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Unitary operation is an essential step for quantum information processing. Wefirst propose an iterative procedure for decomposing a general unitaryoperation without resorting to controlled-NOT gate and single-qubit rotationlibrary. Based on the results of decomposition, we design two compactarchitectures to deterministically implement arbitrary two-qubitpolarization-spatial and spatial-polarization collective unitary operations,respectively. The involved linear optical elements are reduced from 25 to 20and 21 to 20, respectively. Moreover, the parameterized quantum computation canbe flexibly manipulated by wave plates and phase shifters. As an application,we construct the specific quantum circuits to realize two-dimensional quantumwalk and quantum Fourier transformation. Our schemes are simple and feasiblewith the current technology.

Collective unitary evolution with linear optics by Cartan decomposition