We study a series of one-dimensional discrete-time quantum-walk models labeled by half integers $j=1/2, 1, 3/2, ...$, introduced by Miyazaki {\it et al.}, each of which the walker's wave function has $2j+1$ components and hopping range at each time step is $2j$. In long-time limit the density functions of pseudovelocity-distributions are generally given by superposition of appropriately scaled Konno's density function. Since Konno's density function has a finite open support and it diverges at the boundaries of support, limit distribution of pseudovelocities in the $(2j+1)$-component model can have $2j+1$ pikes, when $2j+1$ is even. When $j$ becomes very large, however, we found that these pikes vanish and a universal and monotone convex structure appears around the origin in limit distributions. We discuss a possible route from quantum walks to classical diffusion associated with the $j \to \infty$ limit.
We study a series of one-dimensional discrete-time quantum-walk models labeled by half integers j=1/2,1,3/2,..., introduced by Miyazaki {\it et al.}, each of which the walker's wave function has 2j+1 components and hopping range at each time step is 2j. In long-time limit the density functions of pseudovelocity-distributions are generally given by superposition of appropriately scaled Konno's density function. Since Konno's density function has a finite open support and it diverges at the boundaries of support, limit distribution of pseudovelocities in the (2j+1)-component model can have 2j+1 pikes, when 2j+1 is even. When j becomes very large, however, we found that these pikes vanish and a universal and monotone convex structure appears around the origin in limit distributions. We discuss a possible route from quantum walks to classical diffusion associated with the j→∞ limit.
Quantization of a random-walk model is performed by giving a qudit (a multicomponent wave function) to a walker at site and by introducing a quantum coin, which is a matrix representation of a unitary transformation. In quantum walks, the qudit of the walker is mixed according to the quantum coin at each time step, when the walker hops to other sites. As special cases of the quantum walks driven by high-dimensional quantum coins generally studied by Brun, Carteret, and Ambainis, we study the models obtained by choosing rotation as the unitary transformation, whose matrix representations determine quantum coins. We show that Wigner's (2j+1)-dimensional unitary representations of rotations with half-integers j's are useful to analyze the probability laws of quantum walks. For any value of half-integer j, convergence of all moments of walker's pseudovelocity in the long-time limit is proved. It is generally shown for the present models that, if (2j+1) is even, the probability measure of limit distribution is given by a superposition of (2j+1)∕2 terms of scaled Konno's density functions, and if (2j+1) is odd, it is a superposition of j terms of scaled Konno's density functions and a Dirac's delta function at the origin. For the two-, three-, and four-component models, the probability densities of limit distributions are explicitly calculated and their dependence on the parameters of quantum coins and on the initial qudit of walker is completely determined. Comparison with computer simulation results is also shown.
One-parameter family of discrete-time quantum-walk models on the square lattice, which includes the Grover-walk model as a special case, is analytically studied. Convergence in the long-time limit $t\ensuremath{\rightarrow}\ensuremath{\infty}$ of all joint moments of two components of walker's pseudovelocity, Xt/t and Yt/t, is proved and the probability density of limit distribution is derived. Dependence of the two-dimensional limit density function on the parameter of quantum coin and initial four-component qudit of quantum walker is determined. Symmetry of limit distribution on a plane and localization around the origin are completely controlled. Comparison with numerical results of direct computer simulations is also shown.
We study a one-parameter family of discrete-time quantum walk models on Z and Z2 associated with the Hadamard walk. Weak convergence in the long-time limit of all moments of the walker's pseudovelocity on Z and Z2 is proved. Symmetrization on Z and Z2 is theoretically investigated, leading to the resolution of the Konno-Namiki-Soshi conjecture in the special case of symmetrization of the unbiased Hadamard walk on Z. A necessary condition for the existence of a phenomenon known as localization is given.
Abstract In this paper, we study the properties of lackadaisical quantum walks (LQWs) on a line. This model was first proposed in Wong (2015 J. Phys. A: Math. Theor . 48 435304) as a quantum analogue of lazy random walks where each vertex is attached to τ self-loops. We derive an analytic expression for the localization probability of the walker at the origin after infinite steps, and obtain the peak velocities of the walker. We also calculate the wave function of the walker starting from the origin and obtain a long-time approximation for the entire probability density function. As an application of the density function, we prove that lackadaisical quantum walks spread ballistically for arbitrary τ , and give an analytic solution for the variance of the walker’s probability distribution.
The position density of a "particle" performing a continuous-time quantum walk on the integer lattice, viewed on length scales inversely proportional to the time , converges (as tends to infinity) to a probability distribution that depends on the initial state of the particle. This convergence behavior has recently been demonstrated for the simplest continuous-time random walk [N. Konno, Phys. Rev. E 72, 026113 (2005)]. In this Brief Report, we use a different technique to establish the same convergence for a very large class of continuous-time quantum walks, and we identify the limit distribution in the general case.
The weak convergence theorems of the one- and two-dimensional simple quantum walks, SQW(d),d=1,2, show a striking contrast to the classical counterparts, the simple random walks, SRW(d). In the SRW(d), the distribution of position X(t) of the particle starting from the origin converges to the Gaussian distribution in the diffusion scaling limit, in which the time scale T and spatial scale L both go to infinity as the ratio L/T is kept finite. On the other hand, in the SQW(d), the ratio L/T is kept to define the pseudovelocity V(t)=X(t)/t, and then all joint moments of the components ${V}_{j}(t),1\ensuremath{\leqslant}j\ensuremath{\leqslant}d$, of V(t) converge in the $T=L\ensuremath{\rightarrow}\ensuremath{\infty}$ limit. The limit distributions have novel structures such that they are inverted-bell shaped and their supports are bounded. In the present paper we claim that these properties of the SQW(d) can be explained by the theory of relativistic quantum mechanics. We show that the Dirac equation with a proper ultraviolet cutoff can provide a quantum walk model in three dimensions, where the walker has a four-component qubit. We clarify that the pseudovelocity V(t) of the quantum walker, which solves the Dirac equation, is identified with the relativistic velocity. Since the quantum walker should be a tardyon, not a tachyon, ∣V(t)∣<c, where c is the speed of light, and this restriction (the causality) is the origin of the finiteness of supports of the limit distributions universally found in quantum walk models. By reducing the number of components of momentum in the Dirac equation, we obtain the limit distributions of pseudovelocities for the lower dimensional quantum walks. We show that the obtained limit distributions for the one- and two-dimensional systems have common features with those of SQW(1) and SQW(2).