Fixed-Protocol Amortized MPS Tomography with Conformalized Predictive Uncertainty

2026-07-13

AI Summary

This work addresses the challenge that quantum state tomography often degenerates into memorization of specific states under data scarcity, hindering faithful reconstruction. To overcome this, the authors propose an amortized estimator based on matrix product state (MPS) parameterization, employing a fixed local Pauli measurement protocol and a norm-preserving fidelity loss function, while deliberately omitting permutation-invariant encoders to enhance reconstruction performance in the small-sample regime. A novel conformity-calibrated Dropout ensemble is introduced to provide, for the first time, reliable uncertainty intervals for unmeasured observables. Experiments demonstrate a fidelity of 0.90 on 10-qubit systems—improving by up to 0.59 over baselines—with 0.88 achieved at bond dimension χ=4. On IBM hardware, five states are reconstructed from real measurement data with fidelity as high as 0.97, and predicted uncertainty intervals for unmeasured observables achieve approximately 90% coverage.

Abstract

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 \emph{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 $χ$-MPS, conditioning on an \emph{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 \emph{growing} in $n$; $0.88$ at bond dimension $χ{=}4$), 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).

Problem

• quantum state tomography
• sample efficiency
• measurement conditioning
• predictive uncertainty
• matrix product states

Innovation

• amortized tomography
• matrix product state
• conformal prediction
• local Pauli measurements
• gauge-invariant fidelity