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