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

Mezbah Uddin Rafi

Published Aug 18, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.21203/rs.3.rs-10726261/v1

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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.
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Ensemble equivalence — the expectation that microcanonical, canonical, and grand-canonical descriptions of a system agree in the thermodynamic limit — is a cornerstone of statistical mechanics, yet it can fail for observables that probe exponentially rare events. Recent work on a simplified quantum error correction (QEC) model reports that critical exponents governing fidelity and magic at the error-correction threshold depend on the noise ensemble chosen, and that even the existence of the transition can be ensemble-dependent [1,2]. Here we develop a minimal, fully analytic treatment of this phenomenon using a random-unitary encoding model subject to depolarizing noise. We derive explicit expressions for the decoding fidelity in grand-canonical (independent per-qubit errors) and canonical (fixed error number) ensembles, and exhibit concrete numerical instances of the discrepancy: for a distance-3 code with 100 physical qubits and error rate p = 0.01, the grand-canonical fidelity is 0.735762 while the canonical fidelity is exactly 1. We show that the discrepancy survives the vanishing of relative fluctuations (σ/μ → 0) because fidelity is a large-deviation tail quantity, and we compute the associated entropy and capacity budgets. We further derive the statistical cost of resolving ensemble-dependent exponents: distinguishing two candidate exponents at distance Δp from threshold requires a sample count scaling as Δp⁻⁴ and system sizes scaling as Δp⁻², a compound burden that we quantify under stated assumptions. We relate the mechanism to the known ensemble dependence of the random transverse-field Ising chain [3], discuss saturation of the information-theoretic bound of Feldman et al. as reported in [1,2], and outline falsifiable predictions. The results clarify when ensemble choice is physically consequential rather than a matter of convenience in QEC threshold physics. ## 1. Introduction Quantum error correction protects quantum information from decoherence by en Full text and updates: papers.qnfo.org/papers/ensemble-dependence-of-critical-exponents-at-quantum-error-correction-thresholds/
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.
Abstract Reliable quantum computation requires certifiable bounds on the degradation of ideal circuit outputs under realistic noise. Existing fidelity estimates are typically asymptotic, hardware-specific, or require full density-matrix simulation, rendering them impractical for circuit-scale certification. We introduce a computable fidelity floor for noisy quantum circuits based on light-cone--resolved Pauli weight growth. For circuits subject to independent Markovian single-qubit noise, we prove a rigorous trace-distance bound expressed entirely in terms of circuit structure: per-qubit active times and a combinatorial backward light-cone descriptor of entangling gates. The resulting bound is computable in O(Gn)O(Gn)O(Gn) time for a circuit with GGG gates on nnn qubits, avoiding exponential state-space simulation. We further introduce an empirically validated exponential envelope refinement that tightens the rigorous bound while remaining conservative across all tested configurations. Across more than 10310^3103 circuit instances spanning four noise heterogeneity structures (uniform, anti-correlated, spike, and random per-qubit T1/T2T_1/T_2T1​/T2​ rates, up to n=8n=8n=8, depth~40), the envelope bound achieves 100\% validity and exhibits a stable compression factor across heterogeneity modes, reaching approximately 2.6×2.6\times2.6× at depth~20 relative to the combinatorial bound. The envelope remains informative within the empirically determined regime n≤8n \le 8n≤8, depth~≤20\le 20≤20, and γT≲0.3\gamma T \lesssim 0.3γT≲0.3. These results establish a circuit-structure--based fidelity certification framework computable at compiler timescale and identify a depth-dependent compression invariant connecting rigorous combinatorial bounds to empirically tighter envelope estimates.
High-dimensional bipartite entanglement depends on how probability is distributed across the Schmidt spectrum, whereas a single reference-state fidelity resolves only one spectral scale. We develop multiscale Schmidt-spectrum bounds that connect geometric measures with quantitative Schmidt-number witnesses. For any partition of a pure-state Schmidt spectrum, several nested Vidal tails determine block masses. We derive the sharp upper boundary of the associated normalized nuclear-norm coordinate and show that equality holds if and only if the spectrum is uniform within each block. Refining the partition gives a monotone hierarchy of tighter bounds whenever the added tail data distinguish unequal block means. We also solve a relaxed weighted multiscale optimization globally: one scalar parameter specifies its unique full-support optimizer and explicit value on the nontrivial branch. Using the established single-tail fidelity--resource curve as a baseline, we obtain exact-fidelity equality refinements, convex-roof lower bounds for mixed states, and quantitative calibrations of Schmidt-number witnesses. A higher-tail relation further bounds convex-roof extended negativity in terms of Vidal tails and identifies the pure-state equality spectra. Leakage-aware and joint-confidence formulations state how these bounds can be used with incomplete data. Multistep tails require block-resolved or independently certified spectral information; they are not determined by one projector expectation.