Related papers: Qubo model for the Closest Vector Problem
We present algorithms for the $(1+\epsilon)$-approximate version of the closest vector problem for certain norms. The currently fastest algorithm (Dadush and Kun 2016) for general norms has running time of $2^{O(n)} (1/\epsilon)^n$. We…
In this note we give a polynomial time algorithm for solving the closest vector problem in the class of zonotopal lattices. The Voronoi cell of a zonotopal lattice is a zonotope, i.e. a projection of a regular cube. Examples of zonotopal…
We prove that the $L^2$ CVP distance from a random short ring element to the log-unit lattice of $\Q(\zeta_{2^k})$ converges to $\frac{\pi}{2\sqrt{6}}\sqrt{n}$ as $n=2^{k-1}\to\infty$. We then show that this target lies inside the Voronoi…
In this paper we address the problem of finding well approximating lattices for a given finite set $A$ of points in ${\mathbb R}^n$. More precisely, we search for $\v{o},\v{d_1}, \dots,\v{d_n}\in \mathbb{R}^n$ such that $\v{a}-\v{o}$ is…
Quantum computing poses a threat to contemporary cryptosystems, with advances to a state in which it will cause problems predicted for the next few decades. Many of the proposed cryptosystems designed to be quantum-secure are based on the…
Variational quantum algorithms have been advocated as promising candidates to solve combinatorial optimization problems on near-term quantum computers. Their methodology involves transforming the optimization problem into a quadratic…
In this work, we give provable sieving algorithms for the Shortest Vector Problem (SVP) and the Closest Vector Problem (CVP) on lattices in $\ell_p$ norm ($1\leq p\leq\infty$). The running time we obtain is better than existing provable…
In this paper a variation of the classic vector quantization problem is considered. In the standard formulation, a quantizer is designed to minimize the distortion between input and output when the number of reconstruction points is fixed.…
We introduce a new class of algorithms for finding a short vector in lattices defined by codes of co-dimension $k$ over $\mathbb{Z}_P^d$, where $P$ is prime. The co-dimension $1$ case is solved by exploiting the packing properties of the…
In this paper, we introduce the Maximum Distance Sublattice Problem (MDSP). We observed that the problem of solving an instance of the Closest Vector Problem (CVP) in a lattice $\mathcal{L}$ is the same as solving an instance of MDSP in the…
We show a linear-size reduction from gap Max-2-Lin(2) (a generalization of the gap $\mathrm{Max}$-$\mathrm{Cut}$ problem) to $\gamma\text{-}\mathrm{CVP}_p$ for $\gamma = \mathrm{O}(1)$ and finite $p\geq 1$, as well as a no-go theorem…
A sampling-based optimization method for quadratic functions is proposed. Our method approximately solves the following $n$-dimensional quadratic minimization problem in constant time, which is independent of $n$: $z^*=\min_{\mathbf{v} \in…
The peptide-protein docking problem is an important problem in structural biology that facilitates rational and efficient drug design. In this work, we explore modeling and solving this problem with the quantum-amenable quadratic…
Quadratic unconstrained binary optimization (QUBO) has become the standard format for optimization using quantum computers, i.e., for both the quantum approximate optimization algorithm (QAOA) and quantum annealing (QA). We present a…
Practically relevant problems of quadratic optimization often contain multidimensional arrays of variables interconnected by linear constraints, such as equalities and inequalities. The values of each variable depend on its specific meaning…
Stabilizer states admit compact classical descriptions, but many downstream tasks still require their full amplitude vectors. Since the output itself has size $2^n$, the main algorithmic question is whether one can materialize an $n$-qubit…
In an instance of the minimum eigenvalue problem, we are given a collection of $n$ vectors $v_1,\ldots, v_n \subset {\mathbb{R}^d}$, and the goal is to pick a subset $B\subseteq [n]$ of given vectors to maximize the minimum eigenvalue of…
We give a novel algorithm for enumerating lattice points in any convex body, and give applications to several classic lattice problems, including the Shortest and Closest Vector Problems (SVP and CVP, respectively) and Integer Programming…
We show that, assuming NP $\not\subseteq$ $\cap_{\delta > 0}$DTIME$\left(\exp{n^\delta}\right)$, the shortest vector problem for lattices of rank $n$ in any finite $\ell_p$ norm is hard to approximate within a factor of $2^{(\log n)^{1 -…
We show a $2^{n/2+o(n)}$-time algorithm that finds a (non-zero) vector in a lattice $\mathcal{L} \subset \mathbb{R}^n$ with norm at most $\tilde{O}(\sqrt{n})\cdot \min\{\lambda_1(\mathcal{L}), \det(\mathcal{L})^{1/n}\}$, where…