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

TL;DR

This paper introduces explicit quantum LDPC codes supporting transversal non-Clifford gates with high rate and distance, constructed via products of classical LDPC codes and spectral expanders, enabling efficient magic state distillation.

Contribution

It provides the first known quantum LDPC codes with low-weight stabilizers capable of magic state distillation at arbitrary yield parameters, surpassing previous color code limitations.

Findings

Supports transversal $C^{r-1}Z$ gates with high rate and distance

Constructs classical locally testable codes with multiplication property

Uses product of chain complexes from classical LDPC codes

Abstract

For every integer $r\geq 2$ and every $\epsilon>0$, we construct an explicit infinite family of quantum LDPC codes supporting a transversal $C^{r-1}Z$ gate with length $N$, dimension $K\geq N^{1-\epsilon}$, distance $D\geq N^{1/r}/\operatorname{poly}(\log N)$, and stabilizer weight $w\leq\operatorname{poly}(\log N)$. 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 $\gamma=\log(N/K)/\log(D)>0$. 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.