  Recently, several groups have investigated quantum analogues of random walk
algorithms, both on a line and on a circle. It has been found that the quantum
versions have markedly different features to the classical versions. Namely,
the variance on the line, and the mixing time on the circle increase
quadratically faster in the quantum versions as compared to the classical
versions. Here, we propose a scheme to implement the quantum random walk on a
line and on a circle in an ion trap quantum computer. With current ion trap
technology, the number of steps that could be experimentally implemented will
be relatively small. However, we show how the enhanced features of these walks
could be observed experimentally. In the limit of strong decoherence, the
quantum random walk tends to the classical random walk. By measuring the degree
to which the walk remains `quantum', this algorithm could serve as an important
benchmarking protocol for ion trap quantum computers.
