Decomposing large unitaries into multimode devices of arbitrary size

TL;DR

This paper generalizes the decomposition of large unitaries into smaller components, proposing that using larger multimode devices ($m>2$) can be more resource-efficient and error-tolerant than traditional $2\times2$ decompositions, enabling scalable quantum information processing.

Contribution

It introduces a method to decompose unitaries into $m\times m$ multimode devices, improving efficiency and error tolerance over standard $2\times2$ methods.

Findings

Larger multimode devices reduce resource costs.

Decomposition into $m\times m$ devices improves error tolerance.

Enables scalable quantum tasks like Boson sampling and quantum Fourier transform.

Abstract

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.