One of the challenges of quantum computers in the near- and mid- term is the limited number of qubits we can use for computations. Finding methods that achieve useful quantum improvements under size limitations is thus a key question in the field. In this vein, it was recently shown that a hybrid classical-quantum method can help provide polynomial speed-ups to classical divide-and-conquer algorithms, even when only given access to a quantum computer much smaller than the problem itself. In this work, we study the hybrid divide-and-conquer method in the context of tree search algorithms, and extend it by including quantum backtracking, which allows better results than previous Grover-based methods. Further, we provide general criteria for polynomial speed-ups in the tree search context, and provide a number of examples where polynomial speed ups, using arbitrarily smaller quantum computers, can be obtained. We provide conditions for speedups for the well known algorithm of DPLL, and we prove threshold-free speed-ups for the PPSZ algorithm (the core of the fastest exact Boolean satisfiability solver) for well-behaved classes of formulas. We also provide a simple example where speed-ups can be obtained in an algorithm-independent fashion, under certain well-studied complexity-theoretical assumptions. Finally, we briefly discuss the fundamental limitations of hybrid methods in providing speed-ups for larger problems.

翻译：量子计算机在短期至中期面临的一大挑战是可用于计算的有效量子比特数量受限。因此，在规模限制下实现有用量子改进的方法成为该领域的关键问题。近期研究表明，即使只能使用远小于问题规模的量子计算机，混合经典-量子方法仍能为经典分治算法提供多项式加速。本研究在树搜索算法背景下深入探讨混合分治方法，通过引入量子回溯机制实现优于以往基于Grover方法的效果。我们进一步提出树搜索场景中获得多项式加速的通用准则，并给出多个可采用任意小型量子计算机实现多项式加速的实例。针对广为人知的DPLL算法，我们给出了加速条件；同时证明对于良好结构化公式类，PPSZ算法（最快精确布尔可满足性求解器的核心）可获得无阈值加速。此外，在特定的复杂度理论假设条件下，我们展示了一个与算法无关的加速简化实例。最后，我们简要论述混合方法在更大规模问题上实现加速的根本性局限。