Hybrid divide-and-conquer approach for tree search algorithms

TL;DR

This paper explores a hybrid classical-quantum divide-and-conquer approach to tree search algorithms, demonstrating polynomial speed-ups with smaller quantum computers and analyzing the conditions and limitations of such methods.

Contribution

It extends hybrid divide-and-conquer methods by incorporating quantum backtracking, providing new criteria for speed-ups, and analyzing specific algorithms like DPLL and PPSZ.

Findings

Quantum backtracking improves results over Grover-based methods.

Polynomial speed-ups are achievable with arbitrarily smaller quantum computers.

Threshold-free speed-ups are proven for the PPSZ algorithm.

Abstract

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.