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Quantum Algorithms for Lattice Problems
per 2024/555 Quantum Algorithms for Lattice Problems Abstract We show a polynomial time quantum algorithm for solving the learning with errors problem (LWE) with certain polynomial modulus-noise ratios. Combining with the reductions from lattice problems to LWE shown by Regev [J.ACM 2009], we obtain polynomial time quantum algorithms for solving the decisional shortest vector problem (GapSVP) and the shortest independent vector problem (SIVP) for all $n$-dimensional lattices within approximation factors of $\tilde{\Omega}(n^{4.5})$.
First, we introduce Gaussian functions with complex variances in the design of quantum algorithms. In particular, we exploit the feature of the Karst wave in the discrete Fourier transform of complex Gaussian functions. Using those techniques, we first convert the LWE instance into quantum states with purely imaginary Gaussian amplitudes, then convert purely imaginary Gaussian states into classical linear equations over the LWE secret and error terms, and finally solve the linear system of equations using Gaussian elimination.
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