Near-Asymptotically-Good Quantum Codes with Transversal CCZ Gates and Sublinear-Weight Parity-Checks
Louis Golowich, Venkatesan Guruswami
TL;DR
The paper tackles the long-standing challenge of constructing quantum codes that support fault-tolerant non-Clifford gates with low-weight parity-check measurements. It introduces near-optically optimal quantum codes built as products of algebraically structured classical codes (notably Reed-Solomon), achieving linear dimension and distance with sublinear locality for transversal CCZ gates, and provides an efficient decoder for these codes. By employing high-dimensional product-expansion and (punctured) tensor Reed-Solomon code structures, the authors obtain both $O( ext{√}N)$ locality with $q=O( ext{√}N)$ alphabet and, via alphabet reduction, $ ilde{O}( ext{√}N)$ locality with constant alphabet; they also push to $O(N^{1/3})$ locality in higher-order products at the cost of larger alphabets. A novel multivariate Prony-type decoding algorithm for dual tensor RS codes underpins the decoder, and higher-order product-expansion results resolve conjectures about linear-dimension, linear-distance codes with sublinear locality. These constructions advance practical fault-tolerant quantum computing by enabling transversal non-Clifford gates with feasible check weights and provide a versatile framework for decoding and gate implementation in algebraically structured quantum codes.
Abstract
It is a major challenge to construct good quantum codes supporting fault-tolerant (e.g. transversal) non-Clifford gates with low-weight parity-check measurements. In this paper, we construct the first known quantum codes with linear dimension and distance supporting transversal non-Clifford gates that have sublinear locality (i.e. parity-check weight). Specifically, we construct codes with transversal $CCZ$ gates that have dimension and distance $Θ(N)$ and locality $O(\sqrt{N})$, where $N$ denotes the block length. We furthermore design an efficient decoding algorithm for these codes. The alphabet size of these codes is $q=Θ(\sqrt{N})$, but it can be reduced to a constant (e.g. $q=2$) while incurring a polylogarithmic loss in other parameters. We also show how to decrease the locality to $O(N^{1/3})$, albeit with a larger alphabet size and slightly lower distance. We construct these codes as products of classical codes with appropriate algebraic structure. While our quantum codes are subsystem codes with non-commuting gauge operators, we show they nevertheless permit error correction from noisy syndrome measurements. As byproducts, we prove multiple technical results of independent interest. In particular, our efficient decoder can be viewed as a new multivariate generalization of Prony's method for reconstructing a function from partial access to its Fourier transform. Meanwhile, our distance analysis involves new connections to the classical study of maximally recoverable codes. Our results on product codes also resolve a conjecture of Bravyi & Hastings (2014) in the large-alphabet regime, by providing a new construction of quantum codes with dimension and distance $Θ(N)$ and locality $N^ε$ for arbitrary $ε>0$.