Fine Asymptotics for 1D Quantum Walks Joint work with Tatsuya Tate August 26, 2011 at Sendai Toshikazu Sunada (Meiji University)

What’s a quantum walk ? The notion of quantum walks is a quantum version of classical random walks. ○ A classical walk: depending on random control by an external device such as the flip of a coin or the cast of a dice ○ A quantum walk: directly linked to the probabilistic nature of states in a quantum system concerned

The probabilistic nature of quantum mechanics

History A proto-idea of quantum walks is seen in the theory of path integrals initiated by R.P.Feynman. Its intense study started in Y. Aharonov, L. Davidovich, N. Zagury Mainly aiming at the design of fast algorithms by means of quantum computing.

Discretization

Probability

Comparison with the classical walk on Z

In the quantum case, the probability distribution has intense oscillations.

Weak limit

Asymptotics in the allowed region

Remarks

Asymptotics around the wall

Asymptotics in the hidden region

Large deviation in the hidden region

Idea

References ◎ Quantum walks J. Kempe: Quantum random walks － an introductory over view, Contemporary Phys. 50 (2009), N. Konno: A new type of limit theorems for the one-dimensional quantum random walk, J. Math. Soc. Japan, 57 (2005), ◎ Standard realizations M. Kotani and T. Sunada, Spectral geometry of crystal lattices, Contemporary Math. 338 (2003), M. Kotani and T. Sunada: Standard realizations of crystal lattices via harmonic maps, Transaction A.M.S, 353 (2000), 1-20 ◎ Method of stationary phase L. Hormander: The Analysis of Linear Partial Differential Operators I, Springer-Verlag, 1983 ◎ Large deviations M. Kotani and T. Sunada: Large deviation and the tangent cone at infinity of a crystal lattice, Math. Z. 254 (2006),

A generalset-up

Remarks

Quantum walks on topological crystals

Example １（ Scalar case)

Example 2 （ Scalar case)

Example 3 （ Scalar case)

Integral formula

Main Results

Standard realizations

A non-standard realization Standard realization

Ideas