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 1993.
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), 335-359
N. Konno: A new type of limit theorems for the one-dimensional quantum
random walk, J. Math. Soc. Japan, 57 (2005), 1179-1195
◎ Standard realizations
M. Kotani and T. Sunada, Spectral geometry of crystal lattices, Contemporary
Math. 338 (2003), 271-305
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, SpringerVerlag, 1983
◎ Large deviations
M. Kotani and T. Sunada: Large deviation and the tangent cone at infinity of a
crystal lattice, Math. Z. 254 (2006), 837-870
A generalset-up
Remarks
Quantum walks on topological crystals
Example１（Scalar case)
Example 2（Scalar case)
Example 3（Scalar case)
Integral formula
Main Results
Standard realizations
A non-standard realization
Standard realization
Ideas