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Quantum and Classical Algorithms for Bounded Distance Decoding

In this paper, we provide a comprehensive overview of a recent debate over the quantum versus classical solvability of bounded distance decoding (BDD). Specifically, we review the work of Eldar and Hallgren [EH22], [Hal21] demonstrating a quantum algorithm solving $λ_1 2^{-Ω(\sqrt{k \log q})}$-BDD in polynomial time for lattices of periodicity $q$, finite group rank $k$, and shortest lattice vector length $λ_1$. Subsequently, we prove the results of [DvW21a], [DvW21b] with far greater detail and elaboration than in the original work. Namely, we show that there exists a deterministic, classical algorithm achieving the same result.