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. Our method is a fixed-protocol amortized matrix-product-state (MPS) estimator: one fixed informative protocol mapped directly to MPS cores, 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 learned generative prior over the same cores, inverted by measurement-guided posterior sampling, is what makes the control bite in the first place, and we retain it only as a complementary route (Approach A, deferred to the appendix). 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 ($\approx 0.95$, up to $+0.59$ over prior-only, decisively passing a shuffled-measurement control). A dropout ensemble, conformally recalibrated, gives $\approx 90\%$-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). Our prior-only and shuffled controls are a necessary integrity check we argue learned-QST work should adopt.