Holographic Quantum Error-Correcting Codes

Norbert Levente Kis

We establish a quantitative equivalence between the holographic duality of theAdS/CFT correspondence and the theory of quantum error-correcting codes (QECC).By identifying the code subspace with the low-energy bulk effective field theoryand the physical Hilbert space with the boundary CFT, we show that the RyuTakayanagi formula for entanglement entropy directly determines the code distanced —the minimal weight of an undetectable error. We prove a holographic Hammingbound: d ≤ Area(γmin)/(4Gℏln2), with equality for perfect tensor networks. Thegravitational constant G thus quantifies the error-correcting capacity of spacetime.A testable prediction is derived: the code distance for a black hole horizon saturates the bound, explaining the unitarity of Hawking evaporation as a property of amaximum-distance separable code. We further show that the entanglement wedgereconstruction theorem is equivalent to the Knill–Laflamme conditions for QECC

M.A. Steinberg

Tamara Kohler, Toby S. Cubitt

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. \n \nIt 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.

Igor Devetak, Todd A. Brun, Min-Hsiu Hsieh

Tamara Kohler, Toby S. Cubitt

A bstract 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.

Wen-Ching Li