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

1 Introduction

• A terminal register retains every qubit discarded during binary MERA analysis, enabling a joint-event certificate that accounts for the top state and multiscale errors.The exact reconstruction identity is F = p0Ftop, with layerwise and local discarded-weight diagnostics requiring no independence assumption.Source: §1 · pp. 2–3
• All 60 held-out eight-qubit TFIM reconstructions exceeded 0.99 fidelity under the frozen schedule.The schedule was selected using calibration fields and seeds before held-out fields were opened.Source: §1 · p. 3

2 Related Work

• Prior MERA-learning methods use sequential conditional measurements, while this work asks how one joint terminal record can certify inverse-circuit reconstruction with finite shots and calibrated implementation mismatch.The paper distinguishes this certification question from property prediction and target-specific fidelity estimation.Source: §2 · p. 3

3.1 Target Family

• The target family is pure open-chain ground states of the open-boundary transverse-field Ising model, with exact diagonalization supplying targets and oracle observables for these simulations.Training relies on measurement estimates except in explicitly labeled noiseless diagnostics.Source: §3.1 · p. 4

3.2 Binary MERA analysis circuit

• Binary MERA alternates boundary disentanglers and isometry dilations, discarding one qubit per block while leaving previously discarded wires untouched by later layers.For n = 8, the analysis proceeds 8 → 4 → 2 → 1, leaving seven discarded qubits and one top qubit.Source: §3.2 · p. 4

3.3 Reconstruction

• Inverse-circuit reconstruction uses the retained top state together with the discarded-register all-zero event to relate the reconstructed state to the learned circuit.For p0 > 0, the top state is conditioned on the discarded bits being zero.Source: §3.3 · p. 5

4.1 Exact certificate hierarchy

• The reconstruction fidelity factors exactly as F = p0Ftop, and the hierarchy bounds infidelity using global, layerwise, or local discarded-register failure without independence assumptions.Layer and local certificates are interpretable diagnostics; the bounds also extend to observable error through trace-distance control.Source: §4.1 · pp. 5–6

4.2 Finite-measurement confidence

• Finite-shot certification combines one-sided confidence bounds on discarded-register failure indicators with a top-state infidelity bound based on accepted X, Y, and Z measurements.The top-state bound is 1 − Ftop ≤ 3a^2, with the trivial infidelity bound used when its right-hand side exceeds one.Source: §4.2 · pp. 6–7

5.1 Exact equivalence of ideal schedules

• Under the causal-discard condition, sequential and joint terminal measurement produce the same complete-record distribution, with outcome-dependent feed-forward excluded.Identical complete records therefore give identical sequential-product and terminal estimators; the paper claims no universal confidence or shot-complexity advantage for estimating the same probability from the same data.Source: §5.1 · p. 7

5.2 Instability of normalized postselection

• Normalized postselection can amplify trace-distance perturbations inversely with acceptance probability, and this inverse-acceptance dependence is necessary in general.The section also relates accepted conditional samples to the raw-preparation budget, with the expected raw preparations depending on prefix acceptance.Source: §5.2 · p. 8

5.3 Noise-aware terminal certificate

• A separately calibrated total-variation implementation budget is added to the finite-shot all-zero certificate, while the reconstructed top-state error remains a separate term.The budget may be justified by bounds on analyzed states or implemented channels; a discarded-register budget alone does not control bias in the top register.Source: §5.3 · p. 9

6.1 Finite-shot learner

• The learner optimizes a directly measurable local certificate from computational-basis samples using SPSA with an Adam-style moment update.Each perturbation direction uses two objective evaluations, and monitoring, top tomography, and evaluation measurements are reported separately from the training budget.Source: §6.1 · p. 9

6.2 Frozen eight-qubit protocol

• The eight-qubit schedule was calibrated and confirmed before opening held-out fields, with success defined as fidelity above 0.99 and no numerical or top-tomography failure.The final certificate uses 10,000 discarded-register shots; the 62 held-out Pauli observables are not used in training.Source: §6.2 · pp. 9–10

6.3 Robustness protocol

• Robustness validation applies noise models to the 60 learned circuits and tests fixed-circuit behavior rather than noisy optimization.The tested models include local depolarizing noise, amplitude weakening, and independent terminal readout flips on discarded bits.Source: §6.3 · p. 10

6.4 Sixteen-qubit fairness extensions

• The sixteen-qubit frontier separates parameter count, aggregate gate exposure, exact-gradient behavior, and causal coverage as distinct architectural matching criteria.The long-range metric is mean absolute error of the distance-averaged connected profile for distances r ≥ 8; the exact-gradient diagnostic uses 20 paired restarts per field and is a fixed-budget best-found frontier, not a certified global optimum. The local control uses nine alternating nearest-neighbor layers, with all 36 held-out pairs having overlapping causal cones; at equal 225 coordinates and 560 million training shots, its aggregate SU(4) shot gate exposure is 2.62 times MERA's.Source: §6.4 · pp. 10–11

7.1 Implementation and theorem verification

• The certificate verifier passed all checks on 60 locked states, with maximum absolute error 1.33 × 10^-15 for F − p0Ftop.The independent adversarial test sections also passed.Source: §7.1 · p. 11

7.2 Locked eight-qubit evaluation

• All 60 locked eight-qubit held-out runs exceeded fidelity 0.99, and every finite-shot certificate covered the true infidelity.Six runs crossed the 0.01 warning threshold; these were certificate non-tightness observations, not reconstruction failures.Source: §7.2 · pp. 11–12, Table 3 · p. 12, Figure 1 · p. 12

7.3 Robustness validation

• Across 1080 circuit-noise cells, calibrated budgets upper-bounded exact terminal-distribution shifts, with no locked soundness violation.The maximum ideal sequential-versus-terminal total variation distance was 5.72 × 10^-17; the validation addressed implementation soundness, not training convergence under noise.Source: §7.3 · p. 12, Figure 2 · p. 13

7.5 Resource-controlled sixteen-qubit results

• At the largest n = 16 gate-exposure endpoint, MERA-PM225 and MPS fidelity and energy differences were unresolved, while MERA had lower long-range and half-chain-entropy errors.The long-range error difference was −0.014582 and the half-chain-entropy error difference was −0.030444; both simultaneous intervals excluded zero.Source: §7.5 · p. 14, Table 5 · p. 14
• The n = 16 finite-shot response was nonmonotone, so the tested schedules do not support a uniform shot-efficiency or monotone-convergence claim.Fidelity and long-range-error comparisons varied across the four exposure endpoints.Source: §7.5 · p. 14, Figure 4 · p. 15
• MERA-PM225 had higher fidelity in 58/60 paired restarts and lower long-range error in all 60, but none of the 120 runs met the prespecified convergence rule.These were fixed-budget best-found results, not certified global optima.Source: §7.5 · p. 14, Table 6 · p. 14
• MERA won all 30 paired comparisons against a causal-cone-complete local control despite 2.62× greater aggregate gate exposure for the local model.The comparison was against one specified shared-parameter local family, not all local circuits.Source: §7.5 · p. 14, Table 7 · p. 15

• The exact factorization makes the all-zero event a fidelity certificate that includes both discarded-register outcomes and the reconstructed top-state contribution.The top-state contribution cannot be neglected.Source: §8 · p. 15
• Ideal sequential and terminal schedules have identical complete distributions under matched causal operations, but normalized postselection can amplify perturbations inversely with prefix acceptance.The terminal global event avoids that normalization, but does not reduce circuit depth or ensure less physical noise; implementation soundness still requires the explicit τD term.Source: §8 · pp. 15–16
• At n = 16, MERA retained lower long-range and entropy errors at the largest matched exposure, while fidelity and energy differences from MPS remained unresolved.The exact-gradient multistart diagnostic and causal-cone-complete local comparison also favored MERA, within their reported study scopes.Source: §8 · p. 16, Figure 5 · p. 16
• Finite-shot performance was nonmonotone, and the exact-gradient multistarts did not meet the stationarity criteria within 5000 steps.The resulting MERA solutions are fixed-budget best-found results, not certified capacity optima.Source: §8 · p. 16

Theorem assumptions

• The reconstruction theorem assumes a pure target, unitary analysis circuit, inverse-circuit reconstruction, and a pure top estimate.Source: §9 · p. 17

Calibration

• The implementation budget τD must be calibrated or independently upper bounded; agreement on discarded registers alone does not certify the top qubit.Source: §9 · p. 17

Experimental scope

• All experiments are classical finite-shot simulations; no hardware noise, drift, compilation, or readout-calibration experiment is claimed.Five confirmation runs per calibration field also provide only a weak population-success statement; the 60-run held-out evaluation gives the stronger reliability estimate.Source: §9 · p. 17

Robustness scope

• Robustness tests certify fixed learned circuits, not convergence of training under noise.Source: §9 · p. 17

Architectural scope

• At n = 16, conclusions are limited to h/J ∈{0.9, 1.0, 1.1}, χ = 2, specified parameterizations, and the tested exposure grid; larger sizes or bond dimensions are not established.Source: §9 · p. 17

Resource comparisons

• Aggregate SU(4)-shot-gate exposure omits routing, parallel depth, readout overhead, and architecture-dependent classical cost, while no comparator matches topology, gate count, depth, and parameter sharing simultaneously.Source: §9 · p. 17

Optimization and metrics

• Zero of 120 exact-gradient runs met the prespecified stationarity rule, so best-found results are neither global-optimum evidence nor a capacity bound; long-range error is also one exact post-training observable family.Source: §9 · p. 17

10 Conclusion

• The terminal-register theory links reconstruction fidelity to one joint terminal event and an independently estimated top state; layer and local marginals give interpretable upper bounds.Ideal complete sequential and terminal records are equivalent, but normalized prefix postselection has inverse-acceptance sensitivity; a calibrated implementation budget preserves soundness under noisy terminal statistics.Source: §10 · p. 17
• At n = 16, MERA-PM225 and MPS have no statistically resolved fidelity or energy difference at the largest approximately matched exposure, while MERA has smaller long-range and entropy errors.Exact-gradient multistarts and the causal-cone-complete local control also reinforce the multiscale result, but the convergence audit prevents overstatement.Source: §10 · p. 17
• Eight-qubit TFIM reconstructions exceeded 0.99 fidelity in all 60 held-out trials, and robustness tests found no locked soundness violation.Source: §10 · p. 17

A Detailed Numerical Results

• Table 8 reports eight-qubit held-out results by field, with ten untouched replicates in each row.Source: Table 8 · p. 18
• Table 9 summarizes pooled paired differences across the gate-exposure grid, with 97.5% stratified paired-bootstrap intervals over three fixed fields.Positive fidelity differences and negative error differences favor MERA-PM225.Source: Table 9 · p. 18
• Manuscript values were generated from frozen JSON/CSV reports, with a machine-readable audit recording cryptographic hashes and automated checks requiring final protocol and theorem-verification flags.Source: A Detailed Numerical Results · p. 18

C Conclusion

• The framework estimates reconstruction fidelity from a joint terminal outcome and top-state fidelity, while layer and local marginals provide looser, interpretable bounds.Ideal complete sequential and terminal records contain the same information, but normalized prefix postselection grows more sensitive as acceptance probability falls; an implementation-error budget keeps the noisy certificate sound.Source: C Conclusion · p. 19
• Across 180 independent n = 16 runs, MERA had better fidelity and held-out long-range correlations than MPS at the same parameter counts, and substantially outperformed the shallow local architecture under matched resources.Source: C Conclusion · p. 19
• All 60 eight-qubit reconstructions exceeded 0.99 fidelity, certificates matched exact identities to machine precision, and no locked robustness test produced a soundness violation.Source: C Conclusion · p. 19

A Detailed Numerical Results

• Table 10 reports eight-qubit held-out results by field across ten untouched replicates per row; Table 11 defines paired resource-frontier differences favoring the challenger when ∆F > 0 and ∆ELR < 0.Source: Table 10 · p. 19, Table 11 · p. 19

B Reproducibility and Claim Audit

• Maximum absolute error was 1.33 × 10−15 across exact factorization checks, and all 60 locked n = 8 reconstructions passed.Single/full-layer round trips also passed, reconstructing teacher states from analysis followed by generation.Source: B Reproducibility and Claim Audit · p. 20
• All listed finite-shot, equivalence, theorem, n = 8, n = 16, and parameter-matched checks passed.The audit records 3/3 screening and 9/9 calibration runs, 15/15 fresh confirmations, 60/60 held-out successes, and 180/180 n = 16 frontier runs; MERA's paired long-range intervals were strictly below zero under both comparison rules.Source: B Reproducibility and Claim Audit · p. 20