Finite-Depth, Finite-Shot Guarantees for Constrained Quantum Optimization via Fejér Filtering

We study finite-layer alternations of the Constraint--Enhanced Quantum Approximate Optimization Algorithm (CE--QAOA), a constraint-aware ansatz that operates natively on block one-hot manifolds. Our focus is on feasibility and optimality guarantees. We show that restricting cost angles to a harmonic lattice exposes a positive Fejér filter acting on the cost-phase unitary U_C(γ)=e^-iγH_C in a cost-dephased reference model (used only for analysis). Under a wrapped phase-separation condition, this yields dimension-free finite-depth and finite-shot lower bounds on the success probability of sampling an optimal solution. In particular, we obtain a ratio-form guarantee \[ q_0 \;\ge\; \frac{x}{1+x}, \qquad x \;=\; (p{+}1)^2 \sin^2(δ/2)\,C_β, \] where q₀ is the single-shot success probability, C_β is the mixer-envelope mass on the optimal set, δ is a phase-gap proxy, and p is the number of layers. Riemann--Lebesgue averaging extends the discussion beyond exact lattice normalization. We conclude by outlining coherent realizations of hardware-efficient positive spectral filters as a main open direction.