  Linear-Optical Passive (LOP) devices and photon counters are sufficient to
implement universal quantum computation with single photons, and particular
schemes have already been proposed. In this paper we discuss the link between
the algebraic structure of LOP transformations and quantum computing. We first
show how to decompose the Fock space of N optical modes in finite-dimensional
subspaces that are suitable for encoding strings of qubits and invariant under
LOP transformations (these subspaces are related to the spaces of irreducible
unitary representations of U(N)). Next we show how to design in algorithmic
fashion
  LOP circuits which implement any quantum circuit deterministically. We also
present some simple examples, such as the circuits implementing a CNOT gate and
a Bell-State Generator/Analyzer.
