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arXiv:2112.12468

Holography, cellulations and error correcting codes

Marika Taylor, Woodward, Charles

Published Dec 23, 2021• 2 citations
arXivHTMLPDF

Abstract

Quantum error correction codes associated with the hyperbolic plane have been explored extensively in the context of the AdS$_3$/CFT$_2$ correspondence. In this paper we initiate a systematic study of codes associated with holographic geometries in higher dimensions, relating cellulations of the spatial sections of the geometries to stabiliser codes. We construct analogues of the HaPPY code for three-dimensional hyperbolic space (AdS$_4$), using both absolutely maximally entangled (AME) and non-AME codes. These codes are based on uniform regular tessellations of hyperbolic space but we note that AME codes that preserve the discrete symmetry of the polytope of the tessellation do not exist above two dimensions. We also explore different constructions of stabiliser codes for hyperbolic spaces in which the logical information is associated with the boundary and discuss their potential interpretation. We explain how our codes could be applied to interesting classes of holographic dualities based on gravity-scalar theories (such as JT gravity) through toroidal reductions of hyperbolic spaces.

DOI
10.48550/arxiv.2112.12468

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Quantum error correction codes associated with the hyperbolic plane have been explored extensively in the context of the AdS3_33​/CFT2_22​ correspondence. In this paper we initiate a systematic study of codes associated with holographic geometries in higher dimensions, relating cellulations of the spatial sections of the geometries to stabiliser codes. We construct analogues of the HaPPY code for three-dimensional hyperbolic space (AdS4_44​), using both absolutely maximally entangled (AME) and non-AME codes. These codes are based on uniform regular tessellations of hyperbolic space but we note that AME codes that preserve the discrete symmetry of the polytope of the tessellation do not exist above two dimensions. We also explore different constructions of stabiliser codes for hyperbolic spaces in which the logical information is associated with the boundary and discuss their potential interpretation. We explain how our codes could be applied to interesting classes of holographic dualities based on gravity-scalar theories (such as JT gravity) through toroidal reductions of hyperbolic spaces.
Holographic quantum error-correcting codes have been proposed as toy models that describe key aspects of the anti-de Sitter/conformal field theory (AdS/CFT) correspondence. In this work, we introduce a versatile framework of Majorana dimers capturing the intersection of stabilizer and Gaussian Majorana states. This picture allows for an efficient contraction with a simple diagrammatic interpretation and is amenable to analytical study of holographic quantum error-correcting codes. Equipped with this framework, we revisit the recently proposed hyperbolic pentagon code (HyPeC). Relating its logical code basis to Majorana dimers, we efficiently compute boundary-state properties even for the non-Gaussian case of generic logical input. The dimers characterizing these boundary states coincide with discrete bulk geodesics, leading to a geometric picture from which properties of entanglement, quantum error correction, and bulk/boundary operator mapping immediately follow. We also elaborate upon the emergence of the Ryu-Takayanagi formula from our model, which realizes many of the properties of the recent bit thread proposal. Our work thus elucidates the connection among bulk geometry, entanglement, and quantum error correction in AdS/CFT and lays the foundation for new models of holography.
Holographic quantum-error correcting codes are models of bulk/boundary dualities such as the anti-de Sitter/conformal field theory (AdS/CFT) correspondence, where a higher-dimensional bulk geometry is associated with the code's logical degrees of freedom. Previous discrete holographic codes based on tensor networks have reproduced the general code properties expected from continuum AdS/CFT, such as complementary recovery. However, the boundary states of such tensor networks typically do not exhibit the expected correlation functions of CFT boundary states. In this work, we show that a new class of exact holographic codes, extending the previously proposed hyperinvariant tensor networks into quantum codes, produce the correct boundary correlation functions. This approach yields a dictionary between logical states in the bulk and the critical renormalization group flow of boundary states. Furthermore, these codes exhibit a state-dependent breakdown of complementary recovery as expected from AdS/CFT under small quantum gravity corrections.
Holographic quantum-error correcting codes are models of bulk/boundary dualities such as the anti-de Sitter/conformal field theory (AdS/CFT) correspondence, where a higher-dimensional bulk geometry is associated with the code's logical degrees of freedom. Previous discrete holographic codes based on tensor networks have reproduced the general code properties expected from continuum AdS/CFT, such as complementary recovery. However, the boundary states of such tensor networks typically do not exhibit the expected correlation functions of CFT boundary states. In this work, we show that a new class of exact holographic codes, extending the previously proposed hyperinvariant tensor networks into quantum codes, produce the correct boundary correlation functions. This approach yields a dictionary between logical states in the bulk and the critical renormalization group flow of boundary states. Furthermore, these codes exhibit a state-dependent breakdown of complementary recovery as expected from AdS/CFT under small quantum gravity corrections.
The unification of quantum mechanics and general relativity remains one of the major open problems of theoretical physics. The Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence provides a valuable theoretical framework for this effort via a holographic duality between a theory of quantum gravity in asymptotically AdS spacetime and a conformal quantum field theory on the lower-dimensional boundary. Here, we implement a toy model of this duality called the HaPPY code, a quantum error-correcting code in the form of a tensor network with hyperbolic entanglement patterns, on a trapped-ion quantum computer. We present the first experimental confirmation of the Faulkner-Lewkowycz-Maldacena formula in this model - a key test of the holographic correspondence. We then enrich it with non-stabilizerness, or magic, and observe entropic precursors expected of emergent gravity. Finally, we present and measure a code construction whose entropic behavior is reminiscent of a highly quantum wormhole. Our experiments illustrate how quantum computers can serve as testbeds for modeling the emergence of spacetime.
The AdS/CFT correspondence conjectures a holographic duality between gravity in a bulk space and a critical quantum field theory on its boundary. Tensor networks have come to provide toy models to understand these bulk-boundary correspondences, shedding light on connections between geometry and entanglement. We introduce a versatile and efficient framework for studying tensor networks, extending previous tools for Gaussian matchgate tensors in 1 + 1 dimensions. Using regular bulk tilings, we show that the critical Ising theory can be realized on the boundary of both flat and hyperbolic bulk lattices, obtaining highly accurate critical data. Within our framework, we also produce translation-invariant critical states by an efficiently contractible tensor network with the geometry of the multiscale entanglement renormalization ansatz. Furthermore, we establish a link between holographic quantum error-correcting codes and tensor networks. This work is expected to stimulate a more comprehensive study of tensor network models capturing bulk-boundary correspondences.
Preskill et. al. suggest a highly interesting representation of holography in terms of quantum error correction codes The idea is that time= constant section of AdS, which is hyperbolic space allowing tessellations, can define tensor networks. So called perfect tensors are building bricks of the tensor networks providing representation for holography and at the same time defining error correcting codes by mapping localized interior states (logical qubits) to highly entangled non-local boundary states (physical qubits). There are three observations that put bells ringing and actually motivated this article.