Efficient classical verification of quantum computations

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.