We investigate the effect of tuned inhomogeneous transverse field in a disordered Ising chain on properties of the quantum phase transition and performance of quantum annealing. We set the strength of the transverse field on a spin in proportion to the coupling strength connecting the spin. This setting of transverse field gives a qualitative difference from previous studies, where the transverse field is homogeneous over spins or random independently of the coupling strength. We find that the dynamical critical exponent is finite in contrast to previous studies where it is infinity. We numerically confirm that the performance of quantum annealing is improved as expected from the Kibble--Zurek mechanism. This work demonstrates an importance of tuning the transverse field on each spin in quantum annealing.
We investigate the effect of tuned inhomogeneous transverse field in a disordered Ising chain on properties of the quantum phase transition and performance of quantum annealing. We set the strength of the transverse field on a spin in proportion to the coupling strength connecting the spin. This setting of transverse field gives a qualitative difference from previous studies, where the transverse field is homogeneous over spins or random independently of the coupling strength. We find that the dynamical critical exponent is finite in contrast to previous studies where it is infinity. We numerically confirm that the performance of quantum annealing is improved as expected from the Kibble--Zurek mechanism. This work demonstrates an importance of tuning the transverse field on each spin in quantum annealing.
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=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 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.
In transverse-field Ising models, disorder in the couplings gives rise to a drastic reduction of the critical energy gap and, accordingly, to an unfavorable, slower-than-algebraic scaling of the density of defects produced when the system is driven through its quantum critical point. By applying Kibble-Zurek theory and numerical calculations, we demonstrate in the one-dimensional model that the scaling of defect density with annealing time can be made algebraic by balancing the coupling disorder with suitably chosen inhomogeneous driving fields. Depending on the tail of the coupling distribution at zero, balancing can be either perfect, leading to the well-known inverse-square law of the homogeneous system, or partial, still resulting in an algebraic decrease but with a smaller, nonuniversal exponent. We also study defect production during an environment-temperature quench of the open variant of the model in which the system is slowly cooled down to its quantum critical point. According to our scaling and numerical results, uncorrelated disorder induces a logarithmic temporal decrease of the defect density, while balanced disorder leads again to an algebraic decay.
We solve the mean-field-like p-spin Ising model under a spatiotemporal inhomogeneous transverse field to study the effects of inhomogeneity on the performance of quantum annealing. We previously found that the problematic first-order quantum phase transition that arises under the conventional homogeneous field protocol can be avoided if the temperature is zero and the local field is completely turned off site by site after a finite time. We show in the present paper that, when these ideal conditions are not satisfied, another series of first-order transitions appear, which prevents us from driving the system while avoiding first-order transitions. Nevertheless, under these nonideal conditions, quantitative improvements can be obtained in terms of narrower tunneling barriers in the free-energy landscape. A comparison with classical simulated annealing establishes a limited quantum advantage in the ideal case, since inhomogeneous temperature driving in simulated annealing cannot remove a first-order transition, in contrast to the quantum case. The classical model of spin-vector Monte Carlo is also analyzed, and we find it to have the same thermodynamic phase diagram as the quantum model in the ideal case, with deviations arising at nonzero temperature.
Simulated quantum annealing (SQA) is a classical computational strategy that emulates a quantum annealing (QA) dynamics through a path-integral Monte Carlo whose parameters are changed during the simulation. Here we apply SQA to the one-dimensional transverse field Ising chain, where previous works have shown that, in the presence of disorder, a coherent QA provides a quadratic speedup with respect to classical simulated annealing, with a density of Kibble-Zurek defects decaying as ${\ensuremath{\rho}}_{\mathrm{KZ}}^{\mathrm{QA}}\ensuremath{\sim}{({log}_{10}\ensuremath{\tau})}^{\ensuremath{-}2}$ as opposed to ${\ensuremath{\rho}}_{\mathrm{KZ}}^{\mathrm{SA}}\ensuremath{\sim}{({log}_{10}\ensuremath{\tau})}^{\ensuremath{-}1}, \ensuremath{\tau}$ being the total annealing time, while for the ordered case both give the same power law ${\ensuremath{\rho}}_{\mathrm{KZ}}^{\mathrm{QA}}\ensuremath{\approx}{\ensuremath{\rho}}_{\mathrm{KZ}}^{\mathrm{SA}}\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}1/2}$. We show that the dynamics of SQA, while correctly capturing the Kibble-Zurek scaling ${\ensuremath{\tau}}^{\ensuremath{-}1/2}$ for the ordered case, is unable to reproduce the QA dynamics in the disordered case at intermediate $\ensuremath{\tau}$. We analyze and discuss several issues related to the choice of the Monte Carlo moves (local or global in space), the time-continuum limit needed to eliminate the Trotter-discretization error, and the long autocorrelation times shown by a local-in-space Monte Carlo dynamics for large disordered samples.
Based on the strong-disorder renormalization group method, a microscopic mechanism of defect formation in the quantum annealing of the random transverse-field Ising chain is proposed, which represents the annealing process as a gradual aggregation of strongly coupled spin clusters. The ferromagnetic ground state of clusters is either preserved or get excited in pairwise fusions of clusters, depending on the effective annealing rate of the fusion, the latter events being responsible for the appearance of defects in the final state. A consequence of the theory is that, although the Griffiths-McCoy phases surrounding the critical point are gapless, they are still effectively gapped from the point of view of quantum annealing. Thereby we provide an explanation of the finiteness of gap outside of the critical point, which was implicit in an early approach to the problem by Kibble-Zurek scaling [Dziarmaga, Phys. Rev. B {\bf 74}, 064416 (2006)]. Furthermore, by identifying the accessible excitations, we refine the functional form of the vanishing of the gap at the critical point. The defect density in the final state is found to decrease with the annealing time τ, as n(τ)∼ln−2(ln2ττ) for large τ. In addition to this, our approach gives access also to the density of defects at intermediate times of the annealing process.
We review quantum phase transitions of spin systems in transverse magnetic fields taking the examples of the spin-1/2 Ising and XY models in a transverse field. Beginning with an overview of quantum phase transitions, we introduce a number of model Hamiltonians. We provide exact solutions in one spatial dimension connecting them to conformal field theoretical studies. We also discuss Kitaev models and some other exactly solvable spin systems. Studies of quantum phase transitions in the presence of quenched randomness and with frustrating interactions are presented in detail. We discuss novel phenomena like Griffiths-McCoy singularities. We then turn to more recent topics like information theoretic measures of the quantum phase transitions in these models such as concurrence, entanglement entropy, quantum discord and quantum fidelity. We then focus on non-equilibrium dynamics of a variety of transverse field systems across quantum critical points and lines. After mentioning rapid quenching studies, we dwell on slow dynamics and discuss the Kibble-Zurek scaling for the defect density following a quench across critical points and its modifications for quenching across critical lines, gapless regions and multicritical points. Topics like the role of different quenching schemes, local quenching, quenching of models with random interactions and quenching of a spin chain coupled to a heat bath are touched upon. The connection between non-equilibrium dynamics and quantum information theoretic measures is presented at some length. We indicate the connection between Kibble-Zurek scaling and adiabatic evolution of a state as well as the application of adiabatic dynamics as a tool of a quantum optimization technique known as quantum annealing. The final section is dedicated to a detailed discussion on recent experimental studies of transverse Ising-like systems.