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

<title>Abstract</title> <p>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.</p>