Proposal for a nonadiabatic geometric gate with an Andreev spin qubit

Motivated by the prospect of scalable device fabrication and circuit design, the search for two-state quantum systems or qubits in solid-state systems has been a major experimental and theoretical endeavor Burkard et al. (2023); Vion et al. (2002); *task2; *task3; *task4; *task5. Among all, Andreev physics-based qubits present a particularly promising route. In the superconducting state at low temperatures, the Fermi-level degrees of freedom are frozen out. Therefore, most of the dissipative mechanisms are eliminated so that the qubit may exhibit long coherence times Bocko et al. (1997); *Ketterson2; Hays et al. (2021); Janvier et al. (2015). In this paper, we study a hybrid structure of a ferromagnetic insulator (FI) connected by a weak link (a normal region N) to an $s$-wave superconductor (S) that accommodates spin-resolved Andreev bound states (ABS) in the weak link [see Fig. 1(a)]. We show that a pair of spin-resolved ABS, when occupied by an electron, creates an Andreev spin qubit (ASQ). On account of a large level spacing in our ASQ, which is due to the exchange field of the FI, the external fields controlling the FI magnetization are strongly coupled to the qubit spin. Due to the strong coupling, we expect that spin-flipping errors originating from weak random fields in the environment, such as in the case of hyperfine coupling to nuclei Khaetskii et al. (2002); *Loss22; *Loss23, will be suppressed.

A hallmark feature of the ABS formed in a Josephson junction is that the occupied levels modify a supercurrent across the junction Chtchelkatchev and Nazarov (2003); Nichele et al. (2020) so that transitions between Andreev levels can be detected by means of transport measurement. This is the physical basis of Andreev level qubits Janvier et al. (2015); Lantz et al. (2002); *Zaz2; *Zaz3. It was also observed that in spin active weak links, e.g., with spin-orbit coupling Chtchelkatchev and Nazarov (2003); Hays et al. (2021), or magnetic impurity Bratus’ et al. (1997); *ASQ2, the spin degeneracy of the levels can be lifted, so that the supercurrent flow would depend on the spin state of the electron trapped in the bound state. In our ASQ, the exchange field of the FI breaks the spin degeneracy of the levels, so that quantum information can be stored in the spin state of the electron Wendin and Shumeiko (2021). Similarly, to detect or read out the qubit state, we study the superconducting current affected by the occupied spin-resolved levels.

Generically, the unwanted qubit coupling to the environment and inaccurate external control can reduce the qubit state coherence time, which hinders the experimental implementation of scalable qubit designs Burkard et al. (2023); Vion et al. (2002); *task2; *task3; *task4. The qubits with gates based on the geometric phases may have the inherently fault-tolerant advantage due to the fact that the geometric gates depend only on some global geometric features of the evolution but independent of evolution details, so that they can be robust against control errors Xiang-Bin and Keiji (2001a); *PRLM2; De Chiara and Palma (2003); *GG2; *GG3; *GG4; Zhao et al. (2017). Moreover, utilizing nonadiabatic gates with high operation speed may reduce exposure time to the environment, rendering a high-fidelity gate Zhang et al. (2020). Here, we will show that our hybrid structure is a natural platform to realize a nonadiabatic geometric gate.

Consider the FI-N-S hybrid structure with the following Bogoliubov–de Gennes Hamiltonian

$\displaystyle\hat{H}_{\text{FINS}}=\begin{pmatrix}H_{0}(p,{\bf m})&i\Delta_{\text{S}}(x)\,\sigma_{y}\\ -i\Delta_{\text{S}}(x)\,\sigma_{y}&-H^{*}_{0}(-p,{\bf m})\end{pmatrix},$

(1)

where $H_{0}(p,{\bf m})=\frac{p^{2}}{2m}-\mu+V(x)+\Delta_{\text{xc}}(x){\bf m}(t)\cdot\boldsymbol{\sigma}$, $\mu$ is the chemical potential, $\Delta_{\text{xc}}(x)=-\Delta_{\text{xc}}\,\Theta(-x)$ is the magnetic exchange field, and $\Delta_{\text{S}}(x)=\Delta_{\text{S}}\,\Theta(x-d_{\text{N}})$ is the superconducting pair potential, both written in terms of the Heaviside step function. The scalar potential $V(x)=V_{0}\,\Theta(-x)+\hbar v_{F}Z\delta(x)$ models the FI electron gap where $V_{0}>(\mu+\Delta_{\text{S}}+\Delta_{\text{xc}})$, and $Z$ parametrizes an interface barrier potential where $v_{F}$ is the Fermi velocity. The unit vector ${\bf m}(t)$ gives the time-dependent direction of the magnetization, and $\boldsymbol{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$ are Pauli matrices operating in spin space.

Our goal is to find the subgap ($|\varepsilon|<\Delta_{\text{S}}$) energy spectrum of bound states, which are confined in the normal region. Imposing a condition for constructive quantum interference in a McMillan-Rowell process for the electrons—four times crossing N with two Andreev conversions as well as two reflections from FI, once as electron and once as hole Eschrig (2018)—results in a transcendental equation

$\displaystyle\varphi_{\sigma}(\varepsilon)+2\theta_{\text{A}}+4\theta_{\text{N}}=2\pi n,\,\,\,n=0,\pm 1,\cdots,$

(2)

whose solutions are the discrete bound states energy levels $\varepsilon$ (measured from $\mu$). Here, $\sigma\in\{\uparrow,\downarrow\}$ is specified with respect to the FI magnetization direction, $\varphi_{\sigma}(\varepsilon)\equiv\vartheta_{\sigma}({\varepsilon})-\vartheta_{-\sigma}(-\varepsilon)$, with $\vartheta_{\sigma}({\varepsilon})$ being the spin-dependent phase acquired by the electron upon reflection from the FIN interface, $\vartheta_{\sigma}(\epsilon)=\vartheta_{\sigma}^{(0)}(\varepsilon)+2\,\text{arg}\big{(}1-iZ-iZe^{-i\vartheta_{\sigma}^{(0)}(\varepsilon)}\big{)}$, where $\vartheta^{(0)}_{\sigma}(\varepsilon)\equiv\vartheta_{\sigma}({\varepsilon,Z=0})$ Eschrig (2018). $\theta_{\text{N}}=d_{\text{N}}\varepsilon/\hbar v_{F}$ where $d_{\text{N}}$ is the normal region thickness. Since for subgap energies the S supports only Cooper pair tunneling into the S, the incident electrons from the N region on the clean N-S interface reflect back as a hole rat (experiencing an Andreev reflection), where the phase change associated with the Andreev reflection is given by $\theta_{\text{A}}=-\arccos{(\varepsilon/\Delta_{\text{S}}})$.

When $d_{\text{N}}$ is smaller than the superconducting coherence length $\xi$ and $\Delta_{\text{xc}}=0$, Eq. (2) can admit a single spin-degenerate positive-energy solution whose degeneracy is removed by increasing $\Delta_{\text{xc}}$ [see Fig. 1(b)]. We point out that an oxide layer formed at the FIN interface or a tunable mismatch between the electronic properties across the interface create an effective ultrathin insulating layer that can greatly affect the probability of electron and hole evanescent penetration into the FI region. To capture the essential effect of this insulating layer, we use an interface barrier potential with strength $Z$, which can be utilized to tune the effective exchange interaction experienced by electrons and holes [see Fig. 1(c)]. Moreover, the number of ABS found in a clean N region depends on its thickness, that is, by increasing $d_{\text{N}}$ the ABS levels are pushed towards the middle of the gap at $\mu$, so that more energy levels begin to appear [see Fig. 1(d)].

Let us consider a hybrid structure with only two positive-energy spin-resolved ABS $(\varepsilon_{1},\varepsilon_{2})$, and ignore the continuum of positive energy levels above the S gap. We define the ground state of the system when these ABS are unoccupied and measure energies with respect to $(\varepsilon_{1}+\varepsilon_{2})/2$. The energies then are $-(\varepsilon_{1}+\varepsilon_{2})/2$ for the ground, $(\varepsilon_{1}-\varepsilon_{2})/2$ and $(\varepsilon_{2}-\varepsilon_{1})/2$ for two spin-$\frac{1}{2}$, and $(\varepsilon_{1}+\varepsilon_{2})/2$ for excited state. The ground and excited states have even (electron number) parity. The odd parity states, on the other hand, correspond to a single electron in the system realizing an ASQ. ABS wave functions with energies $(\varepsilon_{1},\varepsilon_{2})$ may have different spatial components. However, due to the tunneling treatment of the magnetic insulator, we assume that the ABS wave functions differ mainly in their spin character, with the orbital components being essentially identical. The effective ASQ Hamiltonian written in terms of the instantaneous quantization axis ${\bf m}(t)$ reads

$\displaystyle H_{\text{A}}(t)=-\frac{1}{2}\Delta\,{\bf m}(t)\cdot\boldsymbol{\sigma},$

(3)

where $\Delta\equiv\varepsilon_{2}-\varepsilon_{1}>0$. It is clear that the timescale for the coherent manipulation of the qubit, $t$, is set by the lifetime of the odd-parity sector, $\tau_{e}$. In the temperature regime considered in this paper, $k_{\text{B}}T\ll\Delta_{\text{S}}$, the odd-parity sector can be prepared, for example, by microwave irradiation Lundin et al. (1999); Chtchelkatchev and Nazarov (2003), electron injection by means of voltage gates Wendin et al. (1999), or spontaneously when an electron is stochastically trapped in the bound state. The latter mechanism, employed in the recent experiment Hays et al. (2021), is due to ubiquitous nonequilibrium quasiparticles and likely originated from a background stray photons with energy exceeding the threshold for breaking a Cooper pair Houzet et al. (2019). There are more deterministic approaches to preparing the ASQ in odd parity state Kurilovich et al. (2023); Wesdorp et al. (2023); *Wes2. For example, if the junction is irradiated with the microwave of frequency $\omega=(\varepsilon_{1}+\varepsilon_{2})/\hbar$, the microwave drive can break a Cooper pair placing electrons on the Andreev levels $(\varepsilon_{1},\varepsilon_{2})$. Moreover, imposing the condition $\Delta_{\text{S}}-\varepsilon_{2}<\hbar\omega<\Delta_{\text{S}}-\varepsilon_{1}$, would guarantee that the microwave drive can only excite the electron from Andreev level $\varepsilon_{2}$ into the continuum, which results in the odd parity state Kurilovich et al. (2023).

Once an electron is trapped, the probability of thermally activated parity-switching processes contains an exponentially small factor $\text{exp}(-\Delta_{\text{S}}/k_{\text{B}}T)$ for tunneling leading to the long lifetime of the trapped electron in the ABS, which is observed to exceed $\tau_{e}=100$ $\mu$s Zgirski et al. (2011). In general, when there are available subgap states caused by, e.g., spatial variations of the order parameter or impurities Shiba (1968), $\tau_{e}$ will be finite provided the trapped electron can tunnel through the gap. We note that the microscopic details of the hybrid structure can greatly modify the residence time for the trapped electron, which can be determined from the “lifetime matrix” lft . As a result, the timescale for the coherent manipulation of the qubit reads

$\displaystyle\hbar/\Delta_{\text{S}}<t<\tau_{e},$

(4)

where the lower bound is imposed to avoid dynamical mixing with the continuum states when the qubit state is evolving.

Isolated single spins can be coherently manipulated using both electrical and optical techniques Burkard et al. (2023); Kato et al. (2004); *spinc2. In our hybrid structure, however, we accomplish the qubit manipulation by the coherent control of the magnetization dynamics. We study the qubit dynamics when the magnetization precession is both resonant and nonresonant. When resonant, our hybrid structure enhances the Rabi oscillation frequency of the qubit dynamics. When nonresonant, as we discuss, our hybrid structure has the benefit of implementing nonadiabatic single-qubit gates via spintronic techniques. Because of the experimentally achievable high-speed control of the magnetization dynamics Stern et al. (2008); *high-speed2, the nonadiabatic spintronic manipulation of the qubit state renders a shorter qubit evolution time, which is an important advantage in realizing high-fidelity quantum gates Li et al. (2021).

Consider first an FI dynamics at resonance, where the magnetization parametrized as ${\bf m}(t)=(\sin\theta\cos(\omega t+\phi_{0}),\sin\theta\sin(\omega t+\phi_{0}),\cos\theta)$ exhibits precessional motion around the $z$ axis with ferromagnetic resonance (FMR) frequency $\omega=\gamma h_{0}\equiv\omega_{\text{FMR}}$, where $\gamma$ is the gyromagnetic ratio, and $h_{0}$ is an effective the magnetic field in the $z$ direction. Note that $h_{0}$ may contain a static external field, demagnetization field, and other crystalline anisotropy fields Suhl (1955); *cone-dyn2, where the effect of the external magnetic field on the qubit, if nonzero, can easily be incorporated into Eq. (2) as a spin-dependent phase shift in the normal part. The cone angle for the magnetization is determined by a transverse rf (microwave) field, $h_{\text{rf}}$, as $\theta\sim h_{\text{rf}}/\alpha h_{0}$, where $\alpha$ is the dimensionless Gilbert damping constant that parametrizes the inherent spin angular momentum losses of the magnetization dynamics.

The qubit spin, on the other hand, exhibits Rabi oscillations with frequency $\Omega\equiv\Delta\sin\theta/\hbar$ when $\omega=\Delta\cos\theta/\hbar$, which is the electron-spin-resonance (ESR) frequency. Now, matching the ESR and FMR frequencies and assuming $\theta\ll 1$, we get

$\displaystyle\Omega\approx\frac{\gamma h_{\text{rf}}}{\alpha}.$

(5)

For small damping $\alpha\ll 1$, $\Omega\gg\gamma h_{\text{rf}}$. This shows that, for a given microwave stimulus $h_{\text{rf}}$, the magnetization acts as a mediator with the bonus of spatially focusing and intensifying the external field that enhances the qubit Rabi oscillation frequency.

In contrast, when the FI dynamics are nonresonant, $\omega\ll\omega_{\text{FMR}}$, one can control phase $\phi_{0}$ to implement arbitrary single-qubit operations. In this regime, a natural spintronic way to maintain a magnetization precession at $\omega=\omega_{\text{ESR}}$ in the “conical” state is simply by maintaining an out-of-the-plane spin accumulation $\boldsymbol{\mu}$ in an attached normal metal [see Fig. 2(a)]. The normal metal with spin accumulation can exert torque $\boldsymbol{\tau}\sim{\bf m}\times(\boldsymbol{\mu}\times{\bf m}/\hbar-\partial_{t}{\bf m})$ Tserkovnyak and Ochoa (2017) on the magnet at the interface. Note that when $\boldsymbol{\mu}=0$ the resultant torque is the ordinary Gilbert damping endowed by the normal metal Tserkovnyak et al. (2002). The presence of the spin accumulation can balance the damping torque leading to coherent precession of magnetization at $\omega=\omega_{\text{ESR}}$ with $\boldsymbol{\tau}=0$. In this case, the magnitude of spin accumulation controls the frequency $\omega=|\boldsymbol{\mu}|/\hbar$, like in a Josephson relation. As a result, phase changes could be accomplished simply by short pulses of large spin accumulations, which can be within the reach of current spintronic experimental techniques Stern et al. (2008); *high-speed2. To realize the conical state, one could take simply an easy-plane magnet. In a uniaxial crystal, the anisotropy energy contributes to the magnetic free energy with a term $\sim m_{z}^{2}$, which is minimized when the spontaneous magnetization direction lies in the $xy$ plane Lifshitz (1995 - 1980). As a result, the out-of-the-plane angle can be controlled by a normal magnetic field.

As an illustrative application, one can adopt a protocol through which the qubit state accumulates only a geometric phase, i.e., the dynamic phase is zero during the whole evolution. To that end, we follow the protocol presented in Ref. Zhao et al. (2017) and show that our ASQ serves as a natural platform to implement a nonadiabatic geometric gate. In the rotating frame, when $\omega=\omega_{\text{ESR}}$, the ASQ Hamiltonian can be written as $H^{(r)}=-\frac{\hbar}{2}\boldsymbol{\Omega}\cdot\boldsymbol{\sigma}$, where $\boldsymbol{\Omega}=\Omega\,(\cos\phi_{0},\,\sin\phi_{0},\,0)$. Consider now $\boldsymbol{\Omega}$ to undergo a cycle described by the following three fixed orientations of $\boldsymbol{\Omega}$ connected by fast quenches:

	$\displaystyle\phi_{1}$		$\displaystyle=\phi_{0}-\frac{\pi}{2}\hskip 45.52458ptt_{0}<t\leq t_{1},$
	$\displaystyle\phi_{2}$		$\displaystyle=\phi_{0}+\gamma+\frac{\pi}{2}\hskip 28.45274ptt_{1}<t\leq t_{2},$
	$\displaystyle\phi_{3}$		$\displaystyle=\phi_{0}-\frac{\pi}{2}\hskip 45.52458ptt_{2}<t\leq t_{3},$		(6)

where $t_{0}\equiv 0$. As a result, the full qubit evolution operator (or the quantum gate) reads

$\displaystyle U_{g}(t_{3})$

$\displaystyle=e^{-\frac{i}{\hbar}\int_{t_{2}}^{t_{3}}H^{(r)}_{3}\,dt}e^{-\frac{i}{\hbar}\int_{t_{1}}^{t_{2}}H^{(r)}_{2}\,dt}e^{-\frac{i}{\hbar}\int_{0}^{t_{1}}H^{(r)}_{1}\,dt},$

(7)

where $H^{(r)}_{i}=-\frac{\hbar}{2}\boldsymbol{\Omega}_{i}\cdot\boldsymbol{\sigma}$ with $\boldsymbol{\Omega}_{i}=\Omega\,(\cos\phi_{i},\,\sin\phi_{i},\,0)$. Now, if we impose $\Omega t_{1}\equiv\beta$, $\Omega(t_{2}-t_{1})=\pi$, and $\Omega(t_{3}-t_{2})=\pi-\beta$, the two orthogonal states $|g_{\pm}\rangle$,

	$\displaystyle{\bf n}\cdot\boldsymbol{\sigma}|g_{\pm}\rangle$	$\displaystyle=\pm|g_{\pm}\rangle,$		(8)
	$\displaystyle{\bf n}$	$\displaystyle=(-\sin\beta\cos\phi_{0},-\sin\beta\sin\phi_{0},\cos\beta),$

undergo a particular cyclic evolution where the initial and final states are related by a purely geometric phase factor. To see this, one can check that $\langle g_{\pm}(t)|H_{i}^{(r)}|g_{\pm}(t)\rangle=0$ for $i=1,2,$ and $3$, where $|g_{\pm}(t)\rangle=U_{g}(t)|g_{\pm}\rangle$. This implies that the dynamic phase at each stage of the evolution is zero. As a result, the geometric gate is obtained as

$\displaystyle U_{g}(t_{3})$

$\displaystyle=e^{i\gamma{\bf n}\cdot\boldsymbol{\sigma}}=e^{i\gamma}|g_{+}\rangle\langle g_{+}|+e^{-i\gamma}|g_{-}\rangle\langle g_{-}|.$

(9)

In order to elucidate the geometric nature of the angle $\gamma$, we note that the expectation values of spin $\langle g_{+}(t)|\boldsymbol{\sigma}|g_{+}(t)\rangle$ at times $0,t_{1},t_{2}$, and $t_{3}$ are ${\bf n},\hat{z},-\hat{z}$, and ${\bf n}$, respectively, undergoing a cyclic evolution on the Bloch sphere where the subtended solid angle associated with the enclosed path is given by $2\gamma$ [see Fig. 2(b)]. Notice that the case of $\gamma=0$ and $\beta=\pi/2$ is equivalent to Hahn spin echo Hahn (1950); *Hahn2 with a $\pi$-pulse (in the second step). Thus, the protocol given in Eq. (V) can be considered as a generalized spin echo with a $(\pi+\gamma)$-pulse. Since the geometric phase depends on the evolution paths, quantum gates based on the geometric phases are resilient against errors in the evolution details, i.e., control errors  Zhao et al. (2017); Jones et al. (2000); *Jones2; Zhang et al. (2015).

To detect or read out the qubit state one can probe a small bias supercurrent, which requires the usage of another weakly coupled superconducting lead to the FI [see the inset in Fig. 2(c)]. Josephson junctions consisting of an FI Senapati et al. (2011) are believed to have interesting properties, such as Josephson $\pi$ state Kawabata et al. (2010); *pi2. Here, to study the ABS in an S-N-FI-S Josephson junction, we model the FI layer as a thin insulating barrier with spin-dependent parameter $Z_{\sigma}$ Tanaka and Kashiwaya (1997). As a result, right- and left-moving electrons (and holes) get a spin-dependent coupling leading to spin-polarized ABS $\varepsilon_{\sigma}$. The discrete spectrum of these bound states can be related to the scattering matrix of the normal region Beenakker (1992) as

$\displaystyle\text{Det}\Big{[}I-e^{i2\theta_{\text{A}}}r_{A}^{*}s_{e,\sigma}r_{A}s_{h,-\sigma}\Big{]}=0,$

(10)

where $I$ is a two by two identity matrix, $r_{A}=\text{diag}(e^{-i\chi/2},e^{i\chi/2})$, the scattering matrix for electrons (holes) with spin $\sigma$, $s_{e(h),\sigma}$, is given by

$\displaystyle s_{e,\sigma}=t_{\text{N}}\begin{pmatrix}r_{\sigma}t_{\text{N}}^{-1}&t_{\sigma}\\ t_{\sigma}&r_{\sigma}t_{\text{N}}\end{pmatrix}$

(11)

with $t_{\text{N}}=e^{i(\theta_{\text{N}}+k_{F}d_{N})}$, $r_{\sigma}=-iZ_{\sigma}/(1+iZ_{\sigma})$, $t_{\sigma}=1/(1+iZ_{\sigma})$, and $s_{h,\sigma}(\theta_{\text{N}})=s^{*}_{e,\sigma}(-\theta_{\text{N}})$. One can check that these Andreev levels satisfy

$\displaystyle\cos(2\theta_{\text{A}}+2\theta_{\text{N}}+\theta)=\lambda(\chi,\theta_{N}),$

(12)

where $\lambda(\chi,\theta_{N})\equiv\sqrt{R_{\uparrow}R_{\downarrow}}\cos{(2\theta_{\text{N}})}+\text{sign}(Z_{\uparrow}Z_{\downarrow})\sqrt{T_{\uparrow}T_{\downarrow}}\cos{\chi}$ with $R_{\sigma}=r^{*}_{\sigma}r_{\sigma}$ and $T_{\sigma}=t^{*}_{\sigma}t_{\sigma}$ are spin-dependent reflection and transmission probabilities due to the FI layer, $\theta=\theta_{\downarrow}-\theta_{\uparrow}$ with $\theta_{\sigma}=\arctan(1/Z_{\sigma})$ being the phase shift of the reflected electron from the N-FI interface, and $\chi$ is the phase difference between the superconducting leads. Note that for $Z_{\sigma}\gg 1$ Eq. (10) produces a similar constraint to that given in Eq. (2). Figures 2(c)-2(e) show the spectrum of ABS for different normal layer thicknesses. As a consequence of the broken spin degeneracy, the supercurrent can depend on the spin state of the occupied Andreev levels Chtchelkatchev and Nazarov (2003). For simplicity, we consider the limit of $d_{\text{N}}\rightarrow 0$, where spin-dependent supercurrent $I_{\sigma}\sim d\varepsilon_{\sigma}/d\chi$ can be written as

$\displaystyle I_{\sigma}\sim\text{sign}(Z_{\uparrow}Z_{\downarrow})\sqrt{\frac{T_{\uparrow}T_{\downarrow}}{1-\lambda^{2}}}\sin{\Big{(}\frac{\sigma\theta+\arccos{\lambda}}{2}\Big{)}}\,\sin\chi.$

(13)

Here, $\lambda\equiv\lambda(\chi,0)$ and we introduce the sign convention $\sigma=+(-)$ for spin $\uparrow(\downarrow)$. Measurement of the spin-dependent supercurrent, e.g., by coupling the supercurrent to a superconducting microwave resonator Janvier et al. (2015); Wallraff et al. (2004); Hays et al. (2020), can provide a qubit state readout.

By fabricating an array of Josephson junctions, our hybrid structure may be generalized to a multi-ASQ system, where the FI magnetizations provide a single-qubit control knob. In a layout with two qubits whose spatial separation is comparable to $\xi$, the two ABS wave functions with close energies may hybridize and induce an effective exchange interaction between qubits. One way to control this interaction could be to consider a single-channel semiconducting nanowire as the weak link so that a gate voltage might be used to tune the ABS wave functions (via affecting the transmission of the wires) and therefore manipulate the ABS hybridization Kornich et al. (2019); Padurariu and Nazarov (2012). Moreover, in SQUID loops containing these qubits, the interaction between two or multiple qubits can be realized by means of tunable inductive coupling between these SQUID loops Kafri et al. (2017); *talk2. This tunable interaction allows single- or two-qubit operations by preventing unwanted qubit crosstalk. As an immediate application of a tunable exchange interaction, the single-qubit protocol discussed in Eq. (V) can be generalized to a two-qubit geometric gate by periodic manipulation of the exchange interaction Zhang et al. (2020). In a multi-ASQ system, it would be an interesting future study to explore other spin-qubit encodings Burkard et al. (2023); Mishra et al. (2021); *Simon2; *Simon3 that further reduce coupling of the qubit to the environment by storing information in the shared spin space of many qubits.

In this article, we propose a spin qubit in an S-N-FI hybrid structure based on the spin-resolved Andreev-bound states localized in the N region. The qubit state can be coherently controlled by manipulating the FI magnetization dynamics. Our hybrid structure demonstrates the potential for spintronic methods to control the spin qubits. We show that a simple protocol for the magnetization dynamics leads to the realization of a nonadiabatic geometric gate.

This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences under Award No. DE-SC0012190.