Related papers: Transversal non-Clifford gates on qLDPC codes breaking the $\sqrt{N}$ distance barrier and quantum-inspired geometry with $\mathbb{Z}

Transversal non-Clifford gates on qLDPC codes breaking the $\sqrt{N}$ distance barrier and quantum-inspired geometry with $\mathbb{Z}_2$ systolic freedom

URL: http://arxiv.org/abs/2507.15056v1
Date: Sun, 20 Jul 2025 17:35:31 GMT
Title: Transversal non-Clifford gates on qLDPC codes breaking the $\sqrt{N}$ distance barrier and quantum-inspired geometry with $\mathbb{Z}_2$ systolic freedom
Authors: Guanyu Zhu,
Abstract summary: A distance barrier of $Omega(sqrtN) for LDPC stabilizer codes was only recently overcome by a construction achieving an $Omega(sqrtN) distance (arXiv:250375375)<n>The resulting code achieves an $Omega(N2/3) distance (a linear $X$-distance of $Theta(N)$) and a dimension of $Theta(N2/3)$.<n>This new quantum code also inspires the discovery of a family of $3q$-dimensional $mathcal
Score: 0.0
License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
Abstract: Historically, a $\sqrt{N}log^{1/2}(N)$ distance barrier for quantum low-density parity-check (LDPC) codes with $N$ qubits persisted for nearly two decades, until the recent discovery of the fibre-bundle code. An open question is whether such a distance barrier can be broken while preserving the ability to perform transversal non-Clifford gates. In this direction, another long-standing distance barrier of $N^{1/3}$ for LDPC stabilizer codes -- present since the discovery of the 3D color code -- was only recently overcome by a construction achieving an $\Omega(\sqrt{N})$ distance (arXiv:2501.19375). The present work further breaks the $\sqrt{N}$ distance barrier by taking a homological product of three good qLDPC codes, combined with the Freedman-Hastings code-to-manifold mapping and the triple cup product to implement transversal CCZ gates. The resulting code achieves an $\Omega(N^{2/3})$ distance (a linear $X$-distance of $\Theta(N)$) and a dimension of $\Theta(N^{2/3})$, which enables fault-tolerant preparation of $\Theta(N^{1/3})$ independent logical CCZ magic states in a single shot, without distillation (`magic state fountain'). This new quantum code also inspires the discovery of a family of exotic $3q$-dimensional manifolds $\mathcal{M}$, which exhibit both a power-law $\mathbb{Z}_2$-($q$, $2q$)-systolic freedom and $\Theta(vol(\mathcal{M}))$ triple intersection points of $2q$-dimensional submanifolds.

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