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Quantum Algorithms for Lattice Problems

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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 Ω ~ ( 𝑛 4.5 ) . Previously, no polynomial or even subexponential time quantum algorithms were known for solving GapSVP or SIVP for all lattices within any polynomial approximation factors. To develop a quantum algorithm for solving LWE, we mainly introduce two new techniques. 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. Second, we use windowed quantum Fourier transform

Quantum Algorithms for Lattice Problems Paper 2024/555 Quantum Algorithms for Lattice Problems Yilei Chen , Tsinghua University , Shanghai Artificial Intelligence Laboratory , Shanghai Qi Zhi Institute 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

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