Absence of Transport Under a Slowly Varying Potential in Disordered Systems

In the tight-binding random Hamiltonian on Z ^d , we consider the charge transport induced by an electric potential which varies sufficiently slowly in time, and prove that it is almost surely equal to zero at high disorder. In order to compute the charge transport, we adopt the adiabatic approximation and prove a weak form of adiabatic theorem while there is no spectral gap at the Fermi energy.     

Does adiabatic quantum optimization fail for NP-complete problems?

It has been recently argued that adiabatic quantum optimization would fail in solving NP-complete problems because of the occurrence of exponentially small gaps due to crossing of local minima of the final Hamiltonian with its global minimum near the end of the adiabatic evolution. Using perturbation expansion, we analytically show that for the NP-hard problem known as maximum independent set, there always exist adiabatic paths along which no such crossings occur. Therefore, in order to prove that adiabatic quantum optimization fails for any NP-complete problem, one must prove that it is impossible to find any such path in polynomial time.     

Effect of local minima on adiabatic quantum optimization

We present a perturbative method to estimate the spectral gap for adiabatic quantum optimization, based on the structure of the energy levels in the problem Hamiltonian. We show that, for problems that have an exponentially large number of local minima close to the global minimum, the gap becomes exponentially small making the computation time exponentially long. The quantum advantage of adiabatic quantum computation may then be accessed only via the local adiabatic evolution, which requires phase coherence throughout the evolution and knowledge of the spectrum. Such problems, therefore, are not suitable for adiabatic quantum computation.     

Adiabatic Theorem without a Gap Condition

We prove the adiabatic theorem for quantum evolution without the traditional gap condition. All that this adiabatic theorem needs is a (piecewise) twice differentiable finite dimensional spectral projection. The result implies that the adiabatic theorem holds for the ground state of atoms in quantized radiation field. The general result we prove gives no information on the rate at which the adiabatic limit is approached. With additional spectral information one can also estimate this rate.     

Adiabatic evolution under quantum control

One difficulty with adiabatic quantum computation is the limit on the computation time. Here we propose two schemes to speed-up the adiabatic evolution. To apply this controlled adiabatic evolution to adiabatic quantum computation, we design one of the schemes without any explicit knowledge of the instantaneous eigenstates of the final Hamiltonian. Whereas in another scheme, we assume that the ground state of the Hamiltonian is known, and this information can be used to design the control. By these techniques, a linear speed-up proportional to the nonlinearity can be predicted. As an illustration, we study a two-level system driven by a time-dependent magnetic field under the control. The problem of finding an item in an unsorted database by adiabatic evolution is also examined. The physics behind the control scheme is interpreted.     

Transitionless driving on local adiabatic quantum search algorithm

We apply the transitionless driving on the local adiabatic quantum search algorithm to speed up the adiabatic process.By studying quantum dynamics of the adiabatic search algorithm with the equivalent two-level system, we derive the transitionless driving Hamiltonian for the local adiabatic quantum search algorithm. We found that when adding a transitionless quantum driving term H_D(t) on the local adiabatic quantum search algorithm, the success rate is 1 exactly with arbitrary evolution time by solving the time-dependent Schr odinger equation in eigen-picture. Moreover, we show the reason for the drastic decrease of the evolution time is that the driving Hamiltonian increases the lowest eigenvalues to a maximum of ON~(1/2).     

Quantum Adiabatic Evolution for Pattern Recognition Problem

Quantum pattern recognition algorithm for two-qubit systems has been implemented by quantum adiabatic evolution.We will estimate required running time for this algorithm by means of an analytical solution of timedependent Hamiltonian since the time complexity of adiabatic quantum evolution is a limitation on the quantum computing.These results can be useful for experimental implementation.     

Semi-Classical Dynamics in Quantum Spin Systems

We consider two limiting regimes, the large-spin and the mean-field limit, for the dynamical evolution of quantum spin systems. We prove that, in these limits, the time evolution of a class of quantum spin systems is determined by a corresponding Hamiltonian dynamics of classical spins. This result can be viewed as a Egorov-type theorem. We extend our results to the thermodynamic limit of lattice spin systems and continuum domains of infinite size, and we study the time evolution of coherent spin states in these limiting regimes.     

Quantum adiabatic theorem and universal holonomy

We give a simplified proof of the quantum adiabatic theorem for a system of possibly degenerate Hamiltonians by taking Berry's phase into account. We also relate the adiabatic transformation to the parallel transport induced by the holonomy in the universal bundle over a Grassman manifold. The special case of a nondegenerate Hamiltonian is precisely the cyclic quantum evolution studied by Aharanov and AnandanThe author is S. Y. Wu     

Symmetry-Protected Quantum Adiabatic Evolution in Spontaneous Symmetry-Breaking Transitions

Quantum adiabatic evolution describes the dynamical evolution of a slowly driven Hamiltonian. In most systems undergoing spontaneous symmetry-breaking transitions, the symmetry-protected quantum adiabatic evolution can still appear, even when the two lowest eigenstates become degenerate. Here, a general derivation to revisit the symmetry-dependent transition and the symmetry-dependent adiabatic condition (SDAC) is given. Further, based on the SDAC, an adiabatic-parameter-fixed sweeping scheme is used for achieving fast adiabatic evolution, which is more efficient than the linear sweeping scheme. In the limit of small adiabatic parameter, an analytic inequality is obtained for the ground state fidelity only dependent on the adiabatic parameter. The general statements are then demonstrated via two typical systems. Besides, the robustness of the symmetry-dependent adiabatic evolution against weak symmetry-breaking sources is studied. The findings can be tested via the techniques in quantum annealing and may provide promising applications in practical quantum technologies.     

Highly efficient charging and discharging of three-level quantum batteries through shortcuts to adiabaticity

Quantum batteries are energy storage devices that satisfy quantum mechanical principles. How to improve the battery’s performance such as stored energy and power is a crucial element in the quantum battery. Here, we investigate the charging and discharging dynamics of a three-level counterdiabatic stimulated Raman adiabatic passage quantum battery via shortcuts to adiabaticity, which can compensate for undesired transitions to realize a fast adiabatic evolution through the application of an additional control field to an initial Hamiltonian. The scheme can significantly speed up the charging and discharging processes of a three-level quantum battery and obtain more stored energy and higher power compared with the original stimulated Raman adiabatic passage. We explore the effect of both the amplitude and the delay time of driving fields on the performances of the quantum battery. Possible experimental implementation in superconducting circuit and nitrogen–vacancy center is also discussed.     

Adiabatic quantum computing with phase modulated laser pulses

Implementation of quantum logical gates for multilevel systems is demonstrated through decoherence control under the quantum adiabatic method using simple phase modulated laser pulses. We make use of selective population inversion and Hamiltonian evolution with time to achieve such goals robustly instead of the standard unitary transformation language.     

Adiabatic Theorems for Quantum Resonances

We study the adiabatic time evolution of quantum resonances over time scales which are small compared to the lifetime of the resonances. We consider three typical examples of resonances: The first one is that of shape resonances corresponding, for example, to the state of a quantum-mechanical particle in a potential well whose shape changes over time scales small compared to the escape time of the particle from the well. Our approach to studying the adiabatic evolution of shape resonances is based on a precise form of the time-energy uncertainty relation and the usual adiabatic theorem in quantum mechanics. The second example concerns resonances that appear as isolated complex eigenvalues of spectrally deformed Hamiltonians, such as those encountered in the N-body Stark effect. Our approach to study such resonances is based on the Balslev-Combes theory of dilatation-analytic Hamiltonians and an adiabatic theorem for nonnormal generators of time evolution. Our third example concerns resonances arising from eigenvalues embedded in the continuous spectrum when a perturbation is turned on, such as those encountered when a small system is coupled to an infinitely extended, dispersive medium. Our approach to this class of examples is based on an extension of adiabatic theorems without a spectral gap condition. We finally comment on resonance crossings, which can be studied using the last approach.     

Inconsistency in the application of the adiabatic theorem

The adiabatic theorem states that an initial eigenstate of a slowly varying Hamiltonian remains close to an instantaneous eigenstate of the Hamiltonian at a later time. We show that a perfunctory application of this statement is problematic if the change in eigenstate is significant, regardless of how closely the evolution satisfies the requirements of the adiabatic theorem. We also introduce an example of a two-level system with an exactly solvable evolution to demonstrate the inapplicability of the adiabatic approximation for a particular slowly varying Hamiltonian.     

Arrow of time, parity violation, and gravity in generalized quantum and classical dynamics

The quantum mechanics with a stationary non-Hermitian Hamiltonian and a complex evolution parameter, as well as its classical limit with nontrivial correlations have been studied. The corresponding dynamics is shown to be irreversible for the isothermal and adiabatic regimes of quantum and classical evolution. The possibility of a universal relationship between irreversibility and dynamical parity violation in the system has been established. The mechanism of gravity generation by the distribution of correlations in a free theory is demonstrated.     

Semiclassical evolution of the spectral curve in the normal random matrix ensemble as Whitham hierarchy

We continue the analysis of the spectral curve of the normal random matrix ensemble, introduced in an earlier paper. Evolution of the full quantum curve is given in terms of compatibility equations of independent flows. The semiclassical limit of these flows is expressed through canonical differential forms of the spectral curve. We also prove that the semiclassical limit of the evolution equations is equivalent to Whitham hierarchy.     

Landau-Zener Tunneling for Dephasing Lindblad Evolutions

We consider a family of time-dependent dephasing Lindblad generators which model the monitoring of the instantaneous Hamiltonian of a system by a Markovian bath. In this family the time dependence of the dephasing operators is (essentially) governed by the instantaneous Hamiltonian. The evolution in the adiabatic limit admits a geometric interpretation and can be solved by quadrature. As an application we derive an analog of the Landau-Zener tunneling formula for this family.     

Berry phase in open quantum systems: a quantum Langevin equation approach

The evolution of a two level system with a slowly varying Hamiltonian, modeled as a spin 1/2 in a slowly varying magnetic field, and interacting with a quantum environment, modeled as a bath of harmonic oscillators is analyzed using a quantum Langevin approach. This allows to easily obtain the dissipation time and the correction to the Berry phase in the case of an adiabatic cyclic evolution.     

Landau problem with a general time-dependent electric field

The time evolution is studied for the Landau level problem with a general time dependent electric field E(t) in a plane perpendicular to the magnetic field. A general and explicit factorization of the time evolution operator is obtained with each factor having a clear physical interpretation. The factorization consists of a geometric factor (path-ordered magnetic translation), a dynamical factor generated by the usual time-independent Landau Hamiltonian, and a nonadiabatic factor that determines the transition probabilities among the Landau levels. Since the path-ordered magnetic translation and the nonadiabatic factor are, up to completely determined numerical phase factors, just ordinary exponentials whose exponents are explicitly expressible in terms of the canonical variables, all of the factors in the factorization are explicitly constructed. New quantum interference effects are implied by this result. The factorization is unique from the point of view of the quantum adiabatic theorem and provides a seemingly first rigorous demonstration of how the quantum adiabatic theorem (incorporating the Berry phase phenomenon) is realized when infinitely degenerate energy levels are involved. Since the factorization separates the effect caused by the electric field into a geometric factor and a nonadiabatic factor, it makes possible to calculate the nonadiabatic transition probabilities near the adiabatic limit. A formula for matrix elements that determines the mixing of the Landau levels for a general, nonadiabatic evolution is also provided by the factorization.