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

• URL: http://arxiv.org/abs/2410.14662v1
• Date: Fri, 18 Oct 2024 17:52:59 GMT
• Title: Quantum LDPC Codes with Transversal Non-Clifford Gates via Products of Algebraic Codes
• Authors: Louis Golowich, Ting-Chun Lin,
• Abstract summary: We construct an explicit infinite family of quantum LDPC codes supporting a $Cr-1Z$ gate with length $N$, dimension $Kgeq N1-epsilon$, distance $Dgeq N1/r/namepoly(log N)$, and stabilizer weight $wleqoperatorname(log N)$.
• Score: 0.9208007322096533
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• 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.

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