Quantum Rainbow Codes: Achieving Linear Rate, Growing Distance and Transversal Non-Clifford Gates with Generalised Colour Codes

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 a family of codes with transversal non-Clifford gates and parameters $[\![n, \Theta(n), \Theta(log(n))]\!]$. This is the first example of a family of LDPC codes with linear rate, growing distance and transversal non-Clifford gates, which are necessary conditions for the magic-state distillation parameter $\gamma =\textrm{log}_d (n/k)$ to be made arbitrarily small. In contrast to several other constructions that satisfy these requirements, our codes are natively defined on qubits, are LDPC, and have non-Clifford gates implementable by single-qubit (rather than entangling) physical operations, but are not asymptotically good.