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Gil Kalai

Publications and source records attributed to Gil Kalai.

At least 37 records · Page 2Linked to original sources
arXiv2021

Statistical Aspects of the Quantum Supremacy Demonstration

The notable claim of quantum supremacy presented by Google's team in 2019 consists of demonstrating the ability of a quantum circuit to generate, albeit with considerable noise, bitstrings from a distribution that is considered hard to simulate on classical computers. Verifying that the generated data is indeed from the claimed distribution and assessing the circuit's noise level and its fidelity is a purely statistical undertaking. The objective of this paper is to explain the relations between quantum computing and some of the statistical aspects involved in demonstrating quantum supremacy in terms that are accessible to statisticians, computer scientists, and mathematicians. Starting with the statistical analysis in Google's demonstration, which we explain, we study various estimators of the fidelity, and different approaches to testing the distributions generated by the quantum computer. We propose different noise models, and discuss their implications. A preliminary study of the Google data, focusing mostly on circuits of 12 and 14 qubits is discussed throughout the paper.

quant-ph↗
arXiv2021

The Argument against Quantum Computers, the Quantum Laws of Nature, and Google's Supremacy Claims

My 2018 lecture at the ICA workshop in Singapore dealt with quantum computation as a meeting point of the laws of computation and the laws of quantum mechanics. We described a computational complexity argument against the feasibility of quantum computers: we identified a very low-level complexity class of probability distributions described by noisy intermediate-scale quantum computers, and explained why it would allow neither good-quality quantum error-correction nor a demonstration of "quantum supremacy," namely, the ability of quantum computers to make computations that are impossible or extremely hard for classical computers. We went on to describe general predictions arising from the argument and proposed general laws that manifest the failure of quantum computers. In October 2019, "Nature" published a paper describing an experimental work that took place at Google. The paper claims to demonstrate quantum (computational) supremacy on a 53-qubit quantum computer, thus clearly challenging my theory. In this paper, I will explain and discuss my work in the perspective of Google's supremacy claims.

quant-ph↗
arXiv2021

The success probability in Lionel Levine's hat problem is strictly decreasing with the number of players, and this is related to interesting questions regarding Hamming powers of Kneser graphs and independent sets in random subgraphs

Lionel Levine's hat challenge has $t$ players, each with a (very large, or infinite) stack of hats on their head, each hat independently colored at random black or white. The players are allowed to coordinate before the random colors are chosen, but not after. Each player sees all hats except for those on her own head. They then proceed to simultaneously try and each pick a black hat from their respective stacks. They are proclaimed successful only if they are all correct. Levine's conjecture was the success probability tends to zero when the number of players grows. We prove that this success probability is strictly decreasing in the number of players, and present some connections to questions in graph theory.

math.CO↗
arXiv2021

Relative Leray numbers via spectral sequences

Let $\mathbb{F}$ be a fixed field and let $X$ be a simplicial complex on the vertex set $V$. The Leray number $L(X;\mathbb{F})$ is the minimal $d$ such that for all $i \geq d$ and $S \subset V$, the induced complex $X[S]$ satisfies $\tilde{H}_i(X[S];\mathbb{F})=0$. Leray numbers play a role in formulating and proving topological Helly type theorems. For two complexes $X,Y$ on the same vertex set $V$, define the relative Leray number $L_Y(X;\mathbb{F})$ as the minimal $d$ such that $\tilde{H}_i(X[V \setminus σ];\mathbb{F})=0$ for all $i \geq d$ and $σ\in Y$. In this paper we extend the topological colorful Helly theorem to the relative setting. Our main tool is a spectral sequence for the intersection of complexes indexed by a geometric lattice.

math.CO↗
arXiv2020

Periodic Boundary Conditions for Periodic Jacobi Matrices on Trees

We consider matrices on infinite trees which are universal covers of Jacobi matrices on finite graphs. We are interested in the question of the existence of sequences of finite covers whose normalized eigenvalue counting measures converge to the density of states of the operator on the infinite tree. We first of all construct a simple example where this convergence fails and then discuss two ways of constructing the required sequences: with random boundary conditions and through normal subgroups.

math.SP↗
arXiv2019

Intersection patterns of planar sets

Let $\mathcal A=\{A_1,\ldots,A_n\}$ be a family of sets in the plane. For $0 \leq i < n$, denote by $f_i$ the number of subsets $σ$ of $\{1,\ldots,n\}$ of cardinality $i+1$ that satisfy $\bigcap_{i \in σ} A_i \neq \emptyset$. Let $k \geq 2$ be an integer. We prove that if each $k$-wise and $(k+1)$-wise intersection of sets from $\mathcal A$ is empty, or a single point, or both open and path-connected, then $f_{k+1}=0$ implies $f_k \leq cf_{k-1}$ for some positive constant $c$ depending only on $k$. Similarly, let $b \geq 2, k > 2b$ be integers. We prove that if each $k$-wise or $(k+1)$-wise intersection of sets from $\mathcal A$ has at most $b$ path-connected components, which all are open, then $f_{k+1}=0$ implies $f_k \leq cf_{k-1}$ for some positive constant $c$ depending only on $b$ and $k$. These results also extend to two-dimensional compact surfaces.

math.CO↗
arXiv2019

The Argument against Quantum Computers

We give a computational complexity argument against the feasibility of quantum computers. We identify a very low complexity class of probability distributions described by noisy intermediate-scale quantum computers, and explain why it will allow neither good-quality quantum error-correction nor a demonstration of "quantum supremacy." Some general principles governing the behavior of noisy quantum systems are derived. Our work supports the "physical Church thesis" studied by Pitowsky (1990) and follows his vision of using abstract ideas about computation to study the performance of actual physical computers.

quant-ph↗
arXiv2018

Bidding Games and Efficient Allocations

Richman games are zero-sum games, where in each turn players bid in order to determine who will play next [Lazarus et al.'99]. We extend the theory to impartial general-sum two player games called \emph{bidding games}, showing the existence of pure subgame-perfect equilibria (PSPE). In particular, we show that PSPEs form a semilattice, with a unique and natural \emph{Bottom Equilibrium}. Our main result shows that if only two actions available to the players in each node, then the Bottom Equilibrium has additional properties: (a) utilities are monotone in budget; (b) every outcome is Pareto-efficient; and (c) any Pareto-efficient outcome is attained for some budget. In the context of combinatorial bargaining, we show that a player with a fraction of X% of the total budget prefers her allocation to X% of the possible allocations. In addition, we provide a polynomial-time algorithm to compute the Bottom Equilibrium of a binary bidding game.

cs.GT↗
arXiv2018

Quasi-random multilinear polynomials

We consider multilinear Littlewood polynomials, polynomials in $n$ variables in which a specified set of monomials $U$ have $\pm 1$ coefficients, and all other coefficients are $0$. We provide upper and lower bounds (which are close for $U$ of degree below $\log n$) on the minimum, over polynomials $h$ consistent with $U$, of the maximum of $|h|$ over $\pm 1$ assignments to the variables. (This is a variant of a question posed by Erdös regarding the maximum on the unit disk of univariate polynomials of given degree with unit coefficients.) We outline connections to the theory of quasi-random graphs and hypergraphs, and to statistical mechanics models. Our methods rely on the analysis of the Gale-Berlekamp game; on the constructive side of the generic chaining method; on a Khintchine-type inequality for polynomials of degree greater than $1$; and on Bernstein's approximation theory inequality.

math.CO↗
arXiv2018

Turán, involution and shifting

We propose a strengthening of the conclusion in Turán's (3,4)-conjecture in terms of algebraic shifting, and show that its analogue for graphs does hold. In another direction, we generalize the Mantel-Turán theorem by weakening its assumption: for any graph G on n vertices and any involution on its vertex set, if for any 3-set S of the vertices, the number of edges in G spanned by S, plus the number of edges in G spanned by the image of S under the involution, is at least 2, then the number of edges in G is at least the Mantel-Turán bound, namely the number achieved by two disjoint cliques of sizes n/2 rounded up and down.

math.CO↗
arXiv2018

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?

math.CO↗
arXiv2016

A Tverberg type theorem for matroids

Let b(M) denote the maximal number of disjoint bases in a matroid M. It is shown that if M is a matroid of rank d+1, then for any continuous map f from the matroidal complex M into the d-dimensional Euclidean space there exist t \geq \sqrt{b(M)}/4 disjoint independent sets σ_1,\ldots,σ_t \in M such that \bigcap_{i=1}^t f(σ_i) \neq \emptyset.

math.CO↗
arXiv2016

Chvátal's Conjecture and Correlation Inequalities

Chvátal's conjecture in extremal combinatorics asserts that for any decreasing family $\mathcal{F}$ of subsets of a finite set $S$, there is a largest intersecting subfamily of $\mathcal{F}$ consisting of all members of $\mathcal{F}$ that include a particular $x \in S$. In this paper we reformulate the conjecture in terms of influences of variables on Boolean functions and correlation inequalities, and study special cases and variants using tools from discrete Fourier analysis.

math.CO↗
arXiv2016

The Quantum Computer Puzzle (Expanded Version)

Quantum computers are hypothetical devices, based on quantum physics, that would enable us to perform certain computations hundreds of orders of magnitude faster than digital computers. This feature is coined as "quantum supremacy" and one aspect or another of such quantum computational supremacy might be brought about in experiments in the near future: by implementing quantum error-correction, systems of non-interacting bosons, exotic new phases of matter called anyons, quantum annealing, or in various other ways. A main concern regarding the feasibility of quantum computers is that quantum systems are inherently noisy: we cannot accurately control them, and we cannot accurately describe them. We will describe an optimistic hypothesis of quantum noise that would allow quantum computing and a pessimistic hypothesis that wouldn't. The quantum computer puzzle is deciding between these two hypotheses. Here is a brief summary of the author's pessimistic point of view as explained in the paper: understanding quantum computers in the presence of noise requires consideration of behavior at different scales. In the small scale, standard models of noise from the mid-90s are suitable, and quantum evolutions and states described by them manifest a very low-level computational power. This small-scale behavior has far-reaching consequences for the behavior of noisy quantum systems at larger scales. On the one hand, it does not allow reaching the starting points for quantum fault tolerance and quantum supremacy, making them both impossible at all scales. On the other hand, it leads to novel implicit ways for modeling noise at larger scales and to various predictions on the behavior of noisy quantum systems.

quant-ph↗
arXiv2015

On the Correlation of Increasing Families

The classical correlation inequality of Harris asserts that any two monotone increasing families on the discrete cube are nonnegatively correlated. In 1996, Talagrand established a lower bound on the correlation in terms of how much the two families depend simultaneously on the same coordinates. Talagrand's method and results inspired a number of important works in combinatorics and probability theory. In this paper we present stronger correlation lower bounds that hold when the increasing families satisfy natural regularity or symmetry conditions. In addition, we present several new classes of examples for which Talagrand's bound is tight. A central tool in the paper is a simple lemma asserting that for monotone events noise decreases correlation. This lemma gives also a very simple derivation of the classical FKG inequality for product measures, and leads to a simplification of part of Talagrand's proof.

math.CO↗
arXiv2015

Contagious error sources would need time travel to prevent quantum computation

We consider an error model for quantum computing that consists of "contagious quantum germs" that can infect every output qubit when at least one input qubit is infected. Once a germ actively causes error, it continues to cause error indefinitely for every qubit it infects, with arbitrary quantum entanglement and correlation. Although this error model looks much worse than quasi-independent error, we show that it reduces to quasi-independent error with the technique of quantum teleportation. The construction, which was previously described by Knill, is that every quantum circuit can be converted to a mixed circuit with bounded quantum depth. We also consider the restriction of bounded quantum depth from the point of view of quantum complexity classes.

quant-ph↗
arXiv2014

Gaussian Noise Sensitivity and BosonSampling

We study the sensitivity to noise of |permanent(X)|^2 for random real and complex n x n Gaussian matrices X, and show that asymptotically the correlation between the noisy and noiseless outcomes tends to zero when the noise level is ω(1)/n. This suggests that, under certain reasonable noise models, the probability distributions produced by noisy BosonSampling are very sensitive to noise. We also show that when the amount of noise is constant the noisy value of |permanent(X)|^2 can be approximated efficiently on a classical computer. These results seem to weaken the possibility of demonstrating quantum-speedup via BosonSampling without quantum fault-tolerance.

quant-ph↗