Quantum Rainbow Codes

Abstract: We introduce rainbow codes, a novel class of quantum error correcting codes generalising colour codes and pin codes. Rainbow codes can be defined on any $D$-dimensional simplicial complex that admits a valid $(D+1)$-colouring of its $0$-simplices. We study in detail the case where these simplicial complexes are derived from chain complexes obtained via the hypergraph product and, by reinterpreting these codes as collections of colour codes joined at domain walls, show that we can obtain code families with growing distance and number of encoded qubits as well as logical non-Clifford gates implemented by transversal application of $T$ and $T^\dag$. By combining these techniques with the quasi-hyperbolic colour codes of Zhu et al. (arXiv:2310.16982) we obtain families of codes with transversal non-Clifford gates and parameters $[![n,O(n),O(log(n))]!]$ which allow the magic-state yield parameter $\gamma = \log_d(n/k)$ to be made arbitrarily small. In contrast to other recent constructions that achieve $\gamma \rightarrow 0$ our codes are natively defined on qubits, are LDPC, and have logical non-Clifford gates implementable by single-qubit (rather than entangling) physical operations, but are not asymptotically good.