The p-stage Quantum Approximate Optimization Algorithm (QAOAp) is a promising approach for combinatorial optimization on noisy intermediate-scale quantum (NISQ) devices, but its theoretical behavior is not well understood beyond p=1. We analyze QAOA2 for the maximum cut problem (MAX-CUT), deriving a graph-size-independent expression for the expected cut fraction on any D-regular graph of girth >5 (i.e. without triangles, squares, or pentagons).We show that for all degrees D≥2 and every D-regular graph G of girth >5, QAOA2 has a larger expected cut fraction than QAOA1 on G. However, we also show that there exists a 2-local randomized classical algorithm A such that A has a larger expected cut fraction than QAOA2 on all G. This supports our conjecture that for every constant p, there exists a local classical MAX-CUT algorithm that performs as well as QAOAp on all graphs.

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We consider the power of local algorithms for approximately solving Max kXOR, a generalization of two constraint satisfaction problems previously studied with classical and quantum algorithms (MaxCut and Max E3LIN2). In Max kXOR each constraint is the XOR of exactly k variables and a parity bit. On instances with either random signs (parities) or no overlapping clauses and D+1 clauses per variable, we calculate the expected satisfying fraction of the depth-1 QAOA from Farhi et al [arXiv:1411.4028] and compare with a generalization of the local threshold algorithm from Hirvonen et al [arXiv:1402.2543]. Notably, the quantum algorithm outperforms the threshold algorithm for k>4.On the other hand, we highlight potential difficulties for the QAOA to achieve computational quantum advantage on this problem. We first compute a tight upper bound on the maximum satisfying fraction of nearly all large random regular Max kXOR instances by numerically calculating the ground state energy density P(k) of a mean-field k-spin glass [arXiv:1606.02365]. The upper bound grows with k much faster than the performance of both one-local algorithms. We also identify a new obstruction result for low-depth quantum circuits (including the QAOA) when k=3, generalizing a result of Bravyi et al [arXiv:1910.08980] when k=2. We conjecture that a similar obstruction exists for all k.