Lower Bounds for Max-Cut via Semidefinite Programming

Abstract

We compare the performance of a quantum local algorithm to a similar classical counterpart on a well-established combinatorial optimization problem LocalMaxCut. We show that a popular quantum algorithm first discovered by Farhi, Goldstone, and Gutmannn [1] called the quantum optimization approximation algorithm (QAOA) has a computational advantage over comparable local classical techniques on degree-3 graphs. These results hint that even small-scale quantum computation, which is relevant to the current state-of the art quantum hardware, could have significant advantages over comparably simple classical computation.

An explicit vector algorithm for high-girth MaxCut

“…We use the Goemans-Willamson rounding scheme; however, instead of actually solving the SDP, we explicitly construct a vector solution where v i • v j is same for all edges and then round to obtain a cut. Consider the following example from Carlson et al (2020), which is the inspiration for this paper. On a d-regular, triangle-free graph, define for every i, j ∈ V :…”

“…Our approach is inspired by the work of Carlson, Kolla, Li, Mani, Sudakov, and Trevisan (2020), who construct and round explicit vector solutions to find large cuts in graphs with few triangles and in K r -free graphs. Furthermore, it turns out that our algorithm may be viewed as an simplification of the Gaussian wave process, as discussed in Section 3.…”

Lower Bounds for Maximum Weighted Cut

“…the weight of every edge equals 1 (see e.g. [1,3,5,7,15,16]) (In what follows, the weight of every edge is an unweighted graph will be equal to 1) or for graphs with integral weights (see e.g. [2,5]).…”