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 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 inspired by geometrically local topological quantum codes, such as the surface code or color code. |
| Cycle code | A code whose parity-check matrix is obtained from the incidence matrix of a graph over \(\mathbb{F}_2\). This code’s properties are derived from the size two chain complex associated with the graph. Not every binary linear code is homological, but there exist homological families that asymptotically saturate the Hamming bound [1]. |
| Fibonacci code | Quantum-inspired binary linear code 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 number of \(1\)’s. |
| 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 [2; 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 [2,3]. |
| Laplacian code | A binary linear code whose parity-check matrix is the graph Laplacian reduced mod 2. For an undirected graph \(\Gamma\) with degree matrix \(D\) and adjacency matrix \(A\), the parity-check matrix is the symmetric matrix \(H=(D-A)\bmod 2\). |
| 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 [4]. |
| Pinwheel code | A geometrically local binary LDPC code defined on planar graphs obtained from the pinwheel tiling [5]. Both bits and checks live on vertices of the graph. If \(L_N\) is the graph Laplacian at generation \(N\), the undepleted check matrix is \(\tilde H_N=(L_N-\mathbb{I})\bmod 2\), and the actual parity-check matrix \(H_N\) is obtained by removing an evenly spaced fraction of boundary checks. |
| Plaquette Ising code | Classical code defined on a cubic lattice in usually two or three dimensions whose parity checks are applied on the four vertices of each square. |
| Quantum-inspired classical block code | A \(q\)-ary linear code whose construction was inspired by a quantum code. |
| X-cube model code | A foliated type-I fracton CSS code on a cubic lattice with qubits on edges, cube stabilizers, and three cross-shaped vertex stabilizers for each vertex [6]. It supports a subextensive number of logical qubits. |
References
- [1]
- H. Bombin and M. A. Martin-Delgado, “Homological error correction: Classical and quantum codes”, Journal of Mathematical Physics 48, (2007) arXiv:quant-ph/0605094 DOI
- [2]
- L. Spagnoli, A. Roggero, and N. Wiebe, “Fault-tolerant simulation of Lattice Gauge Theories with gauge covariant codes”, Quantum 10, 1968 (2026) arXiv:2405.19293 DOI
- [3]
- A. Rajput, A. Roggero, and N. Wiebe, “Quantum Error Correction with Gauge Symmetries”, (2022) arXiv:2112.05186
- [4]
- 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
- [5]
- J. H. Conway and C. Radin, “Quaquaversal tilings and rotations”, Inventiones Mathematicae 132, 179 (1998) DOI
- [6]
- S. Vijay, J. Haah, and L. Fu, “Fracton topological order, generalized lattice gauge theory, and duality”, Physical Review B 94, (2016) arXiv:1603.04442 DOI