The Quantum Boolean Regressor: A symmetry-exact quantum primitive for optimal Boolean-function selection

We introduce the Quantum Boolean Regressor (QBR), a fixed quantum device indexed by a Reed-Muller code RM(d,n). For each pair (d,n), the QBR coherently evaluates all Boolean functions of degree at most d in superposition and then applies a dataset-independent selection unitary W, optimized once offline, so that projective measurement on the address register outputs a best-fitting Boolean function with maximal dataset-averaged probability. The key structural result is that the natural affine symmetry group AGL(n,2) acts on the address×input Hilbert space. This yields a covariant-optimality principle at the POVM level and identifies the commutant of the AGL(n,2) representation as the canonical search space. In practice, this collapses the optimization from dim(A⊗X)^2 parameters to the commutant dimension and removes symmetry-broken local optima. The paper reports exhaustive covariant optima for all nine feasible small cases, with zero restart variance and POVM covariance errors below 1e-15. The trivial optimum is matched for d=0, while every nontrivial case with d>=1 is improved. The largest completed cases RM(2,3) and RM(3,3) exhibit a near-doubling regime relative to the Tier-1 baseline. A universal degree-1 backbone family is also identified, exact for RM(1,1) and near-optimal thereafter.