Cups and Gates I: Cohomology invariants and logical quantum operations
Nikolas P. Breuckmann, Margarita Davydova, Jens N. Eberhardt, Nathanan Tantivasadakarn
TL;DR
This work develops a general, algebraic framework to construct diagonal logical gates for CSS quantum codes by viewing codes as cochain complexes and employing cohomology invariants, notably cup products, to produce constant-depth, transversal gates. By introducing a Λ-fold cup product (the copy-cup gate) on Λ copies of the same code, the authors achieve logical operations at the Λ-th level of the Clifford hierarchy and outline concrete code families (including toric, hyperbolic, Sipser–Spielman, and group-algebra constructions) that realize these gates with favorable asymptotics. The approach rests on a careful synthesis of simplicial/cochain theory with tensor/balanced product constructions, supplemented by integrals to ensure cohomology invariance and locality of the resulting circuits. The results offer a principled path to high-level, fault-tolerant gates in qLDPC codes, connecting homological algebra to practical quantum fault tolerance and suggesting avenues for further generalizations via other cohomology operations and broader code families.
Abstract
We take initial steps towards a general framework for constructing logical gates in general quantum CSS codes. Viewing CSS codes as cochain complexes, we observe that cohomology invariants naturally give rise to diagonal logical gates. We show that such invariants exist if the quantum code has a structure that relaxes certain properties of a differential graded algebra. We show how to equip quantum codes with such a structure by defining cup products on CSS codes. The logical gates obtained from this approach can be implemented by a constant-depth unitary circuit. In particular, we construct a $Λ$-fold cup product that can produce a logical operator in the $Λ$-th level of the Clifford hierarchy on $Λ$ copies of the same quantum code, which we call the copy-cup gate. For any desired $Λ$, we can construct several families of quantum codes that support gates in the $Λ$-th level with various asymptotic code parameters.