Logical Entangling with Phantom Codes in Hypergraph Products

• URL: http://arxiv.org/abs/2607.12948v1
• Date: Tue, 14 Jul 2026 16:35:36 GMT
• Title: Logical Entangling with Phantom Codes in Hypergraph Products
• Abstract summary: Phantom codes implement every ordered in-block logical CNOT through physical qubit permutations and Pauli-frame updates.<n>This mechanism can coexist with the low-weight stabilizer structure of qLDPC codes.<n>We show that the simplex-repetition family is the unique HGP family satisfying the phantom condition.
• Score: 6.119816361973293
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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