  We propose an efficient scheme for verifying quantum computations in the
`high complexity' regime i.e. beyond the remit of classical computers.
Previously proposed schemes remarkably provide confidence against arbitrarily
malicious adversarial behaviour in the misfunctioning of the quantum computing
device. Our scheme is not secure against arbitrarily adversarial behaviour, but
may nevertheless be sufficiently acceptable in many practical situations. With
this concession we gain in manifest simplicity and transparency, and in
contrast to previous schemes, our verifier is entirely classical. It is based
on the fact that adaptive Clifford circuits on general product state inputs
provide universal quantum computation, while the same processes without
adaptation are always classically efficiently simulatable.
