Coined quantum walks may be interpreted as the motion in position space of a quantum particle with a spin degree of freedom; the dynamics are determined by iterating a unitary transformation which is the product of a spin transformation and a translation conditional on the spin state. Coined quantum walks on the d-dimensional lattice can be treated as special cases of coined quantum walks on d-dimensional Euclidean space. We study quantum walks on d-dimensional Euclidean space and prove that the sequence of rescaled probability distributions in position space associated to the unitary evolution of the particle converges to a limit distribution.
We explore a discrete-time, coined quantum walk on a quantum network where the coherent superposition of walker moves originates from the unitary interaction of the walker coin with the qubit degrees of freedom in the quantum network. The walk dynamics leads to a growth of entanglement between the walker and the network on one hand and, on the other, between the network qubits among themselves. The initial entanglement among the network qubits plays a crucial role in determining the asymptotic values of these entanglement measures and the quantum walk statistics. Specifically, the entanglement entropy of the walker-network state and the negativity of the quantum network-qubit state saturate to values increasing with the initial network entanglement. The asymptotic time-averaged walker-position probability distribution shows increasing localization around the initial walker position with higher initial network entanglement. A potential application of these results as a characterization tool for quantum network properties is suggested.
We consider the discrete time unitary dynamics given by a quantum walk on the lattice
Zd performed by a quantum particle with internal degree of freedom, called coin state, according to the following iterated rule: a unitary update of the coin state takes place, followed by a shift on the lattice, conditioned on the coin state of the particle. We study the large time behavior of the quantum mechanical probability distribution of the position observable in
Zd when the sequence of unitary updates is given by an i.i.d. sequence of random matrices. When averaged over the randomness, this distribution is shown to display a drift proportional to the time and its centered counterpart is shown to display a diffusive behavior with a diffusion matrix we compute. A moderate deviation principle is also proven to hold for the averaged distribution and the limit of the suitably rescaled corresponding characteristic function is shown to satisfy a diffusion equation. A generalization to unitary updates distributed according to a Markov process is also provided. An example of i.i.d. random updates for which the analysis of the distribution can be performed without averaging is worked out. The distribution also displays a deterministic drift proportional to time and its centered counterpart gives rise to a random diffusion matrix whose law we compute. A large deviation principle is shown to hold for this example. We finally show that, in general, the expectation of the random diffusion matrix equals the diffusion matrix of the averaged distribution.
It is known that any positive-energy state of a free Dirac particle that is initially highly localized evolves in time by spreading at speeds close to the speed of light. As recently indicated by Strauch, this general phenomenon, and the resulting ``two-horned'' distributions of position probability along any axis through the point of initial localization, can be interpreted in terms of a quantum random walk, in which the roles of ``coin'' and ``walker'' are naturally associated with the spin and translational degrees of freedom in a discretized version of Dirac's equation. We investigate the relationship between these two evolutions analytically and show how the evolved probability density on the
x axis for the Dirac particle at any time
t can be obtained from the asymptotic form of the probability distribution for the position of a ``quantum walker.'' The case of a highly localized initial state is discussed as an example.
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.
Abstract We propose a general framework for quantum walks on d-dimensional spaces. We investigate asymptotic behavior of these walks. We prove that every homogeneous walks with finite degree of freedom have limit distribution. This theorem can also be applied to every crystal lattice. In this theorem, it is not necessary to assume that the support of the initial unit vector is finite. We also pay attention on 1-cocycles, which is related to Heisenberg representation of time evolution of observables. For homogeneous walks with finite degree of freedom, convergence of averages of 1-cocycles associated with the position observable is also proved.