Fig. 2. Approximation ratio versus iteration step.

Each panel corresponds to an SK problem instance of different size: (A) N = 8, (B) N = 24, (C) N = 40, (D) N = 56, and (E) N = 72. A random sampling strategy leads to an average approximation ratio r = 0.5. The performance of the classical greedy baseline is shown by the shaded region with a final average approximation ratio r ≃ 0.848497… for N → + ∞ (see Methods). The quantum-enhanced greedy data show the expectation value of the approximation ratio for 10 randomly generated problem instances (see Fig. 1). The performance at iteration step 0 is that of the truncated one-layer QAOA. The performance at iteration step N is that of the quantum-enhanced algorithm, also reported in Fig. 3. Because we display all individual instances, we omit the error bar for the average case.