  With today's quantum processors venturing into regimes beyond the
capabilities of classical devices [1-3], we face the challenge to verify that
these devices perform as intended, even when we cannot check their results on
classical computers [4,5]. In a recent breakthrough in computer science [6-8],
a protocol was developed that allows the verification of the output of a
computation performed by an untrusted quantum device based only on classical
resources. Here, we follow these ideas, and demonstrate in a first,
proof-of-principle experiment a verification protocol using only classical
means on a small trapped-ion quantum processor. We contrast this to
verification protocols, which require trust and detailed hardware knowledge, as
in gate-level benchmarking [9], or additional quantum resources in case we do
not have access to or trust in the device to be tested [5]. While our
experimental demonstration uses a simplified version [10] of Mahadev's protocol
[6] we demonstrate the necessary steps for verifying fully untrusted devices. A
scaled-up version of our protocol will allow for classical verification,
requiring no hardware access or detailed knowledge of the tested device. Its
security relies on post-quantum secure trapdoor functions within an interactive
proof [11]. The conceptually straightforward, but technologically challenging
scaled-up version of the interactive proofs, considered here, can be used for a
variety of additional tasks such as verifying quantum advantage [8], generating
[12] and certifying quantum randomness [7], or composable remote state
preparation [13].
