We present an extended-algebra (EA) control workflow for fixed-horizon, bounded-amplitude quantum control. The method optimizes a state-transfer trajectory in a polynomial EA, projects the optimized lifted trajectory to native physical controls using a budgeted dictionary of product-formula approximants, and then refines the resulting waveform on the original idealized physical model with gradient ascent pulse engineering (GRAPE). The sub-Riemannian gradient ascent pulse engineering (SR-GRAPE) method and the direct-GRAPE baseline use the same physical Hamiltonian, amplitude bounds, time horizon, discretization, and terminal physical optimizer; the difference is the initialization supplied to the final physical-space optimization. For Greenberger-Horne-Zeilinger (GHZ) preparation with Ising-coupled qubits and local $x\mathrm{and}y$ drives, SR-GRAPE changes which basin is reached. In the five-qubit state-growing benchmark, SR-GRAPE reaches $\mathcal{F}\ge 0.99$ in $25/25$ trials with mean fidelity 0.9917, whereas direct physical-space GRAPE from random feasible initialization remains near the $\mathcal{F}=0.5$ GHZ plateau in all 25 trials. Six-qubit state-growing also gives $25/25$ successes at $\mathcal{F}\ge 0.99$, while direct GRAPE remains at mean fidelity 0.4998 with $0/25$ successes. In a seven-qubit resource-extended setting with $u_{max}=3$, SR-GRAPE again reaches $\mathcal{F}\ge 0.99$ in $25/25$ seeds with mean fidelity 0.9901, whereas the corresponding direct-GRAPE baseline remains at mean fidelity 0.5000 with $0/25$ successes. For the GHZ task, $\mathcal{F}>0.5$ certifies genuine multipartite entanglement; the $\mathcal{F}=0.5$ plateau does not cross this witness threshold.

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• Quantum control