Soft Quantum Kernels with Bias–Variance Optimization on Density Operator Manifolds

Abstract

We introduce the Soft Quantum Kernel, a one-parameter family of positive-semidefinite kernels indexed by a softness parameter and obtained by replacing the pure-state density operator of a standard quantum kernel with the output of a depolarizing channel. The construction traces a convex chord on the manifold of density operators between the pure-state kernel and the maximally mixed state, and admits a closed-form expression that separates an informative component, controlled by , from a uniform regularizing floor. The kernel is shown to satisfy the Mercer property, and an exact bias–variance decomposition of the associated kernel-ridge estimator is derived. The effective degrees of freedom depend continuously on , so that the joint minimization of the expected prediction error over is a biconvex program admitting at least one interior stationary point under standard regularity, in which acts as a spectral regularizer complementary to the Tikhonov penalty. The softness parameter thus admits a transparent statistical interpretation: it shrinks the kernel spectrum multiplicatively while adding a rank-one component, so that the Soft Quantum Kernel is best read as an interpretable spectral regularization scheme rather than as a new kernel class. A computationally free rule fixes from the normalized Shannon entropy of an auxiliary uncertainty distribution and provides a principled entry point for cross-validated refinement. A multi-block extension is cast as a kernel-alignment problem over the unit simplex and admits a closed-form solution in the interior case. The framework is connected to Riemannian optimization on the cone of positive-definite matrices via the Bures–Wasserstein metric, linking the present results to geometric algorithms recently developed in this journal. Hardware validation on a superconducting quantum processor confirms that a Frobenius perturbation of the Gram matrix of order one fifth produces a downstream area-under-curve shift below one percent. We stress that the main experimental claim of the paper concerns the stability of the estimator under physical noise rather than its predictive accuracy; a Monte Carlo study on simulated data, including classical kernel and ensemble baselines, noise-injection experiments, and sensitivity analyses with respect to the initialization and the entropy binning, corroborates this claim in a controlled environment.