Comment on "Scalable Quantum Machine Learning: Trainability, Expressivity and Efficiency": Polynomial Evaluation of the Triplet-Block Readout

Abstract

We examine the classical-cost claim for the triplet-block two-body readout in arXiv:2607.24014v1. The Gaussian-state expansion used there gives an $O(2^{2k/3}\mathrm{poly}(n))$ classical algorithm, but it is not necessary for fixed-body observables. The triplet-block input has an explicitly computable diagonal two-particle reduced density matrix, which passive fermionic linear optics propagates through $\bigwedge^2 W$. This gives a deterministic $O(n^4)$ algorithm for the complete correlator vector $(\langle n_i n_j\rangle)_{i<j}$, independently of $k$ and the fermionic-linear-optics extent. More generally, every number-conserving fixed-$r$-body expectation is polynomially computable whenever the input $r$-particle reduced density matrix is classically available; if that matrix is diagonal, all diagonal correlators are computable in $O(n^{2r})$ time. This invalidates the algorithm-relative exponential-cost conclusion for the supervised two-body readout, without affecting the gradient-variance, barren-plateau, parameter-shift, or sampling-hardness results.