Here is a list of quantum dynamically generated codes.
Code Description
Clifford-deformed surface code (CDSC) A generally non-CSS derivative of the surface code defined by applying a constant-depth Clifford circuit to the original (CSS) surface code. Unlike the surface code, CDSCs include codes whose thresholds and subthreshold performance are enhanced under noise biased towards dephasing. Examples of CDSCs include the XY code, XZZX code, and random CDSCs.
Dynamically-generated quantum error-correcting code Code whose natural definition is in terms of a many-body scaling limit of a local dynamical process. Such processes, which are often non-deterministic, update the code structure and can include random unitary evolution or non-commuting projective measurements.
Floquet code Dynamically-generated stabilizer-based code whose logical qubits are generated through a particular sequence of check-operator measurements such that the number of logical qubits is larger than when the code is viewed as a static subsystem stabilizer code. After each measurement in the sequence, the codespace is a joint \(+1\) eigenspace of an instantaneous stabilizer group (ISG), i.e., a particular stabilizer group corresponding to the measurement. The ISG specifies the state of the system as a Pauli stabilizer state at a particular round of measurement, and it evolves into a (potentially) different ISG depending on the check operators measured in the next step in the sequence. As opposed to subsystem codes, only specific measurement sequences maintain the codespace.
Haar-random code Haar-random codewords are generated in a process involving averaging over unitary operations distributed accoding to the Haar measure. Haar-random codes are used to prove statements about the capacity of a quantum channel to transmit quantum information [1], but encoding and decoding in such \(n\)-qubit codes quickly becomes impractical as \(n\to\infty\).
Honeycomb code Floquet code inspired by the Kitaev honeycomb model [2] whose logical qubits are generated through a particular sequence of measurements.
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.
Low-depth random Clifford-circuit code An encoder for an \([[n,k]]\) quantum error correcting code, is an \(n\)-qubit unitary transformation that takes a \(k\)-qubit state as input (with \(k\leq n\), and the remaining \(n-k\) qubits initialized to \(|0\rangle^{\otimes n-k}\) ) to give a corresponding state in the codespace as the output. An n-qubit quantum circuit with random 2-qubit Clifford gates can act as an encoder into a code with distance \(d\) with high probability, with a size (i.e. number of gates in the circuit) at most \(O(n^2 log n)\)). Noting that two gates acting on disjoint qubits could in fact be executed simultaneously, this is equivalent to the depth (number of time steps in the circuit) being at most \(O(log^3 n)\).
Monitored random-circuit code Error-correcting code arising from a monitored random circuit. Such a circuit is described by a series of intermittant random local projective Pauli measurements with random unitary time-evolution operators. An important sub-family consists of Clifford monitored random circuits, where unitaries are sampled from the Clifford group [5]. When the rate of projective measurements is independently controlled by a probability parameter \(p\), there can exist two stable phases, one described by volume-law entanglement entropy and the other by area-law entanglement entropy. The phases and their transition can be understood from the perspective of quantum error correction, information scrambling, and channel capacities [6][7].
Quantum low-density parity-check (QLDPC) code Family of \([[n,k,d]]\) stabilizer codes for which the number of sites (either qubit or qudit) participating in each stabilizer generator and the number of stabilizer generators that each site participates in are both bounded by a constant as \(n\to\infty\). A geometrically local stabilizer code is a QLDPC code where the sites involved in any syndrome bit are contained in a fixed volume that does not scale with \(n\).
Random-circuit code Code whose encoding is naturally constructed by randomly sampling from a large set of quantum circuits.

References

[1]
M. M. Wilde, “Preface to the Second Edition”, Quantum Information Theory xi. DOI; 1106.1445
[2]
A. Kitaev, “Anyons in an exactly solved model and beyond”, Annals of Physics 321, 2 (2006). DOI; cond-mat/0506438
[3]
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
[4]
M. J. Gullans et al., “Quantum Coding with Low-Depth Random Circuits”, Physical Review X 11, (2021). DOI; 2010.09775
[5]
Y. Li, X. Chen, and M. P. A. Fisher, “Measurement-driven entanglement transition in hybrid quantum circuits”, Physical Review B 100, (2019). DOI; 1901.08092
[6]
S. Choi et al., “Quantum Error Correction in Scrambling Dynamics and Measurement-Induced Phase Transition”, Physical Review Letters 125, (2020). DOI; 1903.05124
[7]
M. J. Gullans and D. A. Huse, “Dynamical Purification Phase Transition Induced by Quantum Measurements”, Physical Review X 10, (2020). DOI; 1905.05195