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Quantum random walks

Quantum random walks. Andre Kochanke. 7/27/2011. Max-Planck-Institute of Quantum Optics. Motivation. Motivation. ?. ?. ?. ?. Overview. Density matrix formalism Randomness in quantum mechanics Transition from classical to quantum walks Experimental realisation.

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Quantum random walks

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  1. Quantum randomwalks Andre Kochanke 7/27/2011 Max-Planck-Institute of Quantum Optics

  2. Motivation

  3. Motivation ? ? ? ?

  4. Overview • Density matrix formalism Randomness in quantummechanics • Transition fromclassicaltoquantumwalks • Experimental realisation

  5. Density matrix approach • Two state system -1 0 1

  6. Density matrix approach • Two state system • Density operator -1 0 1

  7. Density matrix approach • Density operator -1 0 1 Pure state Mixed state

  8. Galton box • Binomial distribution

  9. Galton box • Statistical mixture • First foursteps

  10. Quantum analogy • Used Hilbert space • Specify subspaces -2 0 -3 1 3 -1 2

  11. Quantum analogy • Evolution with shift and coin operators -2 0 -3 1 3 -1 2

  12. Quantum analogy • Evolution with shift and coin operators -2 0 -3 1 3 -1 2

  13. Quantum analogy • Evolution with shift and coin operators -2 0 -3 1 3 -1 2

  14. Quantum analogy • State transformation • Density matrix transformation

  15. Quantum analogy

  16. Quantum analogy 100 steps pc pq pc pq Position pc pq Position Variances Position

  17. Decoherence • Phase shift • Transformed density matrix • Average • Decoherence effect

  18. 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)

  19. Setup CCD Fluorescence picture Objective Dipole traplaser Microwave Cs Microwave M. Karski et al.,Science 325, 174 (2009)

  20. Setup Polarizationsand

  21. Setup Polarizationsand

  22. Results M. Karski et al.,Science 325, 174 (2009)

  23. Results 6 steps Theoretical expectation M. Karski et al.,Science 325, 174 (2009)

  24. Results 6 steps Theoretical expectation M. Karski et al.,Science 325, 174 (2009)

  25. Results Theoretical expectation

  26. Results Theoretical expectation M. Karski et al.,Science 325, 174 (2009)

  27. Results Gaussian fit M. Karski et al.,Science 325, 174 (2009)

  28. Conclusion • The density matrix formalism allows you to describe cassical and quantum behavior • Karskiet 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)

  29. 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 Computationand Quantum Information“, Cambridge University Press, Cambridge (2000)

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