Approximate Bacon-Shor Code and Holography

Understand

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.