MagicCore-LM: Compact Algebraic Interfaces for Learning Near-Clifford Quantum Circuits

Large language models (LLMs) are increasingly applied to specialized scientific problems, often using textual or symbolic descriptions as inputs. However, their general-purpose capabilities do not automatically translate into an understanding of the domain-specific mathematical structure underlying these problems. A faithful description may therefore specify what to compute without exposing the structure needed to learn it effectively. We address this gap in quantum computing, where different gate sequences can implement the same computation while obscuring the algebraic relations that determine measurement outcomes. We introduce MagicCore-LM, a learning interface that uses quantum algebra to translate near-Clifford circuits into compact, unexecuted rotation instructions and transformed measurement queries. The compiler performs exact Clifford transformations, while a relation-aware reader learns how the resulting operation identities, signs, and physical order jointly determine measurement expectations. In a three-seed blind evaluation of eight matched variants, MagicCore reduces in-distribution (IID) mean absolute error by 12.7% relative to raw-gate inputs. Standard set-based readers also achieve lower IID error with Compact inputs, supporting the interface across reader architectures. Deploying the frozen reader with an optimized GPU compiler yields $1.68\times$ and $9.15\times$ end-to-end speedups over Raw on the audited higher-complexity circuits at batch sizes 1 and 32, respectively, on an RTX 4090. These measurements include preprocessing; compiler changes preserve same-batch predictions without retraining. Additional studies examine frame-explicit and propagated-state representations, as well as integration with five pretrained LLMs.