Toy Models of Holographic Duality between local Hamiltonians

TL;DR

This paper develops a new duality model linking local Hamiltonians in the bulk and boundary within holographic quantum error correcting codes, enabling better understanding of emergent geometry and black hole dynamics.

Contribution

It introduces a method to map local Hamiltonians between bulk and boundary in HQECC, enhancing the toy models of holographic duality with dynamic and energy scale features.

Findings

Constructed a bulk-boundary Hamiltonian mapping preserving HQECC features

Demonstrated emergence of bulk geometry as an effective low-energy theory

Modeled dynamic formation of toy black holes within the duality framework

Abstract

Holographic quantum error correcting codes (HQECC) have been proposed as toy models for the AdS/CFT correspondence, and exhibit many of the features of the duality. HQECC give a mapping of states and observables. However, they do not map local bulk Hamiltonians to local Hamiltonians on the boundary. In this work, we combine HQECC with Hamiltonian simulation theory to construct a bulk-boundary mapping between local Hamiltonians, whilst retaining all the features of the HQECC duality. This allows us to construct a duality between models, encompassing the relationship between bulk and boundary energy scales and time dynamics. It also allows us to construct a map in the reverse direction: from local boundary Hamiltonians to the corresponding local Hamiltonian in the bulk. Under this boundary-to-bulk mapping, the bulk geometry emerges as an approximate, low-energy, effective theory living in the code-space of an (approximate) HQECC on the boundary. At higher energy scales, this emergent bulk geometry is modified in a way that matches the toy models of black holes proposed previously for HQECC. Moreover, the duality on the level of dynamics shows how these toy-model black holes can form dynamically.