We propose and study Th-QAOA (pronounced Threshold QAOA), a variation of the Quantum Alternating Operator Ansatz (QAOA) that replaces the standard phase separator operator, which encodes the objective function, with a threshold function that returns a value $1$ for solutions with an objective value above the threshold and a $0$ otherwise. We vary the threshold value to arrive at a quantum optimization algorithm. We focus on a combination with the Grover Mixer operator; the resulting GM-Th-QAOA can be viewed as a generalization of Grover's quantum search algorithm and its minimum/maximum finding cousin to approximate optimization. Our main findings include: (i) we provide intuitive arguments and show empirically that the optimum parameter values of GM-Th-QAOA (angles and threshold value) can be found with $O(\log(p) \times \log M)$ iterations of the classical outer loop, where $p$ is the number of QAOA rounds and $M$ is an upper bound on the solution value (often the number of vertices or edges in an input graph), thus eliminating the notorious outer-loop parameter finding issue of other QAOA algorithms; (ii) GM-Th-QAOA can be simulated classically with little effort up to 100 qubits through a set of tricks that cut down memory requirements; (iii) somewhat surprisingly, GM-Th-QAOA outperforms non-thresholded GM-QAOA in terms of approximation ratios achieved. This third result holds across a range of optimization problems (MaxCut, Max k-VertexCover, Max k-DensestSubgraph, MaxBisection) and various experimental design parameters, such as different input edge densities and constraint sizes.
翻译:我们提出并研究T- QAOA( 宣布阈值 QAOA), QQQaltum Alternated运算 Ansatz (QAOA) 的变异, 取代标准阶段分隔器操作员, 该操作员对目标函数进行编码, 下限函数为解决方案返回值$$, 其目标值高于阈值, 否则为 美元。 我们改变阈值, 以达成量量优化算法。 我们侧重于与 Grover Mixer 运算器的组合; 由此产生的 GM- Th- QAOA, 可以被视为 Grover 量搜索运算算算法的概略化 QA 量位搜索算法, 其最小/ 最大发现表表表表表表表表表表表表表表表显示, GM- TH- QOOOA( 矩志) 的最佳参数值可以通过 $O- liquestal A 的 QA 和 QA 直径A 直径OA 的直径解值, 直径解OA 的直径值在OA 的数值中可以找到其他的解解解算法, 。