  The onset of the era of fully-programmable error-corrected quantum computers
will be marked by major breakthroughs in all areas of science and engineering.
These devices promise to have significant technological and societal impact,
notable examples being the analysis of big data through better machine learning
algorithms and the design of new materials. Nevertheless, the capacity of
quantum computers to faithfully implement quantum algorithms relies crucially
on their ability to prepare specific high-dimensional and high-purity quantum
states, together with suitable quantum measurements. Thus, the unambiguous
certification of these requirements without assumptions on the inner workings
of the quantum computer is critical to the development of trusted quantum
processors. One of the most important approaches for benchmarking quantum
devices is through the mechanism of self-testing that requires a pair of
entangled non-communicating quantum devices. Nevertheless, although computation
typically happens in a localized fashion, no local self-testing scheme is known
to benchmark high dimensional states and measurements. Here, we show that the
quantum self-testing paradigm can be employed to an individual quantum computer
that is modelled as a programmable black box by introducing a noise-tolerant
certification scheme. We substantiate the applicability of our scheme by
providing a family of outcome statistics whose observation certifies that the
computer is producing specific high-dimensional quantum states and implementing
specific measurements.
