  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approximations originally made, such as the Lamb--Dicke approximation.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.

Quantum Physics