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Permanents, bosons and linear optics

The particular complexity of linear quantum optical networks has received certain deserved attention recently due to the possible implications for the theory of quantum computation. Two relevant boson models are discussed in the presented work. The symmetric product of the Hilbert spaces produces a rather abstract model; the second one is obtained by quantization of the harmonic oscillator. In contrast to the considered bosonic processes, the so-called ‘fermionic linear optics’ is effectively simulated on a classical computer. The comparison of the bosonic and fermionic case clarifies the controversy, and the more elaborate oscillator model provides a deeper analogy.