Programmable Quantum Matter: Heralding Large Cluster States in Driven Inhomogeneous Spin Ensembles

Atom-like emitters in solids are promising platforms for quantum sensing and information processing, but inhomogeneities in the emitter fine structure complicate quantum control. We present a framework that leverages this diversity to reduce the resources for generating optically heralded spin cluster states across $N_q$ emitters from the conventional order $O(N_q)$ to $O(1)$ in ensembles of $N_q \sim 10$-$100$. An optimized pulse sequence simultaneously corrects pulse-length and detuning errors, achieving single-qubit gate fidelities exceeding $99.99\%$ for errors (normalized relative to the Rabi drive strength) up to 0.3, while maintaining fidelities above $99\%$ for errors as large as 0.4. Applied as a Carr-Purcell-Meiboom-Gill (CPMG) dynamical decoupling protocol to the dominant noise spectrum of silicon-vacancy centers in diamond, it enhances ensemble coherence times by over $7\times$ compared to interleaved bang-bang based CPMG. For state-of-the-art dilution refrigerators, global resonant optimal decoupling across $N_q$ spins sharply reduces heating, addressing the trade-off between the spin coherence and scaling to $N_q \gg 1$. We further introduce a modified single-photon entanglement protocol with an efficient algorithm for deterministic entanglement compilation. Depending on the decoupling time window, our method yields order $O(10^2$-$10^4)$ more entanglement links than bang-bang sequences, with theoretical guarantees of order $\Omega(N_q)$ unique links, improvable by control tuning. Together, these techniques provide scalable tools - including global control, phase denoising, remote entanglement, and compilation - for robust quantum computing architectures with heterogeneous spin ensembles.