Robust Polynomials and Quantum Algorithms

We define and study the complexity of \emph{robust} polynomials for Boolean functions and the related fault-tolerant quantum decision trees, where input bits are perturbed by noise. We compare several different possible definitions. Our main results are \begin{itemize} \item For every $n$-bit Boolean function $f$ there is an $n$-variate polynomial $p$ of degree \bigO(n) that \emph{robustly} approximates it, in the sense that $p(x)$ remains close to $f(x)$ if we slightly vary each of the $n$ inputs of the polynomial. \item There is an \bigO(n)-query quantum algorithm that \emph{robustly} recovers $n$ noisy input bits. Hence every $n$-bit function can be quantum computed with \bigO(n) queries in the presence of noise. This contrasts with the classical model of Feige~\etal, where functions such as parity need $\Theta(n\log n)$ queries. \end{itemize} We give several extensions and applications of these results

Lower Bounds on Quantum Query Complexity

Shor's and Grover's famous quantum algorithms for factoring and searching show that quantum computers can solve certain computational problems significantly faster than any classical computer. We discuss here what quantum computers_cannot_ do, and specifically how to prove limits on their computational power. We cover the main known techniques for proving lower bounds, and exemplify and compare the methods.Comment: survey, 23 page

Three Puzzles on Mathematics, Computation, and Games

In this lecture I will talk about three mathematical puzzles involving mathematics and computation that have preoccupied me over the years. The first puzzle is to understand the amazing success of the simplex algorithm for linear programming. The second puzzle is about errors made when votes are counted during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure

Learning with Errors is easy with quantum samples

Learning with Errors is one of the fundamental problems in computational learning theory and has in the last years become the cornerstone of post-quantum cryptography. In this work, we study the quantum sample complexity of Learning with Errors and show that there exists an efficient quantum learning algorithm (with polynomial sample and time complexity) for the Learning with Errors problem where the error distribution is the one used in cryptography. While our quantum learning algorithm does not break the LWE-based encryption schemes proposed in the cryptography literature, it does have some interesting implications for cryptography: first, when building an LWE-based scheme, one needs to be careful about the access to the public-key generation algorithm that is given to the adversary; second, our algorithm shows a possible way for attacking LWE-based encryption by using classical samples to approximate the quantum sample state, since then using our quantum learning algorithm would solve LWE