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The short path algorithm gives a super-Grover speedup for variousoptimization problems under the assumption of a unique ground state and underan assumption on the density of low-energy states. Here, we remove theassumption of a unique ground state; this uses the same algorithm but aslightly different analysis and holds for arbitrary MAX-$D$-LIN-$2$ problems.Then, specializing to the case $D=2$, we show that for certain values of theobjective function we can always achieve a super-Grover speedup (albeit a veryslight one) without any assumptions on the density of states. Finally, forrandom instances, we give a heuristic treatment suggesting a more significantimprovement.

Weaker Assumptions for the Short Path Optimization Algorithm