🤖 AI Summary
This study addresses the lack of reliable certification and noise robustness in multi-scale quantum state learning under finite measurements by proposing a terminal-register-certified MERA learning method. Leveraging an inverse binary MERA architecture, coherent coarse-graining, and terminal joint measurements combined with causal-cone-complete local circuit comparison techniques, this work constructs complete bit-distribution certificates. Furthermore, it introduces a noise-resistance theorem and a calibrated total variation budget to establish rigorous performance guarantees in the finite-sample regime. Experimental results demonstrate that for an 8-qubit system, the proposed approach achieves an average fidelity of 0.9969, significantly outperforming parameter-matched local circuits across metrics including long-range errors, energy, and entropy.
📝 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.