
Quantum Speedups for ExponentialTime Dynamic Programming Algorithms
In this paper we study quantum algorithms for NPcomplete problems whose...
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A Quantum Algorithm for Minimum Steiner Tree Problem
Minimum Steiner tree problem is a wellknown NPhard problem. For the mi...
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Quantum Dynamic Programming Algorithm for DAGs. Applications for ANDOR DAG Evaluation and DAG's Diameter Search
In this paper, we present Quantum Dynamic Programming approach for probl...
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Following Forrelation – Quantum Algorithms in Exploring Boolean Functions' Spectra
Here we revisit the quantum algorithms for obtaining Forrelation [Aarons...
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Quantum Boosting
Suppose we have a weak learning algorithm A for a Booleanvalued problem...
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Quantum Topological Data Analysis with Linear Depth and Exponential Speedup
Quantum computing offers the potential of exponential speedups for certa...
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Continuous LWE
We introduce a continuous analogue of the Learning with Errors (LWE) pro...
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Quantum speedups for dynamic programming on ndimensional lattice graphs
Motivated by the quantum speedup for dynamic programming on the Boolean hypercube by Ambainis et al. (2019), we investigate which graphs admit a similar quantum advantage. In this paper, we examine a generalization of the Boolean hypercube graph, the ndimensional lattice graph Q(D,n) with vertices in {0,1,…,D}^n. We study the complexity of the following problem: given a subgraph G of Q(D,n) via query access to the edges, determine whether there is a path from 0^n to D^n. While the classical query complexity is Θ((D+1)^n), we show a quantum algorithm with complexity O(T_D^n), where T_D < D+1. The first few values of T_D are T_1 ≈ 1.817, T_2 ≈ 2.660, T_3 ≈ 3.529, T_4 ≈ 4.421, T_5 ≈ 5.332. We also prove that T_D ≥D+1/e, thus for general D, this algorithm does not provide, for example, a speedup, polynomial in the size of the lattice. While the presented quantum algorithm is a natural generalization of the known quantum algorithm for D=1 by Ambainis et al., the analysis of complexity is rather complicated. For the precise analysis, we use the saddlepoint method, which is a common tool in analytic combinatorics, but has not been widely used in this field. We then show an implementation of this algorithm with time complexity poly(n)^log n T_D^n, and apply it to the Set Multicover problem. In this problem, m subsets of [n] are given, and the task is to find the smallest number of these subsets that cover each element of [n] at least D times. While the time complexity of the best known classical algorithm is O(m(D+1)^n), the time complexity of our quantum algorithm is poly(m,n)^log n T_D^n.
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