Quantum LDPC Codes with Transversal Non-Clifford Gates via Products of Algebraic Codes
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914978568601600 |
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| author | Golowich, Louis Lin, Ting-Chun |
| author_facet | Golowich, Louis Lin, Ting-Chun |
| contents | For every integer $r\geq 2$ and every $ε>0$, we construct an explicit infinite family of quantum LDPC codes supporting a transversal $C^{r-1}Z$ gate with length $N$, dimension $K\geq N^{1-ε}$, 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 $γ=\log(N/K)/\log(D)>0$.
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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_14662 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantum LDPC Codes with Transversal Non-Clifford Gates via Products of Algebraic Codes Golowich, Louis Lin, Ting-Chun Quantum Physics Computational Complexity Information Theory For every integer $r\geq 2$ and every $ε>0$, we construct an explicit infinite family of quantum LDPC codes supporting a transversal $C^{r-1}Z$ gate with length $N$, dimension $K\geq N^{1-ε}$, 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 $γ=\log(N/K)/\log(D)>0$. 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. |
| title | Quantum LDPC Codes with Transversal Non-Clifford Gates via Products of Algebraic Codes |
| topic | Quantum Physics Computational Complexity Information Theory |
| url | https://arxiv.org/abs/2410.14662 |