How Quantum Computers Fail: Quantum Codes, Correlations in Physical Systems, and Noise Accumulation

The feasibility of computationally superior quantum computers is one of the most exciting and clear-cut scientific questions of our time. The question touches on fundamental issues regarding probability, physics, and computability, as well as on exciting problems in experimental physics, engineering, computer science, and mathematics. We propose three related directions towards a negative answer. The first is a conjecture about physical realizations of quantum codes, the second has to do with correlations in stochastic physical systems, and the third proposes a model for quantum evolutions when noise accumulates. The paper is dedicated to the memory of Itamar Pitowsky.Comment: 16 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

Intersections of Leray complexes and regularity of monomial ideals

For a simplicial complex X and a field K, let h_i(X)=\dim \tilde{H}_i(X;K). It is shown that if X,Y are complexes on the same vertex set, then for all k h_{k-1}(X\cap Y) \leq \sum_{\sigma \in Y} \sum_{i+j=k} h_{i-1}(X[\sigma])\cdot h_{j-1}(\lk(Y,\sigma)) . A simplicial complex X is d-Leray over K, if h_i(Y)=0 for all induced subcomplexes Y \subset X and i \geq d. Let L_K(X) denote the minimal d such that X is d-Leray over K. The above theorem implies that if X,Y are simplicial complexes on the same vertex set then L_K(X \cap Y) \leq L_K(X) +L_K(Y). Reformulating this inequality in commutative algebra terms, we obtain the following result conjectured by Terai: If I,J are square-free monomial ideals in S=K[x_1,...,x_n], then reg(I+J) \leq reg(I)+reg(J)-1 where reg(I) denotes the Castelnuovo-Mumford regularity of I.Comment: 9 page