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

2026-03-02

AI Summary

This work investigates performance guarantees for the Constraint-Enhanced Quantum Approximate Optimization Algorithm (CE-QAOA) under limited circuit depth and finite sampling in constrained quantum optimization. By restricting cost angles to harmonic lattices over block-wise one-hot manifolds and introducing a positive Fejér filter-based analytical framework, the study establishes—for the first time—a dimension-independent lower bound on the success sampling probability, expressed as \( q_0 \geq x/(1+x) \). Here, \( x \) depends on the number of layers, a proxy for the phase gap, and the quality of the mixer envelope over the optimal solution set. The analysis reveals the theoretical role of Fejér filtering in constrained quantum optimization and introduces key technical tools—including phase separation conditions and Riemann–Lebesgue averaging—to provide rigorous feasibility and optimality guarantees under finite computational resources.

Technology Category

• Constraint Satisfaction and Optimization: Constraint Optimization
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Quantum Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

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 Fejér filter acting on the cost-phase unitary $U_C(γ)=e^{-iγH_C}$ \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 \[ q_0 \;\ge\; \frac{x}{1+x}, \qquad x \;=\; (p{+}1)^2 \sin^2(δ/2)\,C_β, \] where $q_0$ 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.

Problem

• constrained quantum optimization
• finite-depth
• finite-shot
• feasibility guarantees
• optimality guarantees

Innovation

• Fejér filtering
• constraint-aware QAOA
• finite-depth guarantees
• finite-shot bounds
• dimension-free analysis