  We devise a multiphoton interferometry scheme for sampling a quadratic
function of a specific immanant for any submatrix of a unitary matrix and its
row permutations. The full unitary matrix describes a passive, linear
interferometer, and its submatrix is used when photons enter in and are
detected at subsets of possible input and output channels. Immanants are
mathematical constructs that interpolate between the permanent and determinant;
contrary to determinants and permanents, which have meaningful physical
applications, immanants are devoid of physical meaning classically but here are
shown to be meaningful in a quantum setting. Our quadratic form of immanants is
sampled by injecting vacuum and single photons into interferometer input ports
such that the photon arrival times are entangled, in contrast to previous
methods that control arrival times without entangling. Our method works for any
number of photons, and we solve explicitly the quadratic form for the two-,
three- and four-photon cases.
