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：近中期内，量子计算机所能使用的量子比特数目有限，因此在规模限制下实现有用的量子改进是该领域的一个关键问题。最近的研究表明，即使只有一个比问题本身规模还小的量子计算机可用，混合经典-量子方法仍然可以帮助将传统分治算法的多项式加速。本文研究了混合分治方法在树搜索算法中的应用，并包括了量子回溯方法以实现比之前基于 Grover 的方法更好的结果。我们提供了树搜索背景下多项式加速的普遍标准，并提供了许多例子，其中可以使用比问题本身规模更小的量子计算机实现多项式加速。我们提供了DPLL算法加速的条件，并证明了对于形式良好的公式类别，可以在没有阈值的情况下加速PPSZ算法（最快的精确布尔可满足性求解器的核心部分）。我们还提供了一个简单的例子，证明在某些研究充分的复杂性理论假设下，可以以算法独立的方式实现加速。最后，我们简要讨论了混合方法对大型问题提高速度的根本限制。