Implementing Quantum walks

Transcription

1 Implementing Quantum walks P. Xue, B. C. Sanders, A. Blais, K. Lalumière, D. Leibfried IQIS, University of Calgary University of Sherbrooke NIST, Boulder 1

2 Reminder: quantum walk Quantum walk (discrete) is a generalization of classical random walk with both coin and walker in quantum particles. Classical random walk is used to develop algorithms in computer science. Quantum walk is expected to develop new and faster algorithms based on quadratic speedup of diffusion reduction which is caused by coherence, entanglement between the coin and walker and so on. 2

3 Classical vs quantum walk Running a classical walk on the line results in a probability distribution like: position Whereas running this quantum walk for the same number of steps gives: The standard deviation of the probability distribution is proportional to time for quantum walk while for classical walk to root of time. 3

4 Quantum walk on a circle in phase space: Coin toss and phase kick (original) Initial state: Coin flip---hadamard transformation Conditional phase shift---jc model Quantum walk operation Ref: B. C. Sanders et al., PRA 67, (2003). After N steps: 4

5 Quantum walk via cavity QED Challenges: Coin flip (single qubit rotation on coin state)---drive the atom in the cavity directly by RF pulse or Raman transition Before applying 3 coin flips, the atom is gone. Only after four steps, the signature shows the difference between the quantum walk and classical walk Solve: Fix atom in the cavity---superconducting circuit QED system instead of cavity QED Implement coin flip by driving the resonator instead of driving the charge qubit directly to avoid leaving sweet spot 5

6 Superconducting Circuit QED the stripline resonator Equivalent cavity+2-level atom. Ref: A. Wallraff et al., Nature 431, 162 (2004). 6

7 Indirect coin flip via cavity driving e> In dispersive regime conditional phase shift H: coin flip g> displacement operator (side effects) Effective quantum walk operation: 7

8 The side effect The displacement operator only introduced some side effect. The introduced side effect is that the mean photon number in the cavity changes for each step, equivalent to changing the circle in phase space on which the walk is taking place. Amazing thing is that the kick/hop to different circles does not destroy the quadratic speed up of quantum walk behavior and can be compensated by tuning the frequency of the external field or pulse duration for each step. 8

9 Physical implementation Effect Hadamard by driving cavity: Pulse duration is a function of average photon number of the walker state (strength of electric field) 9

10 10

11 Measure of quantum walk on circles the dominant decoherence is caused by the decay of the resonator. =1 QW =0.5 RW 11

12 Implementation of quantum walk on a line with a single trapped ion Our goal is to propose a scheme for realizing the first singlewalker QW in the laboratory, with the ion's electronic degree of freedom serving as the two-state coin and the motion as the walker's degree of freedom. In contrast to current approaches to developing QW implementations on circles in phase space as we mentioned before, this approach yields a RW-QW transition in unbounded position space, but not being folded back on itself. Although the walk is over position, we show that the experimentally accessible phonon number can be used as a signature for quantum walk. 12

13 a and d are carrier Raman beams--- coin flip on electronic states c and b are displacement Raman beams---desired walker+coin coevolution Without B and Uoff, 13

14 Side effects phonon number dependent displacement operator displacement operator in momentum space and squeezed operator Counting phonons: the motional number distribution has been determined by driving the ion on the first blue sideband and Fourier transforming the atomic population in the > as a function of drive duration 14

15 Controllable decoherence---random phase of coin flip p= QW p=1 RW Mean phonon number can be used as signature of quantum walks QW RW 15

16 Conclusion The RW, which is ubiquitous in physics, chemistry, mathematics, and computer science, underpins Brownian motion and diffusion processes, is used in satisfiability proofs, and is intimately connected with the Wiener measure. Quantization of the RW has led to new quantum algorithms and fascinating physics such as decoherenceinduced diffusion reduction. Our goal is to see the QW realized in the laboratory. However, compromises have to be made to the ideal QW in order to realize the QW experimentally, such as sidestepping the requirement of direct coin flipping in cavity QED and finding an alternative to measuring the position distribution for a quantum walk in an ion trap. Here we discuss how QW can be implemented by making compromises to the ideal QW but demonstrating a true QW in the laboratory. References: 1. P. Xue, B. C. Sanders, A. Blais and K. Lalumière, Phys. Rev. A (2008). 2. P. Xue and B. C. Sanders, New Journal of Physics 10, (2008). 3. P. Xue, B. C. Sanders and D Leibfried, arxiv.org: thanks for your attention 16