Fixed-point Grover Adaptive Search for Quadratic Binary Optimization Problems

We study a Grover-type method for Quadratic Unconstrained Binary Optimization (QUBO) problems. For an $n$-dimensional QUBO problem with $m$ nonzero terms, we construct a marker oracle for such problems with a tuneable parameter, $\Lambda \in \left[ 1, m \right] \cap \mathbb{Z}$. At $d \in \mathbb{Z}_+$ precision, the oracle uses $O (n + \Lambda d)$ qubits, has total depth of $O \left( \tfrac{m}{\Lambda} \log_2 (n) + \log_2 (d) \right)$, and non-Clifford depth of $O \left( \tfrac{m}{\Lambda} \right)$. Moreover, each qubit required to be connected to at most $O \left( \log_2 (\Lambda + d) \right)$ other qubits. In the case of a maximum graph cuts, as $d = 2 \left\lceil \log_2 (n) \right\rceil$ always suffices, the depth of the marker oracle can be made as shallow as $O (\log_2 (n))$. For all values of $\Lambda$, the non-Clifford gate count of these oracles is strictly lower (at least by a factor of $\sim 2$) than previous constructions. Furthermore, we introduce a novel \textit{Fixed-point Grover Adaptive Search for QUBO Problems}, using our oracle design and a hybrid Fixed-point Grover Search, motivated by the works of Boyer et al. and Li et al. This method has better performance guarantees than previous Grover Adaptive Search methods. Some of our results are novel and useful for any method based on Fixed-point Grover Search. Finally, we give a heuristic argument that, with high probability and in $O \left( \tfrac{\log_2 (n)}{\sqrt{\epsilon}} \right)$ time, this adaptive method finds a configuration that is among the best $\epsilon 2^n$ ones.