  Most leading proposals for linear-optical quantum computing (LOQC) use
cluster states, which act as a universal resource for measurement-based
(one-way) quantum computation (MBQC). In ballistic approaches to LOQC, cluster
states are generated passively from small entangled resource states using
so-called fusion operations. Results from percolation theory have previously
been used to argue that universal cluster states can be generated in the
ballistic approach using schemes which exceed the critical threshold for
percolation, but these results consider cluster states with unbounded size.
Here we consider how successful percolation can be maintained using a physical
architecture with fixed physical depth, assuming that the cluster state is
continuously generated and measured, and therefore that only a finite portion
of it is visible at any one point in time. We show that universal LOQC can be
implemented using a constant-size device with modest physical depth, and that
percolation can be exploited using simple pathfinding strategies without the
need for high-complexity algorithms.
