Related papers: Improved Classical and Quantum Algorithms for the …
In this paper, we study quantum algorithms of matrix multiplication from the viewpoint of inputting quantum/classical data to outputting quantum/classical data. The main target is trying to overcome the input and output problem, which are…
Consider the setting where a $\rho$-sparse Rademacher vector is planted in a random $d$-dimensional subspace of $R^n$. A classical question is how to recover this planted vector given a random basis in this subspace. A recent result by…
The NP-hardness of the closest vector problem (CVP) is an important basis for quantum-secure cryptography, in much the same way that integer factorisation's conjectured hardness is at the foundation of cryptosystems like RSA. Recent work…
The closest pair problem is a fundamental problem of computational geometry: given a set of $n$ points in a $d$-dimensional space, find a pair with the smallest distance. A classical algorithm taught in introductory courses solves this…
We consider a version of the nearest-codeword problem on finite fields $\mathbb{F}_q$ using the Manhattan distance, an analog of the Hamming metric for non-binary alphabets. Similarly to other lattice related problems, this problem is…
Compute-and-Forward is an emerging technique to deal with interference. It allows the receiver to decode a suitably chosen integer linear combination of the transmitted messages. The integer coefficients should be adapted to the channel…
$ \newcommand{\SVP}{\mathsf{SVP}} \newcommand{\NP}{\mathsf{NP}} \newcommand{\RTIME}{\mathsf{RTIME}} \newcommand{\RSUBEXP}{\mathsf{RSUBEXP}} \newcommand{\eps}{\epsilon} \newcommand{\poly}{\mathop{\mathrm{poly}}} $We show that unless $\NP…
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…
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…
Finding a good approximation of the top eigenvector of a given $d\times d$ matrix $A$ is a basic and important computational problem, with many applications. We give two different quantum algorithms that, given query access to the entries…
$ \newcommand{\problem}[1]{\ensuremath{\mathrm{#1}} } \newcommand{\SVP}{\problem{SVP}} \newcommand{\ensuremath}[1]{#1} $We prove the following quantitative hardness results for the Shortest Vector Problem in the $\ell_p$ norm ($\SVP_p$),…
We study quantum algorithms for several fundamental string problems, including Longest Common Substring, Lexicographically Minimal String Rotation, and Longest Square Substring. These problems have been widely studied in the stringology…
Brand\~ao and Svore very recently gave quantum algorithms for approximately solving semidefinite programs, which in some regimes are faster than the best-possible classical algorithms in terms of the dimension $n$ of the problem and the…
$ \newcommand{\eps}{\varepsilon} \newcommand{\problem}[1]{\ensuremath{\mathrm{#1}} } \newcommand{\CVP}{\problem{CVP}} \newcommand{\SVP}{\problem{SVP}} \newcommand{\CVPP}{\problem{CVPP}} \newcommand{\ensuremath}[1]{#1} $For odd integers $p…
Lattices have many significant applications in cryptography. In 2021, the $p$-adic signature scheme and public-key encryption cryptosystem were introduced. They are based on the Longest Vector Problem (LVP) and the Closest Vector Problem…
In this paper we consider the closest vector problem (CVP) for lattices $\Lambda \subseteq \mathbb{Z}^n$ given by a generator matrix $A\in \mathcal{M}_{n\times n}(\mathbb{Z})$. Let $b>0$ be the maximum of the absolute values of the entries…
The Quantum Singular Value Transformation (QSVT) is a recent technique that gives a unified framework to describe most quantum algorithms discovered so far, and may lead to the development of novel quantum algorithms. In this paper we…
We study algorithms for solving the problem of constructing a text (long string) from a dictionary (sequence of small strings). The problem has an application in bioinformatics and has a connection with the Sequence assembly method for…
In this work, we study the solution of shortest vector problems (SVPs) arising in terms of learning with error problems (LWEs). LWEs are linear systems of equations over a modular ring, where a perturbation vector is added to the right-hand…
The problem of efficient multiplication of large numbers has been a long-standing challenge in classical computation and has been extensively studied for centuries. It appears that the existing classical algorithms are close to their…