Realizing Logical Diagonal Gates via Transversal Physical Z-Rotations in CSS Codes
Abstract
Calderbank-Shor-Steane (CSS) codes, constructed from nested classical codes C2 ⊆ C1, are typically optimized for good code parameters. However, practical quantum computing equally demands fault-tolerant logical gates. In this work, we characterize nested pairs (C1, C2) whose resulting CSS codes realize a target logical diagonal gate via transversal physical Z-rotations. In doing so, we recover a result of Camps-Moreno et al. that CSS codes can realize only logical single-qubit Z-rotations and multi-qubit controlled Z-rotations via transversal physical Z-rotations. Building on our characterization, we develop the ''appending construction'', that takes as input an $[[n',k']]$ CSS code Q' and a target logical Z-rotation (single-qubit or multi-controlled) UL, and extends Q' by systematically appending n'' physical qubits to obtain an [[n,k]] CSS code Q with n = n'+n'' and k=k'. The target logical gate UL is realized in Q by applying a well-chosen physical transversal Z-rotation to the n'' appended physical qubits. Moreover, any logical gate realized via transversal physical Z-rotations in the input code Q' remains transversally realizable in the extended code Q. The CSS code Q may incur a loss in minimum distance, but the loss can be controlled through the parameter choices made in the construction. By repeatedly applying the appending construction, we can extend any CSS code Q' to obtain a CSS code Q that supports fault-tolerant implementations of multiple desired logical Z-rotations. The cost to be paid for this is the increased physical qubit overhead as the number of target logical gates grows. To illustrate our methodology, we construct CSS code families that transversally realize addressable logical single-qubit and multi-controlled Z-rotations.
Related
- Finite Classification of Transversal Diagonal Gates Beyond Stabilizer Codes
- A note on realizing triples of slice, concordance and Seifert genera
- A purely mechanical system realizing a Coulomb-like interaction
- Realizing graph automorphisms as geometric symmetries of straight line drawings
- Compactly supported real scalar potentials realizing the Hardy uncertainty endpoint for Schrödinger evolutions
BibTeX
@misc{reddy2026realizing,
title = {{Realizing Logical Diagonal Gates via Transversal Physical $Z$-Rotations in CSS Codes}},
author = {K. Sai Mineesh Reddy and Navin Kashyap},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.19094},
eprint = {2608.19094},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.19094},
}
RIS
TY - UNPB
TI - Realizing Logical Diagonal Gates via Transversal Physical Z-Rotations in CSS Codes
AU - K. Sai Mineesh Reddy
AU - Navin Kashyap
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2608.19094
UR - https://arxiv.org/abs/2608.19094
AB - Calderbank-Shor-Steane (CSS) codes, constructed from nested classical codes C_2 ⊆ C_1, are typically optimized for good code parameters. However, practical quantum computing equally demands fault-tolerant logical gates. In this work, we characterize nested pairs (C_1, C_2) whose resulting CSS codes realize a target logical diagonal gate via transversal physical Z-rotations. In doing so, we recover a result of Camps-Moreno et al. that CSS codes can realize only logical single-qubit Z-rotations and multi-qubit controlled Z-rotations via transversal physical Z-rotations. Building on our characterization, we develop the ''appending construction'', that takes as input an $[[n',k']]$ CSS code Q' and a target logical Z-rotation (single-qubit or multi-controlled) U_L, and extends Q' by systematically appending n'' physical qubits to obtain an [[n,k]] CSS code Q with n = n'+n'' and k=k'. The target logical gate U_L is realized in Q by applying a well-chosen physical transversal Z-rotation to the n'' appended physical qubits. Moreover, any logical gate realized via transversal physical Z-rotations in the input code Q' remains transversally realizable in the extended code Q. The CSS code Q may incur a loss in minimum distance, but the loss can be controlled through the parameter choices made in the construction. By repeatedly applying the appending construction, we can extend any CSS code Q' to obtain a CSS code Q that supports fault-tolerant implementations of multiple desired logical Z-rotations. The cost to be paid for this is the increased physical qubit overhead as the number of target logical gates grows. To illustrate our methodology, we construct CSS code families that transversally realize addressable logical single-qubit and multi-controlled Z-rotations.
ER -