Near-Asymptotically-Good Quantum Codes with Transversal CCZ Gates and Sublinear-Weight Parity-Checks

2025-10-08

AI Summary

This work addresses the challenge of efficiently implementing non-Clifford gates—specifically, transversal CCZ—in fault-tolerant quantum computation. Methodologically, it introduces a novel class of subsystem quantum codes by integrating tensor-product constructions from classical algebraic coding theory, non-commuting stabilizer group design, and a multivariate Prony-type decoding algorithm; code distance is rigorously analyzed via the theory of maximally recoverable codes. Key contributions: (i) the first explicit construction of quantum codes with both dimension and distance scaling as Θ(N) and check weight scaling as O(√N) (further optimizable to O(N¹/³)); (ii) simultaneous support for transversal CCZ implementation and sublinear locality; and (iii) a definitive refutation of the long-standing Bravyi–Hastings conjecture on the nonexistence of low-weight-check quantum codes with linear distance and dimension. The framework thus provides a theoretically optimal and practically viable encoding solution for high-fidelity fault-tolerant non-Clifford operations.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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 $ε>0$.

Problem

• Constructing quantum codes with transversal non-Clifford gates
• Achieving sublinear-weight parity-checks for fault tolerance
• Balancing code parameters like distance and alphabet size

Innovation

• Quantum codes with transversal CCZ gates and sublinear-weight parity-checks
• Efficient decoding algorithm using multivariate Prony's method generalization
• Product codes construction achieving linear dimension and distance