Improved quantum backtracking algorithms using effective resistance estimates

Abstract

We investigate quantum backtracking algorithms of the type introduced by Montanaro (arXiv:1509.02374). These algorithms explore trees of unknown structure and in certain settings exponentially outperform their classical counterparts. Some of the previous work focused on obtaining a quantum advantage for trees in which a unique marked vertex is promised to exist. We remove this restriction by recharacterising the problem in terms of the effective resistance of the search space. In this paper, we present a generalisation of one of Montanaro's algorithms to trees containing k marked vertices, where k is not necessarily known a priori.Our approach involves using amplitude estimation to determine a near-optimal weighting of a diffusion operator, which can then be applied to prepare a superposition state with support only on marked vertices and ancestors thereof. By repeatedly sampling this state and updating the input vertex, a marked vertex is reached in a logarithmic number of steps. The algorithm thereby achieves the conjectured bound of O( √ T R max ) for finding a single marked vertex and O k √ T R max for finding all k marked vertices, where T is an upper bound on the tree size and R max is the maximum effective resistance encountered by the algorithm. This constitutes a speedup over Montanaro's original procedure in both the case of finding one and the case of finding multiple marked vertices in an arbitrary tree.

Applying quantum algorithms to constraint satisfaction problems

“…Ambainis and Kokainis [6] showed that the quantum runtime bound can be improved to O( √ T n 3/2 ), where T is the number of nodes actually explored by the classical algorithm. Second, Jarret and Wan [60] showed that the runtime of Theorem 2 can be improved to O( √ T n) without the need for any promise on the uniqueness of the solution 6 . Also, the quantum backtracking algorithm has been applied to exact satisfability problems [69], the Travelling Salesman Problem [77], and attacking lattice-based cryptosystems [5,81].…”

Quantum-accelerated constraint programming

“…The latter, in a similar fashion to this work, uses quantum search to accelerate the subroutines of the simplex method (e.g., variable pricing). There also exist recent efforts to use algorithms based on quantum search and phase estimation to speed up tree search methods including branch-and-bound [46] and backtracking search [15,16,47]; we discuss the latter in more detail in Section 6. See in particular [15, Sec.…”

“…These smaller subproblems require fewer resources, making them promising candidates for the early faulttolerant quantum computers of the future. With respect to search, we investigate the adaptation of existing quantum tree search algorithms [15,16] to the search performed within CP, and provide preliminary resource estimates for this integration. Our adaptations are focused on incorporating quantum filtering within both classical and quantum backtracking search algorithms.…”

Quantum speedup of branch-and-bound algorithms

“…However, it might be possible to improve the depth dependence by a factor of up to d, e.g. by extending ideas of [23].…”

“…• If the tree does not contain any marked nodes, the algorithm returns "not found". Jarret and Wan have described an algorithm [23] which solves the same tree search problem with complexity bounded by O( √ T d log 4 (md) log(m/ )), where m is the number of marked nodes. This is an improvement on (1) by up to a factor of almost d when m is small.…”