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

Bhvain Makwana; Kashyap Patel; Manjunath Joshi; Jaideep Mulherkar

quant-phSep 27, 2026

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

Authors: Bhvain Makwana, Kashyap Patel, Manjunath Joshi, Jaideep Mulherkar

Organizations: Dhirubhai Ambani University, Gandhinagar, Gujarat, India · Georgia Institute of Technology, Atlanta, Georgia, USA

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.

Figures & tables

Method	Structural prior	Full model	Certificate or guarantee	Distinction from this work
Compressed sensing [ 7 ]	Low rank	Yes	Recovery from suitable random measurements	No multiscale inverse circuit or terminal register
Direct fidelity estimation [ 6 ]	Known target	No	Target-overlap estimate	Certifies a supplied target rather than learning MERA
Classical shadows [ 9 ]	Observable-dependent	No	Many-property prediction	Does not by itself reconstruct the inverse circuit
MPS tomography [ 4 , 12 ]	1D low entanglement	Yes	Parent-Hamiltonian/local consistency bounds	Chain geometry rather than multiscale discarded registers
Neural tomography [ 17 ]	Neural ansatz	Yes	Likelihood-based reconstruction	No terminal discarded-weight certificate
Sequential MERA [ 11 , 13 ]	Multiscale entanglement	Yes	Layerwise truncation control	Uses sequential conditional states and proxies

Table 1: Positioning relative to representative state-learning and certification methods. Full model means that the output specifies a generative state or circuit rather than only selected properties.

Architecture	$SU(4)$ blocks	Coordinates	Role
MERA-PM225	26	225	Parameter-shared binary MERA
MPS	15	225	Canonical $\chi=2$ sequential baseline
Local-LC225	68	225	Nine-layer nearest-neighbor control with shared parameters and complete held-out causal coverage

Table 2: Resources in the final

n=16

fairness suite. Coordinates count independent real parameters; physical blocks count

SU(4)

gate occurrences in one circuit execution.

Metric	Mean	Median	Minimum	Maximum	Interpretation
Fidelity	0.996886	0.997276	0.992602	0.999218	60/60 exceed 0.99
Local certificate $\overline{C}_{\mathrm{MERA}}$	0.005965	0.005427	0.001341	0.014464	Sound, sometimes non-tight
Held-out exact MAE	0.008092	0.008014	0.003511	0.013288	62 unused observables
Energy error/site	0.002108	0.001971	0.000322	0.005219	Computed from held-out $X,ZZ$
Certificate slack	0.002851	0.002679	0.000559	0.007065	$\overline{C}_{\mathrm{MERA}}-(1-F)$

Table 3: Locked

n=8

held-out evaluation over 60 runs.

Figure 1: Locked $n=8$ held-out evaluation. Every run exceeds the fidelity gate. Orange certificate points identify the six runs above the prespecified $0.01$ warning threshold; these are certificate non-tightness observations, not reconstruction failures.

Figure 2: Fixed-circuit robustness validation. Exact terminal distribution shifts remain below their conservative calibrated budgets. The validation covers implementation soundness, not convergence of training under noise.

Architecture	Fidelity	Success rate	Certificate	Observable MAE	Energy error
MERA	0.99550	0.950	0.00895	0.00906	0.00319
MPS	0.99765	1.000	0.00444	0.00797	0.00149
TTN	0.97113	0.167	0.04740	0.03106	0.01880

Table 4: Eight-qubit architecture comparison (60 runs per architecture).

Figure 3: Eight-qubit architecture comparison. MPS is slightly better than MERA in fidelity, whereas removing MERA disentanglers to obtain TTN causes a pronounced degradation.

Metric	MERA-PM225	MPS	Paired difference	97.5% interval	Conclusion
Fidelity	0.953529	0.952925	$+0.000604$	$[-0.016347,0.015413]$	Unresolved
Long-range $ZZ$ MAE	0.069314	0.083896	$-0.014582$	$[-0.028112,-0.000340]$	Favors MERA
Entropy error	0.028495	0.058940	$-0.030444$	$[-0.048778,-0.010442]$	Favors MERA
Energy error	0.009409	0.008788	$+0.000621$	$[-0.001672,0.003054]$	Unresolved

Table 5: Largest approximately matched aggregate gate-exposure endpoint at

n=16

: MERA-PM225 minus MPS. Pooled intervals are 97.5% stratified paired-bootstrap intervals over three fixed fields.

$h/J$	$F_{\rm MERA}$	$F_{\rm MPS}$	$\Delta F$	$\Delta E_{\rm LR}$	Fidelity wins
0.9	0.989118	0.979675	$+0.008231$	$-0.061758$	19/20
1.0	0.993167	0.988466	$+0.004284$	$-0.038423$	19/20
1.1	0.997520	0.994560	$+0.003042$	$-0.021442$	20/20

Table 6: Exact-gradient multistart diagnostic. Each field has 20 paired restarts. Values are medians;

\Delta F

and

\Delta E_{\rm LR}

are mean paired differences.

Panel of Figure 4 — Figure 4: Approximately matched aggregate $SU(4)$ -shot-gate exposure. Points show fieldwise means over ten paired replicates; the response is not assumed monotone. At the largest exposure, fidelity is unresolved while long-range error favors MERA-PM225.

Metric	MERA	Local	Paired difference	97.5% interval	Wins
Fidelity	0.967559	0.715802	$+0.251757$	$[0.243706,0.260153]$	30/30
Long-range $ZZ$ MAE	0.058150	0.163972	$-0.105822$	$[-0.116590,-0.095612]$	30/30
Energy error	0.006794	0.050304	$-0.043510$	$[-0.045970,-0.041217]$	30/30
Entropy error	0.016280	0.268855	$-0.252575$	$[-0.280188,-0.225882]$	30/30

Table 7: Causal-cone-complete, parameter-matched control: MERA-PM225 minus Local-LC225. Pooled 97.5% intervals use stratified paired bootstrap resampling over the three fixed fields.

Panel of Figure 5 — Figure 5: Top: exact-gradient 20-restart diagnostic; error bars summarize restart variation and do not certify global optimality. Bottom: the causal-cone-complete local control, which has equal coordinates and greater aggregate gate exposure than MERA-PM225.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

$h/J$	Mean fidelity	Mean $\overline{C}_{\mathrm{MERA}}$	Held-out MAE	Energy error/site	Successes
0.25	0.998379	0.002502	0.005863	0.000679	10/10
0.75	0.995347	0.009513	0.008747	0.002496	10/10
0.90	0.997350	0.005189	0.007346	0.001571	10/10
1.10	0.996484	0.006869	0.008311	0.002409	10/10
1.25	0.996918	0.005644	0.008281	0.002311	10/10
2.00	0.996836	0.006074	0.010000	0.003181	10/10

Appendix

Table 8: Eight-qubit held-out results by field; each row contains ten untouched replicates.

$G$	Fidelity difference	Fidelity interval	Long-range difference	Long-range interval
$2.6\times 10^{7}$	$-0.016781$	$[-0.025979,-0.007736]$	$-0.000857$	$[-0.001670,-0.000066]$
$2.6\times 10^{8}$	$-0.034965$	$[-0.064837,-0.004853]$	$-0.000984$	$[-0.002336,0.000210]$
$2.6\times 10^{9}$	$+0.063084$	$[0.029321,0.095294]$	$-0.044560$	$[-0.063861,-0.028264]$
$8.4\times 10^{9}$	$+0.000604$	$[-0.016347,0.015413]$	$-0.014582$	$[-0.028112,-0.000340]$

Appendix

Table 9: Pooled paired differences across the gate-exposure grid. Positive fidelity and negative error differences favor MERA-PM225. Intervals are 97.5% stratified paired-bootstrap intervals over three fixed fields.

Item	Status	Evidence used in this manuscript
Single/full-layer round trips	Passed	Analysis followed by generation reconstructs teacher states
Certificate hierarchy	Passed	Every one of 60 locked $n=8$ reconstructions
Exact factorization	Passed	Maximum absolute error $1.33\times 10^{-15}$
Finite-shot coverage	Passed	Global, layer, local, and top terms with total confidence at least 0.95
Sequential/terminal equivalence	Passed	Maximum TV $5.72\times 10^{-17}$ on learned circuits
Adversarial theorem tests	Passed	Equivalence, telescoping, postselection, survival, noise-aware sections

Appendix

Table 15

$h/J$	Mean fidelity	Mean $\overline{C}_{\mathrm{MERA}}$	Held-out MAE	Energy error/site	Successes
0.25	0.998379	0.002502	0.005863	0.000679	10/10
0.75	0.995347	0.009513	0.008747	0.002496	10/10
0.90	0.997350	0.005189	0.007346	0.001571	10/10
1.10	0.996484	0.006869	0.008311	0.002409	10/10
1.25	0.996918	0.005644	0.008281	0.002311	10/10
2.00	0.996836	0.006074	0.010000	0.003181	10/10

Appendix

Table 10: Eight-qubit held-out results by field; each row contains ten untouched replicates.

Contrast	$h/J$	$\Delta F$	$\Delta E_{\mathrm{LR}}$	95% CI for $\Delta E_{\mathrm{LR}}$
MERA-225 vs. MPS-225	0.9	0.01430	-0.06325	$[-0.11073,-0.01576]$
MERA-225 vs. MPS-225	1.0	0.02026	-0.03504	$[-0.05282,-0.01727]$
MERA-225 vs. MPS-225	1.1	0.01071	-0.00494	$[-0.01324,0.00336]$
MERA-390 vs. Local-390	0.9	0.30056	-0.26711	$[-0.29234,-0.24189]$
MERA-390 vs. Local-390	1.0	0.12709	-0.07389	$[-0.09218,-0.05560]$
MERA-390 vs. Local-390	1.1	0.05559	-0.01047	$[-0.01531,-0.00562]$

Appendix

Table 11: Fieldwise paired resource-frontier comparisons. We define

\Delta F=F_{\mathrm{challenger}}-F_{\mathrm{baseline}}

and

\Delta E_{\mathrm{LR}}=E_{\mathrm{LR}}^{\mathrm{challenger}}-E_{\mathrm{LR}}^{\mathrm{baseline}}

. Thus,

\Delta F>0

and

\Delta E_{\mathrm{LR}}<0

favor the challenger.

Item	Status	Evidence used in this manuscript
Single/full-layer round trips	Passed	Analysis followed by generation reconstructs teacher states
Certificate hierarchy	Passed	Every one of 60 locked $n=8$ reconstructions
Exact factorization	Passed	Maximum absolute error $1.33\times 10^{-15}$
Finite-shot coverage	Passed	Global, layer, local, and top terms with total confidence at least 0.95
Sequential/terminal equivalence	Passed	Maximum TV $5.72\times 10^{-17}$ on learned circuits
Adversarial theorem tests	Passed	Equivalence, telescoping, postselection, survival, noise-aware sections

Appendix

Table 18

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Optimal Stabilizer Testing and Learning with Limited Quantum Memory

Srinivasan Arunachalam, Louis Schatzki

IBM Research Silicon Valley · Dahlem Center for Complex Quantum Systems Freie Universität Berlin

Qubit Sample Complexity

Open paper AskPDF arXiv

We study stabilizer state testing and learning with limited coherent quantum memory. Here an algorithm sequentially receives copies of an unknown

n

-qubit state, but may keep only

k

qubits of coherent quantum memory between measurements. With unrestricted memory, seminal work of Gross, Nezami and Walter showed how to test

n

-qubit stabilizer states using

6

copies, which is dimension independent, unlike the learning complexity of

Θ(n)

. We show that this testing-vs-learning separation is lost under memory constraints. More concretely we show that (1) The sample complexity of testing stabilizer states in the

k

-qubit memory framework is

Θ(n-k)

. Our upper bound goes via a novel connection to the hidden shift problem and the lower bound is proven using a novel approach to average case bounds on likelihood ratios via combinatorics of the stochastic orthogonal group. (2) The sample complexity of learning stabilizer states with

k

qubits of memory, in the non-adaptive framework, is

Θ(n^2/k)

. As a further application of our techniques, we prove an exponential lower bound for purity testing even when the memory may be left coherent throughout the protocol. Our main results identify coherent quantum memory as the resource enabling the usual separation between stabilizer testing and learning. In particular, even with

k=0.99n

qubits of memory, there is no constant-copy stabilizer tester; furthermore for

k=cn

qubits of memory (for

0< c < 1

), stabilizer testing is as hard as learning, with both requiring

Θ(n)

copies.

Abstract

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