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Paper 1

Homological origin of transversal implementability of logical diagonal gates in quantum CSS codes

Junichi Haruna

Year

2026

Journal

arXiv preprint

DOI

arXiv:2602.14499

arXiv

2602.14499

Transversal Pauli Z rotations provide a natural route to fault-tolerant logical diagonal gates in quantum CSS codes, yet their capability is fundamentally constrained. In this work, we formulate the refinement problem of realizing a logical diagonal gate by a transversal implementation with a finer discrete rotation angle and show that its solvability is completely characterized by the Bockstein homomorphism in homology theory. Furthermore, we prove that the linear independence of the X-stabilizer generators together with the commutativity condition modulo a power of two ensures the existence of transversal implementations of all logical Pauli Z rotations with discrete angles in general CSS codes. Our results identify a canonical homological obstruction governing transversal implementability and provide a conceptual foundation for a formal theory of transversal structures in quantum error correction.

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Paper 2

Tradeoffs on the volume of fault-tolerant circuits

Anirudh Krishna, Gilles Zémor

Year

2025

Journal

arXiv preprint

DOI

arXiv:2510.03057

arXiv

2510.03057

Dating back to the seminal work of von Neumann [von Neumann, Automata Studies, 1956], it is known that error correcting codes can overcome faulty circuit components to enable robust computation. Choosing an appropriate code is non-trivial as it must balance several requirements. Increasing the rate of the code reduces the relative number of redundant bits used in the fault-tolerant circuit, while increasing the distance of the code ensures robustness against faults. If the rate and distance were the only concerns, we could use asymptotically optimal codes as is done in communication settings. However, choosing a code for computation is challenging due to an additional requirement: The code needs to facilitate accessibility of encoded information to enable computation on encoded data. This seems to conflict with having large rate and distance. We prove that this is indeed the case, namely that a code family cannot simultaneously have constant rate, growing distance and short-depth gadgets to perform encoded CNOT gates. As a consequence, achieving good rate and distance may necessarily entail accepting very deep circuits, an undesirable trade-off in certain architectures and applications.

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