Abstract: Historically, a Nlog1/2(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 N1/3 for LDPC stabilizer codes -- present since the discovery of the 3D color code -- was only recently overcome by a construction achieving an Ω(N) distance (arXiv:2501.19375). The present work further breaks the 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 Ω(N2/3) distance (a linear X-distance of Θ(N)) and a dimension of Θ(N2/3), which enables fault-tolerant preparation of Θ(N1/3) 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 M, which exhibit both a power-law Z2-(q, 2q)-systolic freedom and Θ(vol(M)) triple intersection points of 2q-dimensional submanifolds.
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