Universal Quantum Transformer
Classical continuous-space neural networks fundamentally struggle to lock into exact formal rules, whether mathematical, such as modular arithmetic and non-Abelian group algebra, or linguistic, such as systematic compositional generalization. To approximate these discrete logical rules, they often rely on massive parameter scaling, resulting in stochastic instability even after delayed generalization phenomena known as grokking. Here, we introduce the Universal Quantum Transformer (UQT), a novel, quantum-native computing architecture that uses the physical properties of multi-qubit systems as a universal inductive bias for exact algebraic and compositional reasoning. Rather than translating classical neural mechanisms, our framework relies entirely on parameterized geometric phase embedding and $SU(2)$ wave-interference. We demonstrate that an identical quantum attention circuit, operating on a highly compact 5 or 6 qubit substrate with only 551 to 1,650 trainable parameters, exactly learns three highly distinct formal classes: cyclic modular arithmetic ($\mathbb{Z}_{11}$), non-Abelian algebra (the $S_4$ permutation group), and systematic linguistic compositionality (the SCAN language). While standard classical models, including multi-layer perceptrons (MLPs) and Transformers, exhibit stochastic instability at convergence, the UQT achieves mathematically exact, deterministic generalization. We define this stricter regime as crystallization: a step beyond the well-known phenomenon of grokking. Finally, we deploy the UQT on noisy intermediate-scale quantum (NISQ) hardware, achieving 97.5% accuracy on IBM Quantum computers. These results demonstrate that the UQT provides a structurally suited inductive bias for exact formal reasoning that standard classical continuous-space architectures do not natively provide.