Talks

Lattices have allowed us to realize several advanced notions of computation over encrypted data, notably fully homomorphic encryption, attribute-based encryption, as well as fully homomorphic signatures. In this tutorial, we will introduce and derive a simple equation at the core of all of these constructions.

Quantizing 4D geometries leads to discrete area spectra. Such discrete area spectra are also suggested by the holographic principle and entropy counting for black holes.

 Starting with this input of a discrete area spectrum I will construct a path integral for quantum gravity and discuss (quantum) corrections to the GR dynamics that are forced by the discrete area spectra. The resulting model can serve as effective model for the spin foam approach and clarifies  the dynamical principles and underlying key assumptions for spin foams. The considerations also point towards key phenomenological differences to e.g. the ADM quantization scheme, and thus to a way to falsify the key assumption of discrete area spectra. 

 

Observations have shown that nearly all galaxies harbor massive or supermassive black holes at their centers. Gravitational wave (GW) observations of these black holes will shed light on their growth and evolution, and the merger histories of galaxies. Massive and supermassive black holes are also ideal  laboratories for studying strong-field gravity. Pulsar timing arrays (PTAs) use observations of millisecond pulsars to detect low-frequency GWs with frequencies ~1-100 nHz, and can detect GWs emitted by supermassive black hole binaries, which form when two galaxies merge. I will discuss source modeling and detection techniques for PTAs, as well as present limits on nanohertz GWs from the North American Nanohertz Observatory for Gravitational Waves (NANOGrav) collaboration.

 

 

Geometry of a pair of complex Lagrangian submanifolds of a complex symplectic manifold appears in many areas of mathematics and physics,  including exponential integrals in finite and infinite dimensions,  wall-crossing formulas in 2d and 4d, representation theory, resurgence of WKB series and so  on.

In 2014 we started a joint project with Maxim Kontsevich which we  named "Holomorphic Floer Theory" (HFT for short) in order to study all these (and other) phenomena as a part of a bigger picture.

Aim of my talk is to discuss  aspects of HFT related to deformation quantization of complex symplectic manifolds, including the conjectural Riemann-Hilbert correspondence.  Although some parts of this story have been already reported elsewhere, the topic has  many ramifications which have not been discussed earlier.

 

 

It is believed that active quantum error correction will be an essential ingredient to build a scalable quantum computer. The currently favored scheme is the surface code due to its high decoding threshold and efficient decoding algorithm. However, it suffers from large overheads which are even more severe when parity check measurements are subject to errors and have to be repeated. Furthermore, the number of encoded qubits in the surface code does not grow with system size, leading to a sub-optimal use of the physical qubits.

Finally, the decoding algorithm, while efficient, has non-trivial complexity and it is not clear whether it can be implemented in hardware that can keep up with the classical processing.

We present a class of low-density-parity check (LDPC) quantum codes which fix all three of the concerns mentioned above. They were first proposed in [1] and called 4D hyperbolic codes, as their definition is based on four-dimensional, curved geometries. They have the remarkable property that the number of encoded qubits grows linearly with system size, while their distance grows polynomially with system size, i.e. d~n a with 0.1 < a < 0.3. This is remarkable since it was previously conjectured that such codes could not exist [1]. Their structure allows for decoders which can deal with erroneous syndrome measurements, a property called single-shot error correction [2] as well as local decoding schemes [3].

Although [1] analyzed the encoding rate and distance of this code family abstractly, it is a non-trivial task to actually construct them. There is no known efficient deterministic procedure for obtaining small examples. Only single examples of reasonable size had been obtained previously [4]. These previous examples were part of different code families, so that it was not possible to determine a threshold. We succeeded to construct several small examples by utilizing a combination of randomized search and algebraic tools. We analyze the performance of these codes under several different local decoding procedures via Monte Carlo simulations. The decoders all share the property that they can be executed in parallel in O(1) time. Under the phenomenological noise model and including syndrome errors we obtain a threshold of ~5% which to our knowledge is the highest threshold among all local decoding schemes.

[1] A. Lubotzky, A. Guth, Journal Of Mathematical Physics 55, 082202 (2014).
[2] H. Bombin, Physical Review X 5 (3), 031043 (2015).
[3] M. Hastings, QIC 14, 1187 (2014).
[4] V. Londe, A. Leverrier, arXiv:1712.08578 (2017).

 

 

Idempotent (aka Karoubi) completion is used throughout mathematics: for instance, it is a common step when building a Fukaya category. I will explain the n-category generalization of idempotent completion. We call it "condensation completion" because it answers the question of classifying the gapped phases of matter that can be reached from a given one by condensing some of the chemicals in the matter system. From the TFT side, condensation preserves full dualizability. In fact, if one starts with the n-category consisting purely of ℂ in degree n, its condensation completion is equivalent both to the n-category of n-dualizable ℂ-linear (n-1)-categories and to an n-category of lattice condensed matter systems with commuting projector Hamiltonians. This establishes an equivalence between large families of TFTs and of gapped topological phases. Based on joint work with D. Gaiotto.