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 Bartek Czech, Institue for Advanced Study
 Glen Evenbly, University of Sherbrooke
 Martin Ganahl, Perimeter Institute
 Jutho Haegeman, University of Ghent
 Janet Hung, Fudan University
 Robert Leigh, University of Illinois at UrbanaChampaign
 Ashley Milsted, Perimeter Institute
 Robert Myers, Perimeter Institute
 *Tobias Osborne, University of Hannover
 Xiaoliang Qi, Stanford University
 Volker Scholz, Ghent University
 Miles Stoudenmire, University of Calfornia, Irvine
 Jamie Sully, McGill University
 Brian Swingle, MIT, Harvard University & Brandeis University
 Tadashi Takayanagi, Yukawa Institute for Theoretical Physics
 Frank Verstraete, University of Ghent
 Guifre Vidal, Perimeter Institute
 Steven White, University of California, Irvine
*via teleconference
 Javier Arguello, Perimeter Institute
 Ganapathy Baskaran, Institute of Mathematical Sciences Chennai
 Lakshya Bhardwaj, Perimeter Institute
 Arpan Bhattacharyya, Fudan University
 Dean Carmi, Perimeter Institute
 Shira Chapman, Perimeter Institute
 Jordan Cotler, Stanford University
 Bartek Czech, Institute for Advanced Study
 Clement Delcamp, Perimeter Institute
 Bianca Dittrich, Perimeter Institute
 Glen Evenbly, University of Sherbrooke
 Matthew Fishman, California Institute of Technology
 Adrian Franco Rubio, Perimeter Institute
 Adil Gangat, National Taiwan University
 Martin Ganahl, Perimeter Institute
 Jutho Haegeman, University of Ghent
 Muxin Han, Florida Atlantic University
 Markus Hauru, Perimeter Institute
 Joshuah Heath, Boston College
 Michal Heller, Albert Einstein Institute
 Qi Hu, Perimeter Institute
 Janet Hung, Fudan University
 Nick HunterJones, California Institute of Technology
 Robert Jefferson, Perimeter Institute
 Robert Leigh, University of Illinois at UrbanaChampaign
 Adam Lewis, Perimeter Institute
 Shengqiao Luo, Perimeter Institute
 Hugo Marrochio, Perimeter Institute
 Alex May, University of British Columbia
 Roger Melko, Perimeter Institute & University of Waterloo
 Ashley Milsted, Perimeter Institute
 Sebastian Mizera, Perimeter Institute
 Robert Myers, Perimeter Institute
 Xiaoliang Qi, Stanford University
 Jason Pye, University of Waterloo
 Hammam Qassim, Institute for Quantum Computing
 Djordje Radicevic, Perimeter Institute
 Julian Rincon, Perimeter Institute
 Burak Sahinoglu, California Institute of Technology
 Volker Scholz, Ghent University
 Didina Serban, Perimeter Institute
 Andrei Shieber, Perimeter Institute
 Vasudev Shyam, Perimeter Institute
 Joan Simon, University of Edinburgh
 Kevin Slagle, University of Toronto
 Barbara Soda, Perimeter Institute
 Miles Stoudenmire, University of Calfornia, Irvine
 Jamie Sully, McGill University
 Brian Swingle, MIT, Harvard University & Brandeis University
 Tadashi Takayanagi, Yukawa Institute for Theoretical Physics
 Nick Van den Broeck, Perimeter Institute
 Guillaume VerdonAkzam, Institute for Quantum Computing
 Frank Verstraete, University of Ghent
 Guifre Vidal, Perimeter Institute
 Steven White, University of California, Irvine
 Gabriel Wong, University of Virginia
 Shuo Yang, Perimeter Institute
 Beni Yoshida, Perimeter Institute
 Jose Zapata, Centro de Ciencias Matematicas
 Yijian Zou, Perimeter Institute
Tuesday, April 18, 2017
Time 
Event 
Location 
9:00 – 9:30am 
Registration 
Reception 
9:30 – 9:35am 
Guifre Vidal, Perimeter Institute 
Bob Room 
9:35 – 10:35am 
Steven White, University of California 
Bob Room 
10:35 – 11:00am 
Coffee Break 
Bistro – 1^{st} Floor 
11:0012:00pm 
Ashley Milsted, Perimeter Institute 
Bob Room 
12:00 – 2:00pm 
Lunch 
Bistro – 2^{nd} Floor 
2:00 – 2:40pm 
Miles Stoudenmire, University of California 
Bob Room 
2:40 – 3:20pm 
Martin Ganahl, Perimeter Institute 
Bob Room 
3:20 – 3:50pm 
Coffee Break 
Bistro – 1^{st} Floor 
3:50 – 4:30 pm 
Jutho Haegeman, University of Ghent 
Bob Room 
Wednesday, April 19, 2017
Time 
Event 
Location 
9:30 – 10:30am 
Guifre Vidal, Perimeter Institute 
Bob Room 
10:30 – 11:00am 
Coffee Break 
Bistro – 1^{st} Floor 
11:0012:00pm 
Robert Leigh, University of Illinois at UrbanaChampaign 
Bob Room 
12:00 – 2:00pm 
Lunch 
Bistro – 2^{nd} Floor 
2:00 – 2:40pm 
Brian Swingle, 
Bob Room 
2:40 – 3:20pm 
Volkher Scholz, University of Ghent 

3:20 – 3:50pm 
Coffee Break 
Bistro – 1^{st} Floor 
3:50  4:50pm 
Frank Verstraete, University of Ghent 
Bob Room 
5:00 – 6:00pm 
Poster Session 
Atrium 
6:00pm 
Banquet 
Bistro – 2^{nd} Floor 
Thursday, April 20, 2017
Time 
Event 
Location 
9:30 – 10:30am 
Jamie Sully, McGill University 
Bob Room 
10:30 – 11:00am 
Coffee Break 
Bistro – 1^{st} Floor 
11:0012:00pm 
Tadashi Takayanagi, Yukawa Institute for Theoretical Physics 
Bob Room 
12:00 – 2:00pm 
Lunch 
Bistro – 2^{nd} Floor 
2:00 – 3:00pm 
Robert Myers, Perimeter Institute 
Bob Room 
3:00 – 3:30pm 
Coffee Break 
Bistro – 1^{st} Floor 
3:30 – 4:10pm 
Bartek Czech, Institute for Advanced Study 
Bob Room 
Friday, April 21, 2017
Time 
Event 
Location 
9:00 – 10:00am 
Xiaoliang Qi, Stanford University 
Bob Room 
10:00 – 10:30am 
Coffee Break 
Bistro – 1^{st} Floor 
10:30 – 11:10am 
Tobias Osborne, University of Hannover [via teleconference] 
Bob Room 
11:10 – 11:50am 
Janet Hung, Fudan University 
Bob Room 
11:50 – 12:30pm 
Glen Evenbly, University of Sherbrooke 
Bob Room 
12:30pm 
Lunch 
Bistro – 2^{nd} Floor 
Bartek Czech, Institute for Advanced Study
How Tensor Network Renormalization quantifies circuit complexity and why this is a problem of [considerable] gravity
According to a recent proposal, in the AdS/CFT correspondence the circuit complexity of a CFT state is dual to the EinsteinHilbert action of a certain region in the dual spacetime. If the proposal is correct, it should be possible to derive Einstein's equations by varying the complexity in a class of circuits that prepare the requisite CFT state. This talk attempts such a derivation in very special settings: Virasoro descendants of the CFT2 ground state, which are dual to locally AdS3 geometries. By applying Tensor Network Renormalization to the discretized Euclidean path integral that prepares the CFT state, I will justify the recent suggestion by Caputa et al. that the complexity of a path integral is quantified by the Liouville action. The Liouville field specifies the conformal frame in which the path integral is evaluated; in the most efficient / least complexity frame, the Liouville field is closely related to entanglement entropies of CFT2 intervals. Assuming the RyuTakayanagi proposal, the said entanglement entropies are lengths of geodesics living in the dual spacetime. The Liouville equation of motion satisfied by the minimal complexity Liouville field is a geodesicwise rewriting of the nonlinear vacuum Einstein's equations in 3d with a negative cosmological constant. I emphasize that this is very much work in progress; I hope the audience will help me to sharpen the arguments.
Glen Evenbly, University of Sherbrooke
Hyperinvariant tensor networks and holography
I will propose a new class of tensor network state as a model for the AdS/CFT correspondence and holography. This class shall be demonstrated to retain key features of the multiscale entanglement renormalization ansatz (MERA), in that they describe quantum states with algebraic correlation functions, have free variational parameters, and are efficiently contractible. Yet, unlike MERA, they are built according to a uniform tiling of hyperbolic space, without inherent directionality or preferred locations in the holographic bulk, and thus circumvent key arguments made against the MERA as a model for AdS/CFT. Novel holographic features of this tensor network class will be examined, such as an equivalence between the causal cone C[R] and the entanglement wedge E[R] of connected boundary regions R.
Martin Ganahl, Perimeter Institute
Solving Nonrelativistic Quantum Field Theories with continuous Matrix Product States
Since its proposal in the breakthrough paper [F. Verstraete, J.I. Cirac, Phys. Rev. Lett. 104, 190405(2010)], continuous Matrix Product States (cMPS) have emerged as a powerful tool for obtaining nonperturbative ground state and excited state properties of interacting quantum field theories (QFTs) in (1+1)d. At the heart of the cMPS lies an efficient parametrization of manybody wavefunctionals directly in the continuum, that enables one to obtain ground states of QFTs via imaginary time evolution. In the first part of my talk I will give a general introduction to the cMPS formalism. In the second part, I will then discuss a new method for cMPS optimization, based on energy gradient instead of the usual imaginary time evolution. This new method overcomes several problems associated with imaginary time evolution, and allows to perform calculations at much lower cost / higher accuracy than previously possible.
Jutho Haegeman, University of Ghent
Bridging Perturbative Expansions with Tensor Networks
We demonstrate that perturbative expansions for quantum manybody systems can be rephrased in terms of tensor networks, thereby providing a natural framework for interpolating perturbative expansions across a quantum phase transition. This approach leads to classes of tensornetwork states parameterized by few parameters with a clear physical meaning, while still providing excellent variational energies. We also demonstrate how to construct perturbative expansions of the entanglement Hamiltonian, whose eigenvalues form the entanglement spectrum, and how the tensornetwork approach gives rise to order parameters for topological phase transitions.
Janet Hung, Fudan University
Tensor network and (padic) AdS/CFT
We will describe how the reconstruction of a bulk operator can be organised systematically. With a suitable parametrisation, an analogue of the HKLL formula emerges, involving a smearing function satisfying a Klein Gordon equation in the graph. The parametrisation also allows us to read off interaction vertices, and build up loop diagrams systematically. When we interpret the BruhatTits tree as a tensor network, we recover (partially) features of the padic AdS/CFT dictionary discussed recently in the literature.
Robert Leigh, University of Illinois at UrbanaChampaign
Unitary Networks from the Exact Renormalization of Wavefunctionals
The exact renormalization group (ERG) for O(N) vector models at large N on flat Euclidean space admits an interpretation as the bulk dynamics of a holographically dual higher spin gauge theory on AdS_{d+1}. The generating functional of correlation functions of single trace operators is reproduced by the onshell action of this bulk higher spin theory, which is most simply presented in a firstorder (phase space) formalism. This structure arises because of an enormous nonlocal symmetry of free fixed point theories. In this talk, I will review the ERG construction and describe its extension to the RG flow of the wave functionals of arbitrary states of the O(N) vector model at the free fixed point. One finds that the ERG flow of the ground state and a specific class of excited states is implemented by the action of unitary operators which can be chosen to be local. Thus the ERG equations provide a continuum notion of a tensor network. We compare this tensor network with the entanglement renormalization networks, MERA, and cMERA. The ERG tensor network appears to share the general structure of cMERA but differs in important ways.
Ashley Milsted, Perimeter Institute
Emergence of conformal symmetry in critical spin chains
We demonstrate that 1+1D conformal symmetry emerges in critical spin chains by constructing a lattice ansatz Hn for (certain combinations of) the Virasoro generators Ln. The generators Hn offer a new way of extracting conformal data from the low energy eigenstates of the lattice Hamiltonian on a finite circle. In particular, for each energy eigenstate, we can now identify which Virasoro tower it belongs to, as well as determine whether it is a Virasoro primary or a descendant (and similarly for global conformal towers and global conformal primaries/descendants). The central charge is obtained from a simple groundstate expectation value. Nonuniversal, finitesize corrections are the main source of error. We propose and demonstrate the use of periodic Matrix Product States, together with an improved ground state solver, to reach larger system sizes. We uncover that, importantly, the MPS singleparticle excitation ansatz accurately describes all low energy excited states.
Robert Myers, Perimeter Institute
Complexity, Holography & Quantum Field Theory
I will describe some recent work studying proposals for computational complexity in holographic theories and in quantum field theories. In particular, I will discuss some interesting properties of the new gravitational observables and of complexity in the boundary theory.
Tobias Osborne, University of Hannover
Dynamics for holographic codes
In this talk I discuss the problem of introducing dynamics for holographic codes. To do this it is necessary to take a continuum limit of the holographic code. As I argue, a convenient kinematical continuum limit space is given by Jones’ semicontinuous limit. Dynamics are then furnished by a unitary representation of a discrete analogue of the conformal group known as Thompson’s group T. I will describe these representations in detail in the simplest case of a discrete AdS geometry modelled by trees. Consequences such as the ER=EPR argument are then realised in this setup. Extensions to more general tessellations with a MERA structure are possible, and will be (very) briefly sketched.
Xiaoliang Qi, Stanford University
Random tensor networks and holographic coherent states
Arpan Bhattacharyya, Fudan University
AdS/CFT via Tensor Network : Bulk boundary Reconstruction
We will demonstrate , how to reconstruct bulk operator starting form the local boundary using our model of tensor network which is basically using being build form the perfect tensor plus some small perturbations away form it. We will show that it has the similar features as that of HKLL construction thereby making the connection with the holography (AdS/CFT) concrete. Also we will demonstrate the connection between the linear part of the operator reconstruction and the wavelet transformation. Further we will show that the non linear part of the reconstruction has the possibility of giving the "Geodesic Witten diagram ". At last , we will consider the example of padic tree where all these things can be written down explicitly.
Jordan Cotler, Stanford University
cMERA for Interacting Scalar Fields
We upgrade cMERA to a systematic variational ansatz and develop techniques for its application to interacting quantum field theories in arbitrary spacetime dimensions. By establishing a correspondence between the first two terms in the variational expansion and the Gaussian Effective Potential, we can exactly solve for a variational approximation to the cMERA entangler. As examples, we treat scalar ϕ^4 theory and the GrossNeveu model and extract nonperturbative behavior. We also comment on the connection between generalized squeezed coherent states and more generic entanglers.
Matthew Fishman, California Institute of Technology
Improving the Corner Transfer Matrix Renormalization Group Method with Fixed Points
We present an explicitly translationally invariant version of the Corner Transfer Matrix Renormalization Group (CTMRG) method, which allows us to reformulate the method in terms of a set of fixed point equations. This leads to speedups in the convergence time of the algorithm, particularly for systems near criticality. To show the performance of the algorithm, we present various benchmarks for contracting 2D statistical mechanics models as well as 2D quantum models written as projected entangled pair states (PEPS).
Adrian Franco Rubio, Perimeter Institute
Entanglement structure and UV regularization in cMERA
The continuous multiscale entanglement renormalization ansatz or cMERA provides a variational ansatz for the ground state of a quantum field theory. Such states come equipped with an intrinsic length scale that acts as an ultraviolet cutoff. We provide evidence for the existence of this cutoff based on the entanglement structure of a particular family of cMERA states, namely Gaussian states optimized for free bosonic and fermionic CFTs. Our findings reflect that short distance entanglement is not fully present in the ansatz states, thus hinting at ultraviolet regularization.
Adil Gangat, National Taiwan University
Steady States of InfiniteSize Dissipative Quantum Chains via Imaginary Time Evolution
Directly in the thermodynamic limit, we show how to combine imaginary and real time evolution of tensor networks to efficiently and accurately find the nonequilibrium steady states (NESS) of onedimensional dissipative quantum lattices governed by the Lindblad master equation. The imaginary time evolution first bypasses any highly correlated portions of the realtime evolution trajectory by directly converging to the weakly corre lated subspace of the NESS, after which real time evolution completes the convergence to the NESS with high accuracy. We demonstrate the power of the method with the dissipative transverse field quantum Ising chain. We show that a crossover of an order parameter shown to be smooth in previous finitesize studies remains smooth in the thermodynamic limit.
Markus Hauru, Perimeter Institute
Topological conformal defects with tensor network
Qi Hu, Perimeter Institute
Continuous Multiscale Entanglement Renormalization Ansatz
The generalization of the multiscale entanglement renormalization ansatz (MERA) to continuous systems, or cMERA, is a variational ansatz for the ground state of quantum field theories. For a conformal field theory, it can capture the spacetime symmetries of the ground state, and we can extract the conformal data from cMERA.
Adam Lewis, Perimeter Institute
Matrix Product State Simulations of Quantum Fields in an Expanding Universe
The matrix product state (MPS) ansatz makes possible computationallyefficient representations of weakly entangled manybody quantum systems with gapped Hamiltonians near their ground states, notably including massive, relativistic quantum fields on the lattice. No Wick rotation is required to apply the time evolution operator, enabling study of timedependent Hamiltonians. Using free massive scalar field theory on the 1+1 RobertsonWalker metric as a toy example, I present early efforts to exploit this fact to model quantum fields in curved spacetime. We use the ADM formalism to write the appropriate Hamiltonian witnessed by a particular class of normal observers. Possible applications include simulations of gravitational particle production in the presence of interactions, studies of the slicingdependence of entanglement production, and inclusion of the expectation of the stressenergy tensor as a matter source in a numerical relativity simulation.
Alex May, University of British Columbia
Tensor networks for dynamic spacetimes
Existing tensor network models of holography are limited to representing the geometry of constant time slices of static spacetimes. We study the possibility of describing the geometry of a dynamic spacetime using tensor networks. We find it is necessary to give a new definition of length in the network, and propose a definition based on the mutual information. We show that by associating a set of networks with a single quantum state and making use of the mutual information based definition of length, a network analogue of the maximin formula can be used to calculate the entropy of boundary regions.
Hugo Marrochio, Perimeter Institute
Holographic complexity and related progress towards a cMERA realization
Julian Rincon, Perimeter Institute
Continuous matrix product representations for mixed states
The continuous matrix product states (cMPS) is a powerful variational ansatz for the ground state of interacting quantum field theories in 1+1 spacetime dimensions [F. Verstraete, J.I. Cirac, Phys. Rev. Lett. 104, 190405(2010)]. Here we propose a density matrix generalization of the cMPS, the continuous matrix product density operator (cMPDO), and investigate its suitability to represent thermal states and master equation dynamics. We show the existence of the cMPDO by taking the continuum limit of a lattice MPDO and characterize its mathematical properties. For thermal states of field theories, we find that the cMPDO offers an accurate description of their corresponding density matrix. We argue that these results can also be extended for the case of master equation dynamics.
Yijian Zou, Perimeter Institute
Extracting conformal data with periodic boundary matrix product states
We construct Virasoro generators on a finite critical lattice system with the periodic boundary condition, and use them to identify conformal towers. Ground state and excited states corresponding to scaling operators are found with periodic boundary matrix product states. Scaling dimensions and central charge are estimated with high accuracy from finite size scaling.
Hyperinvariant tensor networks and holography
I will propose a new class of tensor network state as a model for the AdS/CFT correspondence and holography. This class shall be demonstrated to retain key features of the multiscale entanglement renormalization ansatz (MERA), in that they describe quantum states with algebraic correlation functions, have free variational parameters, and are efficiently contractible.
Tensor network and (padic) AdS/CFT
We will describe how the reconstruction of a bulk operator can be organised systematically. With a suitable parametrisation, an analogue of the HKLL formula emerges, involving a smearing function satisfying a Klein Gordon equation in the graph. The parametrisation also allows us to read off interaction vertices, and build up loop diagrams systematically. When we interpret the BruhatTits tree as a tensor network, we recover (partially) features of the padic AdS/CFT dictionary discussed recently in the literature.
Dynamics for holographic codes
In this talk I discuss the problem of introducing dynamics for holographic codes. To do this it is necessary to take a continuum limit of the holographic code. As I argue, a convenient kinematical continuum limit space is given by Jones’ semicontinuous limit. Dynamics are then furnished by a unitary representation of a discrete analogue of the conformal group known as Thompson’s group T. I will describe these representations in detail in the simplest case of a discrete AdS geometry modelled by trees. Consequences such as the ER=EPR argument are then realised in this setup.
Random tensor networks and holographic coherent states
Tensor network is a constructive description of manybody quantum entangled states starting from fewbody building blocks. Random tensor networks provide useful models that naturally incorporate various important features of holographic duality, such as the RyuTakayanagi formula for entropyarea relation, and operator correspondence between bulk and boundary. In this talk I will overview the setup and key properties of random tensor networks, and then discuss how to describe quantum superposition of geometries in this formalism.
How Tensor Network Renormalization quantifies circuit complexity and why this is a problem of [considerable] gravity
According to a recent proposal, in the AdS/CFT correspondence the circuit complexity of a CFT state is dual to the EinsteinHilbert action of a certain region in the dual spacetime. If the proposal is correct, it should be possible to derive Einstein's equations by varying the complexity in a class of circuits that prepare the requisite CFT state. This talk attempts such a derivation in very special settings: Virasoro descendants of the CFT2 ground state, which are dual to locally AdS3 geometries.
Complexity, Holography & Quantum Field Theory
I will describe some recent work studying proposals for computational complexity in holographic theories and in quantum field theories. In particular, I will discuss some interesting properties of the new gravitational observables and of complexity in the boundary theory.
Two Continous Approaches to AdS/Tensor Network duality
In this talk, I would like to discuss how we can realize the correspondence between AdS/CFT and tensor network in quantum field theories (i.e. the continous limit). As the first approach I will discuss a possible connection between continuous MERA and AdS/CFT. Next I will introduce the second approach based on the optimization of Euclidean pathintegral, where the strcutures of hyperbolic spaces and entanglement wedges emerge naturally. This second appraoch is closely related to the idea of tensor network renormalization.
Tensor Networks and Holography
Tensor network renormalization and real space Hamiltonian flows
We will review the topic of tensor network renormalization, relate it to real space Hamiltonian flows, and discuss the emergence of matrix product operator algebras as symmetries of the renormalization fixed points.
joint work with Matthias Bal, Michael Marien and Jutho Haegeman
Analytic approaches to tensor networks for field theories
I will discuss analytic approaches to construct tensor network representations of quantum field theories, more specifically conformal field theories in 1+1 dimensions. A key insight is that we should understand how well the tensor network can reproduce the correlation functions of the quantum field theory. Based on this measure of closeness, I will present rigorous results allowing for explicit error bounds which show that both Matrix product states (MPS) as well as the multiscale renormalization Ansatz (MERA) do approximate conformal field theories.
Pages
Scientific Organizers:
 Robert Myers, Perimeter Institute
 Tadashi Takayanagi, Yukawa Institute for Theoretical Physics
 Frank Verstraete, University of Ghent
 Guifre Vidal, Perimeter Institute
 Steven White, University of California, Irvine