Multivariate Multicycle Codes for Complete Single-Shot Decoding

2026-01-26

AI Summary

Existing quantum error-correcting codes struggle to simultaneously achieve high encoding rates, large distances, and full single-shot decodability. This work proposes multivariate multicyclic (MM) codes, constructed from chain complexes of length $t \geq 4$, leveraging Koszul complexes to define the boundary maps of Calderbank–Shor–Steane (CSS) codes. By incorporating meta-checks and high-constraint designs, the framework unifies and generalizes several known code constructions. It presents the first parametrized family that simultaneously offers full single-shot decodability, high rate, large distance, and the potential to support non-Clifford logical gates. Numerical optimization yields new codes—such as [[96,12,8]], [[216,12,12]], and [[486,24,12]]—which significantly outperform existing single-shot decodable CSS codes at practical block lengths.

Technology Category

• Machine Learning: Large Multimodal Models (LMMs)
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Computer Vision: Multi-modal Vision

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG

Abstract

We introduce multivariate multicycle (MM) codes, a new family of quantum error correcting codes that unifies and generalizes bivariate bicycle codes, multivariate bicycle codes, abelian two-block group algebra codes, generalized bicycle codes, trivariate tricycle codes, and n-dimensional toric codes. MM codes are Calderbank-Shor-Steane (CSS) codes defined from length-t chain complexes with $t \ge 4$. The chief advantage of these codes is that they possess metachecks and high confinement that permit complete single-shot decoding, while also having additional algebraic structure that might enable logical non-Clifford gates. We offer a framework that facilitates the construction of long-length chain complexes through the use of Koszul complex. In particular, obtaining explicit boundary maps (parity check and metacheck matrices) is particularly straightforward in our approach. This simple but very general parameterization of codes permitted us to efficiently perform a numerical search, where we identify several MM code candidates that demonstrate these capabilities at high rates and high code distances. Examples of new codes with parameters $[[n,k,d]]$ include $[[96, 12, 8]]$, $[[96, 44, 4]]$ $[[144, 40, 4]]$, $[[216, 12, 12]]$, $[[360, 30, 6]]$, $[[384, 80, 4]]$, $[[486, 24, 12]]$, $[[486, 66, 9]]$ and $[[648, 60, 9]]$. Notably, our codes achieve confinement profiles that surpass all known single-shot decodable quantum CSS codes of practical blocksize.

Problem

• quantum error correcting codes
• single-shot decoding
• CSS codes
• code distance
• high rate

Innovation

• multivariate multicycle codes
• single-shot decoding
• high confinement
• Koszul complex
• quantum CSS codes