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A multiphoton collective phase is a multiphoton-scattering feature thatcannot be reduced to a sequence of two-photon scattering events, and thethree-photon "triad phas" is the smallest nontrivial example. Observing ahigher-order collective phase is experimentally challenging, and only triad andfour-photon tetrad collective phases have been observed. We introduce a schemeto make higher-order multiphoton collective-phase observations feasible bydesigning a sparse interferometer, which significantly reduces complexitycompared with the current best scheme for observing a multiphoton collectivephase. Specifically, our scheme reduces the optical depth from logarithmic toconstant and reduces the number of beam splitters from $O(n\log n)$ to linearscaling with respect to the collective-phase order $n$. As constant depthreduces loss and dispersion to a fixed rate regardless of collective-phaseorder, a major obstacle to observing large-scale collective phases is removed.

Sparse interferometry for measuring multiphoton collective phase