Here is a list of holographic codes.
Code Description
Concatenated Steane code A member of the family of \([[7^m,1,3^m]]\) CSS codes, each of which is a recursive level-\(m\) concatenatenation of the Steane code. This family is one of the first to admit a concatenated threshold [1–5].
Conformal-field theory (CFT) code Approximate code whose codewords lie in the low-energy subspace of a conformal field theory, e.g., the quantum Ising model at its critical point [6,7]. Its encoding is argued to perform source coding (i.e., compression) as well as channel coding (i.e., error correction) [6].
Five-qubit perfect code Five-qubit cyclic stabilizer code that is the smallest qubit stabilizer code to correct a single-qubit error.
Heptagon holographic code Holographic tensor-network code constructed out of a network of encoding isometries of the Steane code. Depending on how the isometry tensors are contracted, there is a zero-rate and a finite-rate code family.
Holographic code Block quantum code whose features serve to model aspects of the AdS/CFT holographic duality and, more generally, quantum gravity.
Holographic hybrid code Holographic tensor-network code constructed out of alternating isometries of the five-qubit and \([[4,1,1,2]]\) Bacon-Shor codes.
Holographic tensor-network code Quantum Lego code whose encoding isometry forms a holographic tensor network, i.e., a tensor network associated with a tiling of 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 number of layers emanating form the central point of the tiling is the radius of the code.
Hyperinvariant tensor-network (HTN) code Holographic tensor-network error-detecting code constructed out of a hyperinvariant tensor network [8], i.e., a MERA-like network admitting a hyperbolic geometry. The network is defined using two layers A and B, with constituent tensors satisfying isometry conditions (a.k.a. multitensor constraints).
Kim-Preskill-Tang (KPT) code A quantum error-correcting code that protects the encoded interior of a black hole from computationally bounded exterior observers. Under the assumption that the Hawking radiation emitted by an old black hole is pseudorandom, there exists a subspace of the radiation system that encodes the black hole interior, entangled with the late outgoing Hawking quanta. The logical operators of this code commute with efficient operations acting on the radiation, protecting the interior up to corrections exponentially small in the black hole's entropy.
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 Holographic code constructed out of a network of hexagonal perfect tensors that tesselates hyperbolic space. The code serves as a minimal model for several aspects of the AdS/CFT holographic duality [9] and potentially a dF/CFT duality [10]. It has been generalized to higher dimensions [11] and to include gauge-like degrees of freedom on the links of the tensor network [12,13]. All boundary global symmetries must be dual to bulk gauge symmetries, and vice versa [14].
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 [15,16] or other low-rank SYK models [17,18].
Six-qubit-tensor holographic code Holographic tensor-network code constructed out of a network of encoding isometries of the \([[6,1,3]]\) six-qubit stabilizer code. The structure of the isometry is similar to that of the heptagon holographic code since both isometries are rank-six tensors, but the isometry in this case is neither a perfect tensor nor a planar-perfect tensor.
Surface-code-fragment (SCF) holographic code Holographic tensor-network code constructed out of a network of encoding isometries of the \([[5,1,2]]\) rotated surface code. The structure of the isometry is similar to that of the HaPPY code since both isometries are rank-six tensors. In the case of the SCF holographic code, the isometry is only a planar-perfect tensor (as opposed to a perfect tensor).
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.
\([[5,1,2]]\) rotated surface code Rotated surface code on one rung of a ladder, with one qubit on the rung, and four qubits surrounding it.
\([[6,1,3]]\) Six-qubit stabilizer code One of two six-qubit distance-three codes that are unique up to equivalence [19], with the other code a trivial extension of the five-qubit code [20]. Stabilizer generators and logical Pauli operators are presented in Ref. [20].
\([[7,1,3]]\) Steane code A \([[7,1,3]]\) self-dual CSS code that is the smallest qubit CSS code to correct a single-qubit error [20]. The code is constructed using the classical binary \([7,4,3]\) Hamming code for protecting against both \(X\) and \(Z\) errors.

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