Quantum random walks
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Quantum random walks
Andre Kochanke
Max-Planck-Institute of Quantum Optics 7/27/2011
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Motivation
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Motivation
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?
??
?
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Overview
• Density matrix formalismRandomness in quantum mechanics
• Transition from classical to quantum walks
• Experimental realisation
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Density matrix approach
• Two state system 12
12
-1 10
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Density matrix approach
• Two state system
• Density operator
12
12
0-1 1
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Density matrix approach
• Density operator
Pure state Mixed state
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12
0-1 1
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Galton box
• Binomial distribution
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Galton box
• Statistical mixture
• First four steps
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Quantum analogy
• Used Hilbert space
• Specify subspaces
0-1-2-3 1 2 3
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Quantum analogy
• Evolution with shift and coin operators
0-1-2-3 1 2 3
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Quantum analogy
• Evolution with shift and coin operators
0-1-2-3 1 2 3
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Quantum analogy
• Evolution with shift and coin operators
0-1-2-3 1 2 3
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Quantum analogy
• State transformation
• Density matrix transformation
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Quantum analogy
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Quantum analogy
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Position
pc
pq
Variances
pc
pq
pc
pq
Position
Position
100 steps
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• Phase shift• Transformed density matrix
• Average
• Decoherence effect
Decoherence
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Different realisations
• C. A. Ryan et al., “Experimental implementation of a discrete-time quantum random walk on an NMR quantum-information processor”, PRA 72, 062317 (2005)
• M. Karski et al., “Quantum Walk in Position Space with Single Optically Trapped Atoms”, Science 325, 174 (2009)
• A. Schreiber et al., “Photons Walking the Line: A Quantum Walk with Adjustable Coin Operations”, PRL 104, 050502 (2010)
• F. Zähringer et al., “Realization of a Quantum Walk with One and Two Trapped Ions”, PRL 104, 100503 (2010)
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SetupCCD
Microwave
Dipole trap laser
Objective
Fluorescence picture
Cs
jF = 4;mF =4i
jF = 3;mF =3i
Mic
row
ave
M. Karski et al., Science 325, 174 (2009)
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Setup
Polarizations and
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Setup
Polarizations and
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Results
M. Karski et al., Science 325, 174 (2009)
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Results
M. Karski et al., Science 325, 174 (2009)
Theoretical expectation
6 steps
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Results
Theoretical expectationM. Karski et al., Science 325, 174 (2009)
6 steps
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Results
Theoretical expectation
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Results
Theoretical expectationM. Karski et al., Science 325, 174 (2009)
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Results
Gaussian fitM. Karski et al., Science 325, 174 (2009)
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Conclusion
• The density matrix formalism allows you to describe cassical and quantum behavior
• Karski et al. showed how to prepare a quantum walk with delocalized atoms
• The quantum random walk is not random at all
M. Karski et al., Science 325, 174 (2009)
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References
• C. A. Ryan et al., “Experimental implementation of a discrete-time quantum random walk on an NMR quantum-information processor”, PRA 72, 062317 (2005)
• M. Karski et al., “Quantum Walk in Position Space with Single Optically Trapped Atoms”, Science 325, 174 (2009)
• SOM for “Quantum Walk in Position Space with Single Optically Trapped Atoms”, Science 325, 174 (2009)
• A. Schreiber et al., “Photons Walking the Line: A Quantum Walk with Adjustable Coin Operations”, PRL 104, 050502 (2010)
• F. Zähringer et al., “Realization of a QuantumWalk with One and Two Trapped Ions”, PRL 104, 100503 (2010)
• M. Karksi, „State-selective transport of single neutral atoms”, Dissertation, Bonn (2010)• C. C. Gerry and P. L. Knight, „Introductory Quantum Optics“, Cambridge University Press,
Cambridge (2005)• M. A. Nielsen and I. A. Chuang, „Quantum Computation and Quantum Information“,
Cambridge University Press, Cambridge (2000)