Limit distributions of two-dimensional quantum walks

Abstract

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 → ∞ of all joint moments of two components of walker's pseudovelocity, X t /t and Y t /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.

Implementation of a spatial two-dimensional quantum random walk with tunable decoherence

“…Along these lines, two-dimensional QRWs provide a powerful tool for modeling complex quantum information and energy transport systems [19,20]. Their realization presents a challenge because of the need for four-level coin operation [21][22][23]. One way to overcome this drawback is to make use of different degrees of freedom of photons, such as polarization and orbital angular momentum, as has been shown in Ref.…”

Entropy rate of message sources driven by quantum walks

“…(34). Despite the exponential scaling of the precision and the dimension with respect to the number of the iterations k, we found that our method converge much faster than mere brute force simulations reconstructing the joint probability distribution in Eq.…”

“…(44) allows us to approximate the scaling of the entropy rate. For the approximation we use the weak limit theory of quantum walks [34]. For high number of steps (high w's) the symmetric probability distribution of a 1D Hadamard QW can be approximated (16) ) and the unbiased CW (circles) on the cycle graph with 16 vertices.…”

Analytical expression for variance of homogeneous-position quantum walk with decoherent position

“…Quantum walks have been investigated from different points of views such as their propagation speed [10][11][12], entanglement between coin and position of the walker [13][14][15], quantum walks in higher dimensions [16] and quantum walks as an entanglement generator [17].…”