Among various algorithms designed to exploit the specific properties of quantum computers with respect to classical ones, the quantum adiabatic algorithm is a versatile proposition to find the minimal value of an arbitrary cost function (ground state energy). Random optimization problems provide a natural testbed to compare its efficiency with that of classical algorithms. These problems correspond to mean field spin glasses that have been extensively studied in the classical case. This paper reviews recent analytical works that extended these studies to incorporate the effect of quantum fluctuations, and presents also some original results in this direction.
Dynamic correlation and response functions of classical and quantum systems in thermal equilibrium are connected by fluctuation-dissipation theorems, which allow an alternative definition of their (unique) temperature. Motivated by this fundamental property, we revisit the issue of thermalization of closed many-body quantum systems long after a sudden quench, focussing on the non-equilibrium dynamics of the Ising chain in a critical transverse field. We show the emergence of distinct observable-dependent effective temperatures, which rule out Gibbs thermalization in a strict sense but might still have a thermodynamic meaning.Introduction. The development of experimental techniques which prevent dissipation in quantum many-body systems has triggered increasing interest in the nonequilibrium dynamics of such closed systems. The unitary non-equilibrium dynamics of a system initially prepared in a state which is not an eigenstate of its Hamiltonian is called a quantum quench. Basic questions as to whether a stationary state is reached and how this can be characterized naturally arise. These questions have been addressed in a number of simple models, including the one-dimensional systems reviewed in Refs. [1,2]. Early studies led to the following picture: Non-integrable systems should eventually reach a thermal stationary state characterized by a Gibbs distribution with a single temperature. Integrable systems, instead, are not expected to thermalize but their asymptotic stationary state should nonetheless be described by a so-called generalized Gibbs ensemble (GGE) with one effective temperature for each conserved quantity [3][4][5][6]. Interestingly enough, depending on the specific quantity and the system's parameters a Gibbs ensemble turns out to capture anyhow some relevant features of the non-equilibrium dynamics of integrable systems [7]. In particular, observables that are non-local in the quasi-particles display numerically the same relaxation scales as in equilibrium with a suitable effective temperature, at least for small quenches [6,7]. Local quantities instead do not, with possible exceptions for quenches at criticality.Our purpose is to revisit the debated issue of thermalization in closed quantum systems with tools developed for the study of classical and quantum dissipative glassy systems. The analysis of thermalization in closed quantum systems focused so far on the property that expectation values of quantities -such as (i) the conserved energy and two-point correlation functions depending on either (ii) one or (iii) two times -should behave, at long times, as the corresponding averages calculated on suitable statistical ensembles. However, an equally im-
The Eigenstate Thermalization Hypothesis (ETH) implies a form for the matrix elements of local operators between eigenstates of the Hamiltonian, expected to be valid for chaotic systems. Another signal of chaos is a positive Lyapunov exponent, defined on the basis of Loschmidt echo or outof-time-order correlators. For this exponent to be positive, correlations between matrix elements unrelated by symmetry, usually neglected, have to exist. The same is true for the peak of the dynamic heterogeneity length χ4, relevant for systems with slow dynamics. These correlations, as well as those between elements of different operators, are encompassed in a generalized form of ETH.
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