arXiv subjects

Guilherme Adamatti Bridi

Publications and source records attributed to Guilherme Adamatti Bridi.

3 recordsLinked to original sources

arXiv2026

On the Role of Problem Structure in Single-Layer QAOA

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.

quant-ph↗

arXiv2025

The only Class 0 Flower snark is the smallest

Graph pebbling is a game played on graphs with pebbles on their vertices. A pebbling move removes two pebbles from one vertex and places one pebble on an adjacent vertex. The pebbling number is the smallest $t$ so that from any initial configuration of $t$ pebbles it is possible, after a sequence of pebbling moves, to place a pebble on any given target vertex. Graphs whose pebbling number is equal to the number of vertices are called Class~$0$ and provide a challenging set of graphs that resist being characterized. In this note, we answer a question recently proposed by the pioneering study on the pebbling number of snark graphs: we prove that the smallest Flower snark $J_3$ is Class~$0$, establishing that $J_3$ is in fact the only Class~$0$ Flower snark.

math.CO↗

arXiv2024

Analytical results for the Quantum Alternating Operator Ansatz with Grover Mixer

An important property of QAOA with Grover mixer is that its expectation value is invariant over any permutation of states. As a consequence, the algorithm is independent of the structure of the problem. If, on the one hand, this characteristic raises serious doubts about the capacity of the algorithm to overcome the bound of the unstructured search problem, on the other hand, it can pave the way to its analytical study. In this sense, a prior work introduced a statistical approach to analyze GM-QAOA that results in an analytical expression for the expectation value depending on the probability distribution associated with the problem Hamiltonian spectrum. Although the method provides surprising simplifications in calculations, the expression depends exponentially on the number of layers, which makes direct analytical treatment unfeasible. In this work, we extend the analysis to the more simple context of Grover Mixer Threshold QAOA (GM-Th-QAOA), a variant that replaces the phase separation operator of GM-QAOA to encode a threshold function. As a result, we obtain an expression for the expectation value independent of the number of layers and, with it, we provide bounds for different performance metrics. Furthermore, we extend the analysis to a more general context of QAOA with Grover mixer, which we called Grover-based QAOA. In that framework, which allows the phase separation operator to encode any compilation of the cost function, we generalize all the bounds by using an argument by contradiction with the optimality of Grover's algorithm on the unstructured search problem. As a result, we get the main contribution of this work, an asymptotic lower bound on the quantile achieved by the expectation value that formalizes the notion that the Grover mixer, at most, reflects a quadratic Grover-style speed-up over classical brute force.

quant-ph↗