Phantom codes: Entangling logical qubits without physical gates

Logical entangling gates are a major cost in fault-tolerant quantum computing. We study phantom codes—stabilizer codes whose intrinsic entanglement enables logical entangling gates without additional physical operations, shifting the cost to classical recompilation of the remaining circuit. We exhaustively enumerate all CSS phantom codes up to [[n,k,d]] with n up to 14 and extend the search to n up to 27 using SAT methods, constructing infinite families including higher-distance generalizations of the hypercube codes [[2^D,D,2]] and the [[12,2,4]] Carbon code. For each example, we classify the non-phantom logical operations. End-to-end simulations of many-body dynamics and resource-state preparation at current error rates show that phantom codes can outperform surface codes in realistic regimes. By leveraging latent entanglement within encoded states, phantom codes reframe the compilation layer of fault-tolerant computing, opening opportunities for lower overhead and higher-fidelity logical gates.