  We theoretically investigate quantum interference of two single photons at a
lossy asymmetric beam splitter, the most general passive 2$\times$2 optical
circuit. The losses in the circuit result in a non-unitary scattering matrix
with a non-trivial set of constraints on the elements of the scattering matrix.
Our analysis using the noise operator formalism shows that the loss allows
tunability of quantum interference to an extent not possible with a lossless
beam splitter. Our theoretical studies support the experimental demonstrations
of programmable quantum interference in highly multimodal systems such as
opaque scattering media and multimode fibers.
