Reconfigurable quantum-optical circuits in a complex medium

Programmable optical circuits form a key part of quantum technologies today. As the size of such circuits is increased, maintaining precise control over every individual component becomes challenging. Here we show how embedding an optical circuit in the higher-dimensional space of a large mode-mixer allows us to forgo control over individual elements, while retaining a high degree of programmability over the circuit. Using this approach, we implement high-dimensional linear optical circuits within a commercial multi-mode fibre placed between controllable phase planes. We employ these circuits to manipulate high-dimensional entanglement in up to 7 dimensions, demonstrating their application as fully programmable quantum gates. Furthermore, we show how these circuits turn the multi-mode fibre itself into a generalised multi-outcome measurement device, allowing us to both transport and certify entanglement. Finally, we show how a high circuit fidelity can be achieved with a low circuit depth by harnessing the resource of a high-dimensional mode-mixer. Our work serves as an alternative yet powerful approach for realising precise control over high-dimensional quantum states of light.