Here is a list of Floquet codes and other related dynamical 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. |
| Dynamical code | Dynamically generated stabilizer-based code whose (not necessarily periodic) sequence of few-body measurements implements state initialization, logical gates and error detection. |
| 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 [1,2]. |
| 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 [2,3]. |
| 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. |
| 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 [4]. 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 [5] 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 [6]. The code has also been generalized to arbitrary non-chiral, Abelian topological order [7]. |
| 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 [4]. |
| 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 [8]. The code admits two different measurement schedules, the XYZ ruby schedule and the color-code schedule. |
| X-cube Floquet code | A 3D Floquet code on the truncated cubic honeycomb, built from coupled layers of square-octagon Floquet toric codes. |
References
- [1]
- S. Mandal and N. Surendran, “Exactly solvable Kitaev model in three dimensions”, Physical Review B 79, (2009) arXiv:0801.0229 DOI
- [2]
- A. Dua, N. Tantivasadakarn, J. Sullivan, and T. D. Ellison, “Engineering 3D Floquet Codes by Rewinding”, PRX Quantum 5, (2024) arXiv:2307.13668 DOI
- [3]
- H. Ma, E. Lake, X. Chen, and M. Hermele, “Fracton topological order via coupled layers”, Physical Review B 95, (2017) arXiv:1701.00747 DOI
- [4]
- M. B. Hastings and J. Haah, “Dynamically Generated Logical Qubits”, Quantum 5, 564 (2021) arXiv:2107.02194 DOI
- [5]
- A. Kitaev, “Anyons in an exactly solved model and beyond”, Annals of Physics 321, 2 (2006) arXiv:cond-mat/0506438 DOI
- [6]
- M. Davydova, N. Tantivasadakarn, and S. Balasubramanian, “Floquet Codes without Parent Subsystem Codes”, PRX Quantum 4, (2023) arXiv:2210.02468 DOI
- [7]
- J. Sullivan, R. Wen, and A. C. Potter, “Floquet codes and phases in twist-defect networks”, (2023) arXiv:2303.17664
- [8]
- 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