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arXiv:quant-ph/0408140 · Physical Review E

Limit theorem for continuous-time quantum walk on the line

Norio Konno

Published Aug 12, 2005• 125 citations
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Abstract

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.

DOI
10.1103/physreve.72.026113

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Concerning a discrete-time quantum walk Xt(d){X}_{t}^{(d)}Xt(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: Xt(d)∕t\ensuremath→dx∕\ensuremathπ(1\ensuremath−x2)1\ensuremath−2x2{X}_{t}^{(d)}∕t\ensuremath{\rightarrow}dx∕\ensuremath{\pi}(1\ensuremath{-}{x}^{2})\sqrt{1\ensuremath{-}2{x}^{2}}Xt(d)​∕t\ensuremath→dx∕\ensuremathπ(1\ensuremath−x2)1\ensuremath−2x2​ as t\ensuremath→\ensuremath∞t\ensuremath{\rightarrow}\ensuremath{\infty}t\ensuremath→\ensuremath∞. The present paper shows that a similar type of weak limit theorem is satisfied for a continuous-time quantum walk Xt(c){X}_{t}^{(c)}Xt(c)​ on the line as follows: Xt(c)∕t\ensuremath→dx∕\ensuremathπ1\ensuremath−x2{X}_{t}^{(c)}∕t\ensuremath{\rightarrow}dx∕\ensuremath{\pi}\sqrt{1\ensuremath{-}{x}^{2}}Xt(c)​∕t\ensuremath→dx∕\ensuremathπ1\ensuremath−x2​ as t\ensuremath→\ensuremath∞t\ensuremath{\rightarrow}\ensuremath{\infty}t\ensuremath→\ensuremath∞. These results for quantum walks form a striking contrast to the central limit theorem for symmetric discrete- and continuous-time classical random walks: Yt∕t\ensuremath→e\ensuremath−x2∕2dx∕2\ensuremathπ{Y}_{t}∕\sqrt{t}\ensuremath{\rightarrow}{e}^{\ensuremath{-}{x}^{2}∕2}dx∕\sqrt{2\ensuremath{\pi}}Yt​∕t​\ensuremath→e\ensuremath−x2∕2dx∕2\ensuremathπ​ as t\ensuremath→\ensuremath∞t\ensuremath{\rightarrow}\ensuremath{\infty}t\ensuremath→\ensuremath∞. 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.
Quantum walks can be defined in two quite distinct ways: discrete-time and continuous-time quantum walks (DTQWs and CTQWs). For classical random walks, there is a natural sense in which continuous-time walks are a limit of discrete-time walks. Quantum mechanically, in the discrete-time case, an additional ``coin space'' must be appended for the walk to have nontrivial time evolution. Continuous-time quantum walks, however, have no such constraints. This means that there is no completely straightforward way to treat a CTQW as a limit of a DTQW, as can be done in the classical case. Various approaches to this problem have been taken in the past. We give a construction for walks on ddd-regular, ddd-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(NlogN)O(\sqrt{N}logN)O(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(NlogN)O(\sqrt{N}logN)O(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).
We consider crossovers with respect to the weak convergence theorems from a discrete-time quantum walk (DTQW). We show that a continuous-time quantum walk (CTQW) and discrete- and continuous-time random walks can be expressed as DTQWs in some limits. At first we generalize our previous study [Phys. Rev. A \textbf{81}, 062129 (2010)] on the DTQW with position measurements. We show that the position measurements per each step with probability p∼1/nβp \sim 1/n^\betap∼1/nβ can be evaluated, where nnn is the final time and 0<β<10<\beta<10<β<1. We also give a corresponding continuous-time case. As a consequence, crossovers from the diffusive spreading (random walk) to the ballistic spreading (quantum walk) can be seen as the parameter β\betaβ shifts from 0 to 1 in both discrete- and continuous-time cases of the weak convergence theorems. Secondly, we introduce a new class of the DTQW, in which the absolute value of the diagonal parts of the quantum coin is proportional to a power of the inverse of the final time nnn. This is called a final-time-dependent DTQW (FTD-DTQW). The CTQW is obtained in a limit of the FTD-DTQW. We also obtain the weak convergence theorem for the FTD-DTQW which shows a variety of spreading properties. Finally, we consider the FTD-DTQW with periodic position measurements. This weak convergence theorem gives a phase diagram which maps sufficiently long-time behaviors of the discrete- and continuous-time quantum and random walks.
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.
Open Quantum Random Walks are the exact quantum generalization of Markov chains on finite graphs or on nets. These random walks are typically quantum in their behavior, step by step, but they seem to show up a rather classical asymptotic behavior, as opposed to the quantum random walks usually considered in Quantum Information Theory (such as the well-known Hadamard random walk). Typically, in the case of Open Quantum Random Walks on nets, their distribution seems to always converges to a Gaussian distribution or a mixture of Gaussian distributions. In the case of nearest neighbors homogeneous Open Quantum Random Walks on Z^d we prove such a Central Limit Theorem, in the case where only one Gaussian distribution appears in the limit. Through the quantum trajectory point of view on quantum master equations, we transform the problem into studying a certain functional of a Markov chain on Z^d times the Banach space of quantum states. The main difficulty is that we know nothing about the invariant measures of this Markov chain, even their existence. Surprisingly enough, we are able to produce a Central Limit Theorem with explicit drift and explicit covariance matrix. In a second step we are able to extend our Central Limit Theorem to the case of several asymptotic Gaussians, in the case where the operator coefficients of the quantum walk are block-diagonal in a common basis.