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

## TL;DR

This paper introduces a method to experimentally estimate the effective Hilbert space dimension of quantum devices by analyzing time-fluctuations and thermalization dynamics, providing a practical measure of their quantum computational capacity.

## Contribution

It proposes a novel, experimentally feasible approach to measure the accessible Hilbert space dimension of quantum devices using fluctuation-dissipation relations in chaotic systems.

## Key findings

- Method to extract Hilbert space dimension from decay rates
- Demonstration of fluctuation-dissipation theorem in high-temperature quantum chaos
- Potential for assessing quantum device functionality

## Abstract

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.

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Source: https://tomesphere.com/paper/1906.06206