Quantum computing offers a promising route for tackling hard optimization problems by encoding them as Ising models. However, sparse qubit connectivity requires the use of minor-embedding, mapping logical qubits onto chains of physical qubits, which necessitates stronger intra-chain coupling to maintain consistency. This elevated coupling strength forces a rescaling of the Hamiltonian due to hardware-imposed limits on the allowable ranges of coupling strengths, reducing the energy gaps between competing states, thus, degrading the solver's performance. Here, we introduce a theoretical model that quantifies this degradation. We show that as the connectivity degree increases, the effective temperature rises as a polynomial function, resulting in a success probability that decays exponentially. Our analysis further establishes worst-case bounds on the energy scale degradation based on the inverse conductance of chain subgraphs, revealing two most important drivers of chain strength, \textit{chain volume} and \textit{chain connectivity}. Our findings indicate that achieving quantum advantage is inherently challenging. Experiments on D-Wave quantum annealers validate these findings, highlighting the need for hardware with improved connectivity and optimized scale-aware embedding algorithms.

Quantum annealers, such as the device built by D-Wave Systems, Inc., offer a way to compute solutions of NP-hard problems that can be expressed in Ising or QUBO (quadratic unconstrained binary optimization) form. Although such solutions are typically of very high quality, problem instances are usually not solved to optimality due to imperfections of the current generations quantum annealers. In this contribution, we aim to understand some of the factors contributing to the hardness of a problem instance, and to use machine learning models to predict the accuracy of the D-Wave 2000Q annealer for solving specific problems. We focus on the Maximum Clique problem, a classic NP-hard problem with important applications in network analysis, bioinformatics, and computational chemistry. By training a machine learning classification model on basic problem characteristics such as the number of edges in the graph, or annealing parameters such as D-Wave's chain strength, we are able to rank certain features in the order of their contribution to the solution hardness, and present a simple decision tree which allows to predict whether a problem will be solvable to optimality with the D-Wave 2000Q. We extend these results by training a machine learning regression model that predicts the clique size found by D-Wave.