Here is a list of approximate quantum codes.

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

Approximate quantum error-correcting code (AQECC) | Encodes quantum information so that it is possible to approximately recover that information from noise up to an error bound in recovery. |

Approximate secret-sharing code | A family of \( [[n,k,d]]_q \) CSS codes approximately correcting errors on up to \(\lfloor (n-1)/2 \rfloor\) qubits, i.e., with approximate distance approaching the no-cloning bound \(n/2\). Constructed using a non-degenerate CSS code, such as a polynomial quantum code, and a classical authentication scheme. The code can be viewed as an \(t\)-error tolerant secret sharing scheme. Since the code yields a small logical subspace using large registers that contain both classical and quantum information, it is not useful for practical error correction problems, but instead demonstrates the power of approximate quantum error correction. |

Covariant code | A block code on \(n\) subsystems that admits a group \(G\) of transversal gates. The group has to be finite for finite-dimensional codes due to the Eastin-Knill theorem. Continuous-\(G\) covariant codes, necessarily infinite-dimensional, are relevant to error correction of quantum reference frames [1] and error-corrected parameter estimation. |

Eigenstate thermalization hypothesis (ETH) code | Also called a thermodynamic code [2]. An \(n\)-qubit approximate code whose codespace is formed by eigenstates of a translationally-invariant quantum many-body system which satisfies the Eigenstate Thermalization Hypothesis (ETH). ETH ensures that codewords cannot be locally distinguished in the thermodynamic limit. Relevant many-body systems include 1D non-interacting spin chains, Motzkin chains, or Heisenberg models. |

GNU permutation-invariant code | Permutation-invariant code whose codewords can be expressed as superpositions of Dicke states with coefficients are square-roots of the binomial distribution. |

Gottesman-Kitaev-Preskill (GKP) code | Quantum lattice code for a non-degenerate lattice, thereby admitting a finite-dimensional logical subspace. Codes on \(n\) modes can be constructed from lattices with \(2n\)-dimensional full-rank Gram matrices \(A\). |

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

Landau-level spin code | Approximate quantum code that encodes a qudit in the finite-dimensional Hilbert space of a single spin, i.e., a spherical Landau level. Codewords are approximately orthogonal Landau-level spin coherent states whose orientations are spaced maximally far apart along a great circle (equator) of the sphere. The larger the spin, the better the performance. |

Local Haar-random circuit qubit code | An \(n\)-qubit code whose codewords are a pair of approximately locally indistinguishable states produced by starting with any two orthogonal \(n\)-qubit states and acting with a random unitary circuit of depth polynomial in \(n\). Two states are locally indistinguishable if they cannot be distinguished by local measurements. A single layer of the encoding circuit is composed of about \(n/2\) two-qubit nearest-neighbor gates run in parallel, with each gate drawn randomly from the Haar distribution on two-qubit unitaries. |

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. |

Matrix-product state (MPS) code | Also called a magnon code. An \(n\)-qubit approximate code whose codespace of \(k=\Omega(\log n)\) qubits is efficiently described in terms of matrix product states (MPS) or Bethe ansatz tensor networks. A no-go theorem states that open-boundary MPS that form a degenerate ground-state space of a gapped local Hamiltonian yield codes with distance that is only constant in the number of qubits \(n\), so MPS excitation ansatze have to be used to achieve a distance scaling nontrivially with \(n\). |

Neural network code | An approximate code obtained from a numerical optimization involving a reinforcement learning agent. |

Numerically optimized bosonic code | Bosonic Fock-state code obtained from a numerical minimization procedure, e.g., from enforcing error-correction criteria against some number of losses while minimizing average occupation number. Useful single-mode codes can be determined using basic numerical optimization [3,4], semidefinite-program recovery/encoding optimization [5,6], or reinforcement learning [7,8]. |

Quantum low-weight check (QLWC) code | Member of a family of \([[n,k,d]]\) modular-qudit or Galois-qudit stabilizer codes for which the number of sites participating in each stabilizer generator is bounded by a constant as \(n\to\infty\). |

Qudit-into-oscillator code | Encodes \(K\)-dimensional Hilbert space into \(n\) bosonic modes. |

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 [9,10] or other low-rank SYK models [11,12]. |

Singleton-bound approaching AQECC | Approximate quantum code of rate \(R\) that can tolerate adversarial errors nearly saturating the quantum Singleton bound of \((1-R)/2\). The formulation of such codes relies on a notion of quantum list decoding. Sampling a description of this code can be done with an efficient randomized algorithm with \(2^{-\Omega(n)}\) failure probability. |

Square-lattice GKP code | Single-mode GKP qudit-into-oscillator code based on the rectangular lattice. Its stabilizer generators are oscillator displacement operators \(\hat{S}_q(2\alpha)=e^{-2i\alpha \hat{p}}\) and \(\hat{S}_p(2\beta)=e^{2i\beta \hat{x}}\). To ensure \(\hat{S}_q(2\alpha)\) and \(\hat{S}_p(2\beta)\) generate a stabilizer group that is Abelian, there is a constraint that \(\alpha\beta=2q\pi\) where \(q\) is an integer denoting the logical dimension. |

Squeezed fock-state code | Approximate boosnic code that encodes a qubit into the same Fock state, but one which is squeezed in opposite directions. |

W-state code | Encodes a quantum state of a \(d_L\)-dimensional Hilbert space into \(n\) physical quantum systems, each associated with a Hilbert space of dimension \(d_L+1\). The encoding resembles the structure of the W state [13]: \begin{align} \ket\psi \to \frac{1}{\sqrt{n}}\bigl(\ket{\psi\perp\perp\ldots} + \ket{\perp\psi\perp\ldots} + \cdots + \ket{\perp\perp\ldots\psi}\bigr)\ , \tag*{(1)}\end{align} where on each physical system, \(\ket\perp\) denotes the \((d_L+1)\)-th basis state and \(\ket\psi\) is encoded using the first \(d_L\) basis states. |

\(U(d)\)-covariant approximate erasure code | Covariant code whose construction takes in an arbitrary erasure-correcting code to yield an approximate QECC that is also covariant with respect to the unitary group. |

\([[4,2,2]]\) CSS code | Four-qubit CSS stabilizer code is the smallest qubit stabilizer code to detect a single-qubit error. |

## References

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- P. Faist et al., “Continuous Symmetries and Approximate Quantum Error Correction”, Physical Review X 10, (2020) arXiv:1902.07714 DOI
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- M. H. Michael et al., “New Class of Quantum Error-Correcting Codes for a Bosonic Mode”, Physical Review X 6, (2016) arXiv:1602.00008 DOI
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- V. V. Albert et al., “Performance and structure of single-mode bosonic codes”, Physical Review A 97, (2018) arXiv:1708.05010 DOI
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- K. Noh, V. V. Albert, and L. Jiang, “Quantum Capacity Bounds of Gaussian Thermal Loss Channels and Achievable Rates With Gottesman-Kitaev-Preskill Codes”, IEEE Transactions on Information Theory 65, 2563 (2019) arXiv:1801.07271 DOI
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- P. Leviant et al., “Quantum capacity and codes for the bosonic loss-dephasing channel”, Quantum 6, 821 (2022) arXiv:2205.00341 DOI
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- Z. Wang et al., “Automated discovery of autonomous quantum error correction schemes”, (2021) arXiv:2108.02766
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- Y. Zeng et al., “Approximate Autonomous Quantum Error Correction with Reinforcement Learning”, Physical Review Letters 131, (2023) arXiv:2212.11651 DOI
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- 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
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- Kitaev, Alexei. "A simple model of quantum holography (part 2)." Entanglement in Strongly-Correlated Quantum Matter (2015): 38.
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- J. Kim, X. Cao, and E. Altman, “Low-rank Sachdev-Ye-Kitaev models”, Physical Review B 101, (2020) arXiv:1910.10173 DOI
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- J. Kim, E. Altman, and X. Cao, “Dirac fast scramblers”, Physical Review B 103, (2021) arXiv:2010.10545 DOI
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- W. Dür, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways”, Physical Review A 62, (2000) arXiv:quant-ph/0005115 DOI