Welcome to the home page of the online seminars on **Tensor Networks in High Energy Physics**!

This is a joint initiative of the *Gravity, Quantum Fields and Information* group at the **Albert Einstein Institute** in Potsdam (Michal Heller, Sukhi Singh), **DESY** in Zeuthen (Karl Jansen), the **Max-Planck Institute for Quantum Optics** in Garching (Mari-Carmen Banuls), and the Tensor Network initiative (Stefan Kuhn, Bianca Dittrich, Adam Lewis) at the **Perimeter Institute for Theoretical Physics** in Canada.

Our aim is to provide an online platform for researchers working on this topic all around the globe to present their work from anywhere they like! (Office, home, restaurant, airport, or even the beach.) We hope this seminar series can make a small contribution towards cutting down costs, unnecessary travel, and carbon emissions.

**How does it work?** A link to the virtual seminar room for each talk is sent out to participating groups via our mailing list. Anyone with the link can tune-in remotely to the live stream, ask questions, and participate in discussion.

In addition, the talks are typically recorded and posted on our YouTube channel: https://www.youtube.com/c/GravityQuantumFieldsandInformationAEI, in case you miss the live stream, and/or want to revisit the talk.

If you are interested in being added to the mailing list to receive information (including the link to the virtual seminar room) please contact Sukhi Singh

## Upcoming Seminars

#### 1. Jonathan Sorce (jointly held with QGI seminars, https://gqfi.aei.mpg.de/node/24)

**When:** April 15, 2020 @ 16.00 (GMT +1 hour, Berlin time)

**Title: **Status update on tensor networks and quantum gravity

**Abstract: **In recent years, the "tensor networks" used to describe condensed matter systems have been discovered to share many qualitative features in common with holographic theories of quantum gravity. Studying these features in tensor networks has helped us build intuition for how the corresponding features should work in quantum gravity; most importantly, tensor networks helped us develop a tractable framework for understanding holographic quantum error correction. A little over a year ago (1812.01171), my collaborators and I proved that every geometric state in a holographic theory of quantum gravity can be represented as a tensor network, explaining the success of the "toy model." I will explain this proof, comment on its implications, and discuss recent work from other authors on the tensor network/holography correspondence. The aim of the talk will be to communicate the lessons from tensor networks that should inform our general approach to quantum gravity research; it should be intelligible to an audience with little to no tensor-network experience.

#### 2. Philippe Corboz

**When:** April 24, 2020 @ 15.30 (GMT +1 hour, Berlin time)

**Title: **TBA

**Abstract: **TBA

#### 3. Javier Molina-Vilaplana

**When:** May 08, 2020 @ 15.30 (GMT +1 hour, Berlin time)

**Title:** *Non-Gaussian Entanglement Renormalization for Quantum Fields*

**Abstract: **The multiscale entanglement renormalization ansatz (MERA), which was originally proposed as a variational method to obtain the ground state of spin chains systems, consists of a real space renormalization group technique that, iteratively, removes the quantum correlations between small adjacent regions of space at each length scale. A continuous version of MERA (cMERA) was proposed for free field theories. Motivated, among others, by the conjecture that cMERA is a realization of the AdS/CFT correspondence, a rigorous and (non)perturbative formalism for interacting theories turns out to be essential to advance in this program.

In this seminar, a non-Gaussian cMERA tensor network for interacting quantum field theories (icMERA) is presented. This consists of a continuous tensor network circuit in which the generator of the entanglement renormalization of the wavefunction is nonperturbatively extended with nonquadratic variational terms. The icMERA circuit nonperturbatively implements a set of scale dependent nonlinear transformations on the fields of the theory, which suppose a generalization of the scale dependent linear transformations induced by the Gaussian cMERA circuit.

Here we present these transformations for the case of self-interacting scalar and fermionic field theories. We show how the icMERA tensor network can be fully optimized for the self interacting scalar theory in (1+1) dimensions. This allows us to evaluate, nonperturbatively, the connected parts of the two- and four-point correlation functions.

Our results show that icMERA wavefunctionals encode proper non-Gaussian correlations of the theory, thus providing a new variational tool to study phenomena related with strongly interacting field theories.

Based on: https://arxiv.org/abs/2003.08438

#### 4. Luca Tagliocozzo

**When:** May 22, 2020 @ 15.30 (GMT +1 hour, Berlin time)

**Title: **TBA

**Abstract: **TBA

## Past Seminars

#### 1. Ignacio Cirac (Opening seminar of the series)

**When:** November 8, 2019 @ 15.00 (GMT +1 hour, Berlin time)

**Title:** *Tensor Networks and Lattice Gauge Theories*

**Abstract:** Certain Quantum Many-body states can be efficiently described in terms of tensor networks. Those include Matrix Product States (MPS), Projected Entangled-Pair Etates (PEPS), or the Multi-scale Entanglement Renormalization Ansatz. They play an important role in quantum computing, error correction, or the description of topological order in condensed matter physics, and are widely used in computational physics. In the last years, it has also been realized their suitability to describe Lattice Gauge Theories, at least in the context of MPS in low dimensions. In this talk I will review some of the basic ideas about tensor networks and their applications to lattice gauge theories, and explain current efforts to extend them to higher dimensions using PEPS.

**YouTube link: **https://www.youtube.com/watch?v=hdb82b1kazw&feature=youtu.be

#### 2. Bartlomiej Czech

**When:** December 6, 2019 @ 15.30 (GMT +1 hour, Berlin time)

**Title:** *What does the Chern-Simons formulation of AdS3 gravity tell us about complexity?*

**Abstract:** I will explain how to realize the wavefunction of a CFT2 ground state as a network of Wilson lines in the Chern-Simons formulation of AdS3 gravity. The position and shape of the network encode the scale at which the wavefunction is defined. The structure of the network is that of a Matrix Product State (MPS) whose constituent tensors effect the Operator Product Expansion. A general argument suggests identifying the "density of complexity" of this MPS network with the extrinsic curvature of the bulk cutoff surface, which by the Gauss-Bonnet theorem agrees with the Complexity = Volume proposal. The viewpoint I offer departs from the circuit paradigm of complexity and dispenses with reference states. Instead, recognizing that field theory states are functionals which send observables to their expectation values, I propose to think of state complexity as the algorithmic complexity of constructing such functionals.

**YouTube link: **https://www.youtube.com/watch?v=2p-mo-LdZxw

#### 3. Frank Verstraete

**When:** January 24, 2020 @ 15.30 (GMT +1 hour, Berlin time)

**Title:** *Quantum symmetries in tensor networks*

**Abstract:** Tensor networks and more specifically matrix product operators provide a natural framework for describing nonlocal symmetries in lattice spin systems. It will be argued that those matrix product operators form representations of tensor fusion categories, and that they lead to simple lattice representations of topological and conformal field theories. We will construct algebraic equations defining the topological / conformal sectors, and construct explicitly all excitations using the operator-state correspondence.

**YouTube link: **https://www.youtube.com/watch?v=IHe5YYsEK7k.

#### 4. Simone Montangero

**When:** February 14, 2020 @ 15.30 (GMT +1 hour, Berlin time)

**Title:** *Tensor network methods applied to high energy physics problems*

**Abstract:** We briefly introduce tensor network methods, a classical numerical approach that promises to become a powerful tool to support future quantum simulations and computations, providing guidance, benchmarking and verification of the quantum computation and simulation results. We review some of the latest achievements we obtained: the gauge-invariant formulation of tensor networks and their application to abelian and non-abelian, one- and two-dimensional lattice gauge theories in regimes where Monte Carlo methods efficiency is hindered by the sign problem. Finally, we present the application of tensor network machine learning techniques to the event classification of LHCb simulated data.

**YouTube link: **https://youtu.be/vrZHkyDvYhI

#### 5. Adam G. M. Lewis

**When:** February 28, 2020 @ 15.30 (GMT +1 hour, Berlin time)

**Title:** *Fermionic Hartle-Hawking Vacua From a Staggered Lattice Scheme*

**Abstract:** I will discuss work in collaboration with Guifré Vidal towards simulation of quantum fields in curved spacetimes. We eventually mean to simulate strongly interacting fields, out of equilibrium, coupled to spacetime curvature in various ways. This study concerns the more modest goal of computing renormalized, quadratic expectation values of free Dirac fields installed upon fixed, two dimensional Lorentzian spacetimes. First, we use a staggered-fermion discretization to generate a sequence of lattice theories yielding the desired QFT in the continuum limit. Numerically-computed lattice correlators are then used to approximate, through extrapolation, those in the continuum. Finally, we use so-called point-splitting regularization and Hadamard renormalization to remove divergences, and thus obtain finite, renormalized expectation values of quadratic operators in the continuum. As illustrative applications, we show how to recover the Unruh effect in flat spacetime, how to compute renormalized expectation values in the Hawking-Hartle vacuum of a 2-dimensional "Schwarzschild" black hole, and how to do the same in the Bunch-Davies vacuum of dS2.

**YouTube Link:** https://www.youtube.com/watch?v=Vu538q3G0wY&list=PLaib4I4mFNmWKntxAZcB-EQJ_B-PkWEEL&index=5

#### 6. Germán Sierra

**When:** March 27, 2020 @ 15.30 (GMT +1 hour, Berlin time)

**Title:** *Tensor Networks with infinite bond dimension*

**Abstract:** In the last few years Tensor Networks (TN) have become a standard technique to study the properties of many body systems. Its origins can be traced back to the AKLT model in 1987, and the Density Matrix Renormalization Group in 1992. They were the first examples of Matrix Product States that were actively investigated by the cond-mat community in the 90's. By the turn of the century, the quantum information community understood the success of the MPS in terms of quantum entanglement. This led to new TNs like PEPS and MERA. In all these TNs the dimension of the auxiliary space, that mediates the entanglement between the physical degrees of freedom, is finite. This feature limits the application of MPS to critical systems in 1D whose low energy states violate the area law of the entanglement entropy. In this talk I will introduce TNs whose bond dimension is infinite, that allows us to overcome this problem and describe critical spin chains and Fractional Quantum Hall systems.

**YouTube Link: **https://www.youtube.com/watch?v=iV8r-wBbU60