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arXiv:1711.04074 · Physical Review A

Fast forward of the adiabatic spin dynamics of entangled states

Iwan Setiawan, Bobby Eka Gunara, Shumpei Masuda, Katsuhiro Nakamura

Published Nov 8, 2017• 36 citations
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Abstract

We develop a fast-forward scheme of the adiabatic spin dynamics of quantum entangled states. We settle the quasiadiabatic dynamics by adding the regularization terms to the original Hamiltonian and then accelerate it with the use of a large time-scaling factor. Assuming the experimentally realizable candidate Hamiltonian consisting of the exchange interactions and magnetic field, we solve the regularization terms. These terms, multiplied by the velocity function, give rise to the state-dependent counterdiabatic terms. The scheme needs neither knowledge of full spectral properties of the system nor solving the initial- and boundary-value problem. Our fast forward Hamiltonian generates a variety of state-dependent counterdiabatic terms for each of adiabatic states, which can include the state-independent one. We highlight this fact by using minimum (two-spin) models for a simple transverse Ising model, quantum annealing, and generation of entanglement.

DOI
10.1103/physreva.96.052106

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We develop a fast-forward scheme of the adiabatic spin dynamics of quantum entangled states. We settle the quasiadiabatic dynamics by adding the regularization terms to the original Hamiltonian and then accelerate it with the use of a large time-scaling factor. Assuming the experimentally realizable candidate Hamiltonian consisting of the exchange interactions and magnetic field, we solve the regularization terms. These terms, multiplied by the velocity function, give rise to the state-dependent counterdiabatic terms. The scheme needs neither knowledge of full spectral properties of the system nor solving the initial- and boundary-value problem. Our fast forward Hamiltonian generates a variety of state-dependent counterdiabatic terms for each of adiabatic states, which can include the state-independent one. We highlight this fact by using minimum (two-spin) models for a simple transverse Ising model, quantum annealing, and generation of entanglement.
We propose a scheme of the fast forward of adiabatic spin dynamics in a triangular spin cluster. We settle the quasi-adiabatic spin dynamics (QASD) by adding the regularization terms to the original Hamiltonian and accelerate it with use of a large time-scaling factor which realizes QASD on shortened time scale. Assuming the candidate regularization Hamiltonian consisting of three-body interactions besides the pair-wise exchange interactions and magnetic field, we solved the regularization terms. These terms multiplied by the velocity function give rise to the statedependent counter-diabatic terms (CDTs) for each of adiabatic states. Our fast forward Hamiltonian proves to generate the plural number of state-dependent CDTs due to the driving three-body interactions. Applying this scheme to a simple transverse Ising model, we find 2 CDTs which contains both two-body and three body interaction. The driving two and three-body interaction in the fast-forward scheme guarantees the complete fidelity of accelerated states.
Abstract We propose a scheme of the fast forward of adiabatic spin dynamics in a triangular 3 spin cluster of quantum annealing model and reveal the structure and role of driving pair-wise interaction and three-body interactions. We settle the quasi-adiabatic spin dynamics (QASD) by adding the regularization terms to the original Hamiltonian and accelerate it with use of a large time-scaling factor which realizes QASD on shortened time scale. Assuming the candidate regularization Hamiltonian consisting of three-body interactions besides the pair-wise exchange interactions and magnetic field, we solved the regularization terms. These terms multiplied by the velocity function give rise to the state-dependent counter-diabatic terms (CDTs) for each of adiabatic states. Applying this method to quantum annealing model, we find CDTs which include pair-wise interaction and the 3-body interactions. The driving pairwise and three-body interaction in the fast-forward scheme guarantees the complete fidelity of accelerated states.
Abstract We discussed a method to accelerate an adiabatic quantum dynamics of XY spin model by using the fast-forward method proposed by Masuda and Nakamura. The Accelerated scheme is constructed by adding the driving Hamiltonian to the original Hamiltonian and speeding it up with a large time-scaling factor and an adiabatic parameter that realizes adiabatic quantum dynamics in a shortened time. Accelerated adiabatic dynamics start by assuming the candidate of driving Hamiltonian consists of the pair-wise exchange interaction and magnetic field. The driving Hamiltonian terms multiplied by the velocity function together with the original Hamiltonian give fast-forward driving for adiabatic states. We apply our method to XY spin model by considering three spin systems on the Kagome lattice. In this model, we obtained the XY pair-wise exchange interaction of nearest neighbors and next-nearest neighbors should be added to the original Hamiltonian as a driving interaction to accelerate the adiabatic motion. This pair-wise driving interaction in the fast-forward scheme guarantees the complete fidelity of accelerated states.
The fast-forward scheme of adiabatic quantum dynamics is applied to finite regular spin clusters with various geometries and the nature of driving interactions is elucidated. Fast forward is the quasiadiabatic dynamics guaranteed by regularization terms added to the reference Hamiltonian, followed by a rescaling of time with the use of a large scaling factor. With the help of the regularization terms consisting of pairwise and three-body interactions, we apply the proposed formula [I. Setiawan et al., Phys. Rev. A 96, 052106 (2017)] to regular triangle and open linear chains for N=3N=3N=3 spin systems and to triangular pyramid, square, primary star graph, and open linear chains for N=4N=4N=4 spin systems. The geometry-induced symmetry greatly decreases the rank of the coefficient matrix of the linear algebraic equation for regularization terms. Choosing a transverse Ising Hamiltonian as a reference, we find, for N=3N=3N=3 spin clusters, that the driving interaction consists of only the geometry-dependent pairwise interactions and there is no need for the three-body interaction and, for N=4N=4N=4 spin clusters, that the geometry-dependent pairwise interactions again constitute a major part of the driving interaction, whereas the universal three-body interaction free from the geometry is necessary but plays a subsidiary role. Our scheme predicts the practical driving interaction in accelerating the adiabatic quantum dynamics of structured regular spin clusters.
Abstract We study a scheme of accelerated adiabatic quantum dynamics. This scheme was originally proposed by Masuda-Nakamura. The strategy of combining two opposite idea: infinitely-large timemagnification factor and infinitely-small growth rate of adiabatic parameter was elucidated. We apply the proposed method to two dimensional system with electric field and magnetic field using two dimensional Dirac equation. We settle the quasi-adiabatic dynamics (QAD) by adding the regularization terms to the original vector potential and scalar potential and accelerate it with use of a large time-scaling factor which realizes QAD on shortened time scale. These terms multiplied by the velocity function give the counterdiabatic terms that generate the fast forward potential. The fast forward potential can accelerate the dynamics of the system from iniitial state to the final state with the same condition.
We study a scheme of fast-forward adiabatic quantum dynamics of a (2 + 1) Dirac particle. This scheme was originally proposed by Masuda and Nakamura. In this scheme, we include the adiabatic parameter that maintains the adiabatic motion of the particle and fast forward its motion by introducing a time scaling parameter. The fast forward adiabatic state is obtained by determining the regularization term and driving potential. We introduce the proposed method to the system with the Dirac particle using a (2 + 1) dimensiontime- dependent Dirac equation and obtain the regularization term, the driving scalar potential VFF and the driving vector potential AFF . By tuning the driving electric field, this method can accelerate the adiabatic dynamics of an electron as a Dirac particle trapped in the ground state in the plane xy and an electric field in the x direction and a constant magnetic field in the y direction. This acceleration will preserve the ground state of the wave function from the initial time to the final time.