  We consider quantum variants of Parrondo games on low-dimensional Hilbert
spaces. The two games which form the Parrondo game are implemented as quantum
walks on a small cycle of length $M$. The dimension of the Hilbert space is
$2M$. We investigate a random sequence of these two games which is realized by
a quantum coin, so that the total Hilbert space dimension is $4M$. We show that
in the quantum Parrondo game constructed in this way a systematic win or loss
occurs in the long time limit. Due to entaglement and self-interference on the
cycle, the game yields a rather complex structure for the win or loss depending
on the parameters.
