Here is a list of holographic codes.
| Code | Description |
|---|---|
| Holographic code | Block quantum code whose encoding isometry serves to model aspects of the AdS/CFT holographic duality. Encoding map models radial time evolution for a fixed time slice in Anti de Sitter (AdS) space, mapping operators in the bulk of AdS, represented by logical qudits, onto operators on the boundary of the corresponding Conformal Field Theory (CFT), represented by physical qudits. Encoding can often be represented by a tensor network associated with a tiling of hyperbolic space. See Defn 4.3 of Ref. [1] for a technical formulation. |
| Pastawski-Yoshida-Harlow-Preskill (HaPPY) code | Also known as a hyperbolic pentagon code (HyPeC). Holographic code constructed out of a network of perfect tensors that tesselates hyperbolic space. Physical qubits are associated with uncontracted tensor legs at the boundary of the tesselation, while logical qubits are associated with uncontracted legs in the bulk. The code serves as a minimal model for several aspects of the AdS/CFT holographic duality and potentially a dF/CFT duality [2]. The construction below is described for qubits, but straightforward generalizations exist to modular qudits, oscillators, and rotors [3]. |
| Three-qutrit code | A \([[3,1,2]]_3\) prime-qudit CSS code that is the smallest qutrit stabilizer code to detect a single-qutrit error. with stabilizer generators \(ZZZ\) and \(XXX\). The code defines a quantum secret-sharing scheme and serves as a minimal model for the AdS/CFT holographic duality. It is also the smallest non-trivial instance of a quantum maximum distance separable code (QMDS), saturating the quantum Singleton bound. The codewords are \begin{align} \begin{split} | \overline{0} \rangle &= \frac{1}{\sqrt{3}} (| 000 \rangle + | 111 \rangle + | 222 \rangle) \\ | \overline{1} \rangle &= \frac{1}{\sqrt{3}} (| 012 \rangle + | 120 \rangle + | 201 \rangle) \\ | \overline{2} \rangle &= \frac{1}{\sqrt{3}} (| 021 \rangle + | 102 \rangle + | 210 \rangle)~. \end{split} \tag*{(1)}\end{align} The elements in the superposition of each logical codeword are related to each other via cyclic permutations. |