Pré-Publication, Document De Travail Année : 2024

Time-Space Tradeoffs in Searching for a Path in Welded Trees, Classically or Quantumly

Résumé

Welded trees of depth d consist of a pair of complete binary trees, each of depth d, with their leaves interconnected by an additional cycle or permutation that alternates between the two trees. These welded trees are accessed through queries to a local oracle, which, given a vertex (and optionally the colour of a possible incident edge), returns a neighbouring vertex.

The Finding Exit problem involves finding a root, known as the exit, given the id of the opposite root called the entrance, with a minimal number of oracle queries. In 2003, Childs et al. established an exponential lower bound Ω(2 d/3 ) for classical algorithms and introduced a quantum algorithm with polynomial complexity in d for Finding Exit. The Finding Path To Exit problem requires finding a path from the entrance to the exit in welded trees. However, solving Finding Path To Exit in the quantum context is considered notably more challenging than Finding Exit. This challenge is underscored by a recent lower bound of Ω(2 d/6 ) by Childs et al. in 2022 for a restricted family of quantum algorithms that maintain complete path information from the entrance. Before our work, achieving a quantum speedup over classical methods for the Finding Path To Exit problem was unclear. Our main contribution refutes this by presenting the first quantum algorithm for Finding Path To Exit to the best of our knowledge that surpasses classical methods in finding a path to the exit. We introduce two quantum algorithms for Finding Path To Exit, slightly distinct from the restricted family of algorithms previously proposed by Childs et al. for the lower bound. The first algorithm operates with a complexity of Õ(2 d/2 ) and utilizes the quantum algorithm for Finding Exit once as a primitive. It transforms Finding Path To Exit into a collision problem, subsequently solved classically. Moreover, it finds a shortest path to exit. Our second quantum algorithm tackles the collision problem using fixed-point quantum search techniques introduced by Yoder et al., achieving a complexity of Õ(2 d/3 ) queries. This represents the first provable quantum speedup from the classical lower bound of Ω(2 d/2 ) to a quantum complexity of Õ(2 d/3 ) for Finding Path To Exit. This speed-up was previously unclear due to limitations in interference caused by path memorization, as explored in the lower bound by Childs et al. Other contributions of this paper include a formal proof of the classical lower bound of Ω(2 d/2 ) for Finding Path To Exit and finer analyses of average hitting times for two classical random walks in welded trees.

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Dates et versions

hal-04685710 , version 1 (03-09-2024)
hal-04685710 , version 2 (05-05-2025)

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  • HAL Id : hal-04685710 , version 1

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Yvan Le Borgne, Shrinidhi Teganahally Sridhara. Time-Space Tradeoffs in Searching for a Path in Welded Trees, Classically or Quantumly. 2024. ⟨hal-04685710v1⟩
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