Decomposing large unitaries into multimode devices of arbitrary size

Decomposing complex unitary evolution into a series of constituent components is a cornerstone of practical quantum information processing. While the decompostion of an \(n\times n\) unitary into a series of \(2\times2\) subunitaries is well established (i.e. beamsplitters and phase shifters in linear optics), we show how this decomposition can be generalised into a series of \(m\times m\) multimode devices, where \(m>2\). If the cost associated with building each \(m\times m\) multimode device is less than constructing with \(\frac{m(m-1)}{2}\) individual \(2\times 2\) devices, we show that the decomposition of large unitaries into \(m\times m\) submatrices is is more resource efficient and exhibits a higher tolerance to errors, than its \(2\times 2\) counterpart. This allows larger-scale unitaries to be constructed with lower errors, which is necessary for various tasks, not least Boson sampling, the quantum Fourier transform and quantum simulations.