Quantum Circuit Generation via test-time learning with large language models

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 $Q(\psi)=0.99$ without feedback.

On the more challenging 25-qubit task, feedback and restart mechanisms enable multiple runs to reach $Q(\psi)=1.0$ within 45 oracle calls, while a budget-matched random hill-climbing baseline stalls below $Q(\psi)\approx0.29$.

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.