Quantum Tanner Color Codes on Qubits with Transversal Gates

This work focuses on growing our understanding of how high dimensional expanders (HDX) can be utilized to construct highly performant quantum codes. While asymptotically good qLDPC codes have been constructed on 2D HDX built from products of graphs, these constructions have a number of limitations, like lack of structure useful for fault-tolerant logic. We develop a framework for transversal logical gates that can naturally utilize symmetric non-product simplicial HDX, and we demonstrate a particular code in this framework that offers various advantages over prior constructions. Specifically, we study the generalization of color codes to \emph{Tanner color codes}, which encompass color, pin, and rainbow codes, and should enable constructions with better parameters. We prove an `unfolding' theorem that characterizes the logical operators of the Tanner color code in terms of logical operators from several colored copies of the companion sheaf code. We leverage this understanding of the logical operators to identify a local condition that ensures such a code on a $D$-dimensional complex has a strictly-transversal $\frac{2 \pi}{2^D}$-phase gate on a single block, $\frac{2 \pi}{2^\ell}$-phase gates on subsets of a single block for $\ell<D$, and $C^{D-1}Z$ across $D$ blocks that preserve the code space. We explicitly instantiate our paradigm in every dimension with codes on highly-symmetric expanding coset complexes. These are the first qubit codes explicitly defined on expanding (non-product) simplicial complexes. We investigate in detail the self-dual 2D family, which has large rate $\geq \frac{7}{64}$ and transversal $CZ$, $S$, and $H$ gates, among many other fault-tolerant (generalizations of) fold-transversal gates arising from the symmetry of the complex. We conjecture that it has constant relative distance. We conclude by describing a Floquet variant of this code with check weight 4.