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

Charlie Nation1 and Diego Porras2

1Department of Physics and Astronomy, University of Sussex, Brighton, BN1 9QH, United Kingdom.
2Institute of Fundamental Physics, IFF-CSIC, Calle Serrano 113b, 28006 Madrid, Spain

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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.

In this article we outline a method to exploit the dynamics of chaotic quantum systems to characterize quantum devices, such as quantum computers. We develop a theoretical model of quantum chaos in isolated systems, and propose that by watching the dynamics, one can obtain information on the complexity of the system from simple measurable quantities. There is a lot of interest at the moment in showing that a given quantum device may outperform regular computers in some meaningful way, though the comparison is not always obvious. Here we develop a method of quantifying the complexity of calculated quantum dynamics on such a device, for a comparison to be made. Further, this article contributes to theoretical developments in the thermalization of closed quantum systems, an important problem in the foundations of quantum statistical physics.

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Cited by

[1] Charlie Nation and Diego Porras, "Taking snapshots of a quantum thermalization process: Emergent classicality in quantum jump trajectories", Physical Review E 102 4, 042115 (2020).

[2] Aiman Khan, David Quigley, Max Marcus, Erling Thyrhaug, and Animesh Datta, "Model-Independent Simulation Complexity of Complex Quantum Dynamics", Physical Review Letters 126 15, 150402 (2021).

[3] Phillip Weinberg and Marin Bukov, "QuSpin: a Python package for dynamics and exact diagonalisation of quantum many body systems. Part II: bosons, fermions and higher spins", SciPost Physics 7 2, 020 (2019).

[4] Charlie Nation and Diego Porras, "Non-ergodic quantum thermalization", arXiv:1908.11773.

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