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arXiv:2609.33567

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

Bhvain Makwana, Kashyap Patel, Manjunath Joshi, Jaideep Mulherkar

quant-phcs.LGSubmitted Sep 27, 2026• Announced Sep 29, 2026• 0 citations
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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.

Comments
21 page, 5 figures

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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 560560560 million simulated training measurements per run achieves fidelity above 0.990.990.99 in all 606060 held-out runs, with a mean fidelity of 0.9968860.9968860.996886. 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=16n=16n=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 2.62×2.62\times2.62× greater aggregate gate exposure yet loses all 30 paired comparisons in fidelity, long-range error, energy, and entropy.
Computational complexity reduction is at the basis of a new formulation of many-body quantum states according to tensor network ansatz, originally framed in one-dimensional lattices. In order to include long-range entanglement characterizing phase transitions, the multiscale entanglement renormalization ansatz (MERA) defines a sequence of coarse-grained lattices, obtained by targeting the map of a scale-invariant system into an identical coarse-grained one. The quantum circuit associated with this hierarchical structure includes the definition of causal relations and unitary extensions, leading to the definition of ground subspaces as stabilizer codes. The emerging error correcting codes are referred to logical indices located at the highest hierarchical level and to physical indices yielded by redundancy, framed in the AdS-CFT correspondence as holographic quantum codes with bulk and boundary indices, respectively. In a use-case scenario based on errors consisting of spin erasure, the correction is implemented as the reconstruction of a bulk local operator.
Single-shot measurement learning (SSML) learns a compensation unitary from a one-bit success/failure record and halts after a prescribed run of consecutive successes. We recast SSML as an adaptive estimator on a parameterized sensing manifold and ask what role it can play in quantum-enhanced sensing. First, we show that the terminal run itself furnishes an intrinsic certificate of local alignment: longer terminal runs certify smaller infidelity, and near the optimum this becomes a Fisher-calibrated certificate of parameter error. Second, for compensation-type sensing families, the Bernoulli success/failure record is locally matched to the probe quantum Fisher information (QFI), so SSML preserves the probe's metrological content despite using only one classical bit per copy. In this sense, SSML makes the quantum enhancement carried by the probe operationally available in an online self-terminating protocol. Applied to GHZ/NOON probes of depth mmm, SSML retains the familiar square-root entanglement gain over product probes at fixed total resource, while an ideal multiscale architecture remains compatible with Heisenberg scaling. Monte Carlo simulations of photonic NOON-state phase sensing show the expected near-inverse decay of terminal infidelity with entangled shots, SQL-like total-resource scaling at fixed entanglement depth, the corresponding fixed-resource entanglement gain, the global limitation of a single fringe scale, and the recovery of Heisenberg-compatible behavior under ideal multiscale hand-off. These results identify SSML as a Fisher-preserving, self-certifying estimator layer for quantum-enhanced sensing.
Can a high-quality quantum gate be certified when uncharacterized state-preparation and measurement errors are dominant? Can this be achieved with low experimental overhead? Here, we introduce a sound black-box certification protocol for a single-qubit gate based on a small set of fixed, deterministic sequences. From the data, the protocol derives finite-sample bounds on the gate's rotation eigenvalue, a gauge-invariant property. Its phase reveals the accuracy of the rotation angle, while its modulus quantifies the loss of coherence under repeated gate applications. We implement the protocol on a 40Ca+^{40}\mathrm{Ca}^{+}40Ca+ trapped-ion processor and certify the X\sqrt{\mathrm{X}}X​-gate rotation eigenvalue using 22 00022\,00022000 circuit executions, and demonstrate the robustness of certification to state-preparation and measurement errors by deliberately degrading the readout. Finally, we prove that these spectral constraints imply, up to a physically meaningful unitary change of basis, a rigorous average gate-fidelity lower bound for every time-independent qubit model compatible with the data. In both readout settings, the spectral bounds yield the same fidelity certificate of 99.94(3)%99.94(3)\%99.94(3)% with 99%99\%99% confidence. Our results establish a new standard for quantum-gate certification by combining soundness and experimental efficiency without requiring trusted reference operations, randomized circuits, or model fitting.
We study the deep multiscale entanglement renormalization Ansatz (DMERA) on quantum hardware and the causal cone of a subset of the qubits, which make up the Ansatz. This causal cone spans O(M+logN)O(M+logN)O(M+logN) physical qubits on a quantum device, where MMM and NNN are the subset size and the total number qubits in the Ansatz, respectively. This allows for the determination of the von Neumann entanglement entropy of the NNN-qubit wave function using O(M+logN)O(M+logN)O(M+logN) qubits by diagonalization of the reduced density matrix (RDM). We show this by randomly initializing a 16-qubit DMERA and diagonalizing the resulting RDM of the MMM-qubit subsystem using density-matrix simulation. As an example of practical interest, we also encode the variational ground state of the long-range transverse field Ising model (LRTIM) on eight spins using DMERA. We perform density-matrix simulation with and without noise to obtain entanglement entropies in separate experiments using only four qubits. Finally, we repeat the experiment on the IBM Kyoto back end reproducing simulation results.
Quantum learning from state samples is often benchmarked in a fixed-budget paradigm, relating error to a prescribed number of copies. We instead adopt a stopping-time viewpoint: in minimal-feedback learning, the learning completion can be defined by an online run-length certificate extracted from a one-bit-per-copy record. As an instantiation, we analyze single-shot measurement learning (SSML), introduced in [Phys. Rev. A 98, 052302 (2018)] and [Phys. Rev. Lett. 126, 170504 (2021)], which tunes a unitary and halts after MHM_HMH​ consecutive successes. Viewing the halting as a sequential certification linking the observed counter to infidelity, we derive sample-complexity bounds that separate search (driving success probability toward unity) from certification (run statistics of consecutive successes). The resulting trade-off among MHM_HMH​, dimension ddd, and one-bit reliability clarifies when performance is control-limited versus certificate-limited. With label-flip noise probability qqq, we find a sharp feasibility threshold: once qMH≳1qM_H \gtrsim 1qMH​≳1, the expected halting time grows exponentially, making the learning completion impractical even under ideal control. More broadly, this shows that under severely constrained feedback, the certification can dominate sample complexity and small label noise becomes the information bottleneck. Finally, the near-optimal accuracy enabled by run-length certification aligns with the quantum-state-estimation (and equivalently, no-cloning) limits, expressed in the stopping-time terms.
Abstract We introduce and test two measurement-free training primitives for quantum reservoir computers (QRC). Protocol~A encodes the loss gradient in an auxiliary phase that is fed back coherently before a dissipative reset; Protocol~B realizes gradient descent as relaxation of an engineered open system. Both match exact-gradient training in simulation---the ancilla and reset simulated as channels in a de-risk stage, their validated error model applied at the benchmark stage---on a minimal transverse-field Ising reservoir, with the coherent cost oracle emulated classically and its realization costed. A fully coherent update loop (quantum parameter register, unitary controlled update, no readout anywhere) descends the true landscape, its feedback drift attenuated by exactly 2F−12F-12F−1 with the fixed point unchanged. Benchmarked against classical reservoir computing--- including the memory-augmented hybrid---they improve one-step chaotic prediction, though a thirty-seed study finds no autonomous-prediction advantage; we therefore make no quantum-advantage claim. The study instead addresses three questions that gate this route. First, within the channel model studied, the reset fidelity already demonstrated in measurement-free quantum error correction [Butt \textit{et al.}, Nat.\Commun.\\textbf{17}, 995 (2026)] suffices: reset is not the bottleneck. Second, the binding constraint is coherent cost-oracle precision, set by a finite-difference catastrophic cancellation that the exact parameter-shift rule removes, loosening the oracle-precision target 2.52.52.5-fold in a like-for-like comparison. Third, a fault-tolerant realization---arithmetic and synthesis costs taken from real circuits---needs only ∼ ⁣104\sim\!10^{4}∼104 physical qubits and is runtime-, not qubit-bound. On three free IBM Quantum 156156156-qubit Heron backends at three error-mitigation levels the simulation-optimized state survives device noise (RMSE 0.0160.0160.016--0.0570.0570.057, i.e.$7$--26%26\%26% of the signal spread, correlation ≥0.97\ge0.97≥0.97). All code, seeds, and job identifiers are released.