Riemannian optimization for linear optical problems

This paper addresses the challenge of optimizing passive multiport interferometers, which are central to quantum technologies but often suffer from complex, non-convex optimization landscapes when using standard parameterizations like the Clements decomposition. The author introduces a Riemannian optimization framework that operates directly on the manifold of unitary matrices. By deriving a closed-form analytical formula for the Riemannian gradient of any differentiable function defined over these unitaries, the paper eliminates the need for automatic differentiation or numerical approximations of the gradient.

The core contribution is a theorem providing an analytical gradient for cost functions defined on the unitary group, specifically tailored for linear optical applications. The author demonstrates that this approach, when implemented with Riemannian quasi-Newton methods like BFGS, is orders of magnitude faster than existing optimizers found in the literature. Furthermore, the paper shows that by avoiding parameterizations of the unitary group, the optimization landscape becomes more favorable, leading to higher success rates in finding global optima for heralded state and gate preparations. The author successfully improves upon several success probabilities for NOON states and photon catalysis preparations previously reported in the literature.

Optimizing linear optical circuits is a computationally demanding task that limits the scale of quantum experiments researchers can design. By providing a faster, more robust optimization framework, this work enables the exploration of larger Hilbert spaces, involving more modes and photons. This is particularly relevant for scaling up quantum computing, communication, and metrology applications that rely on heralded state preparation and gate synthesis. The implementation, provided in the QOptCraft library, offers a practical tool for researchers to solve previously intractable optimization problems in linear optics.