  Quantum computers are expected to offer substantial speedups over their
classical counterparts and to solve problems that are intractable for classical
computers. Beyond such practical significance, the concept of quantum
computation opens up new fundamental questions, among them the issue whether or
not quantum computations can be certified by entities that are inherently
unable to compute the results themselves. Here we present the first
experimental verification of quantum computations. We show, in theory and in
experiment, how a verifier with minimal quantum resources can test a
significantly more powerful quantum computer. The new verification protocol
introduced in this work utilizes the framework of blind quantum computing and
is independent of the experimental quantum-computation platform used. In our
scheme, the verifier is only required to generate single qubits and transmit
them to the quantum computer. We experimentally demonstrate this protocol using
four photonic qubits and show how the verifier can test the computer's ability
to perform measurement-based quantum computations.
