Crossovers induced by discrete-time quantum walks

Abstract

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 $p \sim 1/n^\beta$ can be evaluated, where $n$ is the final time and $0<\beta<1$. 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 $\beta$ 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 $n$. 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.

Time averaged distribution of a discrete-time quantum walk on the path

“…We should remark that a relation between CTQW and DTQW on the infinite line is discussed in [15]. We also see more detailed discussion on the relations with respect to its weak limit theorems in [8]. In the case of CTQW determined by graph Laplacian on P n+2 [5], the time averaged distribution is almost the same as the uniform distribution.…”

From Discrete Time Quantum Walk to Continuous Time Quantum Walk in Limit Distribution

“…In this section, we will see the relationship between the continuous time quantum walk (CTQW), which will be defined in the following subsection, and the DTQW according to Ref. [17].…”

A Characterization of the Graphs to Induce Periodic Grover Walk

“…However it is believed that there are no non-trivial representations of the present distribution of QWs by that of past time like Markov chain [22]. QWs are applied to various study fields, for example, a problem of searching marked elements on graphs [6], [9], [10], [11], fundamental physics [13], [14], the limit theorems for its statistical behaviors [15], [16], spectral analysis [19], [20], and photon synthesis [17]. In [6], Szegedy walk was formulated as a natural quantization of reversible Marcov chains, and Szegedy showed that in most cases the quantized walk hits the marked set within the square root of the classical hitting time.…”