Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates

• URL: http://arxiv.org/abs/2408.09254v1
• Date: Sat, 17 Aug 2024 16:54:51 GMT
• Title: Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates
• Authors: Louis Golowich, Venkatesan Guruswami,
• Abstract summary: We construct quantum codes that support $CCZ$ gates over qudits of arbitrary prime power dimension $q$. The only previously known construction with such linear dimension and distance required a growing alphabet size $q$.
• Score: 23.22566380210149
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We construct quantum codes that support transversal $CCZ$ gates over qudits of arbitrary prime power dimension $q$ (including $q=2$) such that the code dimension and distance grow linearly in the block length. The only previously known construction with such linear dimension and distance required a growing alphabet size $q$ (Krishna & Tillich, 2019). Our codes imply protocols for magic state distillation with overhead exponent $\gamma=\log(n/k)/\log(d)\rightarrow 0$ as the block length $n\rightarrow\infty$, where $k$ and $d$ denote the code dimension and distance respectively. It was previously an open question to obtain such a protocol with a contant alphabet size $q$. We construct our codes by combining two modular components, namely, (i) a transformation from classical codes satisfying certain properties to quantum codes supporting transversal $CCZ$ gates, and (ii) a concatenation scheme for reducing the alphabet size of codes supporting transversal $CCZ$ gates. For this scheme we introduce a quantum analogue of multiplication-friendly codes, which provide a way to express multiplication over a field in terms of a subfield. We obtain our asymptotically good construction by instantiating (i) with algebraic-geometric codes, and applying a constant number of iterations of (ii). We also give an alternative construction with nearly asymptotically good parameters ($k,d=n/2^{O(\log^*n)}$) by instantiating (i) with Reed-Solomon codes and then performing a superconstant number of iterations of (ii).

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