arXiv:1109.2662

The Heun differential equation and the Gauss differential equation related to quantum walks

Norio Konno, Takuya Machida, Tohru Wakasa

Published Sep 13, 2011• 6 citations

Abstract

The limit theorems of discrete- and continuous-time quantum walks on the line have been intensively studied. We show a relation among limit distributions of quantum walks, Heun differential equations and Gauss differential equations. Indeed, we derive the second-order Fucksian differential equations which limit density functions of quantum walks satisfy. Moreover, using both differential equations, we discuss a relationship between discrete- and continuous-time quantum walks. Taking suitable limit, we can transform a Heun equation obtained from the limit density function of the discrete-time quantum walk to a Gauss equation given by that of the continuous-time quantum walk.

DOI: 10.48550/arxiv.1109.2662

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arXiv:1109.2662

The Heun differential equation and the Gauss differential equation related to quantum walks

Norio Konno, Takuya Machida, Tohru Wakasa

Published Sep 13, 2011• 6 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.1109.2662

Related papers

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d

-regular,

d

-colorable graphs when the coin flip operator is Hermitian: from a standard DTQW we construct a family of discrete-time walks with a well-defined continuous-time limit on a related graph. One can think of this limit as a ``coined'' continuous-time walk. We show that these CTQWs share some properties with coined DTQWs. In particular, we look at a spatial search by a DTQW over the two-dimensional (2D) torus (a grid with periodic boundary conditions) of size $\sqrt{N}\ifmmode\times\else\texttimes\fi{}\sqrt{N}$, where it was shown that a coined DTQW can search in time

O(\sqrt{N}logN)

, but a standard CTQW takes $\mathrm{\ensuremath{\Omega}}(N)$ time to search for a marked element. The continuous limit of the DTQW search over the 2D torus exhibits the

O(\sqrt{N}logN)

scaling, like the coined walk it is derived from. We also look at the effects of graph symmetry on the limiting walk, and show that the properties are similar to those of the DTQW as shown in Krovi and Brun, Phys. Rev. A 75, 062332 (2007).

Concerning a discrete-time quantum walk

{X}_{t}^{(d)}

with a symmetric distribution on the line, whose evolution is described by the Hadamard transformation, it was proved by the author that the following weak limit theorem holds: ${X}_{t}^{(d)}∕t\ensuremath{\rightarrow}dx∕\ensuremath{\pi}(1\ensuremath{-}{x}^{2})\sqrt{1\ensuremath{-}2{x}^{2}}$ as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. The present paper shows that a similar type of weak limit theorem is satisfied for a continuous-time quantum walk

{X}_{t}^{(c)}

on the line as follows: ${X}_{t}^{(c)}∕t\ensuremath{\rightarrow}dx∕\ensuremath{\pi}\sqrt{1\ensuremath{-}{x}^{2}}$ as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. These results for quantum walks form a striking contrast to the central limit theorem for symmetric discrete- and continuous-time classical random walks: ${Y}_{t}∕\sqrt{t}\ensuremath{\rightarrow}{e}^{\ensuremath{-}{x}^{2}∕2}dx∕\sqrt{2\ensuremath{\pi}}$ as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. The work deals also with the issue of the relationship between discrete and continuous-time quantum walks. This topic, subject of a long debate in the previous literature, is treated within the formalism of matrix representation and the limit distributions are exhaustively compared in the two cases.