Quantum deletion errors remove qubits from a quantum state. Quantum insertion errors add spurious qubits. The two seem like different problems requiring different codes. A deletion shrinks the state; an insertion inflates it. The error geometries point in opposite directions.
Nakamura and Nozaki (arXiv:2602.20635) show that quantum codes designed to correct t deletion errors can also correct any combination of insertions and deletions totaling t. A code that handles t deletions automatically handles t insertions, or any mixture — without being designed for insertions at all.
The mechanism mirrors classical coding theory: the error-correcting capability is defined by disjoint error spheres around codewords. The quantum indel distance — a new metric the authors introduce — characterizes the separation between quantum states under both insertions and deletions simultaneously. If the deletion spheres are disjoint, the indel spheres are disjoint. The geometry that separates deletion errors also separates insertion errors.
The result is not obvious because insertions and deletions have different physical effects on quantum states. A deletion reduces the Hilbert space dimension; an insertion increases it. But at the level of error correction — the level of distinguishability between corrupted states — the two are equivalent up to the same budget. The code doesn't need to know which type of error occurred; it only needs to know that at most t errors happened.
The general observation: when two error types appear opposite but are defined by the same distance metric, correcting one automatically corrects the other. The correction capability is a property of the metric geometry, not of the specific error mechanism. Codes designed for one direction inherit protection in the other for free.