Here is a list of codes related to dynamically generated codes.
| Code | Description |
|---|---|
| 2D DA color code | A 2D dynamical code constructed aperiodically that utilizes measurement sequences to encode logical information with automorphisms of the 2D color code. The code is assembled from short measurement sequences that can realize all 72 automorphisms of the 2D color code. On a stack of \(N\) triangular patches with a Pauli boundary, the code encodes \(N\) logical qubits. |
| 3D DA color code | A 3D dynamical code constructed aperiodically that utilizes measurement sequences to encode logical information with automorphisms of the 3D color code. The code represents the first step towards universal quantum computation with dynamical automorphism codes. |
| Brown-Fawzi Clifford-circuit code | An \([[n,k]]\) stabilizer code whose encoder is a random Clifford circuit of depth of order \(O(\log^3 n)\). |
| Clifford-deformed surface code (CDSC) | A generally non-CSS derivative of the surface code defined by applying a translationally invariant 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. |
| Cluster-state code | A code based on a cluster state (a.k.a. graph state) and often used in measurement-based quantum computation (MBQC) [1,2] (a.k.a. one-way quantum processing), which substitutes the temporal dimension necessary for decoding a conventional code with a spatial dimension. This is done by encoding the computation into the features of the cluster state”s graph. |
| Crystalline-circuit qubit code | Code dynamically generated by constant-depth unitary Clifford circuits defined on a lattice with some crystalline symmetry. A notable example is the circuit defined on a rotated square lattice with vertices corresponding to iSWAP gates and edges decorated by \(R_X[\pi/2]\), a single-qubit rotation by \(\pi/2\) around the \(X\)-axis. This circuit is invariant under space-time translations by a unit cell \((T, a)\) and all transformations of the square lattice point group \(D_4\). |
| Dynamical code | Dynamically generated stabilizer-based code whose (not necessarily periodic) sequence of few-body measurements implements state initialization, logical gates and error detection. |
| Dynamically generated QECC | Block quantum 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 3D fermionic surface code | A 3D Floquet code on a trivalent lattice whose weight-two checks are the \(XX\), \(YY\), and \(ZZ\) edge terms of the 3D Kitaev honeycomb model [3,4]. |
| Floquet 3D surface code | A 3D Floquet code on a truncated cubic honeycomb with pairs of physical qubits on vertices. It is constructed from three stacks of square-octagon Floquet toric codes, coupled by interlayer \(YY\) measurements in a coupled-layer construction [4,5]. |
| Floquet color code | 2D Floquet code on a trivalent 2D lattice whose parent topological phase is the \(\mathbb{Z}_2\times\mathbb{Z}_2\) 2D color-code phase and whose measurements cycle logical quantum information between the nine \(\mathbb{Z}_2\) surface-code condensed phases of the parent phase. The code’s ISG is the stabilizer group of one of the nine surface codes. |
| Fracton Floquet code | 3D Floquet code whose qubits are placed on vertices of a truncated cubic honeycomb. Its weight-two check operators are placed on edges of each truncated cube, while weight-three check operators are placed on each triangle. Its ISG can be that of the X-cube model code or the checkerboard model code. On a three-torus of size \(L_x \times L_y \times L_z\), the code consists of \(n= 48L_xL_yL_z\) physical qubits and encodes \(k= 2(L_x+L_y+L_z)-6\) logical qubits. |
| Fusion-based quantum computing (FBQC) code | Code whose codewords are resource states used in an FBQC scheme. |
| Haar-random qubit code | Haar-random codewords are generated in a process involving averaging over unitary operations distributed according to the Haar measure. Haar-random codes are used to prove statements about the capacity of a quantum channel to transmit quantum information [6], but encoding and decoding in such \(n\)-qubit codes quickly becomes impractical as \(n\to\infty\). |
| Hastings-Haah Floquet code | Dynamical code whose sequence of check-operator measurements is periodic. The original Hastings-Haah construction introduced periodic measurement schedules that dynamically generate logical qubits even when the underlying subsystem code has fewer or no logical qubits [7]. Its basic examples are the 2D honeycomb Floquet code and the 1D ladder Floquet code. |
| Honeycomb Floquet code | 2D Floquet code based on the Kitaev honeycomb model [8] whose logical qubits are generated through a particular sequence of measurements. A CSS version of the code has been proposed which loosens the restriction of which sequences to use [9]. The code has also been generalized to arbitrary non-chiral, Abelian topological order [10]. |
| Hyperbolic Floquet code | Floquet code whose check-operators correspond to edges of a hyperbolic lattice of degree 3. |
| Ladder Floquet code | 1D Floquet code defined on a ladder qubit geometry, with one qubit per vertex. The check operators consist of \(ZZ\) checks on each rung and alternating \(XX\) and \(YY\) checks on the legs. The period-four measurement schedule measures \(ZZ\), \(XX\), \(ZZ\), and \(YY\) in rounds \(0,1,2,3\) mod \(4\), respectively, dynamically generating one logical qubit [7]. |
| 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. |
| Log-depth geometrically local Clifford-circuit code | A random \([[n,k]]\) stabilizer code whose encoder is a random Clifford circuit of depth of order \(O(\log n)\) on a 1D Euclidean geometry. |
| Modular-qudit dynamical code | Dynamically generated stabilizer-based modular-qudit code whose (not necessarily periodic) sequence of few-body measurements implements state initialization, logical gates and error detection. |
| Modular-qudit honeycomb Floquet code | A modular-qudit extension of the honeycomb Floquet code. |
| Monitored random-circuit code | Error-correcting code arising from a monitored random circuit. Such a circuit is described by a series of intermittent 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 [11]. 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 [12,13]. |
| QLDPC code | Member of a family of stabilizer codes for which the number of sites participating in each stabilizer generator and the number of stabilizer generators that each site participates in are both bounded by a constant as \(n\to\infty\). Sometimes, the two parameters are explicitly stated: each site of an \((l,w)\)-regular QLDPC code is acted on by \(\leq l\) generators of weight \(\leq w\). |
| Random stabilizer code | An \(n\)-qubit, modular-qudit, or Galois-qudit stabilizer code whose construction is non-deterministic. Since stabilizer encoders are Clifford circuits, such codes can be thought of as arising from random Clifford circuits. |
| Random-circuit code | Code whose encoding is naturally constructed by randomly sampling from a large set of quantum circuits. Examples include short random Clifford circuits that define good quantum error-correcting codes [14] and monitored random circuits whose mixed phase dynamically generates error-protected subspaces with nonzero channel-capacity density on polynomial timescales [13]. |
| Ruby Floquet code | 2D Floquet code whose qubits are placed on vertices of a ruby tiling, with weight-two Pauli check operators on \(x\)-, \(y\)-, and \(z\)-labeled edges [15]. The code admits two different measurement schedules, the XYZ ruby schedule and the color-code schedule. |
| Spacetime circuit code | Qubit stabilizer code constructed from a Clifford circuit, i.e., a circuit made up of Clifford gates and Pauli measurements, in order to detect and correct circuit faults. The code utilizes redundancy in the measurement outcomes of a circuit to correct circuit faults, which correspond to Pauli errors of the code. |
| X-cube Floquet code | A 3D Floquet code on the truncated cubic honeycomb, built from coupled layers of square-octagon Floquet toric codes. |
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- W. Brown and O. Fawzi, “Short random circuits define good quantum error correcting codes”, 2013 IEEE International Symposium on Information Theory (2013) arXiv:1312.7646 DOI
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- J. C. Magdalena de la Fuente, J. Old, A. Townsend-Teague, M. Rispler, J. Eisert, and M. Müller, “XYZ Ruby Code: Making a Case for a Three-Colored Graphical Calculus for Quantum Error Correction in Spacetime”, PRX Quantum 6, (2025) arXiv:2407.08566 DOI