## Periodic Reporting for period 1 - SensStabComp (Sensitivity, Stability, and Computation)

Reporting period: 2019-06-01 to 2020-11-30

"What is the problem/issue being addressed?

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

The PI continued to develop his theory for why quantum computers are infeasible, and what are the implications to physics, to the theory of computation and to the connection between them.

A major challenge to the PI's theory came with recent experimental claims: the 2019 announcement of ""quantum supremacy"" via random quantum circuits

by a team of researchers from Google (a paper published in ""Nature"") which followed by a 2020 announcement of ""quantum supremacy"" using a photonic system by a team from USTC in Hefei, China (a paper published in ""Science"").

This has led us to an extensive study of mathematical and statistical aspects of NISQ systems in general and those two systems in particular. Regarding the Google claims the PI developed with Rinott and Shoham statistical tools that will enable careful examinations of those claims. The PI 2014 work with Guy Kindler which is central in his overall argument turned out to shed serious doubts on the claims of the groups from China.

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 bring it to empirical test using deep learning methods.

We extended substantially Fourier methods for the study of noise stability and noise sensitivity. This had board 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.

Why is it important for society?

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

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

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

What are the overall objectives?

Regarding quantum computing the objective is to develop and to test my theory for why quantum computers are not possible, and to further explore connections with foundational questions in physics. Regarding noise stability and sensitivity the plan is to explore

further applications and connections outside and within mathematics."

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

The PI continued to develop his theory for why quantum computers are infeasible, and what are the implications to physics, to the theory of computation and to the connection between them.

A major challenge to the PI's theory came with recent experimental claims: the 2019 announcement of ""quantum supremacy"" via random quantum circuits

by a team of researchers from Google (a paper published in ""Nature"") which followed by a 2020 announcement of ""quantum supremacy"" using a photonic system by a team from USTC in Hefei, China (a paper published in ""Science"").

This has led us to an extensive study of mathematical and statistical aspects of NISQ systems in general and those two systems in particular. Regarding the Google claims the PI developed with Rinott and Shoham statistical tools that will enable careful examinations of those claims. The PI 2014 work with Guy Kindler which is central in his overall argument turned out to shed serious doubts on the claims of the groups from China.

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 bring it to empirical test using deep learning methods.

We extended substantially Fourier methods for the study of noise stability and noise sensitivity. This had board 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.

Why is it important for society?

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

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

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

What are the overall objectives?

Regarding quantum computing the objective is to develop and to test my theory for why quantum computers are not possible, and to further explore connections with foundational questions in physics. Regarding noise stability and sensitivity the plan is to explore

further applications and connections outside and within mathematics."

"I. Quantum computers

The Argument against Quantum Computers, by Gil Kalai

This work 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. 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.

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

This work studies quantum computation as a meeting point of the laws of computation and the laws of quantum mechanics. We describe my argument against the feasibility of quantum computers, 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.

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

Summary: 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. The objective of this paper is to study some of the statistical aspects involved in demonstrating quantum supremacy. We study different approaches to testing the distributions generated by the quantum computer, propose different noise models, and discuss their implications. We give a preliminary study of the Google data, focusing mostly on circuits of 12 and 14 qubits.

II. Models of statistical physics

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 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.

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

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 related to the project.

Our researcher in the project, Galina Weinstein explores, as part of our project, foundational questions in physics."

The Argument against Quantum Computers, by Gil Kalai

This work 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. 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.

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

This work studies quantum computation as a meeting point of the laws of computation and the laws of quantum mechanics. We describe my argument against the feasibility of quantum computers, 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.

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

Summary: 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. The objective of this paper is to study some of the statistical aspects involved in demonstrating quantum supremacy. We study different approaches to testing the distributions generated by the quantum computer, propose different noise models, and discuss their implications. We give a preliminary study of the Google data, focusing mostly on circuits of 12 and 14 qubits.

II. Models of statistical physics

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 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.

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

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 related to the project.

Our researcher in the project, Galina Weinstein explores, as part of our project, foundational questions in physics."

"A) Quantum computers

We expect to develop statistical tools for data coming from noisy intermediate scale quantum computers, and

to carefully examine statistical claims for ""quantum supremacy"" (We prefer the term HQCA - ""huge computational quantum advantage"".)

We expect to further develop my theory explaining the failure, in principle, of quantum computers

and to derive physics consequences and connections with foundational questions in physics.

B) Analysis, combinatorics, and probability

We expect to find new powerful hypercontractive inequalities

We expect to find various applications in extremal combinatorics, additive combinatorics, and percolation and other models from statistical physics

C) Noise stability

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 coming from noisy intermediate scale quantum computers, and

to carefully examine statistical claims for ""quantum supremacy"" (We prefer the term HQCA - ""huge computational quantum advantage"".)

We expect to further develop my theory explaining the failure, in principle, of quantum computers

and to derive physics consequences and connections with foundational questions in physics.

B) Analysis, combinatorics, and probability

We expect to find new powerful hypercontractive inequalities

We expect to find various applications in extremal combinatorics, additive combinatorics, and percolation and other models from statistical physics

C) Noise stability

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