Logical Entangling with Phantom Codes in Hypergraph Products
arXiv:2607.12948
The paper identifies the simplex‑repetition family of binary CSS hypergraph product codes as the unique HGP family that satisfies the phantom‑code condition, allowing logical CNOT gates to be performed through qubit permutations and Pauli‑frame updates while preserving low‑weight stabilizers, and shows that this approach outperforms rotated surface‑code baselines in GHZ‑state preparation and Trotterized quantum simulation on neutral‑atom hardware.
Abstract
Logical entangling gates are a major source of physical spacetime overhead in fault-tolerant quantum computation. Phantom codes reduce this cost by implementing every ordered in-block logical CNOT through physical qubit permutations and Pauli-frame updates. Whether this mechanism can coexist with the low-weight stabilizer structure of qLDPC codes is a central question for low-overhead fault-tolerant architectures. We give a deterministic answer within binary CSS hypergraph product (HGP) codes. Up to natural equivalences, the simplex-repetition family is the unique HGP family satisfying the phantom condition. We then evaluate this family under circuit-level noise in logical GHZ-state preparation and Trotterized many-body quantum simulation. The codes retain low-weight stabilizer checks and yield concrete advantages over rotated surface-code baselines in both benchmarks. Reconfigurable neutral-atom arrays offer a natural setting for this approach, supporting nonlocal qLDPC operations while enabling in-block logical CNOTs without additional physical operations. Together, these results make precise how permutation-based logical entangling constrains code design within the HGP framework, demonstrate the circuit-level benefits of the unique family, and guide the search for phantom qLDPC families with better asymptotic parameters for low-overhead fault tolerance on neutral-atom hardware.