Tensor networks for LGT: beyond 1D

• Mari-Carmen Banuls (Max Planck Institute of Quantum Optics)

The suitability of tensor network ansatzes for the description of physically relevant states in one dimensional lattice gauge theories (LGT) has been demonstrated in the last years by a large amount of systematic studies, including abelian and non-abelian LGTs, and including scenarios where traditional Monte Carlo approaches fail due to a sign problem. While this establishes a solid motivation to extend the program to higher dimensions, a similar systematic study in two dimensions using PEPS requires dealing with specific considerations. Besides a larger computational costs associated to the higher spatial dimension, the presence of plaquette terms in LGTs hinders the efficiency of the most up-to-date PEPS algorithms. With a newly developed update strategy, nevertheless, such terms can be treated by the most efficient techniques. We have used this method to perform the first ab initio iPEPS study of a LGT in 2+1 dimensions: a Z3 invariant model, for which we have determined the phase diagram.

Tensor networks for critical systems

• Frank Verstraete (University of Vienna & University of Ghent)

In this talk I will give an overview of tensor network approaches to critical systems. I will discuss entanglement scaling laws, show how PEPS can simulate systems with Fermi surfaces, and present some results for simulating systems in the continuum.

Tensor network models of AdS/qCFT

• Jens Eisert (Free University of Berlin)

"AdS/CFT endows gravity in anti-de Sitter (AdS) spacetime with a dual description in certain conformal field theories (CFTs) with matching symmetries. Tensor networks on regular discretizations of AdS space provide natural toy models of AdS/CFT, but break the continuous bulk symmetries. In this talk, we discuss several aspects of such toy models based on tensor networks. We show that this produces a quasiregular conformal field theory (qCFT) on the boundary and rigorously compute its symmetries, entanglement properties, and central charge bounds, applicable to a wide range of existing models. An explicit AdS/qCFT model with exact fractional central charges is given by holographic quantum error correcting codes based on Majorana dimers. These models also realize the strong disorder renormalization group, resulting in new connections between critical condensed-matter models, exact quantum error correction, and holography. If time allows, we will briefly review other recent group research on using tensor network models in quantum many-body physics including many-body localization and time crystals as well as in probabilistic modelling. Based on arXiv:2004.04173, Phys. Rev. A 102, 042407 (2020), Phys. Rev. Research 1, 033079 (2019), Science Advances 5, eaaw0092 (2019)."

Ancilla qubit wavefunctions for the pseudogap metal phase of the cuprates

• Subir Sachdev (Harvard University)

There is now significant experimental evidence that the physics of the underdoped cuprates is controlled by a metallic state with a Fermi surface whose volume does not equal the Luttinger value. However, there has been no proposed wavefunction for such a state for electrons in a single band. I will describe a wavefunction which involves tracing over 2 layers of ancilla qubits. The proposal also leads to a gauge theory for the transition to the conventional Fermi liquid state found at large doping.

Quantum Cellular Automata, Tensor Networks, and Area Laws

• Ignacio Cirac (Max Planck Institute of Quantum Optics)

Quantum Cellular Automata are unitary maps that preserve locality and respect causality. I will show that in one spatial dimension they correspond to matrix product unitary operators, and that one can classify them in the presence of symmetries, giving rise to phenomenon analogous to symmetry protection. I will then show that in higher dimensions, they correspond to other tensor networks that fulfill an extra condition and whose bond dimension does not grow with the system size. As a result, they satisfy an area law for the entanglement entropy they can create. I will also define other classes of non-unitary maps, the so-called quantum channels, that either respect causality or preserve locality and show that, whereas the latter obey an area law for the amount of quantum correlations they can create, as measured by the quantum mutual information, the former may violate it. Additionally, neither of them can be expressed as tensor networks with a bond dimension that is independent of the system size.

Fun with replicas and holographic tensor networks

• Michael Walter (University of Amsterdam)

A tensor-network approach to fixed-point models of topological phases

• Andreas Bauer (Free University Berlin)

"I will introduce a tensor-network based language for classifying topological phases via fixed-point models. The "models" will be tensor networks formalizing a discrete Euclidean path integral living in a topological space-time, and can be obtained from Hamiltonian models by Trotterizing the imaginary time evolution. Topological fixed-point models are invariant under topology-preserving space-time deformations. Space-time manifolds and homeomorphisms can be combinatorially represented by graph-like "networks", which together with "moves" form a "liquid". The networks can be interpreted as tensor networks, and the moves as equations which determine the fixed-point models. Different combinatorial representations of the same space-times yield new kinds of fixed-point models. Given the limited time, I will stick to very simple examples in 1+1 dimensions for this talk."

Custom Fermionic Codes for Quantum Simulation

• Riley Chien (Dartmouth College)

Quantum Extremal Islands Made Easy: Complexity on the brane

• Shan-Ming Ruan (Perimeter Institute)

We examine holographic complexity in the doubly holographic model to study quantum extremal islands. We focus on the holographic complexity=volume (CV) proposal for boundary subregions in the island phase. Exploiting the Fefferman-Graham expansion of the metric and other geometric quantities near the brane, we derive the leading contributions to the complexity and interpret these in terms of the generalized volume of the island derived from the higher curvature action for the brane gravity. Motivated by these results, we propose a generalization of the CV proposal for higher curvature theories of gravity. Further, we provide two consistency checks of our proposal by studying Gauss-Bonnet gravity and f(R) gravity in the bulk.

Noise-robustness and experimental data of wavelet-DMERA preparation for critical Ising ground state

• Troy Sewell (University of Maryland)

Multi-scale tensor networks offer a way to efficiently represent ground states of critical systems and may be adapted for state-preparation on a quantum computer. The tensor network for a single scale specifies a quantum channel whose fixed-point is a subregion of the approximate critical ground state. The fixed-point of a noisy channel is perturbed linearly in the noise parameter from the ideal state, making local observables stable against errors for these iterative algorithms. We consider the wavelet-designed circuit for the 1+1D critical Ising ground state as a concrete example to numerically test the noise robustness against our error models and compare the smallest instance case with an implementation on a present-day ion-trap quantum computer.

Tensor network methods for quantum chemistry

• Steven White (University of California, Irvine)

The search for applications of quantum computers has highlighted the field of quantum chemistry, where one can also apply tensor network methods. There are several challenges in getting useful results for molecules compared to simulating a model Hamiltonian in condensed matter physics. The first issue is in descretizing continuum space to get a finite Hamiltonian which is amenable to tensor network techniques. Another is the need for high accuracy, particularly in energies, to compare with experiments. I will give an overview of the approaches used in this field, and then focus on our work using grid and wavelet-based discretizations coupled with DMRG methods.

Towards a realistic holographic tensor network: From p-adic CFT to (minimal) CFT2

• Ling-Yan Hung (Fudan University )

"The success of the Ryu-Takayanagi formula suggests a profound connection between the AdS/CFT correspondence and tensor networks. There are since many works on constructing examples, although it is very difficult to make them explicit and quantitative. We will discuss some new progress in the toy example of p-adic CFT where its tensor network dual was previously constructed explicitly [ arXiv:1703.05445 , arXiv:1812.06059, arXiv:1902.01411], and how some analogue of Einstein equation on the graph emerges as we consider RG flow of these CFTs. These progresses inspire us to find a way to construct explicit holographic tensor networks of more realistic CFTs based on their connection with topological models with one higher dimension, at least in CFT2. We will present some preliminary results and works in progress."

Tensor network description of 3D Quantum Gravity and Diffeomorphism Symmetry

• Bianca Dittrich (Perimeter Institute)

In contrast to the 4D case, there are well understood theories of quantum gravity for the 3D case. Indeed, 3D general relativity constitutes a topological field theory (of BF or equivalently Chern-Simons type) and can be quantized as such. The resulting quantum theory of gravity offers many interesting lessons for the 4D case. In this talk I will discuss the quantum theory which results from quantizing 3D gravity as a topological field theory. This will also allow a derivation of a holographic boundary theory, together with a geometric interpretation of the boundary observables. The resulting structures can be interpreted in terms of tensor networks, which provide states of the boundary theory. I will explain how a choice of network structure and bond dimensions constitutes a complete gauge fixing of the diffeomorphism symmetry in the gravitational bulk system. The theory provides a consistent set of rules for changing the gauge fixing and with it the tensor network structure. This provides an example of how diffeomorphism symmetry can be realized in a tensor network based framework. I will close with some remarks on the 4D case and the challenges we face there.

Dimensional Expressivity Analysis for Quantum Circuits

• Tobias Hartung (Deutsches Elektronen-Synchrotron)

"Besides tensor networks, quantum computations (QC) as well use a Hamiltonian formulation to solve physical problems. Although QC are presently very limited, since only small number of qubits are available, they have the principal advantage that they straightforwardly scale to higher dimensions. A standard tool in the QC approach are Variational Quantum Simulations (VQS) which form a class of hybrid quantum-classical algorithms for solving optimization problems. For example, the objective may be to find the ground state of a Hamiltonian by minimizing the energy. As such, VQS use parametric quantum circuit designs to generate a family of quantum states (e.g., states obeying physical symmetries) and efficiently evaluate a cost function for the given set of variational parameters (e.g., energy of the current quantum state) on a quantum device. The optimization is then performed using a classical feedback loop based on the measurement outcomes of the quantum device. In the case of energy minimization, the optimal parameter set therefore encodes the ground state corresponding to the given Hamiltonian provided that the parametric quantum circuit is able to encode the ground state. Hence, the design of parametric quantum circuits is subject to two competing drivers. On one hand, the set of states, that can be generated by the parametric quantum circuit, has to be large enough to contain the ground state. On the other hand, the circuit should contain as few quantum gates as possible to minimize noise from the quantum device. In other words, when designing a parametric quantum circuit we want to ensure that there are no redundant parameters. In this talk, I will consider the parametric quantum circuit as a map from parameter space to the state space of the quantum device. Using this point of view, the set of generated states forms a manifold. If the quantum circuit is free from redundant parameters, then the number of parameters is precisely the dimension of the manifold of states. This leads us to the notion of dimensional expressivity analysis. I will discuss means of analyzing a given parametric design in order to remove redundant parameters as well as any unwanted symmetries (e.g., a gate whose only effect is a change in global phase). Time permitting, I may discuss the manifold of physical states as well since this will allow us to decide whether or not a parametric quantum circuit can express all physical states (thereby ensuring that the ground state can be expressed as well)."

Query complexity and cutoffs in AdS3/CFT2

• Bartek Czech (Tsinghua University)

A quantum state is a map from operators to real numbers that are their expectation values. Evaluating this map always entails using some algorithm, for example contracting a tensor network. I propose a novel way of quantifying the complexity of a quantum state in terms of "query complexity": the number of times an efficient algorithm for computing correlation functions in the given state calls a certain subroutine. I construct such an algorithm for a general "state at a cutoff" in 1+1-dimensional field theory. The algorithm scans cutoff-sized intervals for operators whose expectation values will be computed. It can be written as a Matrix Product State, with individual matrices performing translations in the space of (cutoff-sized) intervals and reading off consecutive operator inputs. If we take the queried subroutine to be a translation in the space of intervals, query complexity counts "how many" intervals the algorithm visits--a notion of distance in the space of intervals. A unique distance function is consistent with the requisite notion of translations; therefore the query complexity of a state at a cutoff is unambiguously defined. In holographic theories, the query complexity evaluates to the integral of the Ricci scalar on a spatial slice enclosed by the bulk cutoff, which in pure AdS3 agrees with the volume proposal but otherwise departs from it.

Thirring model from tensor networks - phase structure and real-time dynamics

• Krzysztof Cichy (Adam Mickiewicz University)

In this talk, we report on our studies of the Thirring model using MPS techniques. The Thirring model is a quantum field theory describing self-interactions of the Dirac field in 1+1 dimensions. It evinces a non-trivial zero-temperature phase structure in the mass-interaction plane, with gapless (critical) and gapped (massive) phases separated by a Berezinskii-Kosterlitz-Thousless-type transition. To investigate this phase structure, we examined the entanglement entropy, the fermion bilinear condensate and two types of correlation functions. We also show our preliminary results of the real-time dynamics of the Thirring model, using variational uniform MPS and time-dependent variational principle. Co-Authors: Mari Carmen Banuls, Max-Planck Institute of Quantum Optics Hao-Ti Hung, National Taiwan University Ying-Jer Kao, National Taiwan University C.-J. David Lin, National Chiao-Tung University Yu-Ping Lin, University of Colorado at Boulder Amit Singh, National Chiao-Tung University David T. L. Tan, National Chiao-Tung University

A measurement-based variational quantum eigensolver

• Luca Dellantonio (University of Waterloo)

In this talk I will speak about the meeting point of two models that have raised interest in the community in the last years. From one side, we looked at measurement-based quantum computing (MBQC), which is an alternative to circuit-based quantum computing. Instead of modifying a state via gates, MBQC achieves the same result by measuring auxiliary qubits in a graph. From the other side, we considered variational quantum eigensolvers (VQEs), that are one of the most successful tools for exploiting quantum computers in the NISQ era. In our work, we present two measurement-based VQE schemes. The first introduces a new approach for constructing variational families. The second provides a translation of circuit-based to measurement-based schemes. Both schemes offer problem-specific advantages in terms of the required resources and coherence times. We apply them, respectively, to the Schwinger model and the two-dimensional Z(2) lattice gauge theory.

Extracting universal data of critical quantum spin chains from periodic uniform matrix product states

• Yijian Zou (X, the Moonshot Factory)

We explain how periodic uniform matrix product states (puMPS) can be used to extract universal data of critical quantum spin chains. We show that puMPS and puMPS Bloch states accurately capture the ground state and low-energy excited states of critical quantum spin chains up to several hundreds of spins. This enables us to extract (i) scaling dimensions, conformal spins of scaling operators, (ii) spectral renormalization group flows between CFTs, (iii) lattice realizations of scaling operators, and operator product expansion coefficients, (iv) emergent symmetries such as superconformal symmetry, and (v) universal tripartite entanglement of the ground state, including entanglement of purification and reflected entropy. We give examples how each of these is achieved and discuss potential applications to open problems. In collaboration with Ash Milsted, Mike Zaletel, Guifre Vidal

Path-integral Optimization and AdS/CFT

• Tadashi Takayanagi (Yukawa Institute for Theoretical Physics)

In this talk we will start with a review of path-integraloptimization, which provides a useful description of non-unitary tensor networks for Euclidean path-integrals in CFTs. We will explain an emergence of AdS geometry in this method and an interpretation as a computational complexity. Next we will give its application to analytical calculations of entanglement of purification, which was quite recently reproduced by numerical calculations. Finally, we would like to present a derivation of a path-integral optimization method directly from the AdS/CFT.

Unconstrained tree tensor simulations for high-dimensional quantum many-body simulations

• Simone Montangero (Padova University)

We present some recent results on the development of efficient unconstrained tree tensor networks algorithms and their application to high-dimensional many-body quantum systems. In particular, we present our results on topological two-dimensional systems, two-dimensional Rydberg atom systems, and two- and three-dimensional lattice gauge theories in presence of fermonic matter. Finally, we present their application to the study of open many-body quantum systems and in particular to the computation of the entanglement of formation in critical many-body quantum systems, resulting in the generalization of the Calabrese-Cardy formula to open systems.

An overview of Wavelets and MERA

• Glen Evenbly (University of Sherbrooke)

The use of wavelet-based constructions has led to significant progress in the analytic understanding of holographic tensor networks, such as the multi-scale entanglement renormalization ansatz (MERA). In this talk I will give an overview of the (past and more recently established) connections between wavelets and MERA, and the discuss the important results that have followed. I will also discuss work currently underway that exploits the wavelet-MERA connection in order to produce new families of wavelets that are optimal for certain tasks, such as image compression.