Tensor Networks: from Simulations to Holography III
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.
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.
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.
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.
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.
"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.
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.
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.