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2024

Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates

Golowich, Louis, Guruswami, Venkatesan

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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 $q$.

Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates · Around

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Universal Quantum Computation with ideal Clifford gates and noisy ancillas
Sergei Bravyi and Alexei Kitaev · 2005
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Magic-state distillation with low overhead
Sergey Bravyi and Jeongwan Haah · 2012
Earlier work this paper cites.
Universal Fault-Tolerant Quantum Computation with Only Transversal Gates and Error Correction
Adam Paetznick and Ben W. Reichardt · 2013
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