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.
Cups and Gates I: Cohomology invariants and logical quantum operations
University of Bristol（布里斯托大学）；California Institute of Technology（加州理工学院）；Institute for Quantum Information and Matter, California Institute of Technology（加州理工学院量子信息与物质研究所）；Massachusetts Institute of Technology（麻省理工学院）；Johannes Gutenberg University of Mainz（美因茨约翰内斯古腾堡大学）
Cups and Gates II: Operads, twists, and addressable intra-code gates
University of Bristol（布里斯托大学）；California Institute of Technology（加州理工学院）；Johannes Gutenberg University of Mainz（美因茨约翰内斯古腾堡大学）；Stony Brook University（石溪大学）
Transversal dimension jump for product qLDPC codes
Institute for Quantum Information and Matter, Caltech（加州理工学院量子信息与物质研究所）；Fu Foundation School of Engineering and Applied Science, Columbia University（哥伦比亚大学富工程与应用科学学院）；AWS Center for Quantum Computing（亚马逊量子计算中心）；Walter Burke Institute for Theoretical Physics, Caltech（加州理工学院沃尔特·伯克理论物理研究所）
Non-Abelian Self-Correcting Quantum Memory and Transversal Non-Clifford Gate beyond the $n^{1/3}$ Distance Barrier
Mani L. Bhaumik Institute for Theoretical Physics（马尼·L·巴乌米克理论物理研究所）；King’s College London（伦敦国王学院）；University of Maryland, College Park（马里兰大学帕克分校）；Institute for Advanced Study（普林斯顿高等研究院）；IBM Quantum, IBM T.J. Watson Research Center（IBM量子，IBM托马斯·J·沃森研究中心）
Clifford Hierarchy Stabilizer Codes: Transversal Non-Clifford Gates and Magic
Institute for Advanced Study（高等研究院）；IBM Quantum, IBM T.J. Watson Research Center（IBM量子，IBM T.J.沃森研究中心）；Kavli Institute of Theoretical Physics（卡弗里理论物理研究所）；King’s College London（伦敦国王学院）
Transversal non-Clifford gates on qLDPC codes breaking the $\sqrt{N}$ distance barrier and quantum-inspired geometry with $\mathbb{Z}_2$ systolic freedom
A topological theory for qLDPC: non-Clifford gates and magic state fountain on homological product codes with constant rate and beyond the $N^{1/3}$ distance barrier
