit rotations and quantum gates
• small decoherence• unity detection efficiency• scalable
J. I. Cirac P. Zoller21121: εεεεε ⊕→− NOTControlled
0111
1101
1010
0000
→
→
→
→
0
1
control bitcontrol bit target bittarget bit
Quantum gate proposal
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Cirac & Zoller gatewith two ions
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→
→
→
→
S S S S
S D S D
D D DS
D D D S
ion 1
motion
ion 2
,S D
,S D
0 0
control qubit
target qubit
SWAP
1ε 2ε
controlcontrol targettarget
Controlled-NOT operation
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→
→
→
→
S S S S
S D S D
D D DS
D D D S
ion 1
motion
ion 2
,S D
,S D
0 0
control qubit
target qubit
|0>, |1>
1ε 2ε
Controlled-NOT operationControlled-NOT operation
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→
→
→
→
S S S S
S D S D
D D DS
D D D S
ion 1
motion
ion 2
,S DSWAP-1
,S D
0 0
control qubit
target qubit
|0>, |1>
1ε 2ε
Controlled-NOT operationControlled-NOT operation
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SWAP and SWAP-1
,0S,1S
,0D,1D
π
SWAP
,0S,1S
,0D,1D
π
SWAP-1
starting with |n=0> phonons,write into and read from the common vibrational mode
π-pulse on blue SB
control bit control bit
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Conditional phase gate
Effect:phase factor of -1for all, except |D,0 >
target bit
2π π2
Composite pulse phase gateI.Chuang, MIT Boston
Rabi frequency:
1+⋅⋅Ω nηBlue SB:
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Composite phase gate (2π rotation)
( ) ( ) ( ) ( )1 1 1 1( , ) , 2 2,0 , 2 2,0R R R R Rθ φ π π π π π π+ + + +=
1
2
3
4
,0 ,1S D↔on2π
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Population of |S,1> - |D,2> remains unaffected
( ) ( ) ( ) ( )1 1 1 1( , ) 2, 2 ,0 2, 2 ,0R R R R Rθ φ π π π π π π+ + + +=
4
3
2
1
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ion 1
motion
ion 2
,S DSWAP-1
,S D
0 0SWAP
Ion 1Ion 1
Ion 2Ion 2
pulse sequence:pulse sequence:
control bit
target bit
laser frequencypulse durationoptical phase
Controlled-NOT operation
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input
output
Fidelity of Cirac-Zoller CNOT
|<Yexp| Yideal >|2F. Schmidt-Kaler et al., Nature 422, 408 (2003)
Fidelity : 73%
M. Riebe et al., PRL 97, 220407 (2006)
Fidelity : 92,6%
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Bichromatic two-qubit gate Milburn, arXiv:quant-ph/9908037.Milburn, Schneider, and James, Fortschr. Phys. 48, 801 (2000).Sörensen and Mölmer, PRL 82, 1971 (1999).Sörensen and Mölmer, PRA 62, 022311 (2000).
The common absorption of red and blue detuned light leads to a coherent evolution |SS> to |DD>. No excitation of |DS> states. Requires only Lamb Dicke limit Bell state with F=83%
Sackett et al., Nature 406, 256 (2000)
|SS0⟩|SS1⟩
|SD0⟩|SD1⟩
|DS0⟩|DS1⟩
|DD0⟩|DD1⟩
...
... ...
...
Optical qubit theory (manual):Roos C.F., New J. Phys. 10 No 1, 2008, 013002
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Probabilities
Mølmer-Sørensen interaction time evolution
Detuning δ = 20 kHz→ gate time 50 µs
p0+p2 = 0.9965(4)
Pulse length τ [µs]
13000 measurements
J. Benhelm, G. Kirchmair, C. F. Roos, R. Blatt, Nature
Physics 4, 463 (2008)
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Fidelity of the created Bell state
A = 0.990(1)p0+p2 = 0.9965(4)
Parity flops
29400 measurements in 35 min
→ F = 99.3(1)%F21=80%Phase φ of analysis pulse
J. Benhelm, G. Kirchmair, C. F. Roos, R. Blatt, Nature
Physics 4, 463 (2008)
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Mølmer-Sørensen interactionEntangle ↔ Disentangle
Detuning δ = 20 kHz→ gate time 50 µspulse shaping 2 µs
Equivalent to17 gate operations
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• Quantengatter
• Verschränkte Zustände
mit 2 bis 8 Ionen
• Teleportation
• Quanten Simulation
• Frequenzstandards
• Photon-Atom Schnittstellen
• Quantensensoren
• Quanten-Phasenübergänge
Experimentelle Höhepunkte von QIPC
MICROTRAPSCALAAQUTE
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Wesentlicher Milestone für STREP-MICROTRAP Partner
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Skalierbarer Quantenprozessor
Segmentierte Mikrofalle erlaubt das Positionieren
vieler Ionen
Laserpulse erzeugen verschränkte Zustände
Vision
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DPG AMOP Düsseldorf | 21.03.2007Seite 30Vision
Skalierbarer Quantenprozessor
Segmentierte Mikrofalle erlaubt das Positionieren
vieler Ionen
Laserpulse erzeugen verschränkte Zustände