Copy-cup Gates in Tensor Products of Group Algebra Codes

TL;DR

This paper characterizes conditions under which classical group algebra codes enable the construction of quantum codes with constant-depth multi-qubit gates, linking graph theory and code properties.

Contribution

It provides a complete characterization of conditions for copy-cup gates in tensor products of group algebra codes, including new results on code distances and examples.

Findings

Conditions for 2- and 3-copy-cup gates are fully determined.

Bivariate bicycle codes do not support pre-orientation for copy-cup gates.

Codes satisfying certain conditions have bounded or higher code distances.

Abstract

We determine conditions on classical group algebra codes so that they have pre-orientation for cup products and copy-cup gates. This defines quantum codes that have constant-depth $\operatorname{CZ}$ and $\operatorname{CCZ}$ gates constructed via tensor products of classical group algebra codes, including hypergraph and balanced products. We show that determining the conditions relies on solving the perfect matching problem in graph theory. Conditions are fully determined for the 2- and 3-copy-cup gates, for group algebra codes up to weight 4, including for codes with odd check weight. These include the bivariate bicycle codes, which we show do not have the pre-orientation for either type of copy-cup gate. We show that abelian weight 4 group algebra codes satisfying the non-associative 3-copy-cup gate necessarily have a code distance of 2, whereas codes that satisfy conditions for the symmetric 3-copy-cup gate can have higher distances, and in fact also satisfy conditions for the 2-copy-cup gate. Finally we find examples of quantum codes from the product of abelian group algebra codes that have inter-code constant-depth $\operatorname{CZ}$ and $\operatorname{CCZ}$ gates.