Automated discovery of heralded ballistic graph state generators for fusion-based photonic quantum computation

Designing photonic circuits that prepare graph states with high fidelity and success probability is a central challenge in linear optical quantum computing. Existing approaches rely on hand-crafted designs or fusion-based assemblies. In the absence of multiplexing/boosting, both post-selected ballistic circuits and sequential fusion builds exhibit exponentially decreasing single-shot yields, motivating automated discovery of higher-success circuits. We present a general-purpose optimization framework for automated photonic circuit discovery using a novel polynomial-based simulation approach, enabling efficient strong simulation and gradient-based optimization. Our framework employs a two-pass optimization procedure: the first pass identifies a unitary transformation that prepares the desired state with perfect fidelity and maximal success probability, while the second pass implements a novel sparsification algorithm that reduces this solution to a compact photonic circuit with minimal beamsplitter count while preserving performance. This sparsification procedure often reveals underlying mathematical structure, producing highly simplified circuits with rational reflection coefficients. We demonstrate our approach by discovering optimized circuits for $3$-, $4$-, and $5$-qubit graph states across multiple equivalence classes. For 4-qubit states, our circuits achieve success probabilities of $2.053 \times 10^{-3}$ to $7.813 \times 10^{-3}$, outperforming the fusion baseline by up to $4.7 \times$. For 5-qubit states, we achieve $5.926 \times 10^{-5}$ to $1.157 \times 10^{-3}$, demonstrating up to $7.5 \times$ improvement. These results include the first known state preparation circuits for certain 5-qubit graph states.