Here is a list of approximate quantum codes.
Code Description
Approximate quantum error-correcting code (AQECC) Stub.
Approximate secret-sharing code A family of \( [[n,k,d]]_{GF(q)} \) 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 Code constructed in a physical space consisting of a tensor product of \(n\) subsystems (e.g., qubits, modular qudits, or Galois qudits) 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 Can be expressed in terms of Dicke states where the logical states are \begin{align} |\overline{\pm}\rangle = \sum_{\ell=0}^{n} \frac{(\pm 1)^\ell}{\sqrt{2^n}} \sqrt{n \choose \ell} |D^m_{g \ell}\rangle~. \end{align} Here, \(m\) is the number of particles used for encoding \(1\) qubit, and \(g, n \leq m\) are arbitrary positive integers. The state \(|D^m_w\rangle\) is a Dicke state -- a normalized permutation-invariant state on \(m\) qubits with \(w\) excitations, i.e., a normalized sum over all basis elements with \(w\) ones and \(m - w\) zeroes.
Gottesman-Kitaev-Preskill (GKP) code Bosonic qudit-into-oscillator code whose stabilizers 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 another constraint that \(\alpha\beta=2k\pi\) where \(k\) is an integer. Codewords can be expressed as equal weight superpositions of coherent states on an infinite lattice, such as a square lattice in phase space with spatial period \(2\sqrt{\pi}\). The exact GKP state is non-normalizable, so approximate constructs have to be considered.
Local Haar-random circuit 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. The above circuit elements act on nearest-neighbor qubits arranged in a line, i.e., a one-dimensional geometry (\(D=1\), while codes for higher-dimensional geometries require \(O(n^{1/D})\)-depth circuits [3]. Follow-up work [4] revealed that optimal code properties require only \(O(\sqrt{n})\)-depth circuits for that case, and \(O(\sqrt{n})\)-depth circuits for a two-dimensional square-lattice geometry.
Multi-mode GKP code Generalization of the GKP code to \(n\) bosonic modes whose stabilizer group is an infinite countable group of oscillator displacement operators.
Qudit-into-oscillator code Encodes \(K\)-dimensional Hilbert space into \(n\) bosonic modes.
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 [5]: \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)\ , \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 Stub.
\([[4,2,2]]\) CSS code Also known as the \(C_4\) code. Four-qubit CSS stabilizer code with generators \(\{XXXX, ZZZZ\} \) and codewords \begin{align} \begin{split} |\overline{00}\rangle = (|0000\rangle + |1111\rangle)/\sqrt{2}~{\phantom{.}}\\ |\overline{01}\rangle = (|0011\rangle + |1100\rangle)/\sqrt{2}~{\phantom{.}}\\ |\overline{10}\rangle = (|0101\rangle + |1010\rangle)/\sqrt{2}~{\phantom{.}}\\ |\overline{11}\rangle = (|0110\rangle + |1001\rangle)/\sqrt{2}~. \end{split} \end{align} This code is the smallest single-qubit error-detecting code. It is also the smallest instance of the toric code, and its various single-qubit subcodes are small planar surface codes.


P. Hayden et al., “Error Correction of Quantum Reference Frame Information”, PRX Quantum 2, (2021). DOI; 1709.04471
P. Faist et al., “Continuous Symmetries and Approximate Quantum Error Correction”, Physical Review X 10, (2020). DOI; 1902.07714
F. G. S. L. Brandão, A. W. Harrow, and M. Horodecki, “Local Random Quantum Circuits are Approximate Polynomial-Designs”, Communications in Mathematical Physics 346, 397 (2016). DOI
M. J. Gullans et al., “Quantum Coding with Low-Depth Random Circuits”, Physical Review X 11, (2021). DOI; 2010.09775
W. Dür, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways”, Physical Review A 62, (2000). DOI; quant-ph/0005115