On construction of evolutionary operator for rectangular linear optical multiport

The work of Knill et. al. (2001) established the possibility of nondeterministic realization of certain quantum logic operations using linear optical elements, ancilla photons and postselection techniques. It was also shown that any discrete unitary operator acting on $N$ optical modes can be implemented by a triangular multiport device constructed from a series of beam splitters and phase shifters (see work of Reck, Zeilinger et. al., 1994). Here, we consider the rectangular linear optical multiport that is used for the probabilistic realization of unitary transformations on $n$ qubits. This kind of linear optical scheme is suitable for probabilistic realization of unitary operators using ancilla photons and projective measurements. Qubits are encoded into the bosonic states of optical modes in two possible polarizations, and a number of ancilla photons and photodetectors are used for postselection of the qubits' state, based on the output of the detectors. We derive a procedure of evolutionary operator calculation for schemes of the considered type and present algorithms for their efficient computation on symmetric state space. We also provide complexities for different algorithms for the computation of evolutionary operator and estimate demands of resources in each case. A destructive Toffoli gate, acting on three qubits, using one ancilla photon and a photodetector, is implemented using schemes of the presented type.