| Summary: | <jats:p>We present a quantum algorithm for approximating maximum independent sets of a graph based on quantum non-Abelian adiabatic mixing in the sub-Hilbert space of degenerate ground states, which generates quantum annealing in a secondary Hamiltonian. For both sparse and dense random graphs <jats:italic>G</jats:italic>, numerical simulation suggests that our algorithm on average finds an independent set of size close to the maximum size <jats:italic>α</jats:italic>(<jats:italic>G</jats:italic>) in low polynomial time. The best classical algorithms, by contrast, produce independent sets of size about half of <jats:italic>α</jats:italic>(<jats:italic>G</jats:italic>) in polynomial time.</jats:p>
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