Abstract: Verification in quantum computations is crucial since quantum systems are extremely vulnerable to the environment. However, verifying directly the output of a quantum computation is difficult since we know that efficiently simulating a large-scale quantum computation on a classical computer is usually thought to be impossible. To overcome this difficulty, we propose a self-testing system for quantum computations, which can be used to verify if a quantum computation is performed correctly by itself. Our basic idea is using some extra ancilla qubits to test the output of the computation. We design two kinds of permutation circuits into the original quantum circuit: one is applied on the ancilla qubits whose output indicates the testing information, the other is applied on all qubits (including ancilla qubits) which is aiming to uniformly permute the positions of all qubits. We show that both permutation circuits are easy to achieve. By this way, we prove that any quantum computation has an efficient self-testing system. In the end, we also discuss the relation between our self-testing system and interactive proof systems, and show that the two systems are equivalent if the verifier is allowed to have some quantum capacity.

Key words: quantum computation, verification, self-testing systems, complexity theory