Here is a list of Floquet codes and other related dynamical codes.
| Code | Description |
|---|---|
| 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 | 3D Floquet code whose physical qubits are placed on vertices of a particular trivalent honeycomb. Its weight-two check operators are those of the 3D Kitaev honeycomb model [1]. Its ISGs are equivalent to that of the 3D fermionic surface code via a constant-depth circuit with ancillas. |
| Floquet 3D surface code | 3D Floquet code whose physical qubit pairs are placed on vertices of a truncated cubic honeycomb. The code is constructed by coupling three 2D surface codes on three interweaving 4.8.8 (square-octagon) tilings oriented along the \(x\), \(y\), and \(z\) axes, respectively, via a coupled layer construction [2]. Its ISG is that of the 3D surface code. |
| 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 | 2D dynamical code whose sequence of check-operator measurements is periodic. The first instance of a dynamical code. |
| Honeycomb Floquet code | 2D Floquet code based on the Kitaev honeycomb model [3] 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 [4]. The code has also been generalized to arbitrary non-chiral, Abelian topological order [5]. |
| 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 \(YY\) check on the legs. |
| X-cube Floquet code | 3D Floquet code whose qubits are placed on vertices of a truncated cubic honeycomb. Its weight-two check operators are placed on various edges. Its ISG can be that of the X-cube model code or that of several decoupled surface codes. A rewinding of the original measurement sequence yields a period-six sequence whose ISGs are those of the X-cube model, some 3D surface codes, and some decoupled surface codes [6]. |
| XYZ ruby Floquet code | 2D Floquet code whose qubits are placed on vertices of a ruby tiling. Its weight-two check operators are placed on various edges. One third of the time during its measurement schedule, its ISG is that of the 6.6.6 color code concatenated with a three-qubit repetition code. Together, all ISGs generate the gauge group of the 3F subsystem code. A Floquet code with the same underlying subsystem code but with a different measurement schedule was developed in Ref. [6]. |
References
- [1]
- S. Mandal and N. Surendran, “Exactly solvable Kitaev model in three dimensions”, Physical Review B 79, (2009) arXiv:0801.0229 DOI
- [2]
- H. Ma, E. Lake, X. Chen, and M. Hermele, “Fracton topological order via coupled layers”, Physical Review B 95, (2017) arXiv:1701.00747 DOI
- [3]
- A. Kitaev, “Anyons in an exactly solved model and beyond”, Annals of Physics 321, 2 (2006) arXiv:cond-mat/0506438 DOI
- [4]
- M. Davydova, N. Tantivasadakarn, and S. Balasubramanian, “Floquet Codes without Parent Subsystem Codes”, PRX Quantum 4, (2023) arXiv:2210.02468 DOI
- [5]
- J. Sullivan, R. Wen, and A. C. Potter, “Floquet codes and phases in twist-defect networks”, (2023) arXiv:2303.17664
- [6]
- A. Dua, N. Tantivasadakarn, J. Sullivan, and T. D. Ellison, “Engineering 3D Floquet Codes by Rewinding”, PRX Quantum 5, (2024) arXiv:2307.13668 DOI