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arXiv · 1807.03758

Weaker Assumptions for the Short Path Optimization Algorithm

M. B. Hastings
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Abstract

The short path algorithm gives a super-Grover speedup for various optimization problems under the assumption of a unique ground state and under an assumption on the density of low-energy states. Here, we remove the assumption of a unique ground state; this uses the same algorithm but a slightly 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 the objective function we can always achieve a super-Grover speedup (albeit a very slight one) without any assumptions on the density of states. Finally, for random instances, we give a heuristic treatment suggesting a more significant improvement.

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M. B. Hastings. 2018-07-10. Weaker Assumptions for the Short Path Optimization Algorithm. https://arxiv.org/abs/1807.03758

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