Here is a list of holographic codes.

Code | Description |
---|---|

Holographic code | Block quantum code whose features (typically, the encoding isometry) serve to model aspects of the AdS/CFT holographic duality. |

Matrix-model code | Multimode-mode Fock-state bosonic approximate code derived from a matrix model, i.e., a non-Abelian bosonic gauge theory with a large gauge group. The model's degrees of freedom are matrix-valued bosons \(a\), each consisting of \(N^2\) harmonic oscillator modes and subject to an \(SU(N)\) gauge symmetry. |

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 [1]. The construction below is described for qubits, but straightforward generalizations exist to modular qudits, oscillators, and rotors [2]. |

Renormalization group (RG) cat code | Code whose codespace is spanned by \(q\) field-theoretic coherent states which are flowing under the renormalization group (RG) flow of massive free fields. The code approximately protects against displacements that represent local (i.e., short-distance, ultraviolet, or UV) operators. Intuitively, this is because RG cat codewords represent non-local (i.e., long-distance) degrees of freedom, which should only be excitable by acting on a macroscopically large number of short-distance degrees of freedom. |

SYK code | Approximate \(n\)-fermionic code whose codewords are low-energy states of the Sachdev-Ye-Kitaev (SYK) Hamiltonian [3,4] or other low-rank SYK models [5,6]. |

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. |

## References

- [1]
- J. Cotler and A. Strominger, “The Universe as a Quantum Encoder”, (2022) arXiv:2201.11658
- [2]
- P. Faist et al., “Continuous Symmetries and Approximate Quantum Error Correction”, Physical Review X 10, (2020) arXiv:1902.07714 DOI
- [3]
- S. Sachdev and J. Ye, “Gapless spin-fluid ground state in a random quantum Heisenberg magnet”, Physical Review Letters 70, 3339 (1993) arXiv:cond-mat/9212030 DOI
- [4]
- Kitaev, Alexei. "A simple model of quantum holography (part 2)." Entanglement in Strongly-Correlated Quantum Matter (2015): 38.
- [5]
- J. Kim, X. Cao, and E. Altman, “Low-rank Sachdev-Ye-Kitaev models”, Physical Review B 101, (2020) arXiv:1910.10173 DOI
- [6]
- J. Kim, E. Altman, and X. Cao, “Dirac fast scramblers”, Physical Review B 103, (2021) arXiv:2010.10545 DOI