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Quantum devices, such as quantum simulators, quantum annealers, and quantumcomputers, may be exploited to solve problems beyond what is tractable withclassical computers. This may be achieved as the Hilbert space available toperform such `calculations' is far larger than that which may be classicallysimulated. In practice, however, quantum devices have imperfections, which maylimit the accessibility to the whole Hilbert space. We thus determine that thedimension of the space of quantum states that are available to a quantum deviceis a meaningful measure of its functionality, though unfortunately thisquantity cannot be directly experimentally determined. Here we outline anexperimentally realisable approach to obtaining the required Hilbert spacedimension of such a device to compute its time evolution, by exploiting thethermalization dynamics of a probe qubit. This is achieved by obtaining afluctuation-dissipation theorem for high-temperature chaotic quantum systems,which facilitates the extraction of information on the Hilbert space dimensionvia measurements of the decay rate, and time-fluctuations.

Ergodicity probes: using time-fluctuations to measure the Hilbert space dimension