Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates
Louis Golowich, Venkatesan Guruswami
TL;DR
This work addresses fault-tolerant quantum computation by constructing quantum codes that admit transversal non-Clifford gates, specifically CCZ on qudits of dimension $q$ (including qubits with $q=2$). It develops a two-part framework: (i) a classical-to-quantum code transformation that preserves transversal CCZ (and hence enables magic-state distillation), and (ii) an alphabet-reduction via multiplication-friendly concatenation to achieve a constant alphabet. Instantiating this approach with algebraic-geometric codes yields infinite families of quantum codes with $k=Θ(n)$ and $d=Θ(n)$ and overhead $\\gamma=\\log(n/k)/\\log(d) \\rightarrow 0$ as $n\\rightarrow\\infty$, using a constant alphabet $q$; a Reed-Solomon based route provides a nearly asymptotically good alternative with $k,d=\\Ω(n/2^{O(\\log^*n)})$. These results advance the feasibility of constant-alphabet magic-state distillation and transversal non-Clifford fault-tolerance, while leaving open questions about LDPC variants and finite-size performance. Overall, the paper introduces a scalable, modular scheme to obtain asymptotically good quantum codes with transversal CCZ/U gates and analyzes two concrete instantiations with different alphabet-growth profiles.
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 $γ=\log(n/k)/\log(d)\rightarrow 0$ 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).