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

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. Actually, the dimension of the space of quantum states that are available to a quantum device is a meaningful measure of its functionality, but unfortunately this quantity cannot be directly experimentally determined. Here we outline an experimentally realisable approach to obtaining the scaling of the required Hilbert space of such a device to compute such 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. arXiv:1906.06206v1 [quant-ph]

QuSpin: a Python package for dynamics and exact diagonalisation of quantum many body systems. Part II: bosons, fermions and higher spins

“…The reason for this often times relies on the fact that the underlying physics of these problems cannot be explained without taking into consideration the contribution from high-energy states excited during the nonequilibrium process. Some prominent examples of such problems include the study of the many-body localisation (MBL) transition [24,25,26,27,28], the Eigenstate Thermalisation hypothesis [29], ergodicity breaking, thermalization and scrambling [30,31,32], quantum quench dynamics [33], periodically-driven systems [34,35,36,37,38,39,40,41,42], non-demolition measurements in many-body systems [43], long-range quantum coherence [44], dynamics-induced instabilities [45,46,47,48,49,50,51,52], adiabatic and counter-diabatic state preparation [53,54,55,56,57], dynamical [58,59] and topological [60] phase transitions applications of Machine Learning to (non-equilibrium) physics [61,49,62,63,64,65,66], optimal control [67,<...>…”

Taking snapshots of a quantum thermalization process: Emergent classicality in quantum jump trajectories

“…In general, for chaotic systems one may expect this function to be peaked around a certain energy, with a width (E ) that may depend on the energy of the wave function. We showed in [58] that this change in width with energy can in fact be incorporated into our theory. From Eq.…”

“…In this Appendix we outline in brief the RMT methodology developed in Refs. [32,43,49,58], on which our calculations are based. We focus here on making clear the required assumptions on which the calculations rest and refer the reader to the above references for details on the calculations themselves.…”

“…[32] extends and formalizes key features of observables and describes time evolution of observables, and Ref. [58] extends the approach to finite temperatures and applies the method to an application on quantum computers and other devices. Self-averaging of chaotic wave functions is shown and discussed in the Appendixes of Ref.…”

“…Self-averaging of chaotic wave functions is shown and discussed in the Appendixes of Ref. [58]. Reference [49] goes into more detail regarding the assumptions on observables made below, obtaining physical conditions for the fulfillment of the crucial assumptions.…”