  The development of percolation theory was historically shaped by its numerous
applications in various branches of science, in particular in statistical
physics, and was mainly constrained to the case of Euclidean spaces. One of its
central concepts, the percolation transition, is defined through the appearance
of the infinite cluster, and therefore cannot be used in compact spaces, such
as the Hilbert space of an N-qubit system. Here we propose its generalization
for the case of a random space covering by hyperspheres, introducing the
concept of a ``maximal cluster". Our numerical calculations reproduce the
standard power-law relation between the hypersphere radius and the cover
density, but show that as the number of qubits increases, the exponent quickly
vanishes (i.e., the exponentially increasing dimensionality of the Hilbert
space makes its covering by finite-size hyperspheres inefficient). Therefore
the percolation transition is not an efficient model for the behavior of
multiqubit systems, compared to the random walk model in the Hilbert space.
However, our approach to the percolation transition in compact metric spaces
may prove useful for its rigorous treatment in other contexts.

Quantum Physics