We present new advances in achieving exponential quantum speedups for solving optimization problems by low-depth quantum algorithms. Specifically, we focus on families of combinatorial optimization problems that exhibit symmetry and contain planted solutions. We rigorously prove that the 1-step Quantum Approximate Optimization Algorithm (QAOA) can achieve a success probability of $\Omega(1/\sqrt{n})$, and sometimes $\Omega(1)$, for finding the exact solution in many cases. Furthermore, we construct near-symmetric optimization problems by randomly sampling the individual clauses of symmetric problems, and prove that the QAOA maintains a strong success probability in this setting even when the symmetry is broken. Finally, we construct various families of near-symmetric Max-SAT problems and benchmark state-of-the-art classical solvers, discovering instances where all known classical algorithms require exponential time. Therefore, our results indicate that low-depth QAOA could achieve an exponential quantum speedup for optimization problems.

翻译：我们提出了在利用低深度量子算法求解优化问题时实现指数级量子加速的新进展。具体而言，我们关注具有对称性且包含植入解的组合优化问题族。我们严格证明了对于许多情况，一步量子近似优化算法（QAOA）在寻找精确解时能够达到$\Omega(1/\sqrt{n})$的成功概率，有时甚至可达$\Omega(1)$。此外，我们通过对对称问题的各个子句进行随机采样来构造近对称优化问题，并证明了即使对称性被破坏，QAOA在此设定下仍能保持较高的成功概率。最后，我们构造了多种近对称Max-SAT问题族，并对最先进的经典求解器进行了基准测试，发现了所有已知经典算法均需要指数时间求解的实例。因此，我们的结果表明低深度QAOA有望在优化问题上实现指数级量子加速。