Abstract
We explicitly construct a class of holographic quantum error correction codes with non-trivial centers in the code subalgebra. Specifically, we use the Bacon-Shor codes and perfect tensors to construct a gauge code (or a stabilizer code with gauge-fixing), which we call the holographic hybrid code. This code admits a local log-depth encoding/decoding circuit, and can be represented as a holographic tensor network which satisfies an analog of the Ryu-Takayanagi formula and reproduces features of the sub-region duality. We then construct approximate versions of the holographic hybrid codes by “skewing” the code subspace, where the size of skewing is analogous to the size of the gravitational constant in holography. These approximate hybrid codes are not necessarily stabilizer codes, but they can be expressed as the superposition of holographic tensor networks that are stabilizer codes. For such constructions, different logical states, representing different bulk matter content, can “back-react” on the emergent geometry, resembling a key feature of gravity. The locality of the bulk degrees of freedom becomes subspace-dependent and approximate. Such subspace-dependence is manifest from the point of view of the “entanglement wedge” and bulk operator reconstruction from the boundary. Exact complementary error correction breaks down for certain bipartition of the boundary degrees of freedom; however, a limited, state-dependent form is preserved for particular subspaces. We also construct an example where the connected two-point correlation functions can have a power-law decay. Coupled with known constraints from holography, a weakly back-reacting bulk also forces these skewed tensor network models to the “large N limit” where they are built by concatenating a large N number of copies.
Highlights
The AdS/CFT correspondence is a concrete implementation of the holographic principle [1, 2], which connects a d-dimensional conformal field theory (CFT) on Minkowski spacetime with a theory of quantum gravity in d + 1 dimensional asymptotically anti-de Sitter (AdS) space [3, 4]
These include the breakdown of exact complementary recovery in the hybrid code, the subspace-dependence of entanglement wedge and bulk operator reconstruction, and back-reactions where different logical states can lead to different “semi-classical geometries” in the bulk
For the quantum gravity audience who have been following the developments in AdS/CFT and quantum error correction codes, the hybrid code that we construct out of the BaconShor code and the perfect tensor is in many ways similar to a version of the HaPPY code
Summary
The AdS/CFT correspondence is a concrete implementation of the holographic principle [1, 2], which connects a d-dimensional conformal field theory (CFT) on Minkowski spacetime with a theory of quantum gravity in d + 1 dimensional asymptotically anti-de Sitter (AdS) space [3, 4]. Generalizations are made in different directions by [22,23,24,25,26] These tensor network models, other than making geometric connections with AdS/CFT, provide a graphical approach to understanding the construction and properties of quantum error correction codes. In addition to preserving certain properties of the reference code, the generalized code displays some key features of gravity, consistent with our expectations in holography when one considers higher order corrections and quantum extremal surfaces These include the breakdown of exact complementary recovery in the hybrid code, the subspace-dependence of entanglement wedge and bulk operator reconstruction, and back-reactions where different logical states can lead to different “semi-classical geometries” in the bulk. We provide detailed procedures on how the check matrices of the new codes are generated from the gluing operations
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