Quantum random walks

Andre Kochanke

Max-Planck-Institute of Quantum Optics 7/27/2011

2

Motivation

Motivation

3

?

??

?

4

Overview

• Density matrix formalismRandomness in quantum mechanics

• Transition from classical to quantum walks

• Experimental realisation

5

Density matrix approach

• Two state system 12

12

-1 10

6

Density matrix approach

• Two state system

• Density operator

12

12

0-1 1

7

Density matrix approach

• Density operator

Pure state Mixed state

12

12

0-1 1

8

Galton box

• Binomial distribution

9

Galton box

• Statistical mixture

• First four steps

10

Quantum analogy

• Used Hilbert space

• Specify subspaces

0-1-2-3 1 2 3

11

Quantum analogy

• Evolution with shift and coin operators

0-1-2-3 1 2 3

12

Quantum analogy

• Evolution with shift and coin operators

0-1-2-3 1 2 3

13

Quantum analogy

• Evolution with shift and coin operators

0-1-2-3 1 2 3

14

Quantum analogy

• State transformation

• Density matrix transformation

15

Quantum analogy

Quantum analogy

16

Position

pc

pq

Variances

pc

pq

pc

pq

Position

Position

100 steps

17

• Phase shift• Transformed density matrix

• Average

• Decoherence effect

Decoherence

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

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)

20

Setup

Polarizations and

21

Setup

Polarizations and

22

Results

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

23

Results

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

Theoretical expectation

6 steps

24

Results

Theoretical expectationM. Karski et al., Science 325, 174 (2009)

6 steps

25

Results

Theoretical expectation

26

Results

Theoretical expectationM. Karski et al., Science 325, 174 (2009)

27

Results

Gaussian fitM. Karski et al., Science 325, 174 (2009)

28

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)

29

30

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)