16 research outputs found
Quantum Walks with Non-Orthogonal Position States
Quantum walks have by now been realized in a large variety of different
physical settings. In some of these, particularly with trapped ions, the walk
is implemented in phase space, where the corresponding position states are not
orthogonal. We develop a general description of such a quantum walk and show
how to map it into a standard one with orthogonal states, thereby making
available all the tools developed for the latter. This enables a variety of
experiments, which can be implemented with smaller step sizes and more steps.
Tuning the non-orthogonality allows for an easy preparation of extended states
such as momentum eigenstates, which travel at a well-defined speed with low
dispersion. We introduce a method to adjust their velocity by momentum shifts,
which allows to investigate intriguing effects such as the analog of Bloch
oscillations.Comment: 5 pages, 4 figure
Quantum walk of a trapped ion in phase space
We implement the proof of principle for the quantum walk of one ion in a
linear ion trap. With a single-step fidelity exceeding 0.99, we perform three
steps of an asymmetric walk on the line. We clearly reveal the differences to
its classical counterpart if we allow the walker/ion to take all classical
paths simultaneously. Quantum interferences enforce asymmetric, non-classical
distributions in the highly entangled degrees of freedom (of coin and position
states). We theoretically study and experimentally observe the limitation in
the number of steps of our approach, that is imposed by motional squeezing. We
propose an altered protocol based on methods of impulsive steps to overcome
these restrictions, in principal allowing to scale the quantum walk to several
hundreds of steps.Comment: 4 pages, 4 figure
Experimental simulation and limitations of quantum walks with trapped ions
We examine the prospects of discrete quantum walks (QWs) with trapped ions.
In particular, we analyze in detail the limitations of the protocol of
Travaglione and Milburn (PRA 2002) that has been implemented by several
experimental groups in recent years. Based on the first realization in our
group (PRL 2009), we investigate the consequences of leaving the scope of the
approximations originally made, such as the Lamb--Dicke approximation. We
explain the consequential deviations from the idealized QW for different
experimental realizations and an increasing number of steps by taking into
account higher-order terms of the quantum evolution. It turns out that these
become dominant after a few steps already, which is confirmed by experimental
results and is currently limiting the scalability of this approach. Finally, we
propose a new scheme using short laser pulses, derived from a protocol from the
field of quantum computation. We show that the new scheme is not subject to the
above-mentioned restrictions, and analytically and numerically evaluate its
limitations, based on a realistic implementation with our specific setup.
Implementing the protocol with state-of-the-art techniques should allow for
substantially increasing the number of steps to 100 and beyond and should be
extendable to higher-dimensional QWs.Comment: 29 pages, 15 figue
Quantum walks: a comprehensive review
Quantum walks, the quantum mechanical counterpart of classical random walks,
is an advanced tool for building quantum algorithms that has been recently
shown to constitute a universal model of quantum computation. Quantum walks is
now a solid field of research of quantum computation full of exciting open
problems for physicists, computer scientists, mathematicians and engineers.
In this paper we review theoretical advances on the foundations of both
discrete- and continuous-time quantum walks, together with the role that
randomness plays in quantum walks, the connections between the mathematical
models of coined discrete quantum walks and continuous quantum walks, the
quantumness of quantum walks, a summary of papers published on discrete quantum
walks and entanglement as well as a succinct review of experimental proposals
and realizations of discrete-time quantum walks. Furthermore, we have reviewed
several algorithms based on both discrete- and continuous-time quantum walks as
well as a most important result: the computational universality of both
continuous- and discrete- time quantum walks.Comment: Paper accepted for publication in Quantum Information Processing
Journa