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The development of percolation theory was historically shaped by its numerousapplications in various branches of science, in particular in statisticalphysics, and was mainly constrained to the case of Euclidean spaces. One of itscentral concepts, the percolation transition, is defined through the appearanceof the infinite cluster, and therefore cannot be used in compact spaces, suchas the Hilbert space of an N-qubit system. Here we propose its generalizationfor the case of a random space covering by hyperspheres, introducing theconcept of a ``maximal cluster". Our numerical calculations reproduce thestandard power-law relation between the hypersphere radius and the coverdensity, but show that as the number of qubits increases, the exponent quicklyvanishes (i.e., the exponentially increasing dimensionality of the Hilbertspace makes its covering by finite-size hyperspheres inefficient). Thereforethe percolation transition is not an efficient model for the behavior ofmultiqubit systems, compared to the random walk model in the Hilbert space.However, our approach to the percolation transition in compact metric spacesmay prove useful for its rigorous treatment in other contexts.

Continuous percolation in a Hilbert space for a large system of qubits