On verifying validity of quantum computations for BQP

Recently proposed schemes for verifying quantum computations in the `high complexity' regime i.e. beyond the remit of classical computers, have been based on adaptations of the classical theory of interactive proof systems (IP). These schemes remarkably provide confidence against arbitrarily malicious adversarial behaviour in the misfunctioning of the quantum computing device in the course of the verification protocol. Here we propose a different verification scheme which although not secure against arbitrarily adversarial behaviour, may nevertheless be sufficiently acceptable in many practical situations. Our scheme is far simpler than the IP based schemes, and in contrast to them, 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.