Quantum error correction
Protecting quantum information from noise without looking at it — the precondition for quantum computers becoming useful.
Classical error correction copies and votes. Here that fails: the no-cloning theorem forbids copying, and measuring the qubit destroys the superposition. The solution is to spread a logical state across many physical qubits and measure only syndromes — quantities that reveal whether an error occurred, and where, without revealing which state is stored.
That errors are continuous rather than discrete looks like it should make this hopeless, but the measurement forces every error to become either a bit flip or a phase flip, and both are fixable. The threshold theorem says that if the error rate per gate is below a certain bound, arbitrarily long computations can be made reliable. That bound has been reached in the lab; doing it at scale is the remaining work.