arXiv subjects
Publications and source records attributed to Gil Kalai.
We revisit and extend some topics that we studied in our previous works (Rinott, Kalai and Shoham 2022; Kalai, Rinott and Shoham, 2023,2024) regarding the Google 2019 "quantum supremacy" experiment. We extend our analysis of the prediction based on Google's digital error model (Formula (77)), based on more detailed data provided by Google. We also provide some preliminary analysis for a few other NISQ experiments.
Consider the (almost surely) unique Radon partition of a set of $n$ random Gaussian vectors in $\mathbb R^{n-2}$; choose one of the two parts of this partition uniformly at random, and for $0 \le k \le n$, let $p_k$ denote the probability that it has size $k$. In this paper, we prove strong unimodality results for the distribution $(p_0,\dots,p_n)$.
Let $f\colon \{0,1\}^n\to \{0,1\}$ be a monotone Boolean functions, let $\psi_k(f)$ denote the Shapley value of the $k$th variable and $b_k(f)$ denote the Banzhaf value (influence) of the $k$th variable. We prove that if we have $\psi_k(f) \le t$ for all $k$, then the threshold interval of $f$ has length $\displaystyle O \left(\frac {1}{\log (1/t)}\right)$. We also prove that if $f$ is balanced and $b_k(f) \le t$ for every $k$, then $\displaystyle \max_{k} \psi_k(f) \le O\left(\frac {\log \log (1/t)}{\log(1/t)}\right) $.
We present both bijective and automated approaches to Abel-type sums, dear to Dominique Foata.
Considerable effort in experimental quantum computing is devoted to noisy intermediate scale quantum computers (NISQ computers). Understanding the effect of noise is important for various aspects of this endeavor including notable claims for achieving quantum supremacy and attempts to demonstrate quantum error correcting codes. In this paper we use Fourier methods combined with statistical analysis to study the effect of noise. In particular, we use Fourier analysis to refine the linear cross-entropy fidelity estimator. We use both analytical methods and simulations to study the effect of readout and gate errors, and we use our analysis to study the samples of Google's 2019 quantum supremacy experiment.
The (low soundness) linearity testing problem for the middle slice of the Boolean cube is as follows. Let $\varepsilon>0$ and $f$ be a function on the middle slice on the Boolean cube, such that when choosing a uniformly random quadruple $(x,y,z ,x\oplus y\oplus z)$ of vectors of $2n$ bits with exactly $n$ ones, the probability that $f(x\oplus y \oplus z) = f(x) \oplus f(y) \oplus f(z)$ is at least $1/2+\varepsilon$. The linearity testing problem, posed by David, Dinur, Goldenberg, Kindler and Shinkar, asks whether there must be an actual linear function that agrees with $f$ on $1/2+\varepsilon'$ fraction of the inputs, where $\varepsilon' = \varepsilon'(\varepsilon)>0$. We solve this problem, showing that $f$ must indeed be correlated with a linear function. To do so, we prove a dense model theorem for the middle slice of the Boolean hypercube for Gowers uniformity norms. Specifically, we show that for every $k\in\mathbb{N}$, the normalized indicator function of the middle slice of the Boolean hypercube $\{0,1\}^{2n}$ is close in Gowers norm to the normalized indicator function of the union of all slices with weight $t = n\pmod{2^{k-1}}$. Using our techniques we also give a more general `low degree test' and a biased rank theorem for the slice.
In October 2019, Nature published a paper [6] describing an experimental work that was performed at Google. The paper claims to demonstrate quantum (computational) supremacy on a 53-qubit quantum computer. Since then we have been involved in a long-term project to study various statistical aspects of the Google experiment. In [30] we studied Google's statistical framework that we found to be very sound and offered some technical improvements. This document describes three main concerns (based on statistical analysis) about the Google 2019 experiment. The first concern is that the data do not agree with Google's noise model (or any other specific model). The second concern is that a crucial simple formula for a priori estimation of the fidelity seems to involve an unexpected independence assumption, and yet it gives very accurate predictions. The third concern is about statistical properties of the calibration process.
In October 2019, Nature published a paper describing an experiment that took place at Google. The paper claims to demonstrate quantum (computational) supremacy on a 53-qubit quantum computer. Since September 2019 we have been involved in a long-term project to study various statistical aspects of the Google experiment. We have been trying to gather the relevant data and information in order to reconstruct and verify those parts of the Google experiment that are based on classical computations (except when the required computation is too heavy), and to perform a statistical analysis on the data. This document describes the available data and information for the Google experiment, some main questions in the evaluation of the experiment, and some of our results and plans.
More than ten years ago the author described a parameter $K(\rho )$ for the complexity of $n$-qubit quantum state $\rho$ and raised the conjecture (referred to as "Conjecture C") that when this parameter is superpolynomial in $n$, the state $\rho$ is not experimentally feasible (and will not be experimentally achieved without quantum fault-tolerance). Shortly afterward [6] (arXiv:1204.3404), Steve Flammia and Aram Harrow claimed that the simple easy-to-construct $W$ states are counterexamples to "Conjecture C." We point out that Flammia and Harrow's argument regarding $W$-states is incomplete. Moreover, the emergent picture from experimental progress of the past decade on noisy intermediate scale quantum (NISQ) computers suggests that $W$-states, as simple as they appear, cannot be achieved experimentally by NISQ computers, and can not be constructed without quantum fault-tolerance.
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 is that 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 problems in graph theory: relating the size of the largest independent set in a graph and in a random induced subgraph of it, and bounding the size of a set of vertices intersecting every maximum-size independent set in a graph.
This article focuses on the connection between the possibility of quantum computers, the predictability of complex quantum systems in nature, and the issue of free will.
We study several extensions of the notion of perfect graphs to $k$-uniform hypergraphs.
One of the most important and useful examples in discrete geometry is a finite sequence of points on the moment curve $\gamma(t)=(t,t^2,t^3,\dots ,t^d)$ or, more generally, on a {\it strictly monotone curve} in $\mathbb R^d$. These sequences as well as the ambient curve itself can be described in terms of {\it universality properties} and we will study the question: "What is a universal sequence of oriented and unoriented lines in $d$-space'' We give partial answers to this question, and to the analogous one for $k$-flats. Given a large integer $n$, it turns out that, like the case of points the number of universal configurations is bounded by a function of $d$, but unlike the case for points, there are a large number of distinct universal finite sequences of lines. We show that their number is at least $2^{d-1}-2$ and at most $(d-1)!$. However, like for points, in all dimensions except $d=4$, there is essentially a unique {\em continuous} example of a universal family of lines. The case $d=4$ is left as an open question.
In this paper, we present a variety of problems in the interface between combinatorics and geometry around the theorems of Helly, Radon, Carath\'eodory, and Tverberg. Through these problems we describe the fascinating area of Helly-type theorems, and explain some of its main themes and goals.
We extend the famous Erd\H{o}s-Szekeres theorem to $k$-flats in ${\mathbb{R}^d}$
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.
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.
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.