On the Role of Problem Structure in Single-Layer QAOA

Abstract

Problem structure is regarded as a key factor influencing the performance of the Quantum Approximate Optimization Algorithm (QAOA), yet it remains an informal notion without a precise mathematical characterization. In this work, we address this issue by identifying a mathematical object that captures problem structure as seen from the perspective of the target state amplitude in single-layer QAOA. Specifically, we show that this structure is encoded in a joint probability distribution induced by the unique coarsest partition of the computational basis, determined by problem cost values and mixer-dependent walk counts. The statistical dependence between these classes provides a means of quantifying the available problem structure and identifying necessary conditions for exponential amplification. To assess the extent to which this structure can translate into algorithmic performance, we establish a mean-max non-separation bound relating the parameter-averaged probability of measuring the target configuration to its variational maximum. For NPO-PB problems with polynomially many distinct mixer eigenvalues, these probabilities exhibit the same scaling up to polynomial factors. This allows us to characterize the exponential scaling of optimal amplitude amplification by analyzing the parameter-averaged probability without explicitly optimizing the variational parameters. We then derive an exact expression for the parameter-averaged probability as a matrix inner product involving two objects that separately encode the joint structural distribution and the mixer dynamics. This representation provides a framework for identifying the structural mechanisms governing the amplification of the target-state probability or its exponential suppression relative to the mixer capacity. We demonstrate these mechanisms through analytically tractable examples involving transverse field and Grover mixers.