A Discrete Fourier Transform on Lattices with Quantum Applications

TL;DR

This paper defines a Discrete Fourier Transform on certain Euclidean lattices and leverages quantum computing to develop algorithms for sampling from lattice-based distributions, potentially impacting lattice problem complexity.

Contribution

It introduces a lattice DFT applicable to Systematic Normal Form lattices and connects this to quantum algorithms for distribution sampling.

Findings

Efficient quantum algorithms for sampling from lattice distributions.

Approximation of arbitrary lattices by SysNF lattices for DFT computation.

Potential insights into lattice problems like SVP through eigenvector analysis.

Abstract

In this work, we introduce a definition of the Discrete Fourier Transform (DFT) on Euclidean lattices in $\R^n$, that generalizes the $n$-th fold DFT of the integer lattice $\Z^n$ to arbitrary lattices. This definition is not applicable for every lattice, but can be defined on lattices known as Systematic Normal Form (SysNF) introduced in \cite{ES16}. Systematic Normal Form lattices are sets of integer vectors that satisfy a single homogeneous modular equation, which itself satisfies a certain number-theoretic property. Such lattices form a dense set in the space of $n$-dimensional lattices, and can be used to approximate efficiently any lattice. This implies that for every lattice $L$ a DFT can be computed efficiently on a lattice near $L$. Our proof of the statement above uses arguments from quantum computing, and as an application of our definition we show a quantum algorithm for sampling from discrete distributions on lattices, that extends our ability to sample efficiently from the discrete Gaussian distribution \cite{GPV08} to any distribution that is sufficiently "smooth". We conjecture that studying the eigenvectors of the newly-defined lattice DFT may provide new insights into the structure of lattices, especially regarding hard computational problems, like the shortest vector problem.