Abstract. The most important computational problem on lattices is the shortest vector problem ([Formula: see text]). In this paper, we present new algorithms that improve the state-of-the-art for provable classical/quantum algorithms for [Formula: see text]. We present the following results: (1) A new algorithm for [Formula: see text] that provides a smooth tradeoff between time complexity and memory requirement. For any positive integer [Formula: see text], our algorithm takes [Formula: see text] time and requires [Formula: see text] memory. This tradeoff, which ranges from enumeration ([Formula: see text]) to sieving ([Formula: see text] constant), is a consequence of a new time-memory tradeoff for discrete Gaussian sampling above the smoothing parameter. (2) A quantum algorithm for [Formula: see text] that runs in time [Formula: see text] and requires [Formula: see text] classical memory and [Formula: see text] qubits. In a quantum random access memory (QRAM) model, this algorithm takes only [Formula: see text] time and requires a QRAM of size [Formula: see text], [Formula: see text] qubits and [Formula: see text] classical space. This improves over the previously fastest classical (which is also the fastest quantum) algorithm due to [D. Aggarwal et al., Solving the shortest vector problem in 2 n time using discrete Gaussian sampling: Extended abstract, in Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing (STOC), 2015, pp. 733–742] that has a time and space complexity [Formula: see text]. (3) A classical algorithm for [Formula: see text] that runs in time [Formula: see text] time and [Formula: see text] space. This improves over an algorithm of [Y. Chen, K. Chung, and C. Lai, Quantum Inf. Comput., 18 (2018), pp. 285–306] that has the same space complexity. The time complexity of our classical and quantum algorithms are obtained using a known upper bound on a quantity related to the lattice kissing number, which is [Formula: see text]. We conjecture that for most lattices this quantity is a [Formula: see text]. Assuming that this is the case, our classical algorithm runs in time [Formula: see text], our quantum algorithm runs in time [Formula: see text], and our quantum algorithm in a QRAM model runs in time [Formula: see text]. As a direct application of our result, using the reduction in [L. Ducas, Des. Codes. Cryptogr., 92 (2024), pp. 909–916], we obtain a provable quantum algorithm for the lattice isomorphism problem in the case of the trivial lattice [Formula: see text] ([Formula: see text] LIP ) that runs in time [Formula: see text]. Our algorithm requires a QRAM of size [Formula: see text], [Formula: see text] qubits and [Formula: see text] classical space.

The most important computational problem on lattices is the Shortest Vector Problem (SVP). In this paper, we present new algorithms that improve the state-of-the-art for provable classical/quantum algorithms for SVP. We present the following results. 1) A new algorithm for SVP that provides a smooth tradeoff between time complexity and memory requirement. For any positive integer 4 ≤ q ≤ √n, our algorithm takes q^{13n+o(n)} time and requires poly(n)⋅ q^{16n/q²} memory. This tradeoff which ranges from enumeration (q = √n) to sieving (q constant), is a consequence of a new time-memory tradeoff for Discrete Gaussian sampling above the smoothing parameter. 2) A quantum algorithm that runs in time 2^{0.9533n+o(n)} and requires 2^{0.5n+o(n)} classical memory and poly(n) qubits. This improves over the previously fastest classical (which is also the fastest quantum) algorithm due to [Divesh Aggarwal et al., 2015] that has a time and space complexity 2^{n+o(n)}. 3) A classical algorithm for SVP that runs in time 2^{1.741n+o(n)} time and 2^{0.5n+o(n)} space. This improves over an algorithm of [Yanlin Chen et al., 2018] that has the same space complexity. The time complexity of our classical and quantum algorithms are expressed using a quantity related to the kissing number of a lattice. A known upper bound of this quantity is 2^{0.402n}, but in practice for most lattices, it can be much smaller and even 2^o(n). In that case, our classical algorithm runs in time 2^{1.292n} and our quantum algorithm runs in time 2^{0.750n}.

We study the quantum query complexity of two problems. First, we consider the problem of determining if a sequence of parentheses is a properly balanced one (a Dyck word), with a depth of at most k. We call this the Dyck_{k,n} problem. We prove a lower bound of Ω(c^k √n), showing that the complexity of this problem increases exponentially in k. Here n is the length of the word. When k is a constant, this is interesting as a representative example of star-free languages for which a surprising Õ(√n) query quantum algorithm was recently constructed by Aaronson et al. [Scott Aaronson et al., 2018]. Their proof does not give rise to a general algorithm. When k is not a constant, Dyck_{k,n} is not context-free. We give an algorithm with O(√n(log n)^{0.5k}) quantum queries for Dyck_{k,n} for all k. This is better than the trival upper bound n for k = o({log(n)}/{log log n}). Second, we consider connectivity problems on grid graphs in 2 dimensions, if some of the edges of the grid may be missing. By embedding the "balanced parentheses" problem into the grid, we show a lower bound of Ω(n^{1.5-ε}) for the directed 2D grid and Ω(n^{2-ε}) for the undirected 2D grid. The directed problem is interesting as a black-box model for a class of classical dynamic programming strategies including the one that is usually used for the well-known edit distance problem. We also show a generalization of this result to more than 2 dimensions.