Terminal-Register Certification for Finite-Measurement Learning of Multiscale Quantum States

Abstract:Structured quantum-state learning not only depends on an expressive ansatz but also on an operational certificate that stays meaningful with finite measurements and imperfect implementation. We study pure one dimensional states learning by an inverse binary multiscale entanglement renormalization ansatz (MERA). In the learning procedure, the qubits removed during coarse graining are controlled coherently and measured together at the terminal register. We confirm that an ideal sequential and terminal measurement schedule delivers the same complete bit string distribution under matched causal operations, while normalized postselection can amplify perturbations inversely with prefix acceptance. A noise aware theorem introduces an individual calibrated total variation implementation budget to the finite shot certificate. The protocol is estimated on an open boundary transverse field Ising ground state. A frozen 8-qubit schedule using $560$ million simulated training measurements per run achieves fidelity above $0.99$ in all $60$ held-out runs, with a mean fidelity of $0.996886$. 1080 circuit-noise cells and 6480 confidence-coverage rows are covered by fixed-circuit robustness validation without a locked soundness violation. We then address architectural fairness at $n=16$ using three new studies. In a 120-run exact-gradient multistart diagnostic, MERA has higher fidelity in 58/60 paired restarts and lower long-range error in 60/60, although no run met the prespecified stationarity criterion. Finally, a causal cone-complete, parameter matched local circuit achieves $2.62\times$ greater aggregate gate exposure yet loses all 30 paired comparisons in fidelity, long-range error, energy, and entropy.

Quantum Physics (quant-ph); Machine Learning (cs.LG)

arXiv:2609.33567 [quant-ph]

(or arXiv:2609.33567v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.33567 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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