Discrete-Time Quantum Walk and Quantum Dynamical Simulator Yutaka Shikano Institute for Molecular Science Research Associate Professor
What is my current research interest? 1
Foundation of Thermodynamics and Statistical Mechanics Photophysics Lasing Mechanism Quantum Measurement and Foundations (Quantum) Diffusion Process Optomechanical System 2
Another seminar today! 3
What is the discrete time quantum walk? See the review: J. Comput. Theor. Nanosci. 10, 1558 (2013); arxiv:1307.3384. 4
Discrete Time Random Walk (DTRW) Coin Flip Shift Repeat 5
Discrete Time Quantum Walk (DTQW) (A. Ambainis, E. Bach, A. Nayak, A. Vishwanath, and J. Watrous, in STOC 01 (ACM Press, New York, 2001), pp. 37 49.) Quantum Coin Flip Shift Repeat 6
Example of DTQW Initial Condition Position: n = 0 (localized) Coin: Coin Operator: Hadamard Coin Probability distribution of the n-th cite at t step: Let s see the dynamics of quantum walk by 3 rd step! 7
Example of DTQW -3-2 -1 0 1 2 3 0 1 2 3 step 1/8 5/8 1/8 1/8 prob. Quantum Coherence and Interference 8
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Probability Distribution at the 1000-th step DTRW Unbiased Coin (Left and Right with probability ½) DTQW Initial Coin State Coin Operator 10
Weak Limit Theorem (Limit Distribution) DTRW Central Limit Theorem Prob. 1/2 Prob. 1/2 DTQW (N. Konno, Quantum Information Processing 1, 345 (2002).) Coin operator Initial state Probability density 11 11
Changing the initial coin state Initial Coin State Coin Operator 12
Probability Distribution at the 1000-th step DTRW Unbiased Coin (Left and Right with probability ½) DTQW Initial Coin State Coin Operator 13
Origin? Feynman and Hibbs Quantum mechanics and path integral time=n Transmit Reflect time=n+1 a c time=n Reflect Transmit time=n+1 b d DTQW on line
Experimental feasibility? 15
How to experimentally implement DTQW? -3-2 -1 0 1 2 3 0 1 2 3 step 1/8 5/8 1/8 1/8 prob. Quantum Coherence and Interference 16
Polarized Beam Splitter Phase Shifter
Photon Case Polarization -> Coin Photon mode -> Walker space Photon input Fabio Sciarrino group at Italy
DTQW can be realized!! calcite Andrew White Group at Australia Christine Silberhorn Group at Germany 19
Problem DTQW is useful in physics? 20
Topological Phase and DTQW Idea based on the Floquet method. Quantum Walk One- Step Time Evolution has the dispersion relation. 21
To change the gap!! To change the gap, the unitary operator should be changed!! 22
Space-dependent coin 23
By changing the rotation angle by the position, the band structure of the local effective Hamiltonian is changed. (T. Kitagawa et al., Nat. Comm. 3, 882 (2012).) 24
Continuous Time Quantum Walk (CTQW) Dynamics of discretized Schroedinger Equation. = Tight binding hopping model (E. Farhi and S. Gutmann, Phys. Rev. A 58, 915 (1998)) Limit Distribution (Arcsin Law <- Quantum probability theory) p.d. DTQW 25
Connections in asymptotic behaviors From the viewpoint of the limit distribution, Lattice-size-dependent coin Dirac eq. DTQW Increasing the dimension Schroedinger eq. Time-dependent coin & Re-scale CTQW Continuum Limit YS, J. Comput. Theor. Nanosci. 10, 1558 (2013); arxiv:1307.3384. 26
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Motivation On the conventional discrete time quantum walk, the ballistic transport can be expected. This comes from the quantum transport. The Anderson localization can be also expected. The random coin case can be observed. The DTQW can mimic quantum mechanical dynamics? 28
Inhomogeneous DTQW with Self-Dual Our Model Self-dual model inspired by the Aubry-Andre model In the dual basis, the roles of coin and shift are interchanged. Dual basis 29
What is the self-duality? Shift (Original Base) -> Coin (Dual Base) Coin (Original Base) -> Shift (Dual Base) 30
Probability Distribution at the 1000-th Step Initial Coin state 31
Limit Distribution Theorem (YS and H. Katsura, Phys. Rev. E 82, 031122 (2010)) 32
Aubry-Andre Model and Hofstadter Butterfly The Aubry-Andre model is related to the Hofstadter butterfly, which has a self-similar and fractal structure. The quantum walk operator WC is unitary. The absolute value of the eigenvalues of WC is 1. The eigenvalues of WC and CW are the same. We numerically obtain the distribution of the eigenvalues of CW as a function of. 33
Distribution of Eigenvalues (YS and H. Katsura, Phys. Rev. E 82, 031122 (2010); AIP Conf. Proc. 1363, 151 (2011).) 1. 2. Unitarity 3. Local Gauge transform. 4. 5. Non-degenerate Argument of CW 6. 34
Next Investigation We want to calculate the eigenvalue and eigenstate as the analytical formulation. Contribute to the multi-fractal analysis Contribute to dynamics of our inhomogeneous DTQW 35 Only when alpha is irrational and the specific rational number, we can prove the DTQW is localized. However, otherwise, we do not mention about it?
Summary The DTQW is experimentally realized. Therefore, this is a toy model of the quantum dynamical simulator. For mathematics, the DTQW is sensitive to noise but can be taken as the quantum dynamical simulator in one-dimensional system. 2D is OK. 3D is the open problem. 36