Fixed-Protocol Amortized MPS Tomography with Conformalized Predictive Uncertainty

Quantum state tomography is sample-starved, and the states one prepares live on a narrow, learnable manifold. A k=0 prior-only control shows that on concentrated families a prior estimate is already near-optimal, so ``high fidelity at few measurements'' can be family memorization rather than tomography; genuine measurement-efficiency needs a model that conditions on the measurements and demonstrably uses them. On a shared matrix-product-state (MPS) core parameterization we study two routes. Approach~A learns a generative prior over MPS cores with measurement-guided posterior inference (gold-standard-validated, but whose few-measurement accuracy the control shows is largely the prior). Approach~B, our main proposal, is a fixed-protocol amortized MPS estimator trained once with a gauge-invariant fidelity loss; we deliberately do not rest it on a permutation-invariant set encoder (a plain MLP matches it). The decisive lever is the measurement design: motivated by the fact that local reduced density matrices determine a chi-MPS, conditioning on an informative local Pauli set rather than random strings turns a modest, memorization-prone estimator into a high-fidelity one (≈!0.95, up to +0.59 over prior-only, decisively passing a shuffled-measurement control). A dropout ensemble, conformally recalibrated, gives ≈!90%-coverage intervals -- including for observables never measured, where a shot-based interval does not exist. Quality holds as the system grows (fidelity 0.90 at n=10, gain growing in n; 0.88 at bond dimension chi=4), the parameterization is polynomial (native contraction to 20 qubits), and we close the loop on IBM hardware (5 states at 0.97 from hardware-measured Paulis).