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.

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.

Equations

$$H_0(R){\text{C}}^{(n)}(R)=E_n(R){\text{C}}^{(n)}(R)$$

$${\text{C}}^{(n)}(R)=\left(\begin{array}{c}{C}_1^{(n)}(R)\\ ⋮\\ C_N^{(n)}(R)\end{array}\right),$$

$$R\equiv R(t)=R_0+ϵt$$

$${\Psi }_0^{(n)}(R(t))={\text{C}}^{(n)}(R(t))e^{-\frac{i}{\hslash }{\int }_0^t{E}_n(R(t^{\prime }))d{t}^{\prime }}{e}^{i{\xi }_n(R(t))},$$

$${\xi }_n(R(t))=i{\int }_0^{t}d{t}^{\prime }{\text{C}}^{(n)†}{\partial }_t{\text{C}}^{(n)}=iϵ{\int }_0^{t}d{t}^{\prime }{\text{C}}^{(n)†}{\partial }_R{\text{C}}^{(n)}.$$

$$H_0^{reg}(R(t))=H_0(R(t))+ϵ{\tilde{\mathcal{H}}}_n(R(t)).$$

$$i\hslash \frac{\partial }{\partial t}{\Psi }_0^{(n)}(R(t))=(H_0+ϵ{\tilde{\mathcal{H}}}_n){\Psi }_0^{(n)}(R(t)).$$

$${\tilde{\mathcal{H}}}_n{\text{C}}^{(n)}(R)=i\hslash {\partial }_R{\text{C}}^{(n)}(R)-i\hslash ({\text{C}}^{(n)†}{\partial }_R{\text{C}}^{(n)}){\text{C}}^{(n)}(R),$$

$${\Psi }_{FF}^{(n)}(t)$$

$$\Lambda (t)={\int }_0^t\alpha (t^{\prime })d{t}^{\prime },$$

$$T={\int }_0^{T_{FF}}\alpha (t)dt.$$

$$\alpha (t)=\bar{\alpha }-(\bar{\alpha }-1)\cos (\frac{2\pi }{T_{FF}}t),$$

$$i\hslash {\dot{\Psi }}_{FF}^{(n)}$$

$$i\hslash \frac{\partial {\Psi }_{FF}^{(n)}}{\partial t}$$

$$v(t)=\underset{ϵ\to 0,\bar{\alpha }\to \infty }{\lim }ϵ\alpha (t)=\bar{v}\left(1-\cos \frac{2\pi }{T_{FF}}t\right).$$

$$R(\Lambda (t))$$

$$H_0=J(R(t))\sum _{(i,j)\in N.N.}{\sigma }_i^z{\sigma }_j^z-\frac{1}{2}{B}_x(R(t))\sum _{i=1}^N{\sigma }_i^x,$$

$$\tilde{\mathcal{H}}$$

$$C_1\frac{\partial C_1}{\partial R}+3C_2\frac{\partial C_2}{\partial R}=0,$$

$$-6\tilde{W}{C}_2$$

$$2\tilde{W}{C}_1$$

$$\tilde{W}$$

$$H_{FF}=H_0(R(\Lambda (t)))+v(t)\tilde{\mathcal{H}}(R(\Lambda (t)))$$

$$C_1\frac{\partial C_1}{\partial R}+2C_2\frac{\partial C_2}{\partial R}+C_3\frac{\partial C_3}{\partial R}=0,$$

$$\tilde{\mathcal{H}}=i\left(\begin{array}{cccccccc}0& -{\tilde{W}}_1-{\tilde{W}}_2& -2{\tilde{W}}_1& -{\tilde{W}}_1-{\tilde{W}}_2& 0& 0& 0& 0\\ {\tilde{W}}_1+{\tilde{W}}_2& 0& 0& 0& 0& -{\tilde{W}}_1+{\tilde{W}}_2& 0& 0\\ 2{\tilde{W}}_1& 0& 0& 0& {\tilde{W}}_1-{\tilde{W}}_2& 0& {\tilde{W}}_1-{\tilde{W}}_2& 0\\ {\tilde{W}}_1+{\tilde{W}}_2& 0& 0& 0& 0& -{\tilde{W}}_1+{\tilde{W}}_2& 0& 0\\ 0& 0& -{\tilde{W}}_1+{\tilde{W}}_2& 0& 0& 0& 0& {\tilde{W}}_1+{\tilde{W}}_2\\ 0& {\tilde{W}}_1-{\tilde{W}}_2& 0& {\tilde{W}}_1-{\tilde{W}}_2& 0& 0& 0& 2{\tilde{W}}_1\\ 0& 0& -{\tilde{W}}_1+{\tilde{W}}_2& 0& 0& 0& 0& {\tilde{W}}_1+{\tilde{W}}_2\\ 0& 0& 0& 0& -{\tilde{W}}_1-{\tilde{W}}_2& -2{\tilde{W}}_1& -{\tilde{W}}_1-{\tilde{W}}_2& 0\end{array}\right).$$

$$-2({\tilde{W}}_1+{\tilde{W}}_2)C_2-2{\tilde{W}}_1{C}_3$$

$$({\tilde{W}}_1+{\tilde{W}}_2)C_1+(-{\tilde{W}}_1+{\tilde{W}}_2)C_3$$

$$2{\tilde{W}}_1{C}_1+2({\tilde{W}}_1-{\tilde{W}}_2)C_2$$

$${\tilde{W}}_1$$

$${\tilde{W}}_2$$

$$H_{FF}=H_0(R(\Lambda (t)))+v(t)\tilde{\mathcal{H}}(R(\Lambda (t)))$$

$$C_1\frac{\partial C_1}{\partial R}+4C_2\frac{\partial C_2}{\partial R}+3C_6\frac{\partial C_6}{\partial R}=0.$$

Full text

I INTRODUCTION

Effectively manipulating and optimizing the dynamics of given systems constitutes one of big experimental and theoretical subjects in the current technology. In particular, it is a challenging theme to find suitable driving fields for tailoring a quantum system to rapidly generate a target state from a given initial state. In designing quantum computers, the acceleration of adiabatic quantum dynamics is desirable because the coherence of systems is degraded by their interaction with the environment. Since naive numerical trial-and-error methods are time- and resource-consuming, we must deeply understand relevant quantum dynamics to find useful schemes for such accelerations. In this context, various researches on the way to the shortcut to adiabaticity (STA) have been developed, which include invariant-based inverse engineering 1 ; 2 ; 3 , transitionless counter-diabatic (CD) driving 4 ; 5 ; 6 , fast-forward approach 7 ; 8 ; 9 , and variational methods to generate approximate CD protocols10 ; 11 ; 12 .

The fast-forward theory proposed by Masuda and Nakamura 7 was originally concerned with acceleration of general reference quantum dynamics. This theory was developed to accelerate the adiabatic quantum dynamics by introducing the large time-scaling factor in the quasi-adiabatic dynamics guaranteed by regularization terms added to the reference Hamiltonian 8 ; 9 , and was then used to enhance the quantum tunneling power 13 and to construct the non-equilibrium equation of state under a rapid piston 14 . The relation between the fast-forward approach and other methods was rigorously investigated in 15 .

Recently, we proposed a fast forward scheme of adiabatic spin dynamics 16 . Confining to a single and two spin systems there, we showed the acceleration of Landau-Zener transition and that of a generation of entangled states, as can be shown in other methods 4 ; 5 ; 6 ; 3extra ; 10 ; paul ; stef .

The fast forward scheme of adiabatic quantum dynamics has advantages as addressed by TK1 ; TK2 : (1) No need of writing the driving interaction in the spectral representation with use of full spectral properties of given spin systems. No necessity of worrying about the divergence of the driving interaction due to the level crossing; (2) A great flexibility in choosing the regularization Hamiltonian which leads to the driving interaction. Namely, users can specify the regularization Hamiltonian by themselves so as to satisfy the core equation (see Eq.(8) of this paper). The latter advantage will play an important role when we shall investigate spin clusters of various geometries. However, no technical guide was so far presented in solving the core equation for unknown regularization terms.

Within a framework of the transitionless CD driving 4 ; 5 ; 6 , on the other hand, there exist intensive works on a linear chain of many quantum spins described by the Ising model in a transverse field ex1 ; ex2 ; ex3 ; ex33 and the related model campb , which showed the complicated non-local multi-body CD terms that are hard to achieve in experiment. While a variational method to generate approximate local CD protocols11 ; 12 is being cultivated, it is timely to sharpen the fast-forward approach by showing a guiding principle to manage spin clusters with various geometries on the basis of the proposed formula in 16 .

In this paper the fast forward scheme of adiabatic dynamics is applied to regular spin clusters of various geometries with number of spins $N$ up to 4, i.e., regular triangle and open linear chain for $N=3$ spins, and triangular pyramid, square, primary star graph and open linear chain for $N=4$ spins. (Note: the geometry is irrelevant for systems with $N=1$ and $2$ spins.) Choosing the Hamiltonian for a transverse Ising model as a reference, we shall reveal the nature of driving interactions. In Section II, a brief summary is given on the fast forward scheme of adiabatic quantum spin dynamics. In Section III we propose a candidate regularization Hamiltonian consisting of geometry-dependent pair-wise interactions and a universal 3-body interaction, and describe a method of solving the linear algebraic equation for regularization terms. Sections IV and V are devoted to the analysis of spin clusters of various geometries with $N=3$ and $N=4$, respectively. Summary and discussions are given in Section VI. Appendix A gives matrices for some regularization Hamiltonians.

II Fast-forward scheme of adiabatic spin dynamics

For self-containedness, we shall sketch the fast forward scheme of adiabatic spin dynamics16 . Our strategy is as follows: (i) A given original (reference) Hamiltonian $H_{0}$ is assumed to change adiabatically and to generate a stationary state $\Psi_{0}$, which is an eigenstate of the time-independent Schrödinger equation with the instantaneous Hamiltonian. Then $H_{0}$ is regularized so that $\Psi_{0}$ should satisfy the time-dependent Schrödinger equation (TDSE); (ii) Taking $\Psi_{0}$ as a reference state, we shall rescale time in TDSE with use of the scaling factor $\alpha\left(t\right)$, where the mean value $\bar{\alpha}$ of the infinitely-large time scaling factor $\alpha(t)$ will be chosen to compensate the infinitesimally-small growth rate $\epsilon$ of the quasi-adiabatic parameter and to satisfy $\bar{\alpha}\times\epsilon=finite$.

Consider the Hamiltonian for spin systems to be characterized by a slowly time-changing parameter $R(t)$ such as the exchange interaction, magnetic field, etc. Then we can study the eigenvalue problem for the time-independent Schrödinger equation :

[TABLE]

with

[TABLE]

where

[TABLE]

is the adiabatically-changing parameter with $\epsilon\ll 1$. In Eq.(1), $n$ stands for the quantum number for each eigenvalue and eigenstate. Let us assume

[TABLE]

to be a quasi-adiabatic state, i.e., adiabatically evolving state, where $\xi_{n}$ is the adiabatic phase:

[TABLE]

$\Psi^{(n)}_{0}(R(t))$ in Eq.(4) is not a solution of TDSE. To make it to satisfy the TDSE, we must regularize the Hamiltonian as

[TABLE]

Then TDSE becomes

[TABLE]

Here $\mathcal{\tilde{H}}_{n}$ is the $n$-th state-dependent regularization term. Substituting $\Psi^{(n)}_{0}(R(t))$ in Eq.(4) into the above TDSE, we see the eigenvalue problem in Eq.(1) in order of $O(\epsilon^{0})$, and the algebraic equation for $\mathcal{\tilde{H}}_{n}$,

[TABLE]

in order of $O(\epsilon^{1})$. Equation (8) is the core of the present study. The state in Eq.(4) and TDSE in Eq.(7) are working on a very slow time scale. We shall innovate them so that they can work on a laboratory time scale.

With time $t$ rescaled by the advanced time $\Lambda(t)$, the fast-forward state is introduced as

[TABLE]

where $\Lambda(t)$ is defined by

[TABLE]

with the standard time $t$. $\alpha(t)$ is an arbitrary magnification time-scale factor which satisfies $\alpha(0)$ = 1, $\alpha(t)>1(0<t<T_{FF})$ and $\alpha(t)$ = $1(t\geq T_{FF})$. For a long final time $T$ in the original adiabatic dynamics, we can consider the fast forward dynamics with a new time variable which reproduces the target state $\Psi^{(n)}_{0}(R(T))$ in a shorter final time $T_{FF}$ defined by

[TABLE]

The simplest expression for $\alpha(t)$ in the fast-forward range ($0\leq t\leq T_{FF}$) is given by 8 as :

[TABLE]

where $\bar{\alpha}$ is the mean value of $\alpha(t)$ and is given by $\bar{\alpha}=T/T_{FF}$.

Then by taking the time derivative of $\Psi^{(n)}_{FF}$ in Eq.(II) and using the equalities $\partial_{t}\textbf{C}^{(n)}(R(\Lambda(t)))=\alpha\epsilon\partial_{R}\textbf{C}^{(n)}$ and $\partial_{t}\xi_{n}(R(\Lambda(t)))$ = $i\textbf{C}^{(n)\dagger}\partial_{t}\textbf{C}^{(n)}$ = $i\alpha\epsilon\textbf{C}^{(n)\dagger}\partial_{R}\textbf{C}^{(n)}$, we have

[TABLE]

The first and second terms in the angular bracket on the r.h.s are replaced by $\alpha\epsilon\mathcal{\tilde{H}}_{n}\textbf{C}^{(n)}(R(\Lambda(t)))$ and $H_{0}\textbf{C}^{(n)}(R(\Lambda(t)))$, respectively, by using Eqs.(8) and (1). Using the definition of $\Psi^{(n)}_{FF}(t)$ and taking the asymptotic limit $\bar{\alpha}\rightarrow\infty$ and $\epsilon\rightarrow 0$ under the constraint $\bar{\alpha}\cdot\epsilon\equiv\bar{v}=finite$, we obtain

[TABLE]

Here $v(t)$ is a velocity function available from $\alpha(t)$ in the asymptotic limit:

[TABLE]

Consequently, for $0\leq t\leq T_{FF}$,

[TABLE]

$H^{(n)}_{FF}$ is the fast-forward Hamiltonian and $\mathcal{\tilde{H}}_{n}$ is the regularization term obtained from Eq.(8) to generate the fast-forward scheme in spin system. Eqs. (II) and (14) work on a laboratory time scale.

There is a relationship between our formula for $\mathcal{\tilde{H}}_{n}$ in Eq.(8) and Demirplak-Rice-Berry (DRB)’s formula 4 ; 5 ; 6 for the CD term $\mathcal{H}$. If there is a $n$-independent regularization term $\mathcal{\tilde{H}}$ among $\{\mathcal{\tilde{H}}_{n}\}$, we can define $\mathcal{H}\equiv v(t)\mathcal{\tilde{H}}$ with use of $v(t)=\frac{\partial R(\Lambda(t))}{\partial t}$. Then Eq.(8) gives a solution $\mathcal{H}$ which agrees with DRB’s formula for the CD term (See the proof in 16 ). It should be noted, however, that the above correspondence works well only in the case that we can find $n$-independent regulariztion terms $\mathcal{\tilde{H}}$ among $\{\mathcal{\tilde{H}}_{n}\}$. Using the above notion, one may call $v(t)\mathcal{\tilde{H}}_{n}$ as a state-dependent CD term. Hereafter we shall be concerned with the fast forward of adiabatic dynamics of one of the adiabatic states (i.e., the ground state) and therefore the suffix $n$ in $\mathcal{\tilde{H}}_{n}$ will be suppressed.

III Fast-forward driving interactions for spin clusters of various geometries

To begin with, let us explain the method of solving the linear algebraic equation for unknown regularization terms in Eq. (8). Then in the succeeding Sections, we shall treat regular spin clusters of various geometries with $N$ up to 4, i.e., regular triangle and open linear chain for $N=3$ spins (see Fig.1), and triangular pyramid, square, primary star graph and open linear chain for $N=4$ spins (see Fig.2). Our scheme is free from obtaining all eigenvectors for a given adiabatic Hamiltonian. As shown in the core equation in Eq. (8), we need only information of a single eigenstate, typically of the ground state.

As an original (reference) model, we choose the transverse Ising mode, whose Hamiltonian for $N$ spin systems is written as

[TABLE]

where $J(R(t))=R(t)=R_{0}+\epsilon t$ and $B_{x}(R(t))=B_{0}-R(t)$ with $\epsilon\ll 1$ are adiabatically-changing exchange interaction and transverse magnetic field, respectively. $(i,j)\in N.N.$ means nearest-neighbouring pairs. Using the spin configuration bases, the dimension of Hilbert space is $2^{N}$.

Energy matrix corresponding to the Hamiltonian in Eq.(17) is real symmetric, which makes the eigenstates real, and the ground state is expressed by the real components $\{C_{k}:k=1,\cdots,2^{N}\}$. This, in combination with the fact that the length of the corresponding eigenvector is constant and equal to 1, leads to the conclusion that the adiabatic phase $\xi_{n}$ in Eq.(II) is zero in all spin clusters in the present work. Further, because of the geometrical symmetry of spin clusters in Figs. 1 and 2, some of the components $C_{k}$s are degenerate which reduce the number of independent equations in the core equation in Eq. (8).

As for the unknown regularization term ($\mathcal{\tilde{H}}$) in Eq.(8) , we must impose a form which makes its matrix elements pure imaginary because the right-hand side of Eq.(8) is now pure imaginary. Among several possibilities, we assume the regularization term consisting of pair-wise interactions described by $\tilde{W}_{ij}^{yz}=\tilde{W}_{ij}^{yz}(\epsilon t)$ and 3-body interactions $\tilde{Q}_{ijk}^{xyz}=\tilde{Q}_{ijk}^{xyz}(\epsilon t)$. Other possible contributions such as a single-particle energy due to $y$-component of the magnetic field ($\tilde{B}_{y}$), pair-wise interaction $\tilde{W}_{ij}^{xy}$ and 3-body interaction $\tilde{Q}_{ijk}^{xxy}$ lead to incompatible algebraic equations in Eq.(8), and should be excluded. The candidate for regularization Hamiltonian $\mathcal{\tilde{H}}$ then takes the following form :

[TABLE]

where $(i,j)\in all$ and $(i,j,k)\in all$ mean all possible combinations (not permutations), and are not limited to nearest neighbours. The 3-body interaction here is not brought as a result of the truncation of long-range and multi-body counter-diabatic interactions, but is introduced in advance to make the core equation solvable.

Since regular spin clusters have geometric symmetry, some of the interactions ($\tilde{W}_{ij}^{yz}$) are degenerate as shown in Figs. 1 and 2, and the reduced number of independent interactions should be equal to the number of independent equations in Eq.(8). In the present paper, the 3-body interaction will play a subsidiary role. Below we shall solve the regularization terms and obtain the fast-forward Hamiltonian for spin clusters of various geometries.

IV Regular triangle and open linear 3 spins

In this Section we investigate a regular triangle and open linear 3 spins in Fig. 1. We use the spin configuration bases as $\Ket{1}=\Ket{\uparrow\uparrow\uparrow}$, $\Ket{2}=\Ket{\uparrow\uparrow\downarrow}$, $\Ket{3}=\Ket{\uparrow\downarrow\uparrow}$, $\Ket{4}=\Ket{\downarrow\uparrow\uparrow}$, $\Ket{5}=\Ket{\uparrow\downarrow\downarrow}$, $\Ket{6}=\Ket{\downarrow\uparrow\downarrow}$, $\Ket{7}=\Ket{\downarrow\downarrow\uparrow}$ and $\Ket{8}=\Ket{\downarrow\downarrow\downarrow}$.

IV.1 Regular triangle

In the case of the regular triangle, the eigenvalue for the ground state is $E_{0}=-\sqrt{B_{x}^{2}+2B_{x}J+4J^{2}}-\frac{B_{x}}{2}+J$. We have confirmed in Fig. 3(a) that all eight eigenvalues show no mutual energy crossing in the fast-forward time range where we choose $J(R(\Lambda(t)))\equiv R(\Lambda(t))$ and $B_{x}(R(\Lambda(t)))\equiv B_{0}-R(\Lambda(t))$ with $R(\Lambda(t))$ defined in Eq.(16).

The components of the eigenvector for the ground state are :

$C_{1}=V_{1}\zeta$, $C_{2}=V_{2}\zeta$, $C_{3}=V_{3}\zeta$, $C_{4}=V_{4}\zeta$, $C_{5}=V_{5}\zeta$, $C_{6}=V_{6}\zeta$, $C_{7}=V_{7}\zeta$, $C_{8}=V_{8}\zeta$, where $V_{1}=V_{8}=1$, $V_{2}=V_{3}=V_{4}=V_{5}=V_{6}=V_{7}=\frac{2\sqrt{B_{x}^{2}+2B_{x}J+4J^{2}}+B_{x}+4J}{3B_{x}}$, and $\zeta=\frac{1}{\sqrt{2+6V_{2}^{2}}}$.

Here we see the symmetry: $C_{1}=C_{8}$, $C_{2}=C_{3}=C_{4}=C_{5}=C_{6}=C_{7}$. From $R$-derivative of the normalization ($\sum_{j=1}^{8}C_{j}^{2}=2C_{1}^{2}+6C_{2}^{2}=1$), we see

[TABLE]

and then the adiabatic phase $\xi=0$.

As for the regularization Hamiltonian for the regular triangle, we can proceed without having recourse to the 3-body interaction. Three $\tilde{W}_{ij}^{yz}$s should be identical due to the triangular symmetry in Fig. 1(a). Therefore the unknown pairwise interaction is only one: $\tilde{W}\equiv\tilde{W}_{ij}^{yz}$, independent of the pairs $(i,j)$.

By using the spin configuration bases as above, the regularization Hamiltonian in Eq.(18) is characterized by the matrix elements: $\mathcal{\tilde{H}}_{1j}=-\mathcal{\tilde{H}}_{j1}=-2i\tilde{W}$ with $j=2,3,4$, $\mathcal{\tilde{H}}_{8j}=-\mathcal{\tilde{H}}_{j8}=-2i\tilde{W}$ with $j=5,6,7$ and all other elements $=0$. The explicit expression for $\mathcal{\tilde{H}}$ will help us to solve Eq.(8).

Due to the symmetry of $\{C_{j}\}$, the number of independent equations are only two in Eq.(8) :

[TABLE]

Noting the normalization-assisted relation in Eq.(19), one of the above two equations becomes trivial, and Eq.(IV.1) has the solution:

[TABLE]

The second equality above is due to the normalization condition and Eq.(19). Including the regularization term followed by rescaling of time, the fast forward Hamiltonian is written as

[TABLE]

with $H_{0}$ = $J(R(\Lambda(t)))(\sigma_{1}^{z}\sigma_{2}^{z}+\sigma_{2}^{z}\sigma_{3}^{z}+\sigma_{3}^{z}\sigma_{1}^{z})-\frac{1}{2}(\sigma_{1}^{x}+\sigma_{2}^{x}+\sigma_{3}^{x})B_{x}(R(\Lambda(t)))$, and $v\mathcal{\tilde{H}}$ = v(t)\tilde{W}(R(\Lambda(t)))\big{[}(\sigma_{1}^{y}\sigma_{2}^{z}+\sigma_{1}^{z}\sigma_{2}^{y})+(\sigma_{2}^{y}\sigma_{3}^{z}+\sigma_{2}^{z}\sigma_{3}^{y})+(\sigma_{3}^{y}\sigma_{1}^{z}+\sigma_{3}^{z}\sigma_{1}^{y})\big{]}.

The fast forward Hamiltonian guarantees the fast forward of the adiabatic dynamics of the ground state wave function. Figures 3(b) and 3(c) show the time dependence of the regularization term and that of the wave function, respectively. The wave function starts from the ground state with $J=0$, i.e., $C_{1}=C_{2}=C_{3}=C_{4}=C_{5}=C_{6}=C_{7}=C_{8}=\frac{1}{2\sqrt{2}}$. The initial state is a linear combination of $\Ket{\uparrow\uparrow\uparrow}$, $\Ket{\uparrow\uparrow\downarrow}$, $\Ket{\uparrow\downarrow\uparrow}$, $\Ket{\downarrow\uparrow\uparrow}$, $\Ket{\uparrow\downarrow\downarrow}$, $\Ket{\downarrow\uparrow\downarrow}$, $\Ket{\downarrow\downarrow\uparrow}$ and $\Ket{\downarrow\downarrow\downarrow}$ states. As $J$ is increased from [math] and $B_{x}$ is decreased, the system rapidly changes to the final state, a linear combination of reduced bases $\Ket{\uparrow\uparrow\downarrow}$, $\Ket{\uparrow\downarrow\uparrow}$, $\Ket{\downarrow\uparrow\uparrow}$, $\Ket{\uparrow\downarrow\downarrow}$, $\Ket{\downarrow\uparrow\downarrow}$, and $\Ket{\downarrow\downarrow\uparrow}$. In Fig. 3 (c) the solution $\Psi_{FF}(t)$ of TDSE in Eq.(14) has reproduced the time-rescaled ground state wave function, which means the perfect fidelity of $\Psi_{FF}(t)$ during the fast-forward time range $0\leq t\leq T_{FF}$.

IV.2 Open linear 3 spin chain

In a similar way we can obtain the regularization term and fast-forward Hamiltonian in the case of open linear 3 spin chain. In this case the eigenvalue for the ground state is E_{0}=-\frac{1}{6}\big{(}B_{x}+(\beta+\bar{\beta})-\sqrt{3}i(\beta-\bar{\beta})\big{)}, where \beta=\big{(}18J^{2}B_{x}-8B_{x}^{3}+6Ji\sqrt{48J^{4}+39B_{x}^{2}J^{2}+24B_{x}^{4}}\big{)}^{1/3}. We have confirmed in Fig. 4 (a) that all eight eigenvalues show no mutual energy crossing in the fast-forward time range where we choose $J(R(\Lambda(t)))\equiv R(\Lambda(t))$ and $B_{x}(R(\Lambda(t)))\equiv B_{0}-R(\Lambda(t))$ with $R(\Lambda(t))$ defined in Eq.(16).

The components of the eigenvector for the ground state are :

$C_{1}=C_{8}=V_{1}\zeta$, $C_{2}=C_{4}=C_{5}=C_{7}=V_{2}\zeta$, $C_{3}=C_{6}=V_{3}\zeta$, where $V_{1}={\frac{3{B_{x}}^{2}-8JB_{x}-4B_{x}E_{0}-4E_{0}^{2}-8E_{0}J}{4JB_{x}}}$, $V_{2}=-\frac{1}{2}V_{1}-\frac{2J+E_{0}}{B_{x}}$, $V_{3}=1$, and $\zeta=\frac{1}{\sqrt{2V_{1}^{2}+4V_{2}^{2}+2}}$.

Here we see the symmetry: $C_{1}=C_{8}$, $C_{2}=C_{4}=C_{5}=C_{7}$ and $C_{3}=C_{6}$. From $R$-derivative of the normalization ($\sum_{j=1}^{8}C_{j}^{2}=2C_{1}^{2}+4C_{2}^{2}+2C_{3}^{2}=1$), we see

[TABLE]

and then the adiabatic phase $\xi=0$.

The regularization Hamiltonian for the linear 3 spin system can also be available without using the 3-body interaction. Because of the geometric symmetry seen in Fig. 1(b), $\mathcal{\tilde{H}}$ is then characterized by two independent pairwise interactions: $\tilde{W}_{1}\equiv\tilde{W}_{12}^{yz}=\tilde{W}_{23}^{yz}$ and $\tilde{W}_{2}\equiv\tilde{W}_{31}^{yz}$. $\tilde{W}_{1}$ and $\tilde{W}_{2}$ correspond to the nearest-neighboring (N.N.) and 2nd N.N. interactions, respectively. With use of the spin configuration bases, the matrix form for $\mathcal{\tilde{H}}$ in Eq. (18) is given by

[TABLE]

Due to the symmetry of $\{C_{j}\}$, the number of independent equations in Eq. (8) are three:

[TABLE]

By using Eq.(23), the 3rd line (for example) of the above equation proves trivial. Then Eq.(IV.2), whose coefficient matrix has the rank 2, gives the solution:

[TABLE]

Including the regularization terms followed by rescaling of time, the fast forward Hamiltonian are written as

[TABLE]

with $H_{0}$ = $J(R(\Lambda(t)))(\sigma_{1}^{z}\sigma_{2}^{z}+\sigma_{2}^{z}\sigma_{3}^{z})-\frac{1}{2}(\sigma_{1}^{x}+\sigma_{2}^{x}+\sigma_{3}^{x})B_{x}(R(\Lambda(t)))$, and $v\mathcal{\tilde{H}}$ = v(t)\tilde{W}_{1}(R(\Lambda(t)))\big{[}(\sigma_{1}^{y}\sigma_{2}^{z}+\sigma_{1}^{z}\sigma_{2}^{y})+(\sigma_{2}^{y}\sigma_{3}^{z}+\sigma_{2}^{z}\sigma_{3}^{y})\big{]}+v(t)\tilde{W}_{2}(R(\Lambda(t)))(\sigma_{1}^{y}\sigma_{3}^{z}+\sigma_{1}^{z}\sigma_{3}^{y}). The fast forward Hamiltonian guarantees the fast forward of the adiabatic dynamics of the ground state wave function. Figures 4(b) and 4(c) show the time dependence of the regularization terms and that of the wave function, respectively. The wave function starts from the ground state with $J=0$, i.e., $C_{j}=\frac{1}{2\sqrt{2}}$ for $j=1,\cdots,8$. As $J$ is increased from [math] and $B_{x}$ is decreased, the system rapidly changes to the final state, i.e., a linear combination of reduced bases. In Fig. 4 (c) the solution $\Psi_{FF}(t)$ of TDSE in Eq.(14) has exactly reproduced the time-rescaled ground state wave function.

In case of $N=3$ spin systems, we have obtained the regularization terms and the fast-forward Hamiltonian without having recourse to the 3-body interaction. Of course, we can see regularization terms which include the 3-body interaction: For a regular triangle we can have an extra solution consisting of only the 3-body interaction ($\tilde{Q}$), and for the open linear 3 spin system there can be solutions where $\tilde{Q}\neq 0$ and one of $\tilde{W_{1}}$ and $\tilde{W_{2}}$ is non-vanishing. But these extra solutions are less interesting from the viewpoint of searching for simpler controls. In the case of $N=4$ spin systems in next Section, however, we cannot proceed without the 3-body interaction, although it will play only a subsidiary role.

V triangular pyramid, square, star graph and open linear 4 spin chain

Now we shall investigate regular spin clusters with $N=4$ spins, namely, a triangular pyramid, square, star graph and open linear 4 spin chain in Fig. 2. Their original (reference) and regularization Hamiltonians are already given by Eq.(17) and Eq.(18), respectively, where we put $N=4$.

By using the spin configuration bases, $\Ket{1}=\Ket{\uparrow\uparrow\uparrow\uparrow}$, $\Ket{2}=\Ket{\uparrow\uparrow\uparrow\downarrow}$, $\Ket{3}=\Ket{\uparrow\uparrow\downarrow\uparrow}$, $\Ket{4}=\Ket{\uparrow\downarrow\uparrow\uparrow}$, $\Ket{5}=\Ket{\downarrow\uparrow\uparrow\uparrow}$ $\Ket{6}=\Ket{\uparrow\uparrow\downarrow\downarrow}$, $\Ket{7}=\Ket{\uparrow\downarrow\downarrow\uparrow}$, $\Ket{8}=\Ket{\downarrow\downarrow\uparrow\uparrow}$, $\Ket{9}=\Ket{\downarrow\uparrow\uparrow\downarrow}$, $\Ket{10}=\Ket{\uparrow\downarrow\uparrow\downarrow}$, $\Ket{11}=\Ket{\downarrow\uparrow\downarrow\uparrow}$, $\Ket{12}=\Ket{\downarrow\downarrow\downarrow\uparrow}$, $\Ket{13}=\Ket{\downarrow\downarrow\uparrow\downarrow}$, $\Ket{14}=\Ket{\downarrow\uparrow\downarrow\downarrow}$, $\Ket{15}=\Ket{\uparrow\downarrow\downarrow\downarrow}$, and $\Ket{16}=\Ket{\downarrow\downarrow\downarrow\downarrow}$, the matrix form for original Hamiltonian $H_{0}$ in Eq.(17) can be constructed.

V.1 Triangular pyramid

The eigenvalue of the ground state is $E_{0}=\frac{1}{3}(-(\beta+\bar{\beta})+4J+\sqrt{3}i(\beta-\bar{\beta}))$, where $\beta=(35\,{J}^{3}-18{B_{x}}^{2}J+3i\sqrt{108{J}^{6}+309{B_{x}}^{2}{J}^{4}+3{B_{x}}^{4}{J}^{2}+3{B_{x}}^{6}})^{1/3}.$ For all regular clusters with $N=4$ spins in Fig. 2, as is the case of the previous Section, we have numerically confirmed that there is no level crossing between the ground and 1st excited states in the fast-forward time range. So figures of 16 eigenvalues will be suppressed in this Section.

The components of the eigenvector of the ground state are: $C_{1}=C_{16}=V_{1}\zeta$, $C_{2}=C_{3}=C_{4}=C_{5}=C_{12}=C_{13}=C_{14}=C_{15}=V_{2}\zeta$, and $C_{6}=C_{7}=C_{8}=C_{9}=C_{10}=C_{11}=V_{6}\zeta$. Here $\zeta=(2+8V_{2}^{2}+6V_{6}^{2})^{-1/2}$, $V_{1}=1$, $V_{2}=\frac{(\beta+\bar{\beta})+14J-\sqrt{3}i(\beta-\bar{\beta})}{6B_{x}}$, and $V_{6}=-{\frac{2\left(\beta^{2}+\bar{\beta}^{2}\right)-10J\left(\beta+\bar{\beta}\right)-\left(48J^{2}+15B_{x}^{2}\right)+i\sqrt{3}(2\left(\beta^{2}-\bar{\beta}^{2}\right)+10J\left(\beta-\bar{\beta}\right))}{27{B_{x}}^{2}}}$, where the equality $|\beta|^{2}=13J^{2}+3B_{x}^{2}$ is used.

From $R$-derivative of the normalization ($2C_{1}^{2}+8C_{2}^{2}+6C_{6}^{2}=1$), we see

[TABLE]

If we suppress the 3-body interaction, the regularization Hamiltonian consists of only one pairwise interaction $\tilde{W}\equiv\tilde{W}_{ij}^{yz}$, due to the high symmetry of the triangular pyramid in Fig.2 (a). The corresponding matrix for the regularization term can be written as

[TABLE]

Due to the symmetry of $\{C_{j}\}$, the number of independent equations in Eq.(8) are three:

[TABLE]

While one of the above equations is trivial due to Eq.(28), we need one more unknown variable to make meaningful the algebraic equations in Eq. (V.1). Here we evaluate the contribution of the 3-body interaction. The geometrical symmetry allows a universal 3-body interaction $\tilde{Q}\equiv\tilde{Q}_{ijk}^{xyz}$, independent of all possible 3-body configurations $(i,j,k)$. The inclusion of the 3-body interaction improves some matrix elements of $\mathcal{\tilde{H}}$ in Eq.(29) as follows:

[TABLE]

After the above improvements, the algebraic equations in Eq.(V.1) are revised as:

[TABLE]

where one of the above lines is again trivial because of Eq. (28). Equation (V.1), whose coefficient matrix has the rank 2, gives the solution:

[TABLE]

The fast-forward Hamiltonian is given by

[TABLE]

with

[TABLE]

In the triangular pyramid, $\sum_{(i,j)\in N.N.}$ is equivalent to $\sum_{(i,j)\in all}$. The fast forward Hamiltonian guarantees the fast forward of the adiabatic dynamics of the ground state wave function.

Figures 5 (a) and 6 (a) show the time dependence of regularization terms and that of the wave function, respectively. The wave function starts from the ground state with $J=0$, i.e., $C_{j}=\frac{1}{4}$ for $j=1,\cdots,16$. In Fig. 6 (a) the solution $\Psi_{FF}(t)$ of TDSE in Eq.(14) has exactly reproduced the time-rescaled ground state wave function during the fast-forward time range $0\leq t\leq T_{FF}$.

V.2 Square

The eigenvalue of the ground state is $E_{0}=-\beta_{1}$, where $\beta_{1}=\sqrt{8{J}^{2}+2{B_{x}}^{2}+2\beta_{2}}$ with $\beta_{2}=\sqrt{16{J}^{4}+{B_{x}}^{4}}$. The components of the eigenvector of the ground state are: $C_{1}=C_{16}=V_{1}\zeta$, $C_{2}=C_{3}=C_{4}=C_{5}=C_{12}=C_{13}=C_{14}=C_{15}=V_{2}\zeta$, $C_{6}=C_{7}=C_{8}=C_{9}=V_{6}\zeta$, and $C_{10}=C_{11}=V_{10}\zeta$ with $\zeta=(8+2V_{1}^{2}+4V_{6}^{2}+2V_{10}^{2})^{-1/2}$. Here $V_{1}=\frac{(\beta_{1}-4J)(4J^{2}-B_{x}^{2}+\beta_{2})}{8J^{2}B_{x}}$, $V_{2}=1$, $V_{6}=\frac{(\beta_{1}^{2}-4\beta_{2})\beta_{1}}{16J^{2}B_{x}}$, and $V_{10}=\frac{(\beta_{1}+4J)(4J^{2}-B_{x}^{2}+\beta_{2})}{8J^{2}B_{x}}$. From $R$-derivative of the normalization ($2C_{1}^{2}+8C_{2}^{2}+4C_{6}^{2}+2C_{10}^{2}=1$), we see

[TABLE]

The geometric symmetry of the square spin system in Fig.2 (b) allows two candidates as regularization terms, which are $\tilde{W}_{12}=\tilde{W}_{23}=\tilde{W}_{34}=\tilde{W}_{41}=\tilde{W}_{1}$ and $\tilde{W}_{31}=\tilde{W}_{42}=\tilde{W}_{2}$. $\tilde{W}_{1}$ and $\tilde{W}_{2}$ correspond to N.N. and the second N.N. interactions, respectively. The regularization matrix $\mathcal{\tilde{H}}$ is given in Eq.(49). To add one more unknown variable, we include a contribution of the universal 3-body interaction $\tilde{Q}\equiv\tilde{Q}_{ijk}^{xyz}$. This inclusion requires the same improvement of some matrix elements of $\mathcal{\tilde{H}}$ as in Eq. (V.1).

Due to the symmetry of $\{C_{j}\}$, the number of independent algebraic equations are four:

[TABLE]

Because of Eq.(36), one of the above equations is trivial. Ignoring the second line for example, Eq.(V.2), whose coefficient matrix has the rank 3, gives the solution:

[TABLE]

The fast-forward Hamiltonian is given by Eq.(34), where $v(t)\tilde{\mathcal{H}}(R(\Lambda(t)))$ is now replaced by:

[TABLE]

Figures 5 (b) and 6 (b) show the time dependence of regularization terms and that of wave function, respectively. The wave function starts from the ground state with $J=0$, i.e., $C_{j}=\frac{1}{4}$ for $j=1,\cdots,16$. In Fig. 6 (b) the solution $\Psi_{FF}(t)$ of TDSE in Eq.(14) has exactly reproduced the time-rescaled ground state wave function.

V.3 Primary star graph

The eigenvalue of the ground state is $E_{0}=-\beta$, where $\beta=\sqrt{2{B_{x}}^{2}+5{J}^{2}+2\sqrt{{B_{x}}^{4}+{B_{x}}^{2}{J}^{2}+4{J}^{4}}}$. The components of the eigenvector of the ground state are: $C_{1}=C_{16}=V_{1}\zeta$, $C_{2}=C_{3}=C_{5}=C_{12}=C_{13}=C_{15}=V_{2}\zeta$, $C_{4}=C_{14}=V_{4}\zeta$, and $C_{6}=C_{7}=C_{8}=C_{9}=C_{10}=C_{11}=V_{6}\zeta$ with $\zeta=(6+2V_{1}^{2}+6V_{2}^{2}+2V_{4}^{2})^{-1/2}$. Here $V_{1}=\frac{-J(7{B_{x}}^{2}+3J^{2})+\beta(4{B_{x}}^{2}+3\beta J-{\beta}^{2}+J^{2})}{5J{B_{x}}^{2}}$, $V_{2}=\frac{-2J(9J^{2}-4{B_{x}}^{2})+\beta(4{B_{x}}^{2}-2\beta J-{\beta}^{2}+21J^{2})}{30J^{2}B_{x}}$, $V_{4}=\frac{-2J(J^{2}+4{B_{x}}^{2})-\beta(4{B_{x}}^{2}-2\beta J-{\beta}^{2}+J^{2})}{10J^{2}B_{x}}$, and $V_{6}=1$. From $R$-derivative of the normalization ($2C_{1}^{2}+6C_{2}^{2}+2C_{4}^{2}+6C_{6}^{2}=1$), we see

[TABLE]

The geometric symmetry of the primary star-graph spin system in Fig.2(c) allows two candidates as regularization terms, which are $\tilde{W}_{12}=\tilde{W}_{23}=\tilde{W}_{24}=\tilde{W}_{1}$ and $\tilde{W}_{14}=\tilde{W}_{13}=\tilde{W}_{34}=\tilde{W}_{2}$. $\tilde{W}_{1}$ and $\tilde{W}_{2}$ correspond to N.N. and the 2nd N.N. interactions, respectively. The matrix for regularization term $\mathcal{\tilde{H}}$ can be written in Eq.(50). To add one more unknown variable, we include a contribution of the universal 3-body interaction $\tilde{Q}\equiv\tilde{Q}_{ijk}^{xyz}$. This inclusion requires the same improvement of some matrix elements of $\mathcal{\tilde{H}}$ as in Eq. (V.1). One might have an idea to include two species of 3-body interactions with one among N.N.s and another among the 2nd N.N.s. But this idea results in incompatible equations in Eq.(8) and cannot be acceptable. Due to the symmetry of $\{C_{j}\}$, the number of independent equations are four:

[TABLE]

Because of Eq.(40), one of the above 4 equations becomes trivial. Ignoring the first line for example, Eq.(V.3), whose coefficient matrix has the rank 3, gives the solution:

[TABLE]

The fast-forward Hamiltonian is given by Eq.(34), where $v(t)\tilde{\mathcal{H}}(R(\Lambda(t)))$ is replaced by:

[TABLE]

Figures 5 (c) and 6 (c) show the time dependence of regularization terms and that of wave function, respectively. The wave function starts from the ground state with $J=0$, i.e., $C_{j}=\frac{1}{4}$ for $j=1,\cdots,16$. In Fig. 6 (c) the solution $\Psi_{FF}(t)$ of TDSE in Eq.(14) has exactly reproduced the time-rescaled ground state wave function.

V.4 Open linear 4 spin chain

The eigenvalue of the ground state is $E_{0}=-\frac{\beta_{2}}{\sqrt{3}}$, where $\beta_{2}=\sqrt{2(\beta_{1}+\bar{\beta_{1}})+11J^{2}+4B_{x}^{2}}$ with \beta_{1}=\bigl{(}64{J}^{6}+15{J}^{4}{B_{x}}^{2}+21{B_{x}}^{4}{J}^{2}+8{B_{x}}^{6}+3\sqrt{3}J^{2}B_{x}i\sqrt{128{J}^{6}+93{J}^{4}{B_{x}}^{2}+51{B_{x}}^{4}{J}^{2}+25{B_{x}}^{6}}\bigr{)}^{1/3} and $|\beta_{1}|^{2}=4{B_{x}}^{4}+7{B_{x}}^{2}J^{2}+16J^{4}$. The components of the eigenvector of the ground state are: $C_{1}=C_{16}=V_{1}\zeta$, $C_{2}=C_{5}=C_{12}=C_{15}=V_{2}\zeta$, $C_{3}=C_{4}=C_{13}=C_{14}=V_{3}\zeta$, $C_{6}=C_{8}=V_{6}\zeta$, $C_{7}=C_{9}=V_{7}\zeta$, and $C_{10}=C_{11}=V_{10}\zeta$ with $\zeta=(2+4V_{2}^{2}+4V_{3}^{2}+2V_{6}^{2}+2V_{7}^{2}+2V_{10}^{2})^{-1/2}$. Here $V_{1}=1$, $V_{2}=-\frac{\sqrt{3}{J}^{2}\beta_{2}\left(180{B_{x}}^{2}+144{J}^{2}\right)-\sqrt{3}\beta_{2}^{3}\left(12{B_{x}}^{2}+33{J}^{2}\right)+\sqrt{3}\beta_{2}^{5}-162{J}^{5}}{162{J}^{4}B_{x}}$, $V_{3}=\frac{\sqrt{3}{J}^{2}\beta_{2}\left(180{B_{x}}^{2}+198J^{2}\right)-\sqrt{3}\beta_{2}^{3}\left(12{B_{x}}^{2}+33{J}^{2}\right)+\sqrt{3}\beta_{2}^{5}+324{J}^{5}}{162{J}^{4}B_{x}}$,

$V_{6}=\frac{-\sqrt{3}{J}^{2}\beta_{2}(144{B_{x}}^{2}+81{J}^{2})-J\beta_{2}^{2}(36{B_{x}}^{2}+90{J}^{2})+\sqrt{3}\beta_{2}^{3}(12{B_{x}}^{2}+30{J}^{2})+3J\beta_{2}^{4}-\sqrt{3}\beta_{2}^{5}+243{J}^{5}+648{B_{x}}^{2}{J}^{3}}{216{B_{x}}^{2}{J}^{3}}$,

$V_{7}=-\frac{\sqrt{3}{J}^{2}\beta_{2}\left(144{B_{x}}^{2}+81{J}^{2}\right)-J\beta_{2}^{2}\left(36{B_{x}}^{2}+90{J}^{2}\right)-\sqrt{3}\beta_{2}^{3}\left(12{B_{x}}^{2}+30J^{2}\right)+3J\beta_{2}^{4}+\sqrt{3}\beta_{2}^{5}+243{J}^{5}+324{J}^{3}B_{x}^{2}}{108{B_{x}}^{2}{J}^{3}}$, and

$V_{10}=-\frac{\sqrt{3}J^{2}\beta_{2}\left(144{B_{x}}^{2}-9{J}^{2}\right)+J\beta_{2}^{2}\left(108{B_{x}}^{2}+54{J}^{2}\right)+6\sqrt{3}\beta_{2}^{3}\left(2{B_{x}}^{2}+J^{2}\right)-9J\beta_{2}^{4}-\sqrt{3}\beta_{2}^{5}+648{B_{x}}^{2}{J}^{3}-81{J}^{5}}{648{B_{x}}^{2}{J}^{3}}$.

Since $\beta_{2}$ is real, all components of the ground state are also real. From $R$-derivative of the normalization ($2C_{1}^{2}+4C_{2}^{2}+4C_{3}^{2}+2C_{6}^{2}+2C_{7}^{2}+2C_{10}^{2}=1$), we see

[TABLE]

In case of the open linear 4 spin system in Fig.2(d), the symmetry consideration allows 4 regularization terms which consist of $\tilde{W}_{12}=\tilde{W}_{34}=\tilde{W}_{1},\tilde{W}_{23}=\tilde{W}_{2}$, $\tilde{W}_{13}=\tilde{W}_{24}=\tilde{W}_{3}$, and $\tilde{W}_{14}=\tilde{W}_{4}$. The regularization Hamiltonian $\mathcal{\tilde{H}}$ is given in Eq.(51). To add one more unknown variable, we include a contribution of the universal 3-body interaction $\tilde{Q}\equiv\tilde{Q}_{ijk}^{xyz}$. This inclusion requires the same improvement of some matrix elements of $\mathcal{\tilde{H}}$ as in Eq. (V.1). The idea to include plural species of 3-body interactions results in incompatible equations in Eq.(8) and can not be employed. Due to the symmetry of $\{C_{j}\}$, the number of independent equations are six:

[TABLE]

The constraint in Eq.(44) renders one of the above 6 equations trivial, and Eq.(V.4), whose coefficient matrix has the rank 5, gives the following solution:

[TABLE]

with

[TABLE]

The fast-forward Hamiltonian is given by Eq.(34), where $v(t)\tilde{\mathcal{H}}(R(\Lambda(t)))$ is replaced by:

[TABLE]

Figures 5 (d) and 6 (d) show the time dependence of regularization terms and that of wave function, respectively. The wave function starts from the ground state with $J=0$, i.e., $C_{j}=\frac{1}{4}$ for $j=1,\cdots,16$. In Fig. 6 (d) the solution $\Psi_{FF}(t)$ of TDSE in Eq.(14) has reproduced the time-rescaled ground state wave function, which means the perfect fidelity during the fast-forward time range $0\leq t\leq T_{FF}$.

In this Section, the number of independent equations to determine the pair-wise interactions ($\tilde{W}_{i}$) is varied depending on the symmetry of clusters. To make Eq.(8) solvable, however, these equations always require only one extra unknown 3-body interaction, whose contribution to $\mathcal{\tilde{H}}$ is commonly given in Eq.(V.1) for all spin clusters with $N=4$ spins. Therefore the 3-body interaction ($\tilde{Q}$) here is geometry-independent and played a subsidiary role.

VI Summary and discussions

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. Assuming the regularization terms consisting of pair-wise and 3-body interactions, we applied the core formula in Eq.(8) to regular spin clusters with various geometries, e.g., regular triangle and open linear chain for $N=3$ spin systems, and 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, namely, the rank is determined by the geometric symmetry of the regular spin cluster. 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 pair-wise interactions and there is no need for the 3-body interaction. The regular triangle and open linear 3 spins require, respectively, one and 2 species of the pair-wise driving interactions; (2) for $N=4$ spin clusters, the main part of the driving interaction again consists of pair-wise interactions. The triangular pyramid and open linear 4 spins require, respectively, one and 4 species of the pair-wise driving interactions. On the other hand, two species of the pair-wise driving interactions are necessary for the square and primary star graph. For $N=4$ spin clusters, besides these geometry-dependent pair-wise interactions, we need a common geometry-independent 3-body interaction just to make the core equation in Eq.(8) solvable. The 3-body interaction here plays a subsidiary role. The geometric symmetry of regular spin clusters determines the number of independent species of pair-wise driving interactions, and the clusters with the highest symmetry have only one species of pair-wise driving interaction. Our fast-forward scheme provides a flexible method in designing the practical driving interaction in accelerating the adiabatic quantum dynamics of structured regular spin clusters. The scheme may also be useful in our inventing a variational method for treating much bigger regular clusters.

Acknowledgements.

We are grateful to S. Masuda for valuable discussions in the early stage of the present work. The work is supported by Hibah Disertasi Doktor Kemenristekdikti 2018. The work of B.E.G is supported by PUPT Ristekdikti-ITB 2017-2018.

Appendix A Regularization matrix $\mathcal{\tilde{H}}$ without contributions due to the 3-body interaction

A.1 Square

The matrix for regularization term can be written as

[TABLE]

where $A_{1}$ = $2\tilde{W}_{1}+\tilde{W}_{2}$, $A_{2}$ = $\tilde{W}_{2}$, $A_{3}$ = $2\tilde{W}_{1}-\tilde{W}_{2}$.

A.2 Primary star graph

The matrix for regularization term can be written as

[TABLE]

where $A_{1}$ = $2\tilde{W}_{2}+\tilde{W}_{1}$, $A_{2}$ = $3\tilde{W}_{1}$, $A_{3}$ = $\tilde{W}_{1}$, $A_{4}$ = $\tilde{W}_{1}-2\tilde{W}_{2}$.

A.3 Open linear 4 spins

The matrix for regularization term can be written as

[TABLE]

where $A_{1}$ = $\tilde{W}_{1}+\tilde{W}_{3}+\tilde{W}_{4}$, $A_{2}$ = $\tilde{W}_{1}+\tilde{W}_{2}+\tilde{W}_{3}$, $A_{3}$ = $\tilde{W}_{2}-\tilde{W}_{1}+\tilde{W}_{3}$, $A_{4}$ = $\tilde{W}_{1}-\tilde{W}_{4}+\tilde{W}_{3}$, $A_{5}$ = $\tilde{W}_{1}+\tilde{W}_{2}-\tilde{W}_{3}$, $A_{6}$ = $-\tilde{W}_{1}+\tilde{W}_{4}+\tilde{W}_{3}$, $A_{7}$ = $\tilde{W}_{1}-\tilde{W}_{2}+\tilde{W}_{3}$, $A_{8}$ = $\tilde{W}_{1}+\tilde{W}_{4}-\tilde{W}_{3}$.