Improved Transversal Non-Clifford Gates from Cup Products
Abstract
It is a major challenge in quantum fault-tolerance to obtain low-overhead protocols for performing non-Clifford gates. In this vein, we construct quantum codes with low-weight stabilizers that support transversal (i.e. low-depth) implementations of the non-Clifford Cr−1Z gate, for every constant r≥ 3. In particular, we obtain length-n quantum LDPC codes (with constant-weight stabilizers) of polynomial distance d≥ n(1−ε)/r supporting transversal Cr−1Z gates on a close-to-linear number k≥ n1−ε of disjoint tuples of logical qubits, for arbitrarily small ε>0. Our construction is the first with constant-weight stabilizers that obtains dk≫ n, and as a consequence achieves arbitrarily small magic state overhead exponent γ=log(n/k)/log(d)>0. Comparable prior constructions instead required at least polylogarithmic stabilizer weight. We also show how to obtain linearly many k=Ω(n) logical Cr−1Z gates, though with stabilizer weight and physical circuit depth nε. We show that our transversal gates also support addressing (i.e. targeting) of specific logical qubits. To obtain our codes, we develop a general transformation based on cup products that maps classical codes satisfying a multiplication property to quantum codes with transversal Cr−1Z. We apply this transformation to a new family of classical Tanner codes that we construct from punctured tensor products of algebraic codes.
Related
- Parallelizable and addressable transversal non-Clifford gates on good quantum LDPC codes
- Error-Transparent Code Switching Between Permutation-Invariant Codes with Transversal Clifford and Icosahedral Gates
- Substantially Improved Rate and Distance for Addressable Transversal Phase Gates
- Asymptotically good binary triorthogonal codes and higher-level transversal gates
- Finite Classification of Transversal Diagonal Gates Beyond Stabilizer Codes
BibTeX
@misc{golowich2026improved,
title = {{Improved Transversal Non-Clifford Gates from Cup Products}},
author = {Louis Golowich and Itzhak Tamo and Guanyu Zhu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27801},
eprint = {2609.27801},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27801},
}
RIS
TY - UNPB
TI - Improved Transversal Non-Clifford Gates from Cup Products
AU - Louis Golowich
AU - Itzhak Tamo
AU - Guanyu Zhu
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.27801
UR - https://arxiv.org/abs/2609.27801
AB - It is a major challenge in quantum fault-tolerance to obtain low-overhead protocols for performing non-Clifford gates. In this vein, we construct quantum codes with low-weight stabilizers that support transversal (i.e. low-depth) implementations of the non-Clifford C^r−1Z gate, for every constant r≥ 3. In particular, we obtain length-n quantum LDPC codes (with constant-weight stabilizers) of polynomial distance d≥ n^(1−ε)/r supporting transversal C^r−1Z gates on a close-to-linear number k≥ n^1−ε of disjoint tuples of logical qubits, for arbitrarily small ε>0. Our construction is the first with constant-weight stabilizers that obtains dk≫ n, and as a consequence achieves arbitrarily small magic state overhead exponent γ=log(n/k)/log(d)>0. Comparable prior constructions instead required at least polylogarithmic stabilizer weight. We also show how to obtain linearly many k=Ω(n) logical C^r−1Z gates, though with stabilizer weight and physical circuit depth n^ε. We show that our transversal gates also support addressing (i.e. targeting) of specific logical qubits. To obtain our codes, we develop a general transformation based on cup products that maps classical codes satisfying a multiplication property to quantum codes with transversal C^r−1Z. We apply this transformation to a new family of classical Tanner codes that we construct from punctured tensor products of algebraic codes.
ER -