  Quantum devices, such as quantum simulators, quantum annealers, and quantum
computers, may be exploited to solve problems beyond what is tractable with
classical computers. This may be achieved as the Hilbert space available to
perform such `calculations' is far larger than that which may be classically
simulated. In practice, however, quantum devices have imperfections, which may
limit the accessibility to the whole Hilbert space. We thus determine that the
dimension of the space of quantum states that are available to a quantum device
is a meaningful measure of its functionality, though unfortunately this
quantity cannot be directly experimentally determined. Here we outline an
experimentally realisable approach to obtaining the required Hilbert space
dimension of such a device to compute its time evolution, by exploiting the
thermalization dynamics of a probe qubit. This is achieved by obtaining a
fluctuation-dissipation theorem for high-temperature chaotic quantum systems,
which facilitates the extraction of information on the Hilbert space dimension
via measurements of the decay rate, and time-fluctuations.
