Talks

Introduction to Ising model; review spin-1/2 and matrix representation of states  and operators; programming basics

https://drive.google.com/drive/folders/1L6a1YCdB9EejAUmZ52oLDAeB6oosjwhU?usp=sharing

https://www.dropbox.com/sh/sprq2u40e3leoso/AADhyov350QS9TEDdgtErekCa?dl=0

https://github.com/gsellaroli/pi-summer-school-2020

We investigate putting 2+1 free and holographic theories on a product of time with a curved compact 2-d space. We then vary the geometry of the space, keeping the area fixed, at zero/finite temperature, and measure the Casimir/free energy respectively. I will begin by discussing the free theory for a Dirac fermion or scalar field on deformations of the round 2-sphere. I will discuss how the Dirac theory may arise in physical systems such as monolayer graphene. For small deformations we solve analytically using perturbation theory. For large deformations we use novel numerical methods to compute these energies for specific deformations, including ones that drive the space to become singular. I will give evidence that the round sphere globally maximises the Casimir/free energy, which may imply geometric instabilities for spheres of such mono layer materials, although probably not graphene. We then discuss the analog for a holographic theory by studying its gravity dual. Here we use gravity techniques to analytically prove at zero temperature the round sphere maximises the Casimir energy. We discuss attempts to show the same for free energy at finite temperature. And finally I will report on on-going numerical gravity calculations of the Casimir energy for specific  deformations of the round sphere which yield an unexpected result.

Introduction to path integrals and the semi-classical limit

https://drive.google.com/drive/folders/1L6a1YCdB9EejAUmZ52oLDAeB6oosjwhU?usp=sharing

https://www.dropbox.com/sh/sprq2u40e3leoso/AADhyov350QS9TEDdgtErekCa?dl=0

https://github.com/gsellaroli/pi-summer-school-2020

Overview/definition of symmetry, elements of group theory
https://drive.google.com/drive/folders/1L6a1YCdB9EejAUmZ52oLDAeB6oosjwhU?usp=sharing

https://www.dropbox.com/sh/sprq2u40e3leoso/AADhyov350QS9TEDdgtErekCa?dl=0

https://github.com/gsellaroli/pi-summer-school-2020

Foundations of Quantum Statistical Mechanics - Entanglement

https://drive.google.com/drive/folders/1L6a1YCdB9EejAUmZ52oLDAeB6oosjwhU?usp=sharing

https://www.dropbox.com/sh/sprq2u40e3leoso/AADhyov350QS9TEDdgtErekCa?dl=0

https://github.com/gsellaroli/pi-summer-school-2020

Harlow and Hayden [arXiv:1301.4504] argued that distilling information out of Hawking radiation is computationally hard despite the fact that the quantum state of the black hole and its radiation is relatively un-complex. I will trace this computational difficulty to a geometric obstruction in the Einstein-Rosen bridge connecting the black hole and its radiation. Inspired by tensor network models, I will present a conjecture that relates the computational hardness of distilling information to geometric features of the wormhole - specifically to the exponential of the difference in generalized entropies between the two non-minimal quantum extremal surfaces that constitute the obstruction. Due to its shape, this obstruction was dubbed "Python's Lunch", in analogy to the reptile's postprandial bulge.

Using a definition of bulk diff-invariant observables, we go into the bulk of 2d Jackiw-Teitelboim gravity. By mapping the computation to a Schwarzian path integral, we study exact bulk correlation functions and discuss their physical implications. We describe how the black hole thermal atmosphere gets modified by quantum gravitational corrections. Finally, we will discuss how higher topological effects further modify the spectral density and detector response in the Unruh heat bath. 

Given the large push by academia and industry (e.g., IBM and Google), quantum computers with hundred(s) of qubits are at the brink of existence with the promise of outperforming any classical computer. Demonstration of computational advantages of noisy near-term quantum computers over classical computers is an imperative near-term goal. The foremost candidate task for showing this is Random Circuit Sampling (RCS), which is the task of sampling from the output distribution of a random circuit. This is exactly the task that recently Google experimentally performed on 53-qubits.

Stockmeyer's theorem implies that efficient sampling allows for estimation of probability amplitudes. Therefore, hardness of probability estimation implies hardness of sampling. We prove that estimating probabilities to within small errors is #P-hard on average (i.e. for random circuits), and put the results in the context of previous works.

Some ingredients that are developed to make this proof possible are construction of the Cayley path as a rational function valued unitary path that interpolate between two arbitrary unitaries, an extension of Berlekamp-Welch algorithm that efficiently and exactly interpolates rational functions, and construction of probability distributions over unitaries that are arbitrarily close to the Haar measure.