Citation: Breuckmann, Nikolas P., et al. "Cups and Gates I: Cohomology invariants and logical quantum operations." arXiv preprint arXiv:2410.16250 (2024).
Web: https://arxiv.org/abs/2410.16250
Tags: Non-local-codes, Error-correcting-codes, Higher-dimensional
This paper introduces a general machinery for constructing logical quantum operations in qLDPC codes using cohomology invariants. Most of the results proved are quite specific, referring only to gates obtained from integrating cup products. The highlight of this paper, thus, is the overarching philosophy. They give a general principle which says that homological CSS codes should have a good general theory of logical operators. In fact, there should be a generalization of most topological techniques to the more general homological setting. They conjecture that there is an associated area of study they call "homological QFT" or "HomQFT".
Even though the results of this paper feel very specific and even potentially ad-hoc, a good sign that they're on to something is that a few days after this paper came out there were two other related papers which appeared on arxiv:
> Golowich, Louis, and Ting-Chun Lin. "Quantum LDPC Codes with Transversal Non-Clifford Gates via Products of Algebraic Codes." arXiv preprint arXiv:2410.14662 (2024).
> Lin, Ting-Chun. "Transversal non-Clifford gates for quantum LDPC codes on sheaves." arXiv preprint arXiv:2410.14631 (2024).
The main result of the present paper is as follows. They define a cup-copy operation on layered system by taking the cup product of all the inputs then integrating. They prove an explicit condition under which this cup-copy gate is a cohomology invariant. In the case that it is a cohomology invariant they define an associated logical gate, from a general principe which assigns logical gates to cohomology invariants. Then, then define a finite depth local circuit which implements this logical gate. They then prove that this logical gate is at a high level of the Clifford hierarchy. Thus, under certain conditions, the authors are able to construct fault-tolerant non-Clifford gates!