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

For every integer r ≥ 2 and every ǫ > 0, we construct an explicit infinite family of quantum LDPC codes supporting a transversal C r-1 Z gate with length N , dimension K ≥ N 1-ǫ , distance D ≥ N 1/r / poly(log N ), and stabilizer weight w ≤ 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-1 Z 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-1 Z gates by using the multiplication property to combine local circuits based on the topological structure.