## Periodic Reporting for period 2 - SensStabComp (Sensitivity, Stability, and Computation)

Reporting period: 2020-12-01 to 2022-05-31

We studied noise stability and sensitivity of mathematical systems and models with special emphasis on noisy intermediate quantum (NISQ) computers.

The PI continued to develop his theory regarding the infeasibility of quantum computers.

A major challenge to the PI's theory regarding quantum computers came with recent experimental claims: First, the 2019 announcement

by a team from Google regarding achieving "quantum supremacy" via random quantum circuits.

This was followed by a 2020 announcement of "quantum supremacy" using a photonic system by a team from USTC.

This has led us to an extensive study of mathematical and statistical aspects of NISQ systems in general and of those two systems in particular.

Regarding the Google announcement, the PI developed with Rinott and Shoham statistical tools that will enable careful examinations of the mentioned claims.

The PI's 2014 work with Kindler casts serious doubts on the claims of the group from USTC.

We extended our study of noise stability and noise sensitivity in several directions: a) connections with statistics b) connections with economic and game theory models; c) connections with learning - our theory predicts that noise-stable systems and noisy quantum computers in particular are learnable and we intend to

empirically test this using deep-learning methods. We also extended substantially Fourier methods for the study of noise stability and noise sensitivity. This had broad implications for solving problems in combinatorics and in the interface between probability theory, combinatorics, and the theory of computing. We also explored connections to various models of statistical physics. Of particular importance was the development of novel "hypercontractive" inequalities.

The question if quantum computers are possible is a vastly important scientific and technological question with huge implications on society.

It is also important to put particular empirical claims under scrutiny, and both insights and tools developed for this study may have wider implications.

Understanding the role of views out-of-consensus in science, and the correct way to explore and promote them, is also of value.

Our overall objectives are first, to develop spectral and other methods in the study of noise stability and noise sensitivity and to explore applications and connections outside and within mathematics.

Second, to develop and to test the PI's theory as to why quantum computers are not possible, and to further explore connections with foundational questions in physics.

The PI continued to develop his theory regarding the infeasibility of quantum computers.

A major challenge to the PI's theory regarding quantum computers came with recent experimental claims: First, the 2019 announcement

by a team from Google regarding achieving "quantum supremacy" via random quantum circuits.

This was followed by a 2020 announcement of "quantum supremacy" using a photonic system by a team from USTC.

This has led us to an extensive study of mathematical and statistical aspects of NISQ systems in general and of those two systems in particular.

Regarding the Google announcement, the PI developed with Rinott and Shoham statistical tools that will enable careful examinations of the mentioned claims.

The PI's 2014 work with Kindler casts serious doubts on the claims of the group from USTC.

We extended our study of noise stability and noise sensitivity in several directions: a) connections with statistics b) connections with economic and game theory models; c) connections with learning - our theory predicts that noise-stable systems and noisy quantum computers in particular are learnable and we intend to

empirically test this using deep-learning methods. We also extended substantially Fourier methods for the study of noise stability and noise sensitivity. This had broad implications for solving problems in combinatorics and in the interface between probability theory, combinatorics, and the theory of computing. We also explored connections to various models of statistical physics. Of particular importance was the development of novel "hypercontractive" inequalities.

The question if quantum computers are possible is a vastly important scientific and technological question with huge implications on society.

It is also important to put particular empirical claims under scrutiny, and both insights and tools developed for this study may have wider implications.

Understanding the role of views out-of-consensus in science, and the correct way to explore and promote them, is also of value.

Our overall objectives are first, to develop spectral and other methods in the study of noise stability and noise sensitivity and to explore applications and connections outside and within mathematics.

Second, to develop and to test the PI's theory as to why quantum computers are not possible, and to further explore connections with foundational questions in physics.

I will mention 8 papers (six of them appeared or have been accepted for publication and two were submitted) that represent the main directions and achievements of the research,

I. Quantum computers

1) The Argument against Quantum Computers, by Gil Kalai

2) The Argument against Quantum Computers, the Quantum Laws of Nature, and Google's Supremacy Claims, by Gil Kalai

The first paper gives 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. The 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.

The second paper studies quantum computation as a meeting point of the laws of computation and the laws of quantum mechanics and proposes general laws that manifest the failure of quantum computers. In October 2019, "Nature" published a paper, by a team from Google, claiming to demonstrate quantum (computational) supremacy on a 53-qubit quantum computer. In second paper also explains and discusses Google's supremacy claims.

3) Statistical Aspects of the Quantum Supremacy Demonstration by Yosef Rinott, Tomer Shoham, and Gil Kalai

The 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. We study statistical aspects involved in demonstrating quantum supremacy, different approaches to testing the distributions generated by the quantum computer, and various noise models. The paper gives a preliminary study of the Google data.

II. Models of statistical physics

4) Periodic Boundary Conditions for Periodic Jacobi Matrices on Trees by Nir Avni, Jonathan Breuer, Gil Kalai, Barry Simon

This work considers 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 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.

5) Spectral Fluctuations for the Multi-dimensional Anderson Model, by Yoel Grinshpon, and Moshe White (graduate student in the project)

This paper examines fluctuations of polynomial linear statistics for the Anderson model on Zd for any potential with finite moments.

III. Hypercontractivity and applications

6) Global Hypercontractivity and its Applications, by Peter Keevash, Noam Lifshitz, Eoin Long, Dor Minzer

In this paper and several subsequent papers our graduate student Noam Lifshitz developed a major new theory of hypercontractive inequalities and had substantial progress on a large variety of problems in probabilistic and extremal combinatorics.

IV. Foundation of physics and philosophical aspects related to the project.

7) Is the EHT Black Hole Experiment a new Experiment in the Guise of an Old Experiment? by Galina Weinstein.

Our researcher in the project, Galina Weinstein, explores foundational questions in physics and in particular, relations with black holes.

8) Quantum Computers, Predictability, and Free Will, by Gil Kalai.

In this paper the PI argues that a world devoid of quantum computers supports the possibility of free will.

I. Quantum computers

1) The Argument against Quantum Computers, by Gil Kalai

2) The Argument against Quantum Computers, the Quantum Laws of Nature, and Google's Supremacy Claims, by Gil Kalai

The first paper gives 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. The 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.

The second paper studies quantum computation as a meeting point of the laws of computation and the laws of quantum mechanics and proposes general laws that manifest the failure of quantum computers. In October 2019, "Nature" published a paper, by a team from Google, claiming to demonstrate quantum (computational) supremacy on a 53-qubit quantum computer. In second paper also explains and discusses Google's supremacy claims.

3) Statistical Aspects of the Quantum Supremacy Demonstration by Yosef Rinott, Tomer Shoham, and Gil Kalai

The 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. We study statistical aspects involved in demonstrating quantum supremacy, different approaches to testing the distributions generated by the quantum computer, and various noise models. The paper gives a preliminary study of the Google data.

II. Models of statistical physics

4) Periodic Boundary Conditions for Periodic Jacobi Matrices on Trees by Nir Avni, Jonathan Breuer, Gil Kalai, Barry Simon

This work considers 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 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.

5) Spectral Fluctuations for the Multi-dimensional Anderson Model, by Yoel Grinshpon, and Moshe White (graduate student in the project)

This paper examines fluctuations of polynomial linear statistics for the Anderson model on Zd for any potential with finite moments.

III. Hypercontractivity and applications

6) Global Hypercontractivity and its Applications, by Peter Keevash, Noam Lifshitz, Eoin Long, Dor Minzer

In this paper and several subsequent papers our graduate student Noam Lifshitz developed a major new theory of hypercontractive inequalities and had substantial progress on a large variety of problems in probabilistic and extremal combinatorics.

IV. Foundation of physics and philosophical aspects related to the project.

7) Is the EHT Black Hole Experiment a new Experiment in the Guise of an Old Experiment? by Galina Weinstein.

Our researcher in the project, Galina Weinstein, explores foundational questions in physics and in particular, relations with black holes.

8) Quantum Computers, Predictability, and Free Will, by Gil Kalai.

In this paper the PI argues that a world devoid of quantum computers supports the possibility of free will.

A) Quantum computers

We expect to develop statistical tools for data stemming from noisy intermediate scale quantum computers, and to carefully examine statistical claims for "quantum supremacy".

We expect to further develop my theory explaining the impossibility of quantum computers and to derive physics consequences and connections with foundational questions in physics and the theory of computing.

B) Analysis, combinatorics, and probability

We expect to find new powerful hypercontractive inequalities and various applications in extremal combinatorics, additive combinatorics, and models of statistical physics.

C) Noise stability and sensitivity

We expect to extend the study of noise stability and sensitivity to representations of non commutative groups, to statistics, and to explore further connections.

We expect to develop statistical tools for data stemming from noisy intermediate scale quantum computers, and to carefully examine statistical claims for "quantum supremacy".

We expect to further develop my theory explaining the impossibility of quantum computers and to derive physics consequences and connections with foundational questions in physics and the theory of computing.

B) Analysis, combinatorics, and probability

We expect to find new powerful hypercontractive inequalities and various applications in extremal combinatorics, additive combinatorics, and models of statistical physics.

C) Noise stability and sensitivity

We expect to extend the study of noise stability and sensitivity to representations of non commutative groups, to statistics, and to explore further connections.