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

## TL;DR

This paper applies a fast forward scheme to accelerate adiabatic quantum dynamics in finite regular spin clusters, revealing geometry-dependent interactions and simplifying the required driving terms.

## Contribution

It introduces a method to determine minimal geometry-dependent driving interactions for regular spin clusters, reducing complexity in quantum control.

## Key findings

- For N=3 spins, only pairwise interactions are needed.
- For N=4 spins, pairwise interactions dominate, with a minor universal 3-body term.
- Geometry influences the structure of driving interactions, simplifying control schemes.

## 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. The fast forward is the quasi-adiabatic dynamics guaranteed by regularization terms added to the reference Hamiltonian, followed by a rescaling of time with use of a large scaling factor. With help of the regularization terms consisting of pair-wise and 3-body interactions, we apply the proposed formula (Phys. Rev.A 96, 052106(2017)) to regular triangle and open linear chain for N = 3 spin systems, and to triangular pyramid, square, primary star graph and open linear chain for N = 4 spin systems. The geometry-induced symmetry greatly decreases the rank of coefficient matrix of the linear algebraic equation for regularization terms. Choosing a transverse Ising Hamiltonian as a reference, we find: (1) for N = 3 spin clusters, the driving interaction consists of only the geometry-dependent pairwise interactions and there is no need for the 3-body interaction; (2) for N = 4 spin clusters, the geometry-dependent pair-wise interactions again constitute major part of the driving interaction, whereas the universal 3-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.