An effective Hamiltonian approach to quantum random walk

@article{Sarkar2015AnEH,
  title={An effective Hamiltonian approach to quantum random walk},
  author={Debajyoti Sarkar and Niladri Paul and Kaushik Bhattacharya and Tarun Kanti Ghosh},
  journal={Pramana},
  year={2015},
  volume={88},
  pages={1-14}
}
In this article we present an effective Hamiltonian approach for discrete time quantum random walk. A form of the Hamiltonian for one-dimensional quantum walk has been prescribed, utilizing the fact that Hamiltonians are generators of time translations. Then an attempt has been made to generalize the techniques to higher dimensions. We find that the Hamiltonian can be written as the sum of a Weyl Hamiltonian and a Dirac comb potential. The time evolution operator obtained from this prescribed… 
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References

SHOWING 1-10 OF 33 REFERENCES
On the Relationship Between Continuous- and Discrete-Time Quantum Walk
Quantum walk is one of the main tools for quantum algorithms. Defined by analogy to classical random walk, a quantum walk is a time-homogeneous quantum process on a graph. Both random and quantum
Two-component Dirac-like Hamiltonian for generating quantum walk on one-, two- and three-dimensional lattices
TLDR
The two-component Hamiltonian presented here for quantum walk on different lattices can serve as a general framework to simulate, control, and study the dynamics of quantum systems governed by Dirac-like Hamiltonian.
Experimental implementation of a discrete-time quantum random walk on an NMR quantum-information processor
We present an experimental implementation of the coined discrete-time quantum walk on a square using a three-qubit liquid-state nuclear-magnetic-resonance (NMR) quantum-information processor (QIP).
Quantum Walk in Position Space with Single Optically Trapped Atoms
TLDR
In this experiment, a quantum walk on the line with single neutral atoms is implemented by deterministically delocalizing them over the sites of a one-dimensional spin-dependent optical lattice and its spatial coherence is demonstrated.
Full revivals in 2D quantum walks
Recurrence of a random walk is described by the Polya number. For quantum walks, recurrence is understood as the return of the walker to the origin, rather than the full revival of its quantum state.
Quantum walks on graphs
TLDR
A lower bound on the possible speed up by quantum walks for general graphs is given, showing that quantum walks can be at most polynomially faster than their classical counterparts.
Recurrence properties of unbiased coined quantum walks on infinite d -dimensional lattices
The P\'olya number characterizes the recurrence of a random walk. We apply the generalization of this concept to quantum walks [M. \v{S}tefa\v{n}\'ak, I. Jex and T. Kiss, Phys. Rev. Lett.
One-dimensional quantum walks
TLDR
A quantum analog of the symmetric random walk, which the authors call the Hadamard walk, is analyzed, which has position that is nearly uniformly distributed in the range after steps, in sharp contrast to the classical random walk.
Quantum Walk on the Line
Motivated by the immense success of random walk and Markov chain methods in the design of classical algorithms, we consider_quantum_ walks on graphs. We analyse in detail the behaviour of unbiased
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