Quantum LDPC Codes with Transversal Non-Clifford Gates via Products of Algebraic Codes

For every integer rgeq 2 and every ε>0, we construct an explicit infinite family of quantum LDPC codes supporting a transversal C^r-1Z gate with length N, dimension Kgeq N^1-ε, distance Dgeq N^1/r/operatorname{poly}\(log N\), and stabilizer weight wleqoperatorname{poly}\(log N\). The previous state of the art construction (in most parameter regimes) was the r-dimensional color code, which has only constant dimension K=O(1), and otherwise has the same parameters up to polylogarithmic factors. Our construction provides the first known codes with low-weight stabilizers that are capable of magic state distillation with arbitrarily small yield parameter γ=log(N/K)/log(D)>0. A classical analogue of transversal C^r-1Z gates is given by the multiplication property, which requires component-wise products of classical codewords to belong to another similar code. As a byproduct of our techniques, we also obtain a new construction of classical locally testable codes with such a multiplication property. We construct our codes as products of chain complexes associated to classical LDPC codes, which in turn we obtain by imposing local Reed-Solomon codes on a specific spectral expander that we construct. We prove that our codes support the desired transversal C^r-1Z gates by using the multiplication property to combine local circuits based on the topological structure.