IP ⊆ QIP

Inclusion · quantum setting · folklore

Statement

$\mathbf{IP}\subseteq \mathbf{QIP}$: a classical IP protocol is a QIP protocol whose verifier measures each incoming message in the computational basis and otherwise runs the classical verifier, with the same completeness and soundness — folklore.

Sketch

Measuring each prover message in the computational basis turns any quantum prover into a randomized classical one: the distribution of its next message given the classical transcript so far is a classical strategy, so the soundness of the classical protocol bounds its success.