Approximate Bacon-Shor Code and Holography

Abstract

We construct an explicit and solvable toy model for the AdS/CFT correspondence in the form of an approximate quantum error correction code with a non-trivial center 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 in the form of 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.

Fault-Tolerant Logical Gates in Holographic Stabilizer Codes Are Severely Restricted

“…We also extend this argument to approximate codes such as that in Ref. [9], and to codes whose logical algebras may have non-trivial centers [8]. Finally, in Section IV G we show variants of this argument for other levels of the Clifford hierarchy.…”

“…Instead, it is best understood as an approximate code [5]. One family of toy models [9] attempt to capture this aspect of AdS/CFT in terms of an encoding map that is only approximately equal to an encoding isometry of a stabilizer code. Specifically, the map V : H L → H P is given by…”

“…It was observed in Ref. [5] that this duality has many properties in common with quantum error-correcting codes, which motivated the development of several families of error-correcting codes built out of tensor networks [6][7][8][9][10][11][12][13][14] as toy models for the holographic correspondence. These holographic codes encode information into a series of degrees of freedom which are increasingly non-local, with the degree of non-locality manifesting in the logical system as an additional emergent geometric dimension.…”

Holographic tensor network models and quantum error correction: a topical review

“…Two plausible directions for constructing more general codes is to consider tensors describing more general encoding isometries and to consider tilings that are more complicated than simple regular {n, k} ones. One recent approach combining both directions is presented in reference [145], where the tensors are chosen to represent a Bacon-Shor code which generalizes quantum error-correcting codes by including gauge degrees of freedom. This code is then embedded into an alternating hyperbolic tiling composed of squares and hexagons where the logical qubits are encoded only on the squares while the hexagons contain perfect tensors without bulk degrees of freedom.…”

“…Figure 12.Tensor network generalizations of HaPPY codes, with logical states represented as red dots. Left: an alternating hyperbolic tiling of squares and hexagons used in reference[145] with logical degrees of freedom encoded in a Bacon-Shor code on the squares. Right: the black hole geometry from references[121,146], where a central tensor is removed, leading to additional logical 'horizon' degrees of freedom on the remaining open edges.…”