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Programmable unitary photonic networks that interfere hundreds of modes areemerging as a key technology in energy-efficient sensing, machine learning,cryptography, and linear optical quantum computing applications. In this work,we establish a theoretical framework to quantify error tolerance andscalability in a more general class of "binary tree cascade'' programmablephotonic networks that accept up to tens of thousands of discrete input modes$N$. To justify this scalability claim, we derive error tolerance andconfiguration time that scale with $\log_2 N$ for balanced trees versus $N$ inunbalanced trees, despite the same number of total components. Specifically, weuse second-order perturbation theory to compute phase sensitivity in eachwaveguide of balanced and unbalanced networks, and we compute the statistics ofthe sensitivity given random input vectors. We also evaluate such networksafter they self-correct, or self-configure, themselves for errors in thecircuit due to fabrication error and environmental drift. Our findings haveimportant implications for scaling photonic circuits to much larger circuitsizes; this scaling is particularly critical for applications such as principalcomponent analysis and fast Fourier transforms, which are important algorithmsfor machine learning and signal processing.

Scalable and self-correcting photonic computation using balanced photonic binary tree cascades