Braiding and fusion of Majorana fermions in minimal Kitaev spin liquid
on a single hexagon with $5$ qubits

Introduction: A quantum computer is expected to be a most promising next generation computer[1, 2, 3], which can store $2^{N}$ information in $N$ qubit systems. A topological quantum computation based on Majorana fermions[4, 5, 6, 7, 8, 9, 10, 11] is attractive, where $N$ qubits are constructed from $2N$ Majorana fermions. Majorana fermions are theoretically proposed to emerge in fractional quantum Hall effects[12, 13, 9, 14], topological superconductors[15, 16, 17, 18] and Kitaev spin liquids[19, 20, 21, 22]. The Kitaev topological superconductor model is the simplest fermionic model that hosts Majorana fermions[6]. Recently, the Minimal Kitaev model consisting of only two sites is realized in double quantum dots[23, 24]. In the view point of quantum computation, it is desirable to construct a model hosting many Majorana fermions with the use of smaller number of sites.

The Kitaev spin liquid is one of the prominent exactly solvable models on the honeycomb lattice realizing spin liquid with the emergence of Majorana zero modes[19]. It is defined on the honeycomb lattice, where there are three Ising-type exchange interactions ($\varpropto K_{x},K_{y},K_{z}$) depending on the directions of bonds as shown in Fig.1(a). By representing the spin operator by a combination of Majorana fermion operators, the system turns into a free Majorana fermion model on the honeycomb lattice[19]. The theoretical proposal on the Kitaev spin liquid based on perovskite materials[20] evokes an intensive researches[25, 26, 27]. Experimental signature of Majorana fermions is observed by measuring a half quantization of thermal conductivity[27, 28]. Brading of Majorana states at vortices in the Kitaev spin liquid is theoretically proposed[29]. In addition to perovskite material realization, there are several proposals on the realization of the Kitaev spin liquid model in artificial systems such as qubits[30, 31], trapped ions[32], cold atoms[33, 34] and quantum dots[35]. One of the merit of these systems is that it is possible to construct an extremely small-size system and control model parameters temporally. The simplest system is a single hexagon. It is an interesting problem to study whether the Kitaev spin liquid model hosts Majorana fermions on a single hexagon.

In this paper, we investigate the minimal Kitaev spin liquid on a single hexagon. When $K_{z}=0$, there are 32=2⁵ zero-energy states, which leads to 5 qubits. These qubits are protected by particle-hole symmetry even for $K_{z}\neq 0$. Then, we construct braiding operators in terms of spin operators. A readout process of qubits is performed by the fusion protocol which measures the local spin correlation. An initialization is executed by switching on the Heisenberg interaction and magnetic field term, where only one zero-energy state is present.

Minimal Kitaev spin liquid model: We study the Kitaev spin liquid model in an artificial system. Especially, we analyze the spin $1/2$ system on the single hexagon based on the Kitaev model,

$$\hat{H}_{\text{K}}=\sum_{\alpha=1}^{2}\hat{H}_{\text{K}\alpha}+\hat{H}_{\text{K}z},$$

(1)

with

	$\displaystyle\hat{H}_{\text{K}\alpha}$	$\displaystyle=-K_{x}\sigma_{3\alpha-2}^{x}\sigma_{3\alpha-1}^{x}-K_{y}\sigma_{3\alpha-1}^{y}\sigma_{3\alpha}^{y},$		(2)
	$\displaystyle\hat{H}_{\text{K}z}$	$\displaystyle=-K_{z}\left(\sigma_{3}^{z}\sigma_{4}^{z}+\sigma_{6}^{z}\sigma_{1}^{z}\right),$		(3)

where $\sigma_{i}^{\gamma}$ is the Pauli matrix at the site $i$ with $\gamma=x,y,z$. The Ising-type exchange interaction $K_{\gamma}\sigma_{i}^{\gamma}\sigma_{j}^{\gamma}$ is anisotropic depending on the direction of the link $\gamma$ as illustrated in Fig.1(a). It contains $6$ spins, and hence, there are $2^{6}$ states in total.

The Hamiltonian for the Kitaev quantum spin liquid is rewritten in terms of Majorana fermions by way of the Jordan Wigner transformation[36, 37, 38, 39, 40, 41]. We number the site from 1 to 6 as shown in Fig.1(a). There are relations between the spin operators and the fermion operators: $\sigma_{j}^{-}=\Omega_{j}c_{j}$, $\sigma_{j}^{+}=\Omega_{j}c_{j}^{\dagger}$, $\sigma_{j}^{z}=c_{j}^{\dagger}c_{j}-1/2$ with $\Omega_{j}\equiv\prod\limits_{\ell=1}^{j-1}\exp[i\pi c_{\ell}^{\dagger}c_{\ell}]$ and $\sigma_{j}^{+}\equiv\frac{1}{2}(\sigma_{j}^{x}+i\sigma_{j}^{y})$ and $\sigma_{j}^{-}\equiv\frac{1}{2}(\sigma_{j}^{x}-i\sigma_{j}^{y})$. Here, $c_{j}$ and $c_{j}^{\dagger}$ satisfy the anti-commutation relations, $\{c_{i},c_{\ell}\}=\{c_{i}^{\dagger},c_{\ell}^{\dagger}\}=0$, $\{c_{i},c_{\ell}^{\dagger}\}=\delta_{i\ell}$. Furthermore, we introduce Majorana operators as $\gamma_{2j}^{A}=c_{2j}+c_{2j}^{\dagger}$, $\gamma_{2j}^{B}=(c_{2j}-c_{2j}^{\dagger})/i$, $\gamma_{2j+1}^{A}=(c_{2j+1}-c_{2j+1}^{\dagger})/i$, $\gamma_{2j+1}^{B}=c_{2j+1}+c_{2j+1}^{\dagger}$ for $j=1,2,3$. Then, the Hamiltonians (2) and (3) read

	$\displaystyle\hat{H}_{\text{K}\alpha}$	$\displaystyle=-i\left(K_{x}\gamma_{3\alpha-2}^{A}\gamma_{3\alpha-1}^{A}-K_{y}\gamma_{3\alpha-1}^{A}\gamma_{3\alpha}^{A}\right),$		(4)
	$\displaystyle\hat{H}_{\text{K}z}$	$\displaystyle=-\frac{K_{z}}{4}(\gamma_{3}^{A}\gamma_{3}^{B}\gamma_{4}^{A}\gamma_{4}^{B}+\gamma_{6}^{A}\gamma_{6}^{B}\gamma_{1}^{A}\gamma_{1}^{B}),$		(5)

in the Majorana form.

Particle-hole symmetry: The zero-energy states of the Hamiltonian with particle-hole symmetry are Majorana fermions[15, 16, 17, 18]. We discuss particle-hole symmetry in the Kitaev spin liquid. Particle-hole symmetry acts as $P^{-1}\gamma_{j}^{A}P=\gamma_{j}^{A}$ and $P^{-1}\gamma_{j}^{B}P=\gamma_{j}^{B}$ in terms of Majorana fermion operators, or $P^{-1}c_{j}P=c_{j}^{\dagger}$ and $P^{-1}c_{j}^{\dagger}P=c_{j}$ in terms of fermion operators. In terms of the spin operators, it acts as

	$\displaystyle P^{-1}\sigma_{j}^{x}P$	$\displaystyle=\left(-1\right)^{j-1}\sigma_{j}^{x},\qquad P^{-1}\sigma_{j}^{y}P=\left(-1\right)^{j}\sigma_{j}^{y},$
	$\displaystyle P^{-1}\sigma_{j}^{z}P$	$\displaystyle=-\sigma_{j}^{z},$		(6)

where we have used the relation $P^{-1}\Omega_{j}P=\left(-1\right)^{j-1}\Omega_{j}$.

Under the particle-hole symmetry transformation, the Hamiltonian (1) is mapped to $P^{-1}\hat{H}_{\text{K}}\left(K_{x},K_{y},K_{z}\right)P=-\hat{H}_{\text{K}}\left(K_{x},K_{y},-K_{z}\right)$. Hence, the Hamiltonian has particle-hole symmetry for $K_{z}=0$. We later show that particle-hole symmetry is present even for $K_{z}\neq 0$ in the present model, as is consistant with Fig.1(d).

Minimal Kitaev spin chain models: We analyze the minimal Kitaev spin liquid model where $K_{z}$ is much smaller than $K_{x}$ and $K_{y}$. We first consider the limit $K_{z}=0$ and later include the effect due to $K_{z}\neq 0$.

When we set $K_{z}=0$ in Hamiltonian (1), it is decomposed[8, 37, 41] into two independent Kitaev spin chain models $\hat{H}_{\text{K}\alpha}$. There are $2^{3}$ states because there are 3 spins for each $\alpha$. By exactly diagonalizing $\hat{H}_{\text{K}\alpha}$ for each $\alpha$, we find that there are 4-fold degenerate states with $E_{\text{K1}}=E_{\text{K2}}=\pm\sqrt{K_{x}^{2}+K_{y}^{2}}$. Note that there are no zero-energy states in each minimal Kitaev spin chain model.

However, the combined system has $8\times 8=64$ states made of 32 zero-energy states and 32 nonzero-energy states irrespective of $K_{x}$ and $K_{y}$. The Hamiltonian (4) is rewritten in the form

$$\sum_{\alpha=1}^{2}\hat{H}_{\text{K}\alpha}=\sqrt{K_{x}^{2}+K_{y}^{2}}\left(i\gamma_{2}^{A}\bar{\gamma}_{1}^{A}-i\gamma_{5}^{A}\bar{\gamma}_{4}^{A}\right),$$

(7)

where we have defined new Majorana operators

$$\bar{\gamma}_{1}^{A}=\frac{K_{x}\gamma_{1}^{A}+K_{y}\gamma_{3}^{A}}{\sqrt{K_{x}^{2}+K_{y}^{2}}},\qquad\ \bar{\gamma}_{4}^{A}=\frac{K_{x}\gamma_{4}^{A}+K_{y}\gamma_{6}^{A}}{\sqrt{K_{x}^{2}+K_{y}^{2}}}.$$

(8)

Since the Hamiltonian (7) does not contain Majorana operators $\gamma_{3}^{A}$, $\gamma_{6}^{A}$, $\gamma_{j}^{B}$ with $j=1,2,\cdots,6$, we have $\left[\hat{H}_{\text{K}},\gamma_{3}^{A}\right]=\left[\hat{H}_{\text{K}},\gamma_{6}^{A}\right]=0$ and $\left[\hat{H}_{\text{K}},\gamma_{j}^{B}\right]=0$. Hence, there are 8 free Majorana fermions, from which we construct 4 fermion operators as $f_{1}^{A}\equiv\left(\gamma_{3}^{A}-i\gamma_{6}^{A}\right)/2$ and $f_{j}^{B}=\left(\gamma_{2j-1}^{B}-i\gamma_{2j}^{B}\right)/2$ with $j=1,2,3$. In addition, we introduce 2 fermion operators $f_{2}^{A}\equiv\left(\bar{\gamma}_{1}^{A}-i\gamma_{2}^{A}\right)/2$ and $f_{3}^{A}\equiv\left(\bar{\gamma}_{4}^{A}-i\gamma_{5}^{A}\right)/2$. The Hamiltonian (7) is rewritten in terms of these fermion operators as

$$\sum_{\alpha=1}^{2}\hat{H}_{\text{K}\alpha}=2\sqrt{K_{x}^{2}+K_{y}^{2}}\left(\hat{n}_{2}^{A}-\hat{n}_{3}^{A}\right),$$

(9)

where we have defined the number operators $\hat{n}_{2}^{A}\equiv f_{2}^{A\dagger}f_{2}^{A}=\left(i\gamma_{2}^{A}\bar{\gamma}_{1}^{A}+1\right)/2$ and $\hat{n}_{3}^{A}\equiv f_{3}^{A\dagger}f_{3}^{A}=\left(i\gamma_{5}^{A}\bar{\gamma}_{4}^{A}+1\right)/2$. In the similar way, we define $\hat{n}_{j}^{A}=f_{j}^{A\dagger}f_{j}^{A}$ and $\hat{n}_{j}^{B}=f_{j}^{B\dagger}f_{j}^{B}$ with $j=1\sim 3$. We consider the Hilbert space where $\hat{n}_{j}^{A}$ and $\hat{n}_{j}^{B}$ take the eigenvalues $0$ and $1$. We take $f_{2}^{A}$ to be a free fermion. Then, $f_{3}^{A}$ is determined by the zero-energy condition of the Hamiltonian (9). As a result, the Kitaev spin liquid model on the single hexagon contains 10 free Majorana fermions, or 5 qubits defined by $\left|n_{2}^{A}n_{1}^{A}n_{3}^{B}n_{2}^{B}n_{1}^{B}\right\rangle$.

Braiding: The basic operation on qubits is braiding defined[5] by $\mathcal{B}_{\alpha\beta}=\exp\left[\frac{\pi}{4}\gamma_{\beta}\gamma_{\alpha}\right]$. It is generalized[11] to an arbitrary angle such that $\mathcal{B}_{\alpha\beta}\left(\theta\right)=\exp\left[\theta\gamma_{\beta}\gamma_{\alpha}\right]$. The unitary dynamics under the Hamiltonian $H$ reads $U=\exp\left[-iHt/\hbar\right]$. The generalized brading is executed by setting $-iHt/\hbar=\theta\gamma_{\beta}\gamma_{\alpha}$. Generalized braiding operators for $B$ Majorana fermions are rewritten in terms of spin operators as $\exp\left[\theta\gamma_{2j-1}^{B}\gamma_{2j}^{B}\right]=\exp\left[i\theta\sigma_{2j-1}^{y}\sigma_{2j}^{y}\right]$ for $j=1,2,3$, and $\exp\left[\theta\gamma_{2j}^{B}\gamma_{2j+1}^{B}\right]=\exp\left[i\theta\sigma_{2j}^{x}\sigma_{2j+1}^{x}\right]$ for $j=1,2$. Generalized braiding operators for $A$ Majorana fermions are rewritten in terms of spin operators as

	$\displaystyle\exp\left[\theta\bar{\gamma}_{3j-2}^{A}\gamma_{3j-1}^{A}\right]$
	$\displaystyle=\exp\left[\frac{\theta}{\sqrt{K_{x}^{2}+K_{y}^{2}}}\left(K_{x}\gamma_{3j-2}^{A}\gamma_{3j-1}^{A}-K_{y}\gamma_{3j-1}^{A}\gamma_{3j}^{A}\right)\right]$
	$\displaystyle=\exp\left[\frac{i\theta}{\sqrt{K_{x}^{2}+K_{y}^{2}}}\left(K_{x}\sigma_{3j-2}^{x}\sigma_{3j-1}^{x}-K_{y}\sigma_{3j-1}^{y}\sigma_{3j}^{y}\right)\right]$		(10)

for $j=1,2$. Generalized braiding operators consisting of $A$ and $B$ Majorana fermions are rewritten in terms of spin operators as

	$\displaystyle\exp\left[\theta\gamma_{2j-1}^{A}\gamma_{2j-1}^{B}\right]$	$\displaystyle=\exp\left[-2i\theta\sigma_{2j-1}^{z}\right],$		(11)
	$\displaystyle\exp\left[\theta\gamma_{2j}^{A}\gamma_{2j}^{B}\right]$	$\displaystyle=\exp\left[-2i\theta\sigma_{2j}^{z}\right].$		(12)

Hence, it is possible to execute braiding by temporally controlling the spin Hamiltonian.

Fusion: In order to readout the information of qubits based on Majorana fermions, the fusion protocol is used, where the fermion number constructed from Majorana fermions are observed. The fusion is a pair annihilation process of two Majorana fermions, which results in a single fermion ($n_{j}=1$) or a vacuum ($n_{j}=0$), and hence, the qubit $n_{j}$ can be readout. The fermion numbers of Majorana fermions are expressed in terms of spin operators as

	$\displaystyle\hat{n}_{j}^{B}$	$\displaystyle\equiv f_{j}^{B\dagger}f_{j}^{B}=-i\gamma_{2j-1}^{B}\gamma_{2j}^{B}/2=-\sigma_{2j-1}^{y}\sigma_{2j}^{y},\quad j=1,2,3,$
	$\displaystyle\hat{n}_{2}^{A}$	$\displaystyle\equiv f_{2}^{A\dagger}f_{2}^{A}=-i\bar{\gamma}_{1}^{A}\gamma_{2}^{A}=-i\frac{K_{x}\gamma_{1}^{A}+K_{y}\gamma_{3}^{A}}{\sqrt{K_{x}^{2}+K_{y}^{2}}}\gamma_{2}^{A}$
		$\displaystyle=-\frac{K_{x}}{\sqrt{K_{x}^{2}+K_{y}^{2}}}\sigma_{1}^{x}\sigma_{2}^{x}+\frac{K_{y}}{\sqrt{K_{x}^{2}+K_{y}^{2}}}\sigma_{2}^{y}\sigma_{3}^{y}.$		(13)

Hence, the fusion is executed by measuring the local spin correlation $\sigma_{j}^{x}\sigma_{j+1}^{x}$ and $\sigma_{j}^{y}\sigma_{j+1}^{y}$. On the other hand, it is difficult to readout $\hat{n}_{1}^{A}$ because the number operator $\hat{n}_{1}^{A}=-i\gamma_{3}^{A}\gamma_{6}^{A}$ cannot be represented by a local spin correlation operator.

Nonzero $K_{z}$: We next consider the realistic case with $K_{z}\neq 0$. There are conserved quantities[36, 37, 38, 39, 40, 41] known as the $Z_{2}$ gauge fields in the Hamiltonian (1). They are real variables $\hat{u}_{34}\equiv i\gamma_{3}^{B}\gamma_{4}^{B}$ and $\hat{u}_{61}\equiv i\gamma_{6}^{B}\gamma_{1}^{B}$, satisfying $\left[\hat{H}_{\text{K}},\hat{u}_{34}\right]=\left[\hat{H}_{\text{K}},\hat{u}_{61}\right]=0$ and $\hat{u}_{34}^{2}=\hat{u}_{61}^{2}=1$. The Hilbert space is decomposed into the subspaces, where $\hat{u}_{34}$ and $\hat{u}_{61}$ take eigenvalues $\pm 1$. In these subspaces, because the Hamiltonian (5) becomes in terms of Majorana fermions as in

$$\hat{H}_{\text{K}z}=-i\frac{K_{z}}{4}(\hat{u}_{34}\gamma_{3}^{A}\gamma_{4}^{A}+\hat{u}_{61}\gamma_{6}^{A}\gamma_{1}^{A}),$$

(14)

particle-hole symmetry is present. Hence, the Majorana states are particle-hole symmetry protected even for $K_{z}\neq 0$. Furthermore, it is possible to diagonalize exactly the Hamiltonian.

We show the energy spectrum as a function of $K_{z}/K$ with $K_{x}=K_{y}=K$ in Fig.1(c). There are 24 zero-energy states, 8 states with $E=\pm\sqrt{K_{x}^{2}+K_{y}^{2}+K_{z}^{2}}$ and other 24 states as shown in Fig.1(d). The energy spectrum is symmetric with respect to $E=0$ as shown in Fig.1(c). For $\left|K_{z}/K\right|\ll 1$, 32 states are almost degenerate and it is possible to use 5 qubits even for $K_{z}\neq 0$.

Initialization: In quantum computation, it is necessary to prepare one unique quantum state as an initial state. For this purpose, we introduce the Heisenberg interaction[42, 43, 44] together with the magnetic field $B_{z}$ along $z$ direction at the initial stage,

$$\hat{H}_{J}=J\sum_{\left\langle i,j\right\rangle}\sum_{\gamma=x,y,z}\sigma_{i}^{\gamma}\sigma_{j}^{\gamma}-B_{z}\sum_{j=1}^{6}\sigma_{j}^{z},$$

(15)

where we have set $K_{x}=K_{y}=K_{z}\equiv K$. The Hamiltonian $\hat{H}=\hat{H}_{\text{K}}+\hat{H}_{J}$ with Eq.(1) is analytically diagonalizable for the zero-energy state, and we find it given by

	$\displaystyle\left|\psi_{0}\right\rangle\propto$	$\displaystyle\left|010010\right\rangle+\left|010101\right\rangle+\left|011001\right\rangle+\left|110001\right\rangle$
		$\displaystyle-(\left|001100\right\rangle+\left|100100\right\rangle+\left|101000\right\rangle+\left|101011\right\rangle).$		(16)

The state $\left|\psi_{0}\right\rangle$ is used for the initialization process of the qubits.

Kitaev spin liquid on connected hexagons: It is possible to generalize the Kitaev spin liquid model on a single hexagon to that on connected $L$ hexagons, where there are $4L+2$ spins. It is illustrated in the case of $L=2$ and $3$ in Fig.2(a) and (d). The Hamiltonian reads

	$\displaystyle\hat{H}_{\text{K}}$	$\displaystyle=-\sum_{j=1}^{L}(K_{x}\sigma_{2j-1}^{x}\sigma_{2j}^{x}+K_{y}\sigma_{2j}^{y}\sigma_{2j+1}^{y})$
		$\displaystyle-\sum_{j=1}^{L}(K_{x}\sigma_{2j-1+2L+1}^{x}\sigma_{2j+2L+1}^{x}+K_{y}\sigma_{2j+2L+1}^{y}\sigma_{2j+1+2L+1}^{y})$
		$\displaystyle-K_{z}\sum_{j=1}^{L+1}\sigma_{2j-1}^{z}\sigma_{4L+4-2j}^{z}.$		(17)

It is rewritten in terms of Majorana fermions as

	$\displaystyle\hat{H}_{\text{K}}=$	$\displaystyle-i\sum_{j=1}^{L}\left(K_{x}\gamma_{2j-1}^{A}\gamma_{2j}^{A}-K_{y}\gamma_{2j}^{A}\gamma_{2j+1}^{A}\right)$
		$\displaystyle-i\sum_{j=1}^{L}(K_{x}\gamma_{2j-1+2L+1}^{A}\gamma_{2j+2L+1}^{A}$
		$\displaystyle\qquad\qquad-K_{y}\gamma_{2j+2L+1}^{A}\gamma_{2j+1+2L+1}^{A})$
		$\displaystyle-i\sum_{j=1}^{L+1}\frac{K_{z}}{4}u_{2j-1,4L+4-2j}\gamma_{2j-1}^{A}\gamma_{4L+4-2j}^{A},$		(18)

where the $\mathbb{Z}_{2}$ gauge fields are given by

$$u_{2j-1,4L+4-2j}\equiv i\gamma_{2j-1}^{B}\gamma_{4L+4-2j}^{B},\quad j=1,\cdots,L+1.$$

(19)

We first consider the case $K_{z}=0$, where the system of hexagons is decomposed into two chains. The system is particle-hole symmetric. The energy spectrum is shown in Fig.2(b) and (e) for the case of $L=2$ and $3$. There are $2^{3L+2}$-fold degenerate zero-energy states. It is understood as follows. There are $2L+1$ free $B$ Majorana fermions because $\gamma_{j}^{B}$ does not appear in the Hamiltonian of each chain. On the other hand, there is one free $A$ Majorana fermion according to the Lieb theorem dictating the number of the zero-energy states in the bipartite system (18). Accordingly, the number of one type of sites is $L+1$, while that of the other type of sites is $L$. Hence, the difference is $1$ in each chain, implying the presence of one free $A$ Majorana fermion in Hamiltonian (18). Hence, the number of free Majorana fermions is $2L+2$ in total, which results in the $2^{L+1}$-fold degeneracy in the energy spectrum. In addition, the diagonalization of the quadratic Hamiltonian (18) for $\gamma_{j}^{A}$ gives different eigenenergies each other. Hence, one Kitaev spin chain model with length $L$ has $2^{L}$ different states with $2^{L+1}$-fold degeneracy. They produces $2^{3L+2}=2^{L}\times 2^{L+1}\times 2^{L+1}$-fold degenerate zero energy states in total. There emerge $2^{3L+2}$ zero-energy states, and hence, $3L+2$ qubits are constructed.

We next consider the case $K_{z}\neq 0$. The energy spectrum is shown in Fig.2(c) and (f) for the case of $L=2$ and $3$, where particle-hole symmetry holds manifestly. This can be shown with the aid of the $Z_{2}$ gauge fields (19) as in the case of the single hexagon. Hence, the Majorana states are particle-hole symmetry protected even for $K_{z}\neq 0$.

Conclusion: In this paper, we have shown that the Kitaev spin liquid model on a single hexagon acts as a 5-qubit system. It is possible to prepare a unique initial state by introducing the Heisenberg interaction together with magnetic field and to execute braiding by controlling the local spin correlation Hamiltonian, while qubits are readout with the use of the fusion protocol by observing local spin correlators. In addition, an arbitrary number of qubits is constructed by using connected hexagons. Our results are more efficient comparing previous results on the emergence of Majorana fermions at in the Kitaev spin liquid model.

This work is supported by CREST, JST (Grants No. JPMJCR20T2) and Grants-in-Aid for Scientific Research from MEXT KAKENHI (Grant No. 23H00171).