Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence

By Fernando Pastawski, B. Yoshida, D. Harlow, John Preskill. 2015

972 citations recorded by the source corpus.

A bstractWe propose a family of exactly solvable toy models for the AdS/CFT correspondence based on a novel construction of quantum error-correcting codes with a tensor network structure. Our building block is a special type of tensor with maximal entanglement along any bipartition, which gives rise to an isometry from the bulk Hilbert space to the boundary Hilbert space. The entire tensor network is an encoder for a quantum error-correcting code, where the bulk and boundary degrees of freedom may be identified as logical and physical degrees of freedom respectively. These models capture key features of entanglement in the AdS/CFT correspondence; in particular, the Ryu-Takayanagi formula and the negativity of tripartite information are obeyed exactly in many cases. That bulk logical operators can be represented on multiple boundary regions mimics the Rindlerwedge reconstruction of boundary operators from bulk operators, realizing explicitly the quantum error-correcting features of AdS/CFT recently proposed in [1].

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/a39d534dcdb8ac4c4a050977686280ac35b0aa08

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. A. Nahum
2. Brian Swingle
3. Nicholas Hunter-Jones
4. J. Ruhman
5. B. Skinner
6. S. Vijay
7. Tianci Zhou
8. B. Yoshida

In Atomic and Molecular Physics, and Optics

1. A. Nahum
2. D. Harlow
3. Brian Swingle
4. D. Stanford
5. S. Shenker
6. R. Myers
7. Nicholas Hunter-Jones
8. S. Vijay

Citations in the corpus

• Measurement and Entanglement Phase Transitions in All-To-All Quantum Circuits, on Quantum Trees, and in Landau-Ginsburg Theory (2020, background): “This minimal cut is a unifying heuristic [34, 50, 64, 68–70] for the entanglement of various objects, depending on how we choose A and Ā.”
• Matrix product states and projected entangled pair states: Concepts, symmetries, theorems (2020, methodology): “More recently, tensor networks have been used to construct toy models of holographic principles in hyperbolic geometries (Hayden et al., 2016; Pastawski et al., 2015; Swingle, 2012).”
• Absolutely maximally entangled states, combinatorial designs, and multiunitary matrices (2015, neutral): “There is yet another surprising connection between AME states and holography [24,25].”
• Hand-waving and interpretive dance: an introductory course on tensor networks (2016, background): “In addition to being a concrete realisation of the holographic principle, the MERA/CFT duality provides avenues towards designing quantum error correcting codes [95].”
• Chaos and complexity by design (2016, background): “Examples of stabilizer states include ground states of the toric code and the perfect tensors used in construction of holographic quantum error-correcting codes [56].”
• Jerusalem Lectures on Black Holes and Quantum Information (2014, background): “The regime of validity of this construction of local bulk fields has recently been reinterpreted in the language of quantum error correction, a beautiful subject that is unfortunately beyond the scope of these notes [106, 107].”
• The ITensor Software Library for Tensor Network Calculations (2020, background): “ing holographic dualities between physical theories [43, 44] to computing high-dimensional integrals [22,45].”
• Replica wormholes and the black hole interior (2019, result): “ce both have at entanglement spectra at leading order. However, our calculations suggest that the connection to random tensor networks specically (as opposed, for example, to perfect tensor networks [99]) goes much deeper than might previously suspected, with non-perturbative instanton corrections t”
• Lessons from the information paradox (2020, background): “To address these paradoxes, we review how the black-hole interior can be con- structed in AdS/CFT.”
• Chaos, complexity, and random matrices (2017, methodology): “Motivated by tensor network descriptions of the AdS/CFT correspondence [42, 43], random local quantum circuits have been used as a toy model of the Einstein-Rosen bridge and the dynamics of the two-sided AdS black hole [15].”
• The holographic entropy cone (2015, methodology): “This is rather suggestive in view of the recently proposed connections between holography and quantum error correcting codes [33, 49], and it is likely that this observation can be proved for arbitrary n by generalizing the construction of [48].”
• Wormholes, emergent gauge fields, and the weak gravity conjecture (2015, background): “Based on recent developments [12, 70, 71], I am optimistic this will happen soon.”
• The Ryu–Takayanagi Formula from Quantum Error Correction (2016, methodology): “Why should we treat this entropy differently before and after the collapse [83]? One possible resolution of all this would be to avoid considering a code subspace that contains all of the microstates in a fixed energy band, but this is somewhat unsatisfying, especially since in section six of [3] it”
• Holographic Entanglement Entropy (2016, methodology): “ conjecture as was originally argued using ideas borrowed from quantum error correction by Almheiri, Dong, and Harlow (ADH) [173] and beautifully illustrated in toy models built using tensor networks [174] (which has come to be known as the HaPPY code after the authors). Building on these ideas and ”