Entangling logical qubits without physical operations

Abstract

Fault-tolerant logical entangling gates are essential for scalable quantum computing but are limited by the error rates and overheads of physical two-qubit gates and stabilizer measurements. We introduce phantom codes, a class of codes that attain the minimal possible cost: all in-block logical entangling operations reduce to qubit permutations that can be absorbed at the compilation stage, yielding zero physical-gate overhead and perfect logical fidelity. Our first main result is the discovery of such codes via four mechanisms. By exhaustively enumerating all 2.71x1010inequivalent CSS codes up to n=14 we find all codes to this scale and then identify additional instances up to n=21 using SAT methods. We then use quantum Reed-Muller constructions to find higher k instances and a "binzarization-and-concatenation" scheme to achieve higher d. Second, through end-to-end noisy simulations, we demonstrate scalable advantages of phantom codes over the surface code across multiple tasks. This includes a ~207x improvement in logical fidelity for Trotterized many-body quantum simulation at current physical error rates, with comparable qubit counts and a 22% preselection rate; the advantage persists from 8 to $64$ logical qubits. These results establish phantom codes as a viable architectural route to fault-tolerant quantum computation with scalable benefits for workloads with dense local entangling structure.