I will mention eight directions and some noteworthy achievements of the research.
I. Quantum computers - the argument against quantum computers
We gave 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.
II. Statistical analysis of data coming from noisy intermediate quantum computers
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.
III. Hypercontractivity and applications
In the paper "Global Hypercontractivity and its Applications," by Keevash, Lifshitz, Long, and Minzer,
and several subsequent papers a major new theory of hypercontractive inequalities and had substantial progress on a large variety of problems in probabilistic and extremal combinatorics.
IV. Social choice theory
Maskin conjectured that for three and more candidates a certain relaxation of Arrow's famous condition,
allows only the Borda's rule. Gabriel Gendler found a counterexample to Maskin's conjecture and proved that the assertion of the conjecture holds true when there are four or more candidates.
The PI and Lifshitz established sharp connections between the Shapley value - a power measure for voting rules - and the "sharp threshold" behavior.
V. Foundation of physics and computation and philosophical aspects related to the project.
Both noise and computation are related to foundational questions in physics ranging from the question of "What Nature computes?" to paradoxes about black holes. Galina Weinstein, a senior researcher in our project, explored foundational and historical questions in physics and, in particular, relations with black holes. In a paper from 2022, the PI argued that a world devoid of quantum computers supports the possibility of free will.
There was substantial research in three additional directions: VI. the study of models of statistical physics, VII. theoretical computer science, and VIII. Gaussian models in discrete geometry.
