Rank-Aware Conditional Synthesis: Feasible Quantum Generative Modeling on Matrix Product State Manifolds

Matrix Product States (MPSs) have become an indispensable symmetry-based representation for simulating quantum systems on near-term hardware by constraining entanglement entropy through a fixed bond dimension χ. This study identifies a critical “rank explosion” phenomenon that destabilizes this low-rank manifold during conditional quantum diffusion processes. We empirically demonstrate that the introduction of conditional guidance—essential for semantic control—injects global correlations that drive the effective Schmidt rank to increase by 4× (from χ=4 to 16), saturating the simulation limits and necessitating quantum circuits with approximately 1.8×103 Controlled-NOT (CNOT) gates. Such circuit depths fundamentally exceed the operational coherence budgets of Noisy Intermediate-Scale Quantum (NISQ) devices. To mitigate this structural instability, we propose Rank-Aware Conditional Synthesis (RACS), a sampling framework that maintains the latent trajectory within a prescribed MPS manifold through step-wise manifold projection and time-shift error correction. Experimental results on real-world semantic data reveal that RACS reduces reconstruction error, or Mean Squared Error (MSE) by 30.8% and enhances latent trajectory smoothness by 36.8% compared to conventional post hoc truncation. At a fixed hardware-efficient rank of χ=4, RACS achieves a +4.8% fidelity gain and exhibits superior robustness against depolarizing noise. By resolving the tension between conditional expressivity and entanglement constraints, RACS provides a principled, hardware-aware methodology for high-fidelity quantum generative modeling.