  Quantum annealing (QA) holds promise for optimization problems in quantum
computing, especially for combinatorial optimization. This analog framework
attracts attention for its potential to address complex problems. Its
gate-based homologous, QAOA with proven performance, has brought lots of
attention to the NISQ era. Several numerical benchmarks try to classify these
two metaheuristics however, classical computational power highly limits the
performance insights. In this work, we introduce a new parametrized version of
QA enabling a precise 1-local analysis of the algorithm. We develop a tight
Lieb-Robinson bound for regular graphs, achieving the best-known numerical
value to analyze QA locally. Studying MaxCut over cubic graph as a benchmark
optimization problem, we show that a linear-schedule QA with a 1-local analysis
achieves an approximation ratio over 0.7020, outperforming any known 1-local
algorithms.
