Deploying large language models (LLMs) as optimizers for black-box scientific design problems requires efficient test-time refinement under expensive evaluations and without training data. We propose a \emph{memory-augmented test-time optimization} framework that combines episodic memory of high-scoring candidates, score-difference feedback, and restart-from-best sampling to improve iterative search. We evaluate the approach on quantum circuit synthesis, where the objective is to maximize the Meyer--Wallach (MW) global entanglement measure under an exponentially expensive black-box oracle. On 20-qubit circuits, the framework achieves $Q(ψ)=0.99$ without feedback. On the more challenging 25-qubit task, feedback and restart mechanisms enable multiple runs to reach $Q(ψ)=1.0$ within 45 oracle calls, while a budget-matched random hill-climbing baseline stalls below $Q(ψ)\approx0.29$. These results demonstrate that memory and evaluator feedback substantially improve the sample efficiency of LLM-based black-box optimization and establish quantum circuit synthesis as a challenging benchmark for test-time optimization.

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