A new type of limit theorems for the one-dimensional quantum random walk
By Norio Konno. 2002
233 citations recorded by the source corpus.
In this paper we consider the one-dimensional quantum random walk X^{varphi} _n at time n starting from initial qubit state varphi determined by 2 times 2 unitary matrix U. We give a combinatorial expression for the characteristic function of X^{varphi}_n. The expression clarifies the dependence of it on components of unitary matrix U and initial qubit state varphi. As a consequence of the above results, we present a new type of limit theorems for the quantum random walk. In contrast with the de Moivre-Laplace limit theorem, our symmetric case implies that X^{varphi}_n /n converges in distribution to a limit Z^{varphi} as n to infty where Z^{varphi} has a density 1 / pi (1-x^2) sqrt{1-2x^2} for x in (- 1/sqrt{2}, 1/sqrt{2}). Moreover we discuss some known simulation results based on our limit theorems.
Subfields: Artificial Intelligence, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/ab19699a45f9502289e41d1c4ddfcf2de9516fa0
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Citations in the corpus
- Quantum walks: a comprehensive review (2012, methodology): “2.2.8), and Konno has proved that the return probability of a one-dimensional discrete-time quantum walk can be written in terms of elliptic integrals [256].”
- QUANTUM WALKS AND THEIR ALGORITHMIC APPLICATIONS (2003, unknown): “A “coin flip transformation” C C n, 0 = a n, 0 + b n, 1 , C n, 1 = c n, 0 + d n, 1 .”