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arXiv:2605.11557

Wavelet Variance Equipartition as a Threshold for World-Model Quality and Quantum Kernel TN-Simulability

Chon-Fai Kam, Xavier Cadet, Miloud Bessafi, Frederic Cadet

quant-phSubmitted May 12, 2026• Announced May 13, 2026• 0 citations
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Abstract

While world models learn compact representations of complex environments, they lack a physics-grounded metric to assess the structural fidelity of their latent spaces. We identify the wavelet scaling exponent $α$ as a critical diagnostic, proposing optimal representations satisfy variance equipartition ($α\approx 1/2$) -- mirroring Kolmogorov's inertial range. We establish $α= 1/2$ as a sharp transition boundary for the classical simulability of amplitude-encoded quantum kernels. Using tensor-network theory, we prove latents with $α> 1/2$ reside in an area-law phase admitting efficient classical emulation, while $α< 1/2$ triggers a volume-law phase where the Matrix Product State bond dimension $χ$ grows exponentially with qubit count $n$. Analyzing pre-trained VideoMAE latents reveals a dichotomy: spatial tokens approach the equipartition limit ($α\approx 0.423$), but permutation-invariant feature channels exhibit unstructured disorder ($α\approx -0.123$). This forces real-world latents deep into the volume-law phase, providing a data-driven necessary condition for simulation hardness. Finally, we apply Weingarten calculus to derive the exact variance of the scrambled transition probability under a 2-design ensemble. We prove this variance scales strictly as $\Var[X] = Θ(d^{-2})$. We confirm this numerically with a log-log slope of $-1.881$ ($R^2 = 0.999$), identifying a formidable shot-noise wall demanding a measurement budget of $M = Ω(d^2)$ that constrains quantum machine learning scalability.

Related papers

Trained quantum Kolmogorov--Arnold networks can dequantize, and a discrete-logarithm encoding need not: a measurement-based map of where quantum advantage can live

Hikaru Wakaura

Abstract Quantum machine-learning models are only useful if a classical computer cannot cheaply imitate them. We test this directly for quantum Kolmogorov–Arnold networks (QKANs)—neural networks whose tunable “activation functions” are small quantum circuits—by asking whether each model can be contracted as an effcient tensor network (TN), the standard tool for classical quantum simulation. We make the test practical with a task-aware yardstick: the bond dimension a classical surrogate needs to copy the model’s output at the accuracy the model itself reaches. We find a clear ladder. The amplitude-encoded Chebyshev QKAN uses only logarithmically many qubits and is exactly classical. The variational QKAN, built from all-to-all controlled rotations, looks hard at random settings but, once trained to genuinely learn a smooth task (R2 = 0.30–0.62), is copied by a constant-bond-dimension surrogate at the sizes (n ≤ 10) we test. Noise pushes it toward classicality near p ≈ 0.03–0.05, echoing noisy random circuits. Every generic attempt to force hardness—a global measurement, a deeper entangler, or an interaction-based (IQP) feature map—instead destroys trainability through exponentially small gradients (a “barren plateau”). The escape is a discrete-logarithm feature map: its kernel variance is flat (0.07±0.01) from 4 to 14 qubits while a generic IQP kernel decays as 2 −n , and a support-vector machine with it learns a classically hard concept to ≥ 99% accuracy while generic and classical kernels stay at chance—hardness placed in number theory stays both classically hard and shot-effciently learnable. The work provides a concrete, measurement-based criterion for when a quantum model is reproducible by a low-bond- dimension surrogate, and a design rule: seek advantage in structured, provably hard encodings rather than in deep entanglement.

•Jul 1, 2026•0 citations

Quantum Kolmogorov--Arnold networks are classically reproducible at their attained accuracy, and a discrete-logarithm encoding need not be: a measurement-based map of where quantum advantage can live

Hikaru Wakaura

Abstract Quantum machine-learning models are useful only if classical computers cannot cheaply imitate them. We test this directly for quantum Kolmogorov--Arnold networks (QKANs)---neural networks whose tunable ``activation functions'' are small quantum circuits---by asking whether each model can be contracted as an efficient tensor network (TN). Our task-aware yardstick is the bond dimension a classical surrogate needs to copy the model's output at the accuracy the model itself reaches. We find a clear ladder. The amplitude-encoded Chebyshev QKAN uses only logarithmically many qubits and is exactly classical. The variational QKAN, built from all-to-all controlled rotations, looks hard at random settings but generalizes only weakly once trained (held-out $R^2\!\le\!0.27$) and is copied at that attained accuracy by a low-bond-dimension surrogate ($n\!\le\!10$); at fixed strict tolerance the required bond dimension grows near-exponentially. Depolarizing noise erases it (purity collapse) near $p\!\approx\!0.03$--$0.05$ at the sizes we reach. Every generic attempt to force hardness---a global measurement or an interaction-based (IQP) feature map---instead destroys trainability through exponentially small gradients (a ``barren plateau''). The escape is a discrete-logarithm feature map: its kernel variance is flat from $4$ to $14$ qubits while a generic IQP kernel concentrates exponentially, and a support-vector machine with it learns a classically hard concept to $\ge99\%$ accuracy while generic and classical kernels stay at chance; wired into the KAN architecture, it fits discrete-log targets to $R^2\!\approx\!1$. Hardness placed in number theory stays classically hard, shot-efficient, and learnable. The work provides a measurement-based reproducibility criterion and a design rule: seek advantage in structured, provably hard encodings rather than in deep entanglement.

•Oct 20, 2026•0 citations

Empirical Evaluation of Quantum Kernel Methods for Classification: A Comparative Study with Classical Kernels

Ronak Bhalgami

Quantum kernel methods are the most concrete near-term proposal for quantum machine learning on classical data, yet published benchmarks disagree on whether they outperform tuned classical kernels, and the disagreement cannot be resolved from the literature because the comparisons are not protocol-controlled. We argue that the obstacle is one of measurement, not theory: the usual setup fixes the quantum feature map's data scaling while grid-searching the classical kernel's hyperparameters, recomputes nothing per fold, and reports cost in simulator time. We therefore treat the comparison itself as the object of study and impose three controls. The bandwidth is tuned on the same grid budget as the classical $γ$, the kernel is recomputed inside every fold from fold-fitted preprocessing, and cost is counted in hardware circuit evaluations. On top of these we run a factorial design of 324 cells in which dimensionality, training size, class separability and data structure are explicit covariates, together with four standard datasets, a shots-versus-gate-noise ablation, and a cost sweep. The dominant effect we measure is not quantum: tuning the feature map's bandwidth is worth 0.120 accuracy on synthetic data and 0.339 on real data, exceeding every quantum-classical difference we observe. Tuned quantum kernels are then statistically indistinguishable from a tuned RBF on all four real datasets. The advantage on synthetic data is confined almost entirely to data generated by the feature map itself, and it shrinks with dimensionality (-0.0045 per qubit) while growing with sample size. Finite sampling costs an order of magnitude more accuracy than gate and readout noise, and a fidelity kernel needs $5\times10^{7}$ circuits at n = 10,000. Protocol variance is thus sufficient to explain the field's contradictory conclusions.

quant-pharXiv:2610.42029•Nov 2, 2026

Classical Simulability from Operator Entanglement Scaling

Neil Dowling

Local-operator entanglement (LOE) quantifies the nonlocal structure of Heisenberg operators and serves as a diagnostic of many-body chaos. We provide rigorous bounds showing when an operator can be well-approximated by a matrix-product operator (MPO), given asymptotic scaling of its LOE $α$-Rényi entropies. Specifically, we prove that a volume law scaling for $α\geq 1$ implies that the operator cannot be approximated efficiently as an MPO while faithfully reproducing all expectation values. On the other hand, if we restrict to correlations over a relevant sub-class of (ensembles of) states, then logarithmic scaling of the $α< 1$ entropies implies MPO simulability. This result covers a range of relevant quantities, including infinite temperature autocorrelation functions, out-of-time-ordered correlators, and average-case expectation values over ensembles of computational basis states. Beyond this regime, we provide numerical evidence together with a random matrix model to argue that this simulability result also typically holds for arbitrary states. Our results put on firm footing the heuristic expectation that a low operator entanglement implies efficient tensor network representability, extending celebrated foundational results from the theory of matrix-product states and providing a formal link between quantum chaos and classical simulability.

quant-pharXiv:2603.05656•Physical Review A•Mar 9, 2026•0 citations

Exact Bounds on Exponential Concentration for Block-Product Quantum Fidelity Kernels

Mezbah Uddin Rafi

Abstract We characterize the achievable second-moment concentration exponent for block-product fidelity quantum kernels whose per-block ensembles have a maximally mixed first moment. For n = mb qubits partitioned into m independent b-qubit blocks, with no entangling gate crossing a block boundary, the exponent c, defined by E[κ²] = 2⁻ᶜⁿ, satisfies c ∈ [1, c(b)] for every finite b, where c(b) = [(b−1) + log₂(2ᵇ+1)]/b increases monotonically from log₂ 3 ≈ 1.585 at b=1 to 2 as b→∞. The lower endpoint is attained exactly by digitizing (basis-encoding) block ensembles, proved here; the upper endpoint follows from the optimal second frame potential of state 2-designs (Nakata et al.) [8], which we apply rather than re-derive. We validate the resulting scaling against idealized Haar-random blocks and finite-depth parameterized circuits, with propagated standard errors, obtaining relative error from −2.9% to +0.14% for block sizes b=2 through 6, tightening as b grows. We identify a structural limitation of single-qubit (b=1) blocks: a continuous data parameter spans only a one-dimensional subset of the Bloch sphere, which has zero measure with respect to the Haar measure, so no purely data-driven single-qubit encoding can realize an exact 2-design; a depth study shows the corresponding purity converging non-monotonically, consistent with this limitation. We further distinguish two notions of practical resolvability — the shot-noise threshold implied by the variance exponent, and the finite-shot consequence for pairwise similarity ranking — report each together with its associated agreement criterion, and calibrate the relationship between them on finite-depth circuits across a shot-budget-by-qubit-count grid (180 points, median heuristic error 0.017). Finally, we relate this result to adjacent constructions that also avoid concentration through different mechanisms — covariant kernels, permutation-equivariant barren-plateau-free kernels, and a Rydberg-blockade many-body kernel — each of which lies outside the hypothesis class considered here.

•Aug 18, 2026•0 citations

Finite-size resource scaling for learning quantum phase transitions with fidelity-based support vector machines

Aaqib Ali, Giovanni Scala, Cosmo Lupo +1 more

Quantum kernels offer a valid procedure for learning quantum phase transitions on quantum processing devices, yet issues on the scalability of the learning strategy in connection with the symmetry of the critical model have not been clarified. We derive a link between model symmetry and fidelity-kernel resource scaling. We quantify the measurement resources required to estimate fidelity-based quantum kernels for many-body ground states while preserving the structure of the resulting Gram matrix under finite-shot sampling. Crucially, we show that increasing symmetry in the underlying spin model systematically amplifies these shot requirements. Moving from the $\mathbb{Z}_2$-symmetric Ising/XY regimes to the $U(1)$-symmetric XX (and XXZ) regimes leads to stronger kernel concentration and therefore substantially larger shot costs under the same bounds. We consider a tunable one-dimensional spin-$\tfrac{1}{2}$ Hamiltonian spanning the transverse-field Ising, XY, XX, and XXZ limits, and define the kernel as the ground-state fidelity. Kernel entries are estimated using a SWAP-test estimator with $S$ shots, and we adapt the ensemble spread and concentration-avoidance shot bounds to obtain practical shot requirements in terms of the interquartile range of kernel values and a representative kernel magnitude. For the free-fermion XY/XX family, we use the closed-form Bogoliubov-angle fidelity, while for the interacting XXZ chain we compute fidelities by exact diagonalization and benchmark shot-noise effects. Our symmetry-aware bounds provide a pragmatic procedure for physics-informed quantum machine learning.

quant-pharXiv:2603.18211•Mar 20, 2026•0 citations
While world models learn compact representations of complex environments, they lack a physics-grounded metric to assess the structural fidelity of their latent spaces. We identify the wavelet scaling exponent ααα as a critical diagnostic, proposing optimal representations satisfy variance equipartition (α≈1/2α\approx 1/2α≈1/2) -- mirroring Kolmogorov's inertial range. We establish α=1/2α= 1/2α=1/2 as a sharp transition boundary for the classical simulability of amplitude-encoded quantum kernels. Using tensor-network theory, we prove latents with α>1/2α> 1/2α>1/2 reside in an area-law phase admitting efficient classical emulation, while α<1/2α< 1/2α<1/2 triggers a volume-law phase where the Matrix Product State bond dimension χχχ grows exponentially with qubit count nnn. Analyzing pre-trained VideoMAE latents reveals a dichotomy: spatial tokens approach the equipartition limit (α≈0.423α\approx 0.423α≈0.423), but permutation-invariant feature channels exhibit unstructured disorder (α≈−0.123α\approx -0.123α≈−0.123). This forces real-world latents deep into the volume-law phase, providing a data-driven necessary condition for simulation hardness. Finally, we apply Weingarten calculus to derive the exact variance of the scrambled transition probability under a 2-design ensemble. We prove this variance scales strictly as \Var[X]=Θ(d−2)\Var[X] = Θ(d^{-2})\Var[X]=Θ(d−2). We confirm this numerically with a log-log slope of −1.881-1.881−1.881 (R2=0.999R^2 = 0.999R2=0.999), identifying a formidable shot-noise wall demanding a measurement budget of M=Ω(d2)M = Ω(d^2)M=Ω(d2) that constrains quantum machine learning scalability.
Abstract Quantum machine-learning models are only useful if a classical computer cannot cheaply imitate them. We test this directly for quantum Kolmogorov–Arnold networks (QKANs)—neural networks whose tunable “activation functions” are small quantum circuits—by asking whether each model can be contracted as an effcient tensor network (TN), the standard tool for classical quantum simulation. We make the test practical with a task-aware yardstick: the bond dimension a classical surrogate needs to copy the model’s output at the accuracy the model itself reaches. We find a clear ladder. The amplitude-encoded Chebyshev QKAN uses only logarithmically many qubits and is exactly classical. The variational QKAN, built from all-to-all controlled rotations, looks hard at random settings but, once trained to genuinely learn a smooth task (R2 = 0.30–0.62), is copied by a constant-bond-dimension surrogate at the sizes (n ≤ 10) we test. Noise pushes it toward classicality near p ≈ 0.03–0.05, echoing noisy random circuits. Every generic attempt to force hardness—a global measurement, a deeper entangler, or an interaction-based (IQP) feature map—instead destroys trainability through exponentially small gradients (a “barren plateau”). The escape is a discrete-logarithm feature map: its kernel variance is flat (0.07±0.01) from 4 to 14 qubits while a generic IQP kernel decays as 2 −n , and a support-vector machine with it learns a classically hard concept to ≥ 99% accuracy while generic and classical kernels stay at chance—hardness placed in number theory stays both classically hard and shot-effciently learnable. The work provides a concrete, measurement-based criterion for when a quantum model is reproducible by a low-bond- dimension surrogate, and a design rule: seek advantage in structured, provably hard encodings rather than in deep entanglement.
Abstract Quantum machine-learning models are useful only if classical computers cannot cheaply imitate them. We test this directly for quantum Kolmogorov--Arnold networks (QKANs)---neural networks whose tunable ``activation functions'' are small quantum circuits---by asking whether each model can be contracted as an efficient tensor network (TN). Our task-aware yardstick is the bond dimension a classical surrogate needs to copy the model's output at the accuracy the model itself reaches. We find a clear ladder. The amplitude-encoded Chebyshev QKAN uses only logarithmically many qubits and is exactly classical. The variational QKAN, built from all-to-all controlled rotations, looks hard at random settings but generalizes only weakly once trained (held-out R2 ⁣≤ ⁣0.27R^2\!\le\!0.27R2≤0.27) and is copied at that attained accuracy by a low-bond-dimension surrogate (n ⁣≤ ⁣10n\!\le\!10n≤10); at fixed strict tolerance the required bond dimension grows near-exponentially. Depolarizing noise erases it (purity collapse) near p ⁣≈ ⁣0.03p\!\approx\!0.03p≈0.03--0.050.050.05 at the sizes we reach. Every generic attempt to force hardness---a global measurement or an interaction-based (IQP) feature map---instead destroys trainability through exponentially small gradients (a ``barren plateau''). The escape is a discrete-logarithm feature map: its kernel variance is flat from 444 to 141414 qubits while a generic IQP kernel concentrates exponentially, and a support-vector machine with it learns a classically hard concept to ≥99%\ge99\%≥99% accuracy while generic and classical kernels stay at chance; wired into the KAN architecture, it fits discrete-log targets to R2 ⁣≈ ⁣1R^2\!\approx\!1R2≈1. Hardness placed in number theory stays classically hard, shot-efficient, and learnable. The work provides a measurement-based reproducibility criterion and a design rule: seek advantage in structured, provably hard encodings rather than in deep entanglement.
Quantum kernel methods are the most concrete near-term proposal for quantum machine learning on classical data, yet published benchmarks disagree on whether they outperform tuned classical kernels, and the disagreement cannot be resolved from the literature because the comparisons are not protocol-controlled. We argue that the obstacle is one of measurement, not theory: the usual setup fixes the quantum feature map's data scaling while grid-searching the classical kernel's hyperparameters, recomputes nothing per fold, and reports cost in simulator time. We therefore treat the comparison itself as the object of study and impose three controls. The bandwidth is tuned on the same grid budget as the classical γγγ, the kernel is recomputed inside every fold from fold-fitted preprocessing, and cost is counted in hardware circuit evaluations. On top of these we run a factorial design of 324 cells in which dimensionality, training size, class separability and data structure are explicit covariates, together with four standard datasets, a shots-versus-gate-noise ablation, and a cost sweep. The dominant effect we measure is not quantum: tuning the feature map's bandwidth is worth 0.120 accuracy on synthetic data and 0.339 on real data, exceeding every quantum-classical difference we observe. Tuned quantum kernels are then statistically indistinguishable from a tuned RBF on all four real datasets. The advantage on synthetic data is confined almost entirely to data generated by the feature map itself, and it shrinks with dimensionality (-0.0045 per qubit) while growing with sample size. Finite sampling costs an order of magnitude more accuracy than gate and readout noise, and a fidelity kernel needs 5×1075\times10^{7}5×107 circuits at n = 10,000. Protocol variance is thus sufficient to explain the field's contradictory conclusions.
Local-operator entanglement (LOE) quantifies the nonlocal structure of Heisenberg operators and serves as a diagnostic of many-body chaos. We provide rigorous bounds showing when an operator can be well-approximated by a matrix-product operator (MPO), given asymptotic scaling of its LOE ααα-Rényi entropies. Specifically, we prove that a volume law scaling for α≥1α\geq 1α≥1 implies that the operator cannot be approximated efficiently as an MPO while faithfully reproducing all expectation values. On the other hand, if we restrict to correlations over a relevant sub-class of (ensembles of) states, then logarithmic scaling of the α<1α< 1α<1 entropies implies MPO simulability. This result covers a range of relevant quantities, including infinite temperature autocorrelation functions, out-of-time-ordered correlators, and average-case expectation values over ensembles of computational basis states. Beyond this regime, we provide numerical evidence together with a random matrix model to argue that this simulability result also typically holds for arbitrary states. Our results put on firm footing the heuristic expectation that a low operator entanglement implies efficient tensor network representability, extending celebrated foundational results from the theory of matrix-product states and providing a formal link between quantum chaos and classical simulability.
Abstract We characterize the achievable second-moment concentration exponent for block-product fidelity quantum kernels whose per-block ensembles have a maximally mixed first moment. For n = mb qubits partitioned into m independent b-qubit blocks, with no entangling gate crossing a block boundary, the exponent c, defined by E[κ²] = 2⁻ᶜⁿ, satisfies c ∈ [1, c(b)] for every finite b, where c(b) = [(b−1) + log₂(2ᵇ+1)]/b increases monotonically from log₂ 3 ≈ 1.585 at b=1 to 2 as b→∞. The lower endpoint is attained exactly by digitizing (basis-encoding) block ensembles, proved here; the upper endpoint follows from the optimal second frame potential of state 2-designs (Nakata et al.) [8], which we apply rather than re-derive. We validate the resulting scaling against idealized Haar-random blocks and finite-depth parameterized circuits, with propagated standard errors, obtaining relative error from −2.9% to +0.14% for block sizes b=2 through 6, tightening as b grows. We identify a structural limitation of single-qubit (b=1) blocks: a continuous data parameter spans only a one-dimensional subset of the Bloch sphere, which has zero measure with respect to the Haar measure, so no purely data-driven single-qubit encoding can realize an exact 2-design; a depth study shows the corresponding purity converging non-monotonically, consistent with this limitation. We further distinguish two notions of practical resolvability — the shot-noise threshold implied by the variance exponent, and the finite-shot consequence for pairwise similarity ranking — report each together with its associated agreement criterion, and calibrate the relationship between them on finite-depth circuits across a shot-budget-by-qubit-count grid (180 points, median heuristic error 0.017). Finally, we relate this result to adjacent constructions that also avoid concentration through different mechanisms — covariant kernels, permutation-equivariant barren-plateau-free kernels, and a Rydberg-blockade many-body kernel — each of which lies outside the hypothesis class considered here.
Quantum kernels offer a valid procedure for learning quantum phase transitions on quantum processing devices, yet issues on the scalability of the learning strategy in connection with the symmetry of the critical model have not been clarified. We derive a link between model symmetry and fidelity-kernel resource scaling. We quantify the measurement resources required to estimate fidelity-based quantum kernels for many-body ground states while preserving the structure of the resulting Gram matrix under finite-shot sampling. Crucially, we show that increasing symmetry in the underlying spin model systematically amplifies these shot requirements. Moving from the Z2\mathbb{Z}_2Z2​-symmetric Ising/XY regimes to the U(1)U(1)U(1)-symmetric XX (and XXZ) regimes leads to stronger kernel concentration and therefore substantially larger shot costs under the same bounds. We consider a tunable one-dimensional spin-12\tfrac{1}{2}21​ Hamiltonian spanning the transverse-field Ising, XY, XX, and XXZ limits, and define the kernel as the ground-state fidelity. Kernel entries are estimated using a SWAP-test estimator with SSS shots, and we adapt the ensemble spread and concentration-avoidance shot bounds to obtain practical shot requirements in terms of the interquartile range of kernel values and a representative kernel magnitude. For the free-fermion XY/XX family, we use the closed-form Bogoliubov-angle fidelity, while for the interacting XXZ chain we compute fidelities by exact diagonalization and benchmark shot-noise effects. Our symmetry-aware bounds provide a pragmatic procedure for physics-informed quantum machine learning.