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We investigate the time dependence of the survival probability of quantum walks governed by Fibonacci walks with phase parameters on the trapped two-dimensional lattice. We have shown that the survival probability of the quantum walk decays with time obey to the stretched exponential law for all initial states of walkers. We have also shown that stretched exponential decay parameter β can be arranged by phase parameter combination. Obtained numerical results show that phase parameters can be used as a control parameter to determine the decay rate of the survival probability of the quantum walk.

In recent years, the quantum walk on the trapped lattice has been intensively investigated due to importance applications in quantum information and computing. Therefore, many theoretical and experimental studies have been carried out to understand the effect of the trapping states on the quantum walk. For example, Agliari [

It can be seen from the brief review that, in these studies, the focus is on how the traps affect the distribution of the survival probability of the usual quantum walk governed by Hadamard, Grover or Fourier coin operators. Recently, the quantum walk with phase parameters has been considered by a number of authors, see [

This paper is organized as follows. In Section 2, we define the quantum walk on the square lattice with phase parameters. We also present in this section the time evolution for a single-particle walk as well as multi-particle walk in which the particles are distinguishable. In Section 3, we derive the survival probability using the exact numeration method of [

We consider the discrete-time quantum walk on the square lattice with boundary conditions. The coin space of a single walker is given by where corresponds to the chirality states left, right, up, and down respectively. In this paper we will put,

The position space is given by. The Hilbert space of the total system is given by. Recall that the evolution of the walk is given by, where is the shift operator, is the identity operator and is the coin operator governing the walk. In this paper we will take, where is a two-dimensional generalization of the Fibonacci operator and is the diagonal phase adjustment. In particular, diagonal phase adjustment can be defined as, where;

Denote the total number of sites on the lattice by, then the shift operator is given by,

We shall write the wave function of the walker at time as,

with. We define the state of the particle by,

where for represent probability amplitudes of the particle at site at time, depending on the chirality state of the particle. In terms of the initial state, the evolution of the walk is given by. The density operator of the quantum walk is given by. In terms of the density operator we can write the probability distribution of the walker at time as,

We should note that can be written as,

which is standard in papers on the quantum walk instead of the representation above involving the density operator.

On the other hand, single particle quantum walks can be easily generalized multi-particle quantum walks if the particles are assumed as uncorrelated, non-interacting and distinguishable. The multi-particle walk has been investigated by the author of the present paper [16,17]. The case where indistinguishability plays a role has also been considered. For simplicity in this study we consider walkers which are non-interacting distinguishable particles and they are initially uncorrelated. If we assume the particles are distinguishable and are initially uncorrelated the Hilbert space is given by

or

where is the Hilbert space of the walker for. The evolution of the -particle walk in which indistinguishability does not play a role is given by, where is the same for each walker. The initial state of the particles is written as

Alternatively, Equation (10) is given by a tensor product of the single walker initial states as

where for is the particle state with chirality, and position (recall we are assuming that there are sites on the square lattice). In terms of, the evolution of the walk is given by. Using the reduced density operator we can write the probability distribution of a single walker at time by

where. in Equation (12) represents the probability of a single walker at site at time when the walker starts from site at. The set is the set of transition probabilities from to of a single particle, which is the same interpretation given by Equation (6).

In [

Here enumerates a particular independent initial configuration of the system. Note that or, where is the number of traps on the square lattice and is the trapping density. We take the lattice sites at. If site is a trapping site, we let, so that the summation above is not restricted to the untrapped sites. We also assume that the initial configurations on the walkers are such that the walkers occupy all the untrapped sites. To account for randomization we calculate the mean survival probability,

where denotes the number of different configurations. Recall we assumed the walkers are distinguishable and uncorrelated, so the evolution of the Mparticle walk is equivalent to a single-particle walk with an ensemble of initial configurations. So by quantum computation of the single-particle distribution, the classical survival probability is useful.

In this section we numerically analyze the dynamics of the survival probability in the two-dimensional multiparticle quantum walk with phase parameters. Firstly, we investigate the survival probability for three different initial states with fixed phase parameters. Initial chirality states of untrapped cites are chosen as in the first case, , in the second case sequentially and in the last case, randomly chosen either of or. We call them left, sequential and mixed initializations respectively. In numeric simulations, the number of sites on the square lattice is taken to be. To account for random distribution of traps, a statistical configurational average of the mean survival probability is calculated over different independent realizations of the initial system with different configurations. We should remark that the coin operator governing the walk is given by, where is a two-dimensional generalization of the Fibonacci operator and is the diagonal phase adjustment (see Section 2 for details).

Survival probability of QW are plotted in Figures 1(a)-(c) for left, sequential and mixed initializations and for different trap densities and with arbitrary fixed phase parameters. Figures are plotted in the double logarithmic scale. We see that the survival probability decreases in time and is less at higher trap densities. All different initializations show same behavior and the survival probability is seen to exhibit a linear dependence on time in the double logarithmic scale. When we fit the lines to the curves in the time regime, line fits obey the Kohlrausch-Williams-Watts stretched exponential function [

where exponent determines the decay rate of the sur-

vival probability.

In order to further investigate and clarify the phase parameter dependence of the decay parameter we will consider different phase combination for quantum walk on the two dimensional randomly trapped lattice. However here we have considered only left initialization since it has been shown in Figures 1 and 2 that the stretched exponential parameter of survival probability almost linearly increases with increasing trap density for all initializations at fixed phase parameters. For four different combination of the parameter, the time dependence of the survival probability of quantum walk has been plotted in Figures 3(a)-(d) in the double logarithmic scale for several trap densities. For the trap den-

in the cases of the left, sequential and mixed initializations.

; (d).

sities and, we have chosen different phase parameter combinations in

In

figure that the decay parameter dramatically changes due to phase combination. For all different phase parameters the decay parameter almost linearly increases with increasing trap density.

These results show that the phase variables of the coin operator of walkers can be used as an control parameter to adjust the decay rate of the survival probability of the quantum walk on the trapped lattice for all initial states of walkers.

In this study, the coin operator was taken to be a parametrization of the usual quantum walk governed by a two-dimensional generalization of the Fibonacci operator and by adapting the exact numeration method of Havlin et al. [

This work was partially supported by the Istanbul University under Project No: 28432.