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arXiv:2112.08490 · Physical Review A

Three phases of quantum annealing: Fast, slow, and very slow

Artur Soriani, Pierre Nazé, Marcus V. S. Bonança, Bartłomiej Gardas, Sebastian Deffner

Published Apr 14, 2022• 17 citations
arXivHTMLPDF

Abstract

Currently, existing quantum annealers have proven themselves as viable technology for the first practical applications in the noisy-intermediate-scale-quantum era. However, to fully exploit their capabilities, a comprehensive characterization of their finite-time excitations is instrumental. To this end, we develop a phase diagram for driven Ising chains, from which the scaling behavior of the excess work can be read off as a function of process duration and system size. ``Fast'' processes are well described by the Kibble-Zurek mechanism; ``slow'' process are governed by effective Landau-Zener dynamics; and ``very slow'' processes can be approximated with adiabatic perturbation theory.

DOI
10.1103/physreva.105.042423

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Currently, existing quantum annealers have proven themselves as viable technology for the first practical applications in the noisy-intermediate-scale-quantum era. However, to fully exploit their capabilities, a comprehensive characterization of their finite-time excitations is instrumental. To this end, we develop a phase diagram for driven Ising chains, from which the scaling behavior of the excess work can be read off as a function of process duration and system size. Fast'' processes are well described by the Kibble-Zurek mechanism; slow'' process are governed by effective Landau-Zener dynamics; and ``very slow'' processes can be approximated with adiabatic perturbation theory.
The quantum Ising chain has shortcuts to adiabaticity when operated with weak processes. However, when exactly do the non-equilibrium effects of the Kibble-Zurek mechanism, inherent to the system, appear in the optimal protocols in such a context? I propose here that such contrasting difference occurs due to the manner by which one measures the excitation spent energy of the system. Therefore, in this work, I made a qualitative analysis of a quantum annealing procedure of the time-averaged excess work, where the system acquires as a diverging decorrelation time the heuristic Kibble-Zurek mechanism relaxation time. Four important effects are then observed: the absence of shortcuts to adiabaticity, the pausing effect around the critical point in the optimal protocol when the Kibble-Zurek mechanism holds, the persistence of the time-averaged work to avoid slowly-varying regime even for large switching times, and diverging fluctuations of the time-averaged work. In the end, by comparing the excess and the time-averaged excess works, I conclude that this last one is not useful to measure the excitation spent energy in weak processes, although brings an intuition to what happens in the strong driving case.
We present here our study of the adiabatic quantum dynamics of a random Ising chain across its quantum critical point. The model investigated is an Ising chain in a transverse field with disorder present both in the exchange coupling and in the transverse field. The transverse field term is proportional to a function $\ensuremath{\Gamma}(t)$ which, as in the Kibble-Zurek mechanism, is linearly reduced to zero in time with a rate ${\ensuremath{\tau}}^{\ensuremath{-}1}$, $\ensuremath{\Gamma}(t)=\ensuremath{-}t∕\ensuremath{\tau}$, starting at $t=\ensuremath{-}\ensuremath{\infty}$ from the quantum disordered phase $(\ensuremath{\Gamma}=\ensuremath{\infty})$ and ending at t=0t=0t=0 in the classical ferromagnetic phase $(\ensuremath{\Gamma}=0)$. We first analyze the distribution of the gaps, occurring at the critical point ${\ensuremath{\Gamma}}_{c}=1$, which are relevant for breaking the adiabaticity of the dynamics. We then present extensive numerical simulations for the residual energy Eres{E}_{\mathrm{res}}Eres​ and density of defects ${\ensuremath{\rho}}_{k}$ at the end of the annealing, as a function of the annealing inverse rate $\ensuremath{\tau}$. Both the average ${E}_{\mathrm{res}}(\ensuremath{\tau})$ and ${\ensuremath{\rho}}_{k}(\ensuremath{\tau})$ are found to behave logarithmically for large $\ensuremath{\tau}$, but with different exponents, $[{E}_{\mathrm{res}}(\ensuremath{\tau})∕L{]}_{\mathrm{av}}\ensuremath{\sim}1∕{\mathrm{ln}}^{\ensuremath{\zeta}}(\ensuremath{\tau})$ with $\ensuremath{\zeta}\ensuremath{\approx}3.4$, and $[{\ensuremath{\rho}}_{k}(\ensuremath{\tau}){]}_{\mathrm{av}}\ensuremath{\sim}1∕{\mathrm{ln}}^{2}(\ensuremath{\tau})$. We propose a mechanism for $1∕{\mathrm{ln}}^{2}\ensuremath{\tau}$ behavior of $[{\ensuremath{\rho}}_{k}{]}_{\mathrm{av}}$ based on the Landau-Zener tunneling theory and on a Fisher's-type real-space renormalization group analysis of the relevant gaps. The model proposed shows therefore a paradigmatic example of how an adiabatic quantum computation can become very slow when disorder is at play, even in the absence of any source of frustration.
We study errors of quantum annealing and simulated annealing to highlight better performance of quantum annealing over simulated annealing. Quantum annealing and simulated annealing perform optimization through a quantum adiabatic evolution and a quasi-static evolution respectively. In both methods, dynamics across a phase transition plays a crucial role. The Kibble-Zurek mechanism is known as an underlying physics of defect formation during a time-evolution across a phase transition. We apply an argument for the Kibble-Zurek mechanism to the error generation of quantum annealing and simulated annealing. We show that, for the disordered Ising chain, the kink density and residual energy per spin of quantum annealing decay as (ln τ ) −2 and (ln τ ) −4 respectively, whereas those of simulated annealing decay as (ln τ ) −1 and (ln τ ) −2 with the annealing rate −1/ τ . These results imply better performance of quantum annealing. We also develop our theory for a two-dimensional spin-glass model.
Current-generation quantum annealers have already proven to be successful problem solvers. Yet quantum annealing is still very much in its infancy, with suboptimal applicability. For instance, to date it is still an open question which annealing protocol causes the fewest diabatic excitations for a given eigenspectrum, and even whether there is a universally optimal strategy. Therefore, in this paper, we report analytical and numerical studies of the diabatic excitations arising from nonlinear protocols applied to the transverse field Ising chain, the exactly solvable model that serves as a quantum annealing playground. Our analysis focuses on several driving schemes that inhibit or facilitate the dynamic phases discussed in a previous work. Rather remarkably, we find that the paradigmatic Kibble-Zurek behavior can be suppressed with ``pauses'' in the evolution, both for crossing and for stopping at the quantum critical point of the system.
The Kibble-Zurek mechanism captures universality when a system is driven through a continuous phase transition. Here we study the dynamical aspect of quantum phase transitions in the Ising Field Theory where the quantum critical point can be crossed in different directions in the two-dimensional coupling space leading to different scaling laws. Using the Truncated Conformal Space Approach, we investigate the microscopic details of the Kibble-Zurek mechanism in terms of instantaneous eigenstates in a genuinely interacting field theory. For different protocols, we demonstrate dynamical scaling in the non-adiabatic time window and provide analytic and numerical evidence for specific scaling properties of various quantities. In particular, we argue that the higher cumulants of the excess heat exhibit universal scaling in generic interacting models for a slow enough ramp.
We study the dynamical scaling properties of the two-dimensional fully frustrated Ising model by simulated annealing. The model has a highly degenerate, critical ground state with well known equilibrium properties. To study its out-of-equilibrium behavior, we develop an annealing protocol that is convenient for extending the Kibble-Zurek mechanism to the case of an exponentially divergent correlation length in the limit of the temperature T→Tc=0T \to T_c = 0T→Tc​=0. We find excellent agreement with the scaling forms derived, including a dependence on the initial temperature that is not present in the conventional Kibble-Zurek mechanism. The dynamic exponent zzz is consistent among all quantities studied; fits of its dependence on the initial temperature give z=1.99(2)z = 1.99(2)z=1.99(2) (susceptibility), z=2.072(8)z = 2.072(8)z=2.072(8) (excess energy), and z=2.09(3)z = 2.09(3)z=2.09(3) (Edwards-Anderson order parameter). These values are close to z=2z=2z=2, which is consistent with previous studies of relaxation after instantaneous quenches and of time dependent correlations in equilibrium, including the estimate z=2.004(9)z = 2.004(9)z=2.004(9) from equilibrium autocorrelation functions.