Here is a list of classical codes whose constructions were motivated by prior quantum codes.
| Code | Description |
|---|---|
| Classical fractal liquid code | Member of a family of \([L^D,O(L^{D-1}),O(L^{D-\epsilon})]_p\) linear codes on \(D\)-dimensional square lattices of side length \(L\) and for some prime \(p\) and \(\epsilon > 0\) that is based on \(p\)-ary generalizations of the Sierpinski triangle. |
| Classical topological code | Classical code defined on a two-dimensional lattice and derived from a geometrically local stabilizer code, such as the surface code or color code. |
| Cycle code | A code whose parity-check matrix forms the incidence matrix of a graph. This code's properties are derived from the size two chain complex associated with the graph. |
| Fibonacci code | The code is defined on an \(L\times L/2\) lattice with one bit on each site, where \(L=2^N\) for an integer \(N\geq 2\). The codewords are defined to satisfy the condition that, for each lattice site \((x,y)\), the bits on \((x,y)\), \((x+1,y)\), \((x-1,y)\) and \((x,y+1)\) (where the lattice is taken to be periodic in both directions) contain an even numbers of \(1\)'s. The codewords can be generated using a one-dimensional cellular automaton of length \(L\) (periodic). The \(2^L\) possible initial states correspond to the \(2^L\) codewords. For each generation, the state of each cell is the xor sum of that cell and its two neighbors in the previous generation. After \(L/2-1\) generations, the entire history generated by the automaton corresponds to a codeword, where the initial state is the first row of the lattice, the first generation is the second row, etc. |
| Gauss' law code | An \([m+Dm,Dm,3]\) linear binary code for \(m\geq 3^D\), defined by the Gauss' law constraint of a \(D\)-dimensional fermionic \(\mathbb{Z}_2\) gauge theory [1; Thm. 1]. The code can be re-phrased as a distance-one stabilizer code whose stabilizers consist of gauge-group elements. It can be concatenated to form a stabilizer code for fault-tolerant quantum simulation of the underlying gauge theory [1,2]. |
| Newman-Moore code | Member of a family of \([L^2,O(L),O(L^{\frac{\log 3}{\log 2}})]\) binary linear codes on \(L\times L\) square lattices that form the ground-state subspace of a class of exactly solvable spin-glass models with three-body interactions. The codewords resemble the Sierpinski triangle on a square lattice, which can be generated by a cellular automaton [3]. |
| Plaquette Ising code | Classical code defined on a two-dimensional square lattice whose parity checks are applied on the four vertices of each square. |
| Quantum-inspired classical block code | A block code of length \(n\) whose construction was inspired by a quantum code. |
| X-cube model code | A foliated type-I fracton code supporting a subextensive number of logical qubits. Variants include the membrane-coupled [4], twice-foliated [5], and several generalized [6] X-cube models. |
References
- [1]
- L. Spagnoli, A. Roggero, and N. Wiebe, “Fault-tolerant simulation of Lattice Gauge Theories with gauge covariant codes”, (2024) arXiv:2405.19293
- [2]
- A. Rajput, A. Roggero, and N. Wiebe, “Quantum Error Correction with Gauge Symmetries”, (2022) arXiv:2112.05186
- [3]
- D. R. Chowdhury, S. Basu, I. S. Gupta, and P. P. Chaudhuri, “Design of CAECC - cellular automata based error correcting code”, IEEE Transactions on Computers 43, 759 (1994) DOI
- [4]
- H. Ma, E. Lake, X. Chen, and M. Hermele, “Fracton topological order via coupled layers”, Physical Review B 95, (2017) arXiv:1701.00747 DOI
- [5]
- W. Shirley, K. Slagle, and X. Chen, “Fractional excitations in foliated fracton phases”, Annals of Physics 410, 167922 (2019) arXiv:1806.08625 DOI
- [6]
- T. Rakovszky and V. Khemani, “The Physics of (good) LDPC Codes II. Product constructions”, (2024) arXiv:2402.16831