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arXiv:1906.04433 · Physical Review A

Fast-forward approach to adiabatic quantum dynamics of regular spin clusters: Nature of geometry-dependent driving interactions

Iwan Setiawan, Bobby Eka Gunara, S. K. Avazbaev, Katsuhiro Nakamura

Published Jun 24, 2019• 16 citations
arXivHTMLPDF

Abstract

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 $N=3$ spin systems and to triangular pyramid, square, primary star graph, and open linear chains for $N=4$ 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 $N=3$ 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 $N=4$ 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.

DOI
10.1103/physreva.99.062116

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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 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.
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 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.
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.
Quantum computers have data in the form of quantum bits (qubits), which are 0 and 1 simultaneously. This 0 and 1 state is the electron spin state on the quantum computer, namely spin up and spin down. Qubits are very sensitive to environmental disturbances, so it is necessary to accelerate the adiabatic dynamics so that the system's coherence is not disturbed by environmental interactions. These spin dynamics can be accelerated using the fast forward method, which uses an additional phase to drive the spin adiabatically. The adiabatic state is a state where the dynamics of a system have the same characteristics as the initial state after the dynamics take place. This research was conducted by qualitatively reviewing physical theory and conducting literature studies on previous research on the fast-forward method and the Landau-Majorana-Zener system. The results of this study obtained wave function solutions, regularization terms, and additional Hamiltonians in the Landau-Majorana-Zener system to accelerate adiabatic spin dynamics so that spin dynamics on quantum computers are not disturbed by the environment. After adding additional Hamiltonian adiabatic dynamics in the Landau-Majorana Zener system can be maintained but requires considerable energy. The magnetic field needed to create the same graph as the wave function graph before adding Hamiltonian, B0 = 100 Tesla. The energy required to accelerate the adiabatic spin dynamics is large, so the cost is also quite large. Therefore, efforts are needed to minimize the energy required to accelerate the adiabatic spin dynamics.
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.