Time-continuous quantum error correction — correcting errors as they happen rather than in discrete rounds — would be ideal for protecting quantum information. The standard approach in circuit quantum electrodynamics: couple qubits to a measurement device via dispersive interactions that approximate parity measurements. This should continuously monitor the error syndrome and correct in real time.
It doesn't work, and the reason is fundamental (arXiv:2603.02106). The parity measurement requires a three-body interaction (two data qubits coupled through a measurement mode). Approximating this with two-body couplings — each qubit coupled independently to the measurement device — makes it impossible to simultaneously suppress measurement backaction on both the logical and error subspaces. You can protect one or the other, not both.
The structural insight: the obstacle is not engineering but algebra. Two-body couplings cannot implement the three-body interaction required for continuous parity monitoring without corrupting the logical information. The approximation is not “good enough for now” — it is structurally incapable of the task. The fix must be architectural: either use systems with native three-body interactions, or switch to erasure-based encodings where the error subspace doesn't need protection. The limitation generalizes beyond circuit QED to any platform approximating higher-order interactions with lower-order ones.