Finite-Depth, Finite-Shot Guarantees for Constrained Quantum Optimization via Fej\'er Filtering

Abstract

We study finite-layer alternations of the \emph{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 Fej\'er filter acting on the cost-phase unitary $U_C(\gamma)=e^{-i\gamma H_C}$ \emph{in a cost-dephased reference model (used only for analysis)}. Under a wrapped phase-separation condition, this yields \emph{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 \[ q_0 \;\ge\; \frac{x}{1+x}, \qquad x \;=\; (p{+}1)^2 \sin^2(\delta/2)\,C_\beta, \] where $q_0$ is the single-shot success probability, $C_\beta$ is the mixer-envelope mass on the optimal set, $\delta$ is a phase-gap proxy, and $p$ is the number of layers. A Coherent equivalent is proved subsequently and a Riemann--Lebesgue averaging extends the discussion beyond exact lattice normalization. We conclude by outlining coherent realizations of near-term-hardware-efficient positive spectral filters as a main open direction for this framework.