2026-08-14 –, Room 3
We introduce multivariate multicycle (MM) codes, a new family of quantum error correcting (QEC) codes that unifies and generalizes many of the established families of QEC codes and possess record-breaking capabilities by measure of confinement for single-shot decoding. Our work is enabled by Open Source Computer Algebra Research (OSCAR.jl) which provides capabilities for computations in commutative and homological algebra and QuantumClifford.jl, an open source package for stabilizer tableaux algebra which provides tools for working with graph states and vast array of error correction codes and capabilities.
The property of single-shot decoding is a crucial requirement for low-overhead error correction and it is one of the hallmarks of fault-tolerant quantum error correction, however very few codes possess this property. The concept of single-shot decoding was recognized as important early in the history of our field [Bombin2015], but significant progress did not happen until recently [Campbell2019], with the introduction of “metachecks”, i.e. “checks on checks” that are solely meant to detect measurement errors in the syndrome itself. Only in 2020 [Quintavalle2020], a universal measure of the “single-shot” capability was well defined, namely “the confinement profile”, which can be used to compare the resilience of a code to measurement errors (the main impediment to single-shot performance). By the dawn of 2026, only a few families of “complete single-shot” QEC codes exist, namely abelian-multicycle codes (AMCs) and 4D homological product codes that include 4D toric and 4D surface codes. The code construction we present changes this by providing a very simple principled technique for generating metachecks and gives us instances of codes with record breaking confinement.
Our Multivariate Multicycle codes are CSS codes defined from length-t chain complexes with t ≥ 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. Moreover, many known families of codes can be expressed simply as special cases of the construction we have discovered. To put this into perspective, n-dimensional Toric, AMC, bivariate bicycle, trivariate tricycle, symmetric cyclic hypergraph product (C2), repeated cyclic hypergraph product (CxR), multivariate bicycle, generalized bicycle, abelian two-block group algebra, Haah’s cubic codes, and “square” La-Cross codes are easily recovered as subfamilies of our codes. 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.
[Bombin2015]: Bombín, H., 2015. Single-shot fault-tolerant quantum error correction. Physical Review X, 5(3), p.031043.
[Campbell2019]: Campbell, E.T., 2019. A theory of single-shot error correction for adversarial noise. Quantum Science and Technology, 4(2), p.025006.
[Quintavalle2020]: Quintavalle, A.O., Vasmer, M., Roffe, J. and Campbell, E.T., 2021. Single-shot error correction of three-dimensional homological product codes. PRX Quantum, 2(2), p.020340.
Feroz Ahmed Mian is a PhD student at the University of Massachusetts Amherst. He studies quantum error correction and fault-tolerance in quantum computation.