Tensor Network

Computing vacuum expectation values is paramount in studying Quantum Field Theories (QFTs) since they provide relevant information for comparing the underlying theory with experimental results. However, unless the ground state of the system is explicitly known, such computations are very difficult and Monte Carlo simulations generally run months to years on state-of-the-art high performance computers. Additionally, there are various physically interesting situations, in which most numerical methods currently in use are not applicable at all (e.g., the early universe or setting requiring Lorentzian backgrounds). Thus, new algorithms are required to address such problems in QFT. In recent joint work with K. Jansen (NIC, DESY Zeuthen), I have shown that zeta-functions of Fourier integral operators can be applied to regularize vacuum expectation values with Euclidean and Lorentzian backgrounds and that these zeta-regularized vacuum expectation values are in fact physically meaningful. In order to prove physicality, we introduced a discretization scheme which is accessible on a quantum computer. Using this discretization scheme, we can efficiently approximate ground states on a quantum device and henceforth compute vacuum expectation values. Furthermore, the Fourier integral operator $\zeta$-function approach is applicable to Lattice formulations in Lorentzian background. 

We compute the partition function of a massive free boson in a square lattice using a tensor network algorithm. We introduce a singular value decomposition (SVD) of continuous matrices that leads to very accurate numerical results. It is shown the emergence of a CDL fixed point structure. In the massless limit, we reproduce the results of conformal field theory including a precise value of the central charge.

Adiabatic evolution is a common strategy for manipulating quantum states. However, it is inherently slow and therefore susceptible to decoherence. Shortcuts to adiabaticity are methods of achieving faster adiabatic evolution, in order to maintain high fidelity in the presence of decoherence and noise. In this talk I will review recent progress on counter-diabatic (CD) driving for many-body systems. In particular, we will discuss a variational principle that allows to systematically compute approximate CD Hamiltonians. Two recent experiments will be discussed. 

In the first part of the talk, I will mention the ongoing numerical efforts using lattice calculations to understand the holographic dualities relating the super Yang-Mills (SYM) theories in various dimensions and their conjectured Type II supergravity theories in the decoupling limit. In the second part, I will discuss the tensor renormalization group study of the SU(2) gauge-Higgs model in two dimensions using the higher-order tensor renormalization group (HOTRG) algorithm and compare the results with the Monte Carlo simulations. 

We discuss some algebraic quantum field theory (AQFT) ingredients that should be useful in defining a tensor network describing a Lorentzian space-time.

We look into toy models that approximate Minkowski space and show how Lorentz boosts are approximately recovered, and obtain Rindler modes that can be compared with the entanglement spectrum. 

We make connections of these approximations of Lorentz boosts with the corner transfer matrices in integrable models, and comment on the discrete realization of the Reeh-Schlieder theorem that governs entanglement in a Hilbert space with lower bounded energy. 

Simplicial complexes naturally describe discrete topological spaces. When their links are assigned a length they describe discrete geometries. As such simplicial complexes have been widely used in quantum gravity approaches that involve a discretization of spacetime. Recently they are becoming increasingly popular to describe complex interacting systems such a brain networks or social networks. In this talk we present non-equilibrium statistical mechanics approaches to model large simplicial complexes. We propose the simplicial complex model of Network Geometry with Flavor (NGF), we explore the hyperbolic nature of its emergent geometry and their relation with Tree Tensor Networks. Finally we reveal the rich interplay between Network Geometry with Flavor and dynamics. We investigate the percolation properties of NGF using the renormalization group finding KTP and discontinuous phase transitions depending on the dimensionality simplex. We also comment on the synchronization properties of NGF and the emergence of frustrated synchronization.

Three fundamental factors determine the quality of a statistical learning algorithm: expressiveness, generalization and optimization.  The classic strategy for handling these factors is relatively well understood.  In contrast, the radically different approach of deep learning, which in the last few years has revolutionized the world of artificial intelligence, is shrouded by mystery.  This talk will describe a series of works aimed at unraveling some of the mysteries revolving expressiveness, arguably the most prominent factor behind the success of deep learning.  I will begin by showing that state of the art deep learning architectures, such as convolutional networks, can be represented as tensor networks -- a computational model commonly employed in quantum physics.  This connection will inspire the use of quantum entanglement for defining measures of data correlations modeled by deep networks.  Next, I will turn to a quantum max-flow / min-cut theorem characterizing the entanglement captured by tensor networks.  This theorem will give rise to new results that shed light on expressiveness in deep learning, and in addition, provide new tools for deep network design.

Works covered in the talk were in collaboration with Yoav Levine, Or Sharir, David Yakira and Amnon Shashua.

In this talk, I will discuss how to assign geometries, such as metric tensors, to certain tensor networks using quantum entanglement and tensor Radon transform. In addition, we show that behaviour similar to linearized gravity can naturally emerge in said tensor networks, provided a modified version of Jacobson's entanglement equilibrium is satisfied. Since the aforementioned properties can be reached without relying on AdS/CFT, the approach also shows promise towards constructing tensor network models for cosmological spacetimes. 

A great deal of progress has been made toward a classification of bosonic topological orders whose microscopic constituents are bosons. Much less is known about the classification of their fermionic counterparts. In this talk I will describe a systematic way of producing fermionic topological orders using the technique of fermion condensation. Roughly, this can be understood as binding a physical fermion to an emergent fermion and condensing the pair. I will discuss the `super pivotal categories' that describe universal properties of these phases and use them to construct exactly solvable string-net models. These string-net models feature conventional anyons and two flavours of vortices. I will show that one of the vortex types is similar to a vortex in a p+ip superconductor binding a Majorana zero mode, and will mention some possible applications.

The construction of trial wave functions has proven itself to be very useful for understanding strongly interacting quantum many-body systems. Two famous examples of such trial wave functions are the resonating valence bond state proposed by Anderson and the Laughlin wave function, which have provided an (intuitive) understanding of respectively spin liquids and fractional Quantum Hall states. Tensor network states are another, more recent, class of such trial wave functions which are based on entanglement properties of local, gapped systems. In this talk I will discuss the use of tensor network states for topological phases, and what we can learn from this approach. I will consider one- and two-dimensional systems, consisting of both spins and fermions. The focus will be on the different connections that can be made using tensor networks, such as connecting theory to numerics, and physical properties to ground state entanglement.