Quantum algorithm for tree size estimation, with applications to backtracking and 2-player games

Ambainis, Andris; Kokainis, Martins

doi:10.48550/arxiv.1704.06774

Cited by 2 publications

(15 citation statements)

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“…Although, for practical speed-ups, they certainly are. We reiterate that all the results that we will present, can accommodate these tree size-estimation based methods of [20].…”

Section: Definition Ii2 (Maximal Branching Number)mentioning

confidence: 83%

“…More recently, Montanaro [3], Ambainis & Kokainis [20], and Jarret & Wan [21] have given quantum-walk based algorithms which allow us to achieve an essentially full quadratic speed-ups in the number of queries to the tree (specially, when trees are exponentially sized). We will refer to this the quantum backtracking method.…”

Section: Quantum Algorithms For Tree Searchmentioning

confidence: 99%

“…The original paper on quantum backtracking implements DPLL with the unit rule [3], and led to a body of work focused on improvements [20,21] and applications [5,6,20,22].…”

Section: Quantum Algorithms For Tree Searchmentioning

confidence: 99%

“…In [20], the authors provide an extension to quantum backtracking, which allows one to estimate the size of the search tree. Moreover, it can also estimate the size of a search space corresponding to a partial traversal of a given classical backtracking algorithm, according to a certain order.…”

Section: Quantum Algorithms For Tree Searchmentioning

confidence: 99%

“…Unlike backtracking, Grover can also not be used as an online algorithm, as discussed earlier in this section. In the majority of this paper, we will be concerned with worst-case times, so the online methods of [20] will not be critical. Although, for practical speed-ups, they certainly are.…”

Section: Definition Ii2 (Maximal Branching Number)mentioning

confidence: 99%

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Hybrid divide-and-conquer approach for tree search algorithms

Rennela¹,

Brand²,

Laarman³

et al. 2020

Preprint

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As we are entering the era of real-world small quantum computers, finding applications for these limited devices is a key challenge. In this vein, it was recently shown that a hybrid classicalquantum 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 still be obtained. This study possible speed-ups for the well known algorithm of DPLL and prove threshold-free speed-ups for the tree search subroutines of the so-called PPSZ algorithm -which is the core of the fastest exact Boolean satisfiability solver -for certain classes of formulas. We also provide a simple example where speed-ups can be obtained in an algorithmindependent 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.

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“…Although, for practical speed-ups, they certainly are. We reiterate that all the results that we will present, can accommodate these tree size-estimation based methods of [20].…”

Section: Definition Ii2 (Maximal Branching Number)mentioning

confidence: 83%

Section: Quantum Algorithms For Tree Searchmentioning

confidence: 99%

“…The original paper on quantum backtracking implements DPLL with the unit rule [3], and led to a body of work focused on improvements [20,21] and applications [5,6,20,22].…”

Section: Quantum Algorithms For Tree Searchmentioning

confidence: 99%

Section: Quantum Algorithms For Tree Searchmentioning

confidence: 99%

Section: Definition Ii2 (Maximal Branching Number)mentioning

confidence: 99%

See 3 more Smart Citations

Hybrid divide-and-conquer approach for tree search algorithms

Rennela¹,

Brand²,

Laarman³

et al. 2020

Preprint

View full text Add to dashboard Cite

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Improved quantum backtracking algorithms using effective resistance estimates

Jarret

Wan

2018

Phys. Rev. A

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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.

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Quantum algorithm for tree size estimation, with applications to backtracking and 2-player games

Cited by 2 publications

References 0 publications

Hybrid divide-and-conquer approach for tree search algorithms

Hybrid divide-and-conquer approach for tree search algorithms

Improved quantum backtracking algorithms using effective resistance estimates

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