Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates

TL;DR

This paper presents a method to construct quantum error-correcting codes supporting transversal non-Clifford gates with linearly growing parameters, enabling efficient magic state distillation protocols with constant alphabet size.

Contribution

It introduces a novel construction of asymptotically good quantum codes supporting transversal CCZ gates using classical algebraic-geometric and Reed-Solomon codes, with a new concatenation scheme.

Findings

Codes support transversal CCZ gates for arbitrary prime power dimensions.

Protocols achieve overhead exponent tending to zero as block length increases.

Construction works with constant alphabet size, including qubits.

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).