**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 Einstein-Hilbert action of a certain region in the dual space-time. 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 Ryu-Takayanagi proposal, the said entanglement entropies are lengths of geodesics living in the dual space-time. The Liouville equation of motion satisfied by the minimal complexity Liouville field is a geodesic-wise rewriting of the non-linear 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

*Hyper-invariant 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 multi-scale 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 Non-relativistic 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 non-perturbative 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 many-body 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 tensor-network 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 tensor-network approach gives rise to order parameters for topological phase transitions.

**Janet Hung**, Fudan University

*Tensor network and (p-adic) 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 Bruhat-Tits tree as a tensor network, we recover (partially) features of the p-adic AdS/CFT dictionary discussed recently in the literature.

**Robert Leigh**, University of Illinois at Urbana-Champaign

*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 on-shell action of this bulk higher spin theory, which is most simply presented in a first-order (phase space) formalism. This structure arises because of an enormous non-local 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 ground-state expectation value. Non-universal, finite-size 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 single-particle 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*

Tensor network is a constructive description of many-body quantum entangled states starting from few-body building blocks. Random tensor networks provide useful models that naturally incorporate various important features of holographic duality, such as the Ryu-Takayanagi formula for entropy-area 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. By introducing quantum link variables, we show that random tensor networks on all geometries form an overcomplete basis of the boundary Hilbert space, such that each boundary state can be mapped to a superposition of (spatial) geometries. We discuss how small fluctuations around each geometry forms a “code subspace” in which bulk operators can be mapped to boundary isometrically. We further compute the overlap between distinct geometries, and show that the overlap is suppressed exponentially in an area law fashion, in consistency with the holographic principle. In summary, random tensor networks on all geometries form an overcomplete basis of “holographic coherent states” which may provide a new starting point for describing quantum gravity physics.

References

[1] Patrick Hayden, Sepehr Nezami, Xiao-Liang Qi, Nathaniel Thomas, Michael Walter, Zhao Yang, JHEP 11 (2016) 009

[2] Xiao-Liang Qi, Zhao Yang, Yi-Zhuang You, arxiv:1703.06533

**Volker Scholz**, Ghent University

*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. In particular, I will discuss the case of Wess-Zumino-Witten models.

based on joint work with Robert Koenig (MPS), Brian Swingle and Michael Walter (MERA)

**Miles Stoudenmire**, University of California, Irvine

*Applying DMRG to Non-relativistic Continuous Systems in 1D and 3D*

The density matrix renormalization group works very well for one-dimensional (1D) lattice systems, and can naively be adapted for non-relativistic continuum systems in 1D by discretizing real space using a grid. I will discuss challenges inherent in this approach and successful applications. Recently, the success of the grid approach for 1D motivated us to extend the approach to 3D by treating the transverse directions with a basis set. This hybrid grid/basis-set approach allows DMRG to scale much better for long molecules and we obtain state-of-the-art results with modest computing resources. A key component of the approach is a powerful algorithm for compressing long-range interactions into a matrix product operator which I will present in some detail.

**James Sully,** Stanford Linear Accelerator Center

*Tensor Networks and Holography*

**Brian Swingle**, MIT, Harvard University & Brandeis University

*Tensor networks and Legendre transforms*

Tensor networks have primarily, thought not exclusively, been used to the describe quantum states of lattice models where there is some inherent discreteness in the system. This raises issues when trying to describe quantum field theories using tensor networks, since the field theory is continuous (or at least the regulator should not play a central role). I'll present some work in progress studying tensor networks designed to directly compute correlation functions instead of the full state. Here the discreteness arises from our choice of where and how to probe the field theory. This approach is roughly analogous to studying a Legendre transform of the state. I'll discuss the properties of such networks and show how to construct them in some cases of interest, including non-interacting fermion field theories. Partly based on work with Volkher Scholz and Michael Walter.

**Tadashi Takayanagi**, Yukawa Institute for Theoretical Physics

*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 path-integral, where the strcutures of hyperbolic spaces and entanglement wedges emerge naturally. This second appraoch is closely related to the idea of tensor network renormalization.

**Frank Verstraete**, University of Ghent

*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

**Guifre Vidal**, Perimeter Institute

*The continuous multi-scale entanglement renormalization ansatz (cMERA)*

The first half of the talk will introduce the cMERA, as proposed by Haegeman, Osborne, Verschelde and Verstratete in 2011 [1], as an extension to quantum field theories (QFTs) in the continuum of the MERA tensor network for lattice systems. The second half of the talk will review recent results [2] that show how a cMERA optimized to approximate the ground state of a conformal field theory (CFT) retains all of its spacetime symmetries, although these symmetries are realized quasi-locally. In particular, the conformal data of the original CFT can be extracted from the optimized cMERA.

[1] J. Haegeman, T. J. Osborne, H. Verschelde, F. Verstraete, Entanglement renormalization for quantum fields, Phys. Rev. Lett, 110, 100402 (2013), arXiv:1102.5524

[2] Q. Hu, G. Vidal, Spacetime symmetries and conformal data in the continuous multi-scale entanglement renormalization ansatz, arXiv:1703.04798

**Steven White,** University of California, Irvine

*Discretizing the many-electron Schrodinger Equation*

Large parts of condensed matter theoretical physics and quantum chemistry have as a central goal discretizing and solving the continuum many-electron Schrodinger Equation. What do we want to get from these calculations? What are key problems of interest? What sort of approaches are used? I'll start with a broad overview of these questions using the renormalization group as a conceptual framework. I'll then progress towards our recent tensor network approaches for the many electron problem, discussing along the way issues of the area law, wavelet techniques and Wilson's related work, wavelets and MERA, and discretizations that combine grids and basis sets.