Appendix B: Holographic Error Correction and Spacetime Robustness

In Volume II “Protocol” of Vector Cosmology VI, we proposed a subversive viewpoint: Physical laws are the checksum algorithms of the cosmic operating system, and the robustness of spacetime originates from the redundant encoding of holographic error-correcting codes.

This appendix will delve into the mathematical core of this theory, introducing the famous HaPPY Code model (proposed by Harlow, Pastawski, Preskill, Yoshida). We will show how to use tensor networks to non-locally encode bulk logical qubits into boundary physical qubits, thus endowing spacetime with astonishing “Antifragility”.

B.1 The HaPPY Code: Tensor Networks as Encoders

To understand how spacetime tolerates errors, we need to construct a specific tensor network model.

The HaPPY code uses the Five-qubit Code as the basic building block, tiling it on a discrete tensor network of hyperbolic geometry (AdS space).

• Basic unit: Each tensor node represents an Isometry $T$. It maps 1 bulk qubit (logical bit) and several auxiliary bits to 5 boundary-direction output bits.

• Holographic encoding: When we stack these nodes layer by layer, forming a fractal structure similar to MERA, the information of a logical qubit located at the center (bulk interior) is dispersed and transmitted to countless physical bits at the outermost layer (boundary).

Mathematical property:

This is a holographic version of a [[n, k, d]] quantum error-correcting code.

$$|{\psi }_{bulk}⟩\stackrel{\text{Encoding}}{\to }|{\Psi }_{boundary}⟩$$

The boundary state $|{\Psi }_{boundary}⟩$ contains all information of the bulk state $|{\psi }_{bulk}⟩$, but no single boundary pixel has complete information.

B.2 Entanglement Wedge and Subregion Duality

The core mechanism of holographic error correction lies in “Subregion Duality”.

This is like: You don’t need to have the entire holographic plate; you only need a small fragment to restore part of the object’s perspective.

In AdS/CFT, this is expressed as the Entanglement Wedge Reconstruction theorem.

• Boundary region $R$: Suppose we can only access a region $R$ on the boundary.

• Entanglement wedge $W_E(R)$: Is the spatial region in the bulk enclosed by the minimal surface (RT surface) of $R$.

Theorem:

Only bulk operators ${\varphi }_{bulk}$ located inside the entanglement wedge $W_E(R)$ can be reconstructed by operators ${\mathcal{O}}_R$ on the boundary region $R$.

$${\varphi }_{bulk}\approx U^{†}{\mathcal{O}}_{R}U$$

Physical meaning:

• Information redundancy: A point deep in the bulk (e.g., a particle near the black hole center) has its information “smeared” across almost the entire boundary. Therefore, to reconstruct it, you need to access more than half of the boundary region.

• Robustness: If you lose data from a small region on the boundary (local error or erasure), objects deep in the bulk are completely unaffected. Because their information is still backed up in other regions of the boundary.

This is why our universe is so robust.

Even if local spacetime pixels “die” due to vacuum fluctuations, as long as the global entanglement structure remains, physical laws can instantly “interpolate” and repair the dead regions through error correction algorithms.

B.3 Radial Causality and Logical Gate Protection

Finally, we look at the protective role of “depth” ($z$-axis).

In HaPPY code networks, each step from the boundary toward the interior is actually passing through a layer of error-correcting logic gates.

• Shallow layers: Represent high-frequency, short-range entanglement. Susceptible to ultraviolet noise interference.

• Deep layers: Represent low-frequency, long-range entanglement. Have undergone “Majority Vote” and “checksum” through layers of logic gates.

Depth is protection.

The deeper into the bulk interior (away from the boundary), the higher the protection level of information, which explains why macroscopic physical laws (low-energy effective theories) are much more stable than microscopic quantum fluctuations.

• Gravity: Here is not just geometric curvature; it is the “cost function” of error-correcting codes.

• When we move an object in deep space, we are actually changing the position of logical bits in the tensor network. This operation requires coordination across a wide range of boundaries, thus manifesting as a “long-range force.”

Conclusion:

Spacetime is not empty. Spacetime is a set of “self-correcting quantum algorithms”.

We exist because the Hamming Distance of this algorithm is large enough to tolerate Planck-scale wild fluctuations without causing logical collapse of the macroscopic world.