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From Discrete Time Quantum Walk to Continuous Time Quantum Walk in Limit Distribution
The discrete time quantum walk defined as a quantum-mechanical analogue of
the discrete time random walk have recently been attracted from various and
interdisciplinary fields. In this review, the weak limit theorem, that is, the
asymptotic behavior, of the one-dimensional discrete time quantum walk is
analytically shown. From the limit distribution of the discrete time quantum
walk, the discrete time quantum walk can be taken as the quantum dynamical
simulator of some physical systems.Comment: This is the invited review paper on the special issue on Theoretical
and Mathematical Aspects of Discrete Time Quantum Walk from Journal of
Computational and Theoretical Nanoscienc
Universal computation by multi-particle quantum walk
A quantum walk is a time-homogeneous quantum-mechanical process on a graph
defined by analogy to classical random walk. The quantum walker is a particle
that moves from a given vertex to adjacent vertices in quantum superposition.
Here we consider a generalization of quantum walk to systems with more than one
walker. A continuous-time multi-particle quantum walk is generated by a
time-independent Hamiltonian with a term corresponding to a single-particle
quantum walk for each particle, along with an interaction term. Multi-particle
quantum walk includes a broad class of interacting many-body systems such as
the Bose-Hubbard model and systems of fermions or distinguishable particles
with nearest-neighbor interactions. We show that multi-particle quantum walk is
capable of universal quantum computation. Since it is also possible to
efficiently simulate a multi-particle quantum walk of the type we consider
using a universal quantum computer, this model exactly captures the power of
quantum computation. In principle our construction could be used as an
architecture for building a scalable quantum computer with no need for
time-dependent control
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