Conceptual Connections around Density Determination in Cellular Automata

Strictly local one-dimensional topological quantum error correction with symmetry-constrained cellular automata

“…this is known as the density classification problem [41,42]. Unfortunately, it can be rigorously shown that perfect majority voting cannot be achieved with binary CAs in any dimension [43].…”

“…This, however, is not a deal breaker for majority-based error correction (both classical and quantum) as long as the erroneously classified instances are rare with respect to the noise channel in question. Motivated by applications for classical error correction, there evolved a vivid field concerned with the construction of approximate density classifiers (e.g., [53,35,54,55,56]) and extensions capable of performing density classification exactly (e.g., [57,58,59,60]), see [42] for a review. This is how we address the problem of finding a local decoder for the MCQC: Lemma 1 allows us to filter the literature of one-dimensional binary CAs for self-dual density classifiers; rewritten in syndrome-delta representation, these could be directly applied as potential Majorana chain decoders.…”

“…To decode the MCQC by means of a CA, implementing a global majority vote by local rules seems a good approach. This task is known as density classification problem [41,42] and has been shown to be unsolvable for binary CAs in any dimension [43]. In Subsec.…”

Representing Families of Cellular Automata Rules

“…This fact stands as a contradiction against the classical definition of the DCT, since in order for it to evolve to an all-black or all-white configuration, it would obviously need to change the number of cells in each state throughout time. This means that DCT is unsolvable when formulated according to its classical definition [2,3].…”

“…In order to better understand how that kind of complex behavior emerges, many explorations have been made in the context of the power implicit in CA rules. For instance, classical benchmark problems have been used for this, including the density classification task [2,3] and the parity problem [4]. The density classification task tries to discover the most frequent bit in the initial configuration of the lattice; the parity problem tries to find the parity of the number of 1s in the initial configuration of the lattice.…”