††thanks: These two authors contributed equally to this work††thanks: These
two authors contributed equally to this work

# Coupled superconducting spin qubits with spin-orbit interaction

Maria Spethmann  Department of Physics, University of Basel, 4056 Basel,
Switzerland  Xian-Peng Zhang  Department of Physics, University of Basel, 4056
Basel, Switzerland  Jelena Klinovaja  Department of Physics, University of
Basel, 4056 Basel, Switzerland  Daniel Loss  Department of Physics, University
of Basel, 4056 Basel, Switzerland

###### Abstract

Superconducting spin qubits, also known as Andreev spin qubits, promise to
combine the benefits of superconducting qubits and spin qubits defined in
quantum dots. While most approaches to control these qubits rely on
controlling the spin degree of freedom via the supercurrent, superconducting
spin qubits can also be coupled to each other via the superconductor to
implement two-qubit quantum gates. We theoretically investigate the
interaction between superconducting spin qubits in the weak tunneling regime
and concentrate on the effect of spin-orbit interaction (SOI), which can be
large in semiconductor-based quantum dots and thereby offers an additional
tuning parameter for quantum gates. We find analytically that the effective
interaction between two superconducting spin qubits consists of Ising,
Heisenberg, and Dzyaloshinskii-Moriya interactions and can be tuned by the
superconducting phase difference, the tunnel barrier strength, or the SOI
parameters. The Josephson current becomes dependent on SOI and spin
orientations. We demonstrate that this interaction can be used for fast
controlled phase-flip gates with a fidelity >99.99%. We propose a scalable
network of superconducting spin qubits which is suitable for implementing the
surface code.

##  I Introduction

Superconducting qubits [1, 2, 3] and spin qubits [4, 5, 6, 7, 8, 9] are
promising candidates to build a quantum computer, as they can be scaled up
using the fabrication techniques of the semiconductor industry. While
superconducting qubits are experimentally more advanced, they typically have a
macroscopic size ($\sim 0.1$ mm [10]). Spin qubits, on the other hand, show
long coherence times [11] and are very compact ($\sim 50$ nm [9]), but the
small size poses a challenge to incorporate all the gates needed to control
the spins individually.

Recently, superconducting spin qubits [12, 13, 14, 15], also known as Andreev
spin qubits, have become a growing research field which combines the concepts
of superconducting qubits and spin qubits. For this, a quantum dot is placed
in a Josephson junction where the normal part hosts the subgap bound states,
the Andreev bound states [16, 17]. The odd-occupancy Andreev bound states
define a spin qubit that couples to the supercurrent [12].

While superconducting spin qubits can be coupled to each other through the
superconducting current (“inductive coupling”) [12], another way of coupling
is by effective wave-function overlap. When two quantum dots couple to the
same superconductor at a distance smaller than the superconducting coherence
length ($\sim 0.7$ $\mu$m [18]), spins located on these dots interact
isotropically via Cooper pairs that split [19, 20, 21, 22, 23]. This length
scale is significantly longer than the direct exchange of regular spin qubits
($\sim 50$ nm [9]), which makes it easier to add control and readout
components. Furthermore, coupling through overlap [24] is tunable by the
superconducting phase difference [19] and may allow for a more complex qubit
connectivity, as discussed below. Importantly, in contrast to inductive
coupling, which is quadratic in spin-orbit interaction (SOI) strength [13],
our spin coupling does not vanish in the absence of the SOI. Experimentally,
directly overlapping Andreev bound states have been demonstrated in
experiments on so-called Andreev molecules [25, 26, 27, 28], where the
coupling is so strong that two Andreev bound states hybridize [29, 30]. The
coupling mechanism is based on crossed Andreev processes, very similar to
Cooper pair splitters [31, 32, 33, 34, 35, 36, 37]. These recent experiments
provide strong motivation for the setup proposed here.

The main aspect we wish to analyze here is how SOI [38] affects the exchange
interaction between superconducting spin qubits. SOI is a desired property of
the quantum dot: superconducting spin qubits in their initial proposal require
SOI to control the spin degree of freedom via the supercurrent [12, 13, 15,
39]. But also regular spin qubits need SOI to perform single-qubit rotations
via electric dipole spin resonance (EDSR) [40, 6], which allows for all-
electrical control. In any case, SOI will likely be present in semiconductor
devices and can even be tuned with gates [41, 42, 43, 44, 45, 46, 47, 48, 49,
50, 51, 52], adding a further control parameter. We here want to answer the
following questions: How does the SOI affect the effective interaction between
superconducting spin qubits? Can this effect be used to implement efficient
quantum gates?

We derive a low-energy spin Hamiltonian and find that the SOI induces some
“twists” in the otherwise isotropic exchange interaction, which results in not
only Heisenberg, Ising, and Dzyaloshinskii-Moriya (DM) interactions [53, 54]
(similar to Refs. [55, 56]), but also effective staggered magnetic fields. By
tuning the SOI strength and the superconducting phase difference, we can
either undo the twist or tune the exchange interaction to be of pure Ising
type, which allows one to realize a fast controlled phase-flip gate for two
qubits. We analyze the second possibility in more detail and find that we only
need to fine-tune one of the parameters (either the superconducting phase or
the SOI strength) to reach a high gate fidelity. Finally, we sketch how to
arrange the superconducting spin qubits in a two-dimensional scalable array,
suitable for the surface code [57].

We organize the remainder of this paper as follows. In Sec. II we introduce
our theoretical model of the quantum dots and the superconductors and derive
the spin-spin interaction between the spin qubits. In Sec. III we show how
this interaction can be used to realize a fast controlled phase-flip gate and
we calculate the gate fidelity. In Sec. IV we outline a scalable design of a
qubit network. Section V summarizes our findings. Finally, Appendix A gives
the details of the perturbation theory, Appendix B shows the results
accounting for an Aharonov-Bohm flux, and Appendix C presents the exact
expressions of the Josephson supercurrent.

Figure 1:  (a) Our model describes two spin-1/2 quantum-dot states (violet)
that are weakly tunnel coupled ($t_{jn}$) to two superconducting leads (green)
such that they form superconducting spin qubits. Due to the presence of SOI,
the tunneling is not spin conserving. (b) The superconducting spin qubits can
also interact with each other via one superconductor only. (c-f) Virtual
tunneling paths that contribute to the spin interaction given in Eq. (6).
Black arrows show the virtual tunneling paths of the electrons. The
contributions from panel (e) depend on the superconducting phase difference
$\varphi$. (g) We propose a scalable architecture of superconducting spin
qubits. The setup consists of parallel nanowires (gray), orthogonal
superconducting stripes (green), and a crossbar design of gates (pink) that
form quantum dots in the nanowires and control the strength of tunnel barriers
separating them from superconductors. When corresponding tunnel barriers are
lowered, the superconducting spin qubits interact pairwise with each other,
see panel (a).

##  II Model and theory

We describe a model of two quantum dots in double nanowires [36, 27, 58, 59,
37] that are tunnel coupled to two superconducting leads, as shown in Fig.
1(a). We consider the limit of weak tunnel coupling, large on-site Coulomb
repulsion, and negative dot detuning from the Fermi energy, such that in the
ground state the dot is occupied by one electron. This parameter choice
describes a superconducting spin qubit [16]111The fixed dot filling also
suppresses quasiparticle poisoning, thus stabilizing the superconducting spin
qubit [99]. (Superconducting spin qubits also exist at different parameter
regimes and in atomic break junctions [61, 62, 63].)

The total Hamiltonian consists of three parts,
$H_{\text{tot}}=H_{D}+H_{L}+H_{T}$. The dot Hamiltonian is

$H_{D}=\sum_{sn}\epsilon
d^{\dagger}_{ns}d_{ns}+C_{d}d^{\dagger}_{n\uparrow}d_{n\uparrow}d^{\dagger}_{n\downarrow}d_{n\downarrow},$
(1)

where $d^{\dagger}_{ns}$ creates an electron on the dot $n\in\\{1,2\\}$ with
spin $s\in\\{\uparrow,\downarrow\\}$ and energy $\epsilon<0$, which is equal
on both dots and tuned by electrostatic gates. Further, we have an on-site
Coulomb repulsion $C_{d}$. We assume a small dot size such that we only take
into account one dot energy level.

The leads are conventional singlet superconductors and are described by the
$s$-wave BCS Hamiltonian:

$H_{L}=\sum_{j\boldsymbol{k}s}\xi_{k}c_{j\boldsymbol{k}s}^{\dagger}c_{j\boldsymbol{k}s}-\sum_{j\boldsymbol{k}}(\Delta_{j}c_{j\boldsymbol{k}\uparrow}^{\dagger}c^{\dagger}_{j\,\shortminus\boldsymbol{k}\downarrow}+\text{H.c.}).$
(2)

Here, $c_{j\boldsymbol{k}s}^{\dagger}$ is the creation operator of lead
$j\in\\{L,R\\}$ with spin $s$, wave vector $\boldsymbol{k}$, and energy
$\xi_{k}$. The pairing potential $\Delta_{j}=\Delta e^{-i\varphi_{j}}$ is
characterized by the superconducting gap $\Delta$ and phase $\varphi_{j}$. The
phase difference between the superconductors is
$\varphi=\varphi_{L}-\varphi_{R}$.

The leads and the dots are tunnel coupled:

$\displaystyle H_{T}$
$\displaystyle=\sum_{jn\boldsymbol{k}}t_{jn}c^{\dagger}_{j\boldsymbol{k}}U_{jn}d_{n}+\text{H.c.},$
(3)

with $d_{n}=\begin{pmatrix}d_{n\uparrow}\\\ d_{n\downarrow}\end{pmatrix}$,
$c_{j\boldsymbol{k}}=\begin{pmatrix}c_{j\boldsymbol{k}\uparrow}\\\
c_{j\boldsymbol{k}\downarrow}\end{pmatrix}$, positive tunnel amplitude
$t_{jn}$, and unitary matrix $U_{jn}\in\text{SU}(2)$. The SOI in
semiconducting nanowires can be strong [64, 43, 49] and causes spin rotation
[65, 66], which is described by the rotation matrix in spin space $U_{jn}$
[67, 38, 68]. We expect this spin rotation matrix to also describe the spin-
orbit mixing in multiorbital dots. Our spin then is the pseudospin of the low-
energy states, and on-site spin-orbit mixing just renormalizes our
phenomenological rotation matrix [69].

Next we want to find the effective spin coupling between the dots. Since at
least four tunneling processes are needed for spin interaction, we perform a
standard fourth-order perturbation expansion in $H_{T}$ [19, 70]. It is valid
in the weak-coupling regime $t_{jn},\Gamma_{j}\ll\Delta,|\epsilon|$ [19, 71],
with $\Gamma_{j}=\pi\rho_{F}t_{j1}t_{j2}$ where $\rho_{F}$ is the normal
density of states per spin of the leads at the Fermi energy. We also work in
the Coulomb blockade regime by assuming that the Coulomb repulsion $C_{d}$ on
the dot is the largest energy scale of the system and forbids double
occupation. The influence of finite Coulomb interaction will be discussed
elsewhere [72]. We start with a basis transformation of operators defined for
the superconducting leads and the second dot:

$\displaystyle\begin{pmatrix}\tilde{c}_{R\boldsymbol{k}\uparrow}\\\
\tilde{c}_{R\boldsymbol{k}\downarrow}\end{pmatrix}$
$\displaystyle=U^{\dagger}_{R1}\begin{pmatrix}c_{R\boldsymbol{k}\uparrow}\\\
c_{R\boldsymbol{k}\downarrow}\end{pmatrix},$
$\displaystyle\begin{pmatrix}\tilde{d}_{2\uparrow}\\\
\tilde{d}_{2\downarrow}\end{pmatrix}$
$\displaystyle=U_{R1}^{\dagger}U_{R2}\begin{pmatrix}d_{2\uparrow}\\\
d_{2\downarrow}\end{pmatrix},$
$\displaystyle\begin{pmatrix}\tilde{c}_{L\boldsymbol{k}\uparrow}\\\
\tilde{c}_{L\boldsymbol{k}\downarrow}\end{pmatrix}$
$\displaystyle=U^{\dagger}_{R1}U_{R2}U^{\dagger}_{L2}\begin{pmatrix}c_{L\boldsymbol{k}\uparrow}\\\
c_{L\boldsymbol{k}\downarrow}\end{pmatrix}.$ (4)

Consequently, in the rotated basis, the electron spin could flip only during
the tunneling between dot $1$ and lead $L$, captured by
$U=U^{\dagger}_{R1}U_{R2}U^{\dagger}_{L2}U_{L1}$. Hereafter, we parametrize
the total SOI effects by the angle $\alpha$ and by the unitary vector
$\mathbf{u}$ defined via $U\equiv e^{i\alpha\mathbf{u}\cdot\mathbf{S}_{1}}$,
which eventually acts on the spin $\mathbf{S}_{1}$ on dot 1 (see details in
Appendix A).

We now sum up all virtual paths with four tunneling processes that contribute
to a spin interaction between the dots [see Figs. 1(c)-1(f) and Appendix A].
Without SOI the spin interaction is isotropic
$\propto\mathbf{S}_{1}\cdot\mathbf{S}_{2}$ [19]. Thus the virtual paths in
which electrons tunnel only to the right [see Fig. 1(d)] or left [see Fig.
1(c)] lead cause twisted spin exchange [73, 74, 75, 76, 55, 77, 78],
$\mathbf{S}_{1}\cdot\tilde{\mathbf{S}}_{2}$ and
$(U^{\dagger}\mathbf{S}_{1}U)\cdot\tilde{\mathbf{S}}_{2}$, respectively. Here
$\tilde{\mathbf{S}}_{2}=U^{\dagger}_{R}\mathbf{S}_{2}U_{R}$ is the spin
operator in the rotated basis given by Eq. (II), where
$U_{R}=U_{R1}^{\dagger}U_{R2}$. We note that one can always eliminate these
twisting effects with single-qubit rotations. The “two-lead paths” generate an
anomalous twisted spin exchange
$e^{i\phi}(U^{\dagger}S^{\nu}_{1}\mathbb{1})\tilde{S}^{\nu}_{2}+\text{H.c.}$
with $S_{n}^{\nu}\in\\{S_{n}^{o},S_{n}^{x},S_{n}^{y},S_{n}^{z}\\}$, where
$S_{n}^{o}=\frac{1}{2}$ and $\phi=\varphi$ [$\phi=0$] correspond to processes
shown in Fig. 1(e) [Fig. 1(f)]. By “anomalous” we mean the left
($U^{\dagger}$) and right ($\mathbb{1}$) twisting matrix of $S^{\nu}_{1}$ are
different, and one cannot eliminate the twisting effect by rotating the spin
basis. We sum up the “left-lead,” “right-lead,” and “two-lead” paths and get
for $\Gamma_{L}=\Gamma_{R}=\Gamma$ and $\Delta\gg|\epsilon|$ 222In this limit,
the “two-lead” path shown in Fig. 1(f) can be neglected, see Appendix A. the
low-energy Hamiltonian

$\displaystyle H$
$\displaystyle=J\mathbf{S}_{1}\\!\cdot\tilde{\mathbf{S}}_{2}+J\left(e^{-i\alpha\mathbf{u}\cdot\mathbf{S}_{1}}\mathbf{S}_{1}e^{i\alpha\mathbf{u}\cdot\mathbf{S}_{1}}\right)\cdot\tilde{\mathbf{S}}_{2}$
(5)
$\displaystyle+2J\cos\varphi\left[\cos\\!\left(\textstyle\frac{\alpha}{2}\right)\,\mathbf{S}_{1}\\!\cdot\tilde{\mathbf{S}}_{2}-\sin\\!\left(\textstyle\frac{\alpha}{2}\right)\mathbf{u}\cdot(\mathbf{S}_{1}\\!\times\tilde{\mathbf{S}}_{2})\right]$
$\displaystyle+J\sin\varphi\sin\left(\textstyle\frac{\alpha}{2}\right)\,\mathbf{u}\cdot\left(\mathbf{S}_{1}-\tilde{\mathbf{S}}_{2}\right),$

with $J\approx\Gamma^{2}/|\epsilon|$. (In Appendix B we present the result
without the restrictions $\Gamma_{L}=\Gamma_{R}$ and $\Delta\gg\epsilon$.) For
$\varphi\neq 0$ or $\pi$, the time-reversal symmetry is broken, and anomalous
twisted spin-exchange results in effective staggered magnetic fields [see the
third line of Eq. (5)]. Here, we dropped a spin-irrelevant function of
$\varphi$. From Eq. (5) we already see a few important features. For the
special case $U\to\mathbb{1}$ and $U_{R}\to\mathbb{1}$, $H$ reduces into an
isotropic Heisenberg-like interaction [19]. For $U\to\mathbb{1}$ and
$U_{R}\neq\mathbb{1}$, $H$ reduces to an ordinary twisted spin exchange [80,
55, 81] consistent with Ref. [23]. For $U\neq\mathbb{1}$ the interaction is
nontrivial. Defining parallel and orthogonal components of a vector as
$x^{\|}\\!=\\!\boldsymbol{u}\\!\cdot\\!\boldsymbol{x}$ and
$\boldsymbol{x}^{\perp}=\boldsymbol{x}-x^{\|}\boldsymbol{u}$, $H$ can be
rewritten as

$H=h(S^{\|}_{1}-\tilde{S}^{\|}_{2})+J_{\|}S^{\|}_{1}\tilde{S}^{\|}_{2}+J_{\perp}\mathbf{S}^{\perp}_{1}\cdot\tilde{\mathbf{S}}^{\perp}_{2}+J_{\text{DM}}(\mathbf{S}_{1}\times\tilde{\mathbf{S}}_{2})^{\|},$
(6)

where staggered magnetic field and spin-exchange constants, respectively, are
given by

$\displaystyle h=J\sin\varphi\sin\textstyle\frac{\alpha}{2},$ (7a)
$\displaystyle J_{\|}=2J(1+\cos\varphi\cos\textstyle\frac{\alpha}{2}),$ (7b)
$\displaystyle
J_{\perp}=2J\cos\textstyle\frac{\alpha}{2}(\cos\varphi+\cos\frac{\alpha}{2}),$
(7c) $\displaystyle
J_{\text{DM}}=-2J\sin\textstyle\frac{\alpha}{2}(\cos\varphi+\cos\frac{\alpha}{2}).$
(7d)

The anisotropy $J_{\|}-J_{\perp}$ and the DM interaction $J_{\text{DM}}$ are
tuned by the SOI parameter $\alpha$, the lead-dot coupling $\Gamma_{j}$, and
the superconducting phase difference $\varphi$. Both $\alpha$ [43] and
$\Gamma_{j}$ [82] can be tuned by the electrostatic gates, while a magnetic
flux can tune $\varphi$ when the superconductors form a loop [83, 84]. To get
a better understanding of the spin interaction we plot $J_{\perp}/J_{\|}$ and
$J_{\text{DM}}/J_{\|}$ as a function of $\varphi$ for different $\alpha$ in
Figs. 2(a) and 2(b). For $\alpha=0$, we reproduce the isotropic interaction
limit. The perpendicular component $J_{\perp}/J_{\|}$ disappears at
$\alpha=\pi$, and one obtains maximum modulation of the DM interaction.
Importantly, the interaction takes on pure Ising form for
$\varphi=\pi\pm\alpha/2$ (for $\varphi\neq 0$ or $\pi$).

Our Hamiltonian $H$ is invariant under the spin rotation around the axis
$\mathbf{u}$, and hence $C=S^{\|}_{1}+\tilde{S}^{\|}_{2}$ is a conserved
quantum number such that the setup is also suitable for singlet-triplet qubits
[85, 86]. Furthermore, our effective Hamiltonian is valid for arbitrary
strengths of the SOI $\alpha$, in contrast to the inductive coupling in Ref.
[13], which only considers weak SOI and vanishes as $\alpha^{2}$ $\to 0$.

We note that the SOI induces a spin-dependent phase shift in the Josephson
current, $\hat{I}=\partial_{\varphi}H$, given by

$\displaystyle\hat{I}=\hat{I}_{s}\sin\varphi+J\sin\textstyle(\frac{\alpha}{2})(S^{\|}_{1}-\tilde{S}^{\|}_{2})\cos\varphi,$
(8)
$\displaystyle\frac{\hat{I}_{s}}{2J}=\cos\textstyle(\frac{\alpha}{2})\left(\frac{1}{4}-\mathbf{S}_{1}\cdot\tilde{\mathbf{S}}_{2}\right)+\sin(\frac{\alpha}{2})(\mathbf{S}_{1}\times\tilde{\mathbf{S}}_{2})^{\|}.$
(9)

The $\cos\varphi$ term enables detection of SOI and the relative spin
orientations of the dots, see Appendix C.

Figure 2: The anisotropy of the spin exchange [Eq. (6)], $J_{\perp}/J_{\|}$,
and the DM interaction, $J_{DM}/J_{\|}$, can be tuned by varying the
superconducting phase difference $\varphi$ and the SOI strength $\alpha$. If
$\varphi=\pi\pm\alpha/2$ ($\alpha\neq 0$), the interaction is of pure Ising
type. We note that $\alpha$ can be varied while keeping the rotated spin basis
on the second dot $\tilde{S}_{2}$ unchanged, e.g. by modulating $U_{Ln}$.

##  III Controlled phase flip gate

We can use the interaction defined in Eq. (6) to realize a controlled phase-
flip gate for two qubits. For $\varphi=\pi\pm\alpha/2$ (for $\varphi\neq 0$ or
$\pi$), we get an Ising interaction

$\displaystyle
2J\sin^{2}(\alpha/2)[S_{1}^{z}S_{2}^{z}\mp\left(\textstyle\frac{1}{2}S_{1}^{z}-\frac{1}{2}S_{2}^{z}\right)],$
(10)

where we choose our quantization $z$ axis along ${\bf u}$. The Ising
interaction enables a controlled phase-flip gate
$U_{\text{CPF}}\equiv\text{diag}(1,1,1,-1)=U(\alpha,\varphi=\pi\pm\frac{\alpha}{2})$
via the evolution [4], defined for $\alpha\neq 0$, as

$\displaystyle U(\alpha,\varphi)=e^{-i\scriptstyle\frac{\pi}{4}}e^{i\pi
S_{n(\varphi)}^{z}}e^{{\scriptstyle\frac{-i\pi}{J_{\|}(\alpha,\varphi)}}H(\alpha,\varphi)},$
(11)

where $n(\varphi)=1$ for $0<\varphi<\pi$ and $n(\varphi)=2$ for
$\pi<\varphi<2\pi$. The controlled phase-flip gate is sufficient for universal
quantum computing, as it is related to the CNOT gate by single-qubit rotations
[4] (single-qubit gates to be explained below). The advantage of using the
Ising interaction instead of the isotropic interaction
$\propto\mathbf{S}_{1}\cdot\mathbf{S}_{2}$ is that the latter requires two
interaction gates, for example, via two square-root-of-SWAP gates [4], while
the Ising interaction only requires one interaction gate. This allows faster
switching times and decreases the error probability of the gate [87]. If we
consider an aluminum superconductor with $\Delta\approx 0.1$ meV [88] and tune
the dot energy to $\epsilon\approx-10\,\mu$eV and $\Gamma\approx 1\,\mu$eV,
the interaction strength becomes $J\approx 0.1\,\mu$eV, which corresponds to a
switching time of $t=\frac{\hbar\pi}{J_{\|}(\alpha,\varphi)}\approx 10$ ns for
$\alpha=\pi$, $\varphi=\frac{\pi}{2}$. This time is more than 1000 times
smaller than the lifetime of a trapped quasiparticle [63, 89, 15]. As noted
above, the interaction becomes Ising-like for $\varphi=\pi\pm\alpha/2$, which
means that it is sufficient to tune either $\varphi$ or $\alpha$, but not
both. This becomes clear when we calculate the fidelity [90]

$F=\left|\frac{1}{4}\textstyle\text{tr}\,U^{\dagger}_{\text{CPF}}U(\alpha,\varphi)\right|^{2},$
(12)

where the parameters in $U(\alpha,\varphi)$ vary around their ideal values. In
Fig. 3 we show the infidelity $1-F$ as a function of $\alpha$ and $\varphi$.
We average over some statistical Gaussian noise of the switching time, the
tunnel ratio $r=\frac{\Gamma_{L}}{\Gamma_{R}}$ (see Appendix B) and the pulse
time of the single-qubit $z$ rotation. By tuning either $\alpha$ or $\varphi$,
we get a fidelity $>99.99\%$.

Figure 3: Infidelity $1-F$ of the controlled phase-flip gate
$U(\alpha,\varphi)$ for varying SOI strength $\alpha$ and superconducting
phase difference $\varphi$ [see Eqs. (11) and (12)]. Only one parameter needs
to be fine-tuned to reach a fidelity $>99.99\%$. We average over statistical
Gaussian noise of switching time
$t=\frac{\hbar\pi}{J_{\|}(\alpha,\varphi)}(1\pm 0.003)$, tunnel ratio
$\frac{\Gamma_{L}}{\Gamma_{R}}=1\pm 0.005$, and $z$-rotation pulse time
$t_{z}=\frac{3\pi\hbar}{h_{z,1}}(1\pm 0.001)$ [with magnetic field $h_{z,1}$,
Eq. (13)], see Appendix B.

##  IV Scalable network

Based on the results from the preceding sections, we next propose a scalable
network of superconducting spin qubits, see Fig. 1(g). The network consists of
parallel nanowires in $x$ direction and orthogonal superconducting stripes in
$z$ direction. A crossbar design [91] of gates confines quantum dots in the
nanowires, controls the tunnel barriers between the dots and the
superconductors, and allows one to address qubits individually. In summary,
the total Hamiltonian of the network is

$H_{\text{net}}=\sum_{ij}H_{ij}+\sum_{i}(h_{x,i}S_{x,i}+h_{z,i}S_{z,i}),$ (13)

where $H_{ij}(\alpha_{ij},\varphi_{ij},\Gamma_{L,ij},\Gamma_{R,ij})$ describes
the interaction between dot $i$ and dot $j$ [see Eq. (6)], and $h_{x/z,i}$ is
the global/effective magnetic field of dot $i$ (including the Bohr magneton
and the corresponding $g$ factors).

Scalable single-qubit gates are implemented in the usual way as for regular
spin qubits [91]. A small global magnetic field in $x$ direction splits the
spin states. The spins can now be flipped via EDSR [92, 93, 94, 49]. An
electric microwave pulse is sent to the tunnel barrier gates: due to SOI the
spins experience an effective magnetic field in the $z$ direction. We assume
that the global magnetic field is small enough not to affect the two-qubit
gates.

For two-qubit gates there are two possibilities in our scheme, depending on
the relative position of the dots. Both possibilities have in common that the
superconducting spin qubits start to interact if barriers are lowered. If two
neighboring quantum dots are positioned on the same nanowire [as in Fig.
1(b)], they only interact through one superconductor and the interaction will
be an “ordinary twisted” interaction. As discussed in the previous section and
in Ref. [23], this means the interaction can be converted into an isotropic
interaction by additional single-qubit rotations or by tuning the spin-orbit
strength with the gates. This allows one to perform two square-root-of-SWAP
gates, equivalent to a CNOT gate [4, 87]. For two spins on neighboring
nanowires [as in Fig. 1(a)] the exchange interaction between them will be
“anomalously twisted”. As discussed in the previous section, this allows one
to perform a controlled phase-flip gate, which is related to a CNOT gate via
single-qubit rotations. The advantage of this interaction is that it is
tunable by the superconducting phase difference and only requires one
interaction gate.

The two-dimensional setup allows error correction codes such as the surface
code [57]. Furthermore, in principle this setup allows for increased qubit
connectivity: When the distance between two dots is within the superconducting
coherence length, also coupling beyond nearest neighbor is possible. Finally,
undesired capacitive coupling between the quantum dots and the superconductors
can be suppressed by detuning the resonant frequency of the superconductor and
the qubit frequency.

##  V Conclusion

We have found an analytic expression of the pairwise interaction of
superconducting spin qubits in the presence of SOI. The interaction consists
of Ising, Heisenberg, and DM interactions and is tunable by the
superconducting phase difference, the spin-orbit parameters, and the tunnel
coupling. We can realize a controlled phase-flip gate and show that a high
fidelity can be reached without the requirement that both the superconducting
phase difference and the SOI parameters are fine-tuned. Based on these
results, we propose a scalable network of superconducting spin qubits where
single-qubit rotations are performed with EDSR and two-qubit gates rely on the
spin interaction we have found. Given the fast progress in the field [15, 27,
28, 58, 95], we believe that our proposal is within experimental reach.

## Acknowledgements

We thank Bence Hetényi and Stefano Bosco for helpful discussions. This project
has received funding from the European Union’s Horizon 2020 Research and
Innovation Program under Grant Agreement No. 862046 and under Grant Agreement
No. 757725 (the ERC Starting Grant). This work was supported by the Swiss
National Science Foundation, NCCR QSIT, and NCCR SPIN, a National Centre of
Competence (or Excellence) in Research, funded by the Swiss National Science
Foundation (Grant No. 51NF40-180604).

##  Appendix A Details of the perturbation theory

In this section we show the details of the perturbation theory that lead to
the spin interaction [see Eq. (5) of the main text]. Our Hamiltonian is given
by Eqs. (1), (2), and (3). First, we perform three subsequent basis
transformations, defined by Eq. (II), which do not change the Hamiltonian of
the dots or superconductors because they are assumed to be rotationally
invariant.We also diagonalize the Hamiltonian of the superconductors via a
Bogoliubov transformation
($\tilde{c}_{j\boldsymbol{k}s}=u_{k}\gamma_{j\boldsymbol{k}s}+sv_{jk}\gamma_{j\shortminus\boldsymbol{k}\shortminus
s}^{\dagger}$, with $u_{k}=\sqrt{\frac{E_{k}+\xi_{k}}{2E_{k}}}$,
$v_{jk}=\sqrt{\frac{E_{k}-\xi_{k}}{2E_{k}}}e^{-i\varphi_{j}}$,
$E_{k}=\sqrt{({\xi_{k}})^{2}+\Delta^{2}}\,$):

$\displaystyle
H_{L}=\sum_{j\boldsymbol{k}\sigma}E_{k}\gamma_{j\boldsymbol{k}\sigma}^{\dagger}\gamma_{j\boldsymbol{k}\sigma}.$
(1)

Excitations from the superconducting ground state are described by Bogoliubov
quasiparticles ($\gamma_{j\boldsymbol{k}s}^{\dagger}$). The tunnel Hamiltonian
becomes (with $\tilde{d}_{1\sigma}=d_{1\sigma}$)

$\displaystyle H_{T}=$
$\displaystyle\sum_{\begin{subarray}{c}jn\boldsymbol{k}s\\\
(j,n)\neq(L,1)\end{subarray}}\left[t_{jn}(u_{k}\gamma_{j\boldsymbol{k}s}^{\dagger}+sv_{jk}^{*}\gamma_{j\shortminus\boldsymbol{k}\shortminus
s})\tilde{d}_{ns}+H.c.\right]$
$\displaystyle+\sum_{\boldsymbol{k}ss^{\prime}}\left[t_{L1}(u_{k}\gamma_{Lks}^{\dagger}+sv_{Lk}^{*}\gamma_{L\shortminus\boldsymbol{k}\shortminus
s})U_{ss^{\prime}}\tilde{d}_{1s^{\prime}}+H.c.\right].$ (2)

We see that the total unitary basis transformation
$U=U_{R1}^{\dagger}U_{R2}U_{L2}^{\dagger}U_{L1}$ now only acts on the spins of
electrons that tunnel between the left superconductor and dot 1.

We now want to calculate the spin-exchange interaction between the dots
mediated by the Cooper pairs in the superconductors. For this, we evaluate the
fourth-order perturbation theory contribution in the weak-coupling limit
$t_{jn},\Gamma_{j}\ll\Delta,|\epsilon|$ [19, 71], with
$\Gamma_{j}=\pi\rho_{F}t_{j1}t_{j2}$ and $\rho_{F}$ being the normal density
of states per spin of the leads at the Fermi energy:

$\displaystyle H=PH_{T}\left(\frac{1-P}{E_{0}-H_{0}}H_{T}\right)^{3}P.$ (3)

Here, the unperturbed Hamiltonian is $H_{0}=H_{L}+H_{D}$ with the ground-state
energy $E_{0}$. The operator $P$ projects to the spin-1/2 low-energy subspace
$\\{|s_{1},s_{2};0\rangle\\}_{s_{1}s_{2}}$, where both dots are occupied with
one electron with spins $s_{1}$ and $s_{2}$ (in the $d_{1s}$,
$\tilde{d}_{2s}$-basis), and both superconductors are in their ground state.
In other words, we evaluate $\langle
s^{\prime}_{1},s^{\prime}_{2};0|H_{\text{eff}}|s_{1},s_{2};0\rangle$, by
summing up virtual tunneling paths. We take the perturbation theory to fourth
order because it is the smallest order at which spin interaction between the
two quantum dots is possible. The first $H_{T}$ will always destroy one
electron on one of the dots (because in the $C_{d}\rightarrow\infty$ limit we
do not allow double occupancy) and create one Bogoliubov quasiparticle in the
superconductors. Thus the first virtual intermediate state will be
$|s_{1},0;\gamma_{j\boldsymbol{k}s}\rangle$ or
$|0,s_{2};\gamma_{j\boldsymbol{k}s}\rangle$. Next, in the limit
$\Delta\gg|\epsilon|$, the electron on the other dot will tunnel to the
superconductor and destroy the Bogoliubov particle again, in total creating a
Cooper pair. The second virtual intermediate state is then $|0,0;0\rangle$.
This process will dominate, as it costs more energy to create a second
Bogoliubov quasiparticle than to remove an electron from the dot. The third
virtual intermediate state involves a Bogoliubov quasiparticle and an
electron, similar to the first virtual intermediate state, and after the
fourth tunneling we get back to our ground state.

For virtual paths that involve the right superconductor only [see Fig. 1(d) of
the main text] one can permute the order of the first two tunneling events and
of the third and fourth tunneling events, giving a total of four virtual
tunneling paths. It turns out that each of them gives the same contribution to
$\langle s^{\prime}_{1},s^{\prime}_{2};0|H_{\text{eff}}|s_{1},s_{2};0\rangle$,
namely,

$\displaystyle
t_{R1}^{2}t_{R2}^{2}\sum_{\boldsymbol{k},\boldsymbol{q}}\frac{u_{k}v^{*}_{Rk}u_{q}v_{Rq}}{(E_{k}-\epsilon)2\epsilon(E_{q}-\epsilon)}s_{1}s_{1}^{\prime}\delta_{s_{1},-s_{2}}\delta_{s_{1}^{\prime},-s_{2}^{\prime}}.$
(4)

We evaluate

$\displaystyle\sum_{\boldsymbol{k},\boldsymbol{q}}$
$\displaystyle\textstyle\frac{|u_{k}v_{Rk}u_{q}v_{Rq}|}{(E_{k}-\epsilon)2\epsilon(E_{q}-\epsilon)}$
(5)
$\displaystyle={\displaystyle\frac{1}{8\epsilon}}\Bigg{(}\sum\limits_{\boldsymbol{k}}\textstyle\frac{1}{\sqrt{\left(\frac{\xi_{k}}{\Delta}\right)^{2}+1}\left(\sqrt{\left(\frac{\xi_{k}}{\Delta}\right)^{2}+1}-\frac{\epsilon}{\Delta}\right)\Delta}\Bigg{)}^{2}$
$\displaystyle={\displaystyle\frac{1}{8\epsilon}}\Bigg{[}\int\limits_{-\Lambda}^{\Lambda}\textstyle\frac{\rho(\xi)}{\sqrt{\left(\frac{\xi}{\Delta}\right)^{2}+1}\left(\sqrt{\left(\frac{\xi}{\Delta}\right)^{2}+1}-\frac{\epsilon}{\Delta}\right)\Delta}\text{d}\xi\Bigg{]}^{2}$
$\displaystyle\approx\frac{\rho_{F}^{2}}{8\epsilon}\Bigg{[}\int\limits_{-\infty}^{\infty}\textstyle\frac{\text{d}x}{\sqrt{x^{2}+1}(\sqrt{x^{2}+1}-\frac{\epsilon}{\Delta})}\Bigg{]}^{2}\overset{\Delta\gg\epsilon}{\approx}-\displaystyle\frac{\rho_{F}^{2}\pi^{2}}{8|\epsilon|}.$

Here, $\Lambda$ is some cut-off energy (of the order of the Fermi energy) and
$\rho(\xi)$ is the density of states per spin, which we assume to be
approximately constant $\rho(\xi)\approx\rho(0)=\rho_{F}$ in the region around
the Fermi energy, which contributes most to the integral. Further, we can
replace
$s_{1}s_{1}^{\prime}\delta_{s_{1},-s_{2}}\delta_{s_{1}^{\prime},-s_{2}^{\prime}}\to-2\boldsymbol{S}_{1}\cdot\tilde{\boldsymbol{S}}_{2}-\frac{1}{2}$,
where the $-\frac{1}{2}$ is an irrelevant constant. The operator
$\boldsymbol{S}_{1}$ is the spin operator of dot $n=1$ and
$\tilde{\boldsymbol{S}}_{2}$ is the spin operator of dot $n=2$ in the rotated
basis [see Eq. (II)]:
$\tilde{\mathbf{S}}_{2}=U_{R2}U^{\dagger}_{R1}\mathbf{S}_{2}U_{R1}^{\dagger}U_{R2}$.
All virtual paths corresponding to processes shown in Fig. 1(d) together give

$\displaystyle\frac{\rho_{F}^{2}\pi^{2}}{|\epsilon|}t_{R1}^{2}t_{R2}^{2}\boldsymbol{S}_{1}\cdot\tilde{\boldsymbol{S}}_{2}.$
(6)

For virtual paths involving the left superconductor only [see Fig. 1(c)] we
can use the result for the right lead and simply rotate the spin operator of
the upper dots by $U$, as the “left-lead-paths” and “right-lead-paths” are
equivalent up to a basis rotation:

$\displaystyle\frac{\rho_{F}^{2}\pi^{2}}{|\epsilon|}t_{L1}^{2}t_{L2}^{2}(U^{\dagger}\boldsymbol{S}_{1}U)\cdot\tilde{\boldsymbol{S}}_{2}.$
(7)

For virtual paths where a Cooper pair transfers from the left to the right
superconductor [see Fig. 1(e)], there exist four permutations of tunneling
paths. Each of them contributes with

$\displaystyle-\frac{\rho_{F}^{2}\pi^{2}}{8|\epsilon|}t_{L1}t_{L2}t_{R1}t_{R2}e^{-i\varphi}(-1)s_{1}s_{2}^{\prime}\delta_{s_{1}-s_{2}}U^{*}_{\shortminus
s_{2}^{\prime}s_{1}^{\prime}}.$ (8)

Using that

$\displaystyle U$
$\displaystyle=\textstyle\cos(\frac{\alpha}{2})+2i\boldsymbol{u}\cdot\boldsymbol{S}_{1}\sin(\frac{\alpha}{2})$
(9)
$\displaystyle=\begin{pmatrix}\cos{(\frac{\alpha}{2})}+iu_{z}\sin{(\frac{\alpha}{2})}&(u_{y}+iu_{x})\sin{(\frac{\alpha}{2})}\\\
(-u_{y}+iu_{x})\sin{(\frac{\alpha}{2})}&\cos{(\frac{\alpha}{2})}-iu_{z}\sin{(\frac{\alpha}{2})}\end{pmatrix},$

we simplify

$\displaystyle s_{1}s_{2}^{\prime}\delta_{s_{2}-s_{1}}U^{*}_{\shortminus
s_{2}^{\prime}s_{1}^{\prime}}\to\,\,$ $\displaystyle\textstyle
2\cos(\frac{\alpha}{2})(\boldsymbol{S}_{1}\cdot\tilde{\boldsymbol{S}}_{2}-\frac{1}{4})+u_{z}\sin(\frac{\alpha}{2})(2S_{1y}\tilde{S}_{2x}-2S_{1x}\tilde{S}_{2y}+iS_{1z}-i\tilde{S}_{2z})$
(10)
$\displaystyle+u_{y}\sin\textstyle(\frac{\alpha}{2})(2S_{1x}\tilde{S}_{2z}-2S_{1z}\tilde{S}_{2x}+iS_{1y}-i\tilde{S}_{2y})+u_{x}\sin(\frac{\alpha}{2})(2S_{1z}\tilde{S}_{2y}-2S_{1y}\tilde{S}_{2z}+iS_{1x}-i\tilde{S}_{2x}).$

When we reverse the tunnel sequence such that Cooper pairs travel from the
right to the left superconductor, this adds the Hermitian conjugate to the
expression above [Eq. (8)]. Multiplying with the permutation factor 4 we get

$\displaystyle\frac{2\rho_{F}^{2}\pi^{2}}{|\epsilon|}t_{L1}t_{L2}t_{R1}t_{R2}\Bigg{\\{}\\!\cos{\varphi}\Big{[}\\!\cos\textstyle(\frac{\alpha}{2})$
$\displaystyle\boldsymbol{S}_{1}\\!\cdot\tilde{\boldsymbol{S}}_{2}\\!-\\!\sin(\textstyle\frac{\alpha}{2})\boldsymbol{u}\cdot\\!(\boldsymbol{S}_{1}\\!\times\\!\tilde{\boldsymbol{S}}_{2})\Big{]}+\frac{1}{2}\sin(\varphi)\sin(\textstyle\frac{\alpha}{2})\boldsymbol{u}\cdot\\!(\boldsymbol{S}_{1}\\!-\\!\tilde{\boldsymbol{S}}_{2})\Bigg{\\}}+g(\varphi),$
(11) $\displaystyle
g(\varphi)=-\frac{\rho_{F}^{2}\pi^{2}}{2|\epsilon|}t_{L1}t_{L2}t_{R1}t_{R2}\cos{\varphi}\cos{\left(\frac{\alpha}{2}\right)}.$

Adding all contributions together, we calculate the effective Hamiltonian in
the spin-1/2 subspace to be

$\displaystyle H=$
$\displaystyle\frac{\rho_{F}^{2}\pi^{2}}{|\epsilon|}\Bigg{(}t_{R1}^{2}t_{R2}^{2}\boldsymbol{S}_{1}\cdot\tilde{\boldsymbol{S}}_{2}+t_{L1}^{2}t_{L2}^{2}(U^{\dagger}\boldsymbol{S}_{1}U)\cdot\tilde{\boldsymbol{S}}_{2}$
(12)
$\displaystyle+2t_{L1}t_{L2}t_{R1}t_{R2}\Big{\\{}\\!\cos{\varphi}\Big{[}\\!\cos(\textstyle\frac{\alpha}{2})\boldsymbol{S}_{1}\\!\cdot\tilde{\boldsymbol{S}}_{2}\\!-\\!\sin(\textstyle\frac{\alpha}{2})\boldsymbol{u}\cdot\\!(\boldsymbol{S}_{1}\\!\times\\!\tilde{\boldsymbol{S}}_{2})\Big{]}+\frac{1}{2}\sin(\varphi)\sin(\textstyle\frac{\alpha}{2})\boldsymbol{u}\cdot\\!(\boldsymbol{S}_{1}\\!-\\!\tilde{\boldsymbol{S}}_{2})\Big{\\}}\Bigg{)}+g(\varphi).$

This equation is Eq. (5) of the main text for $t_{L1}t_{L2}=t_{R1}t_{R2}$.

We note that the spin Hamiltonian in Eq. (6) can be viewed as a form of
Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction [96, 97, 98] between the dot
spins which is mediated by Cooper pairs that virtually split and recombine or
vice versa.

##  Appendix B Ahoronov-Bohm flux

In this section we present a more general result of the effective Hamiltonian
found by using the perturbation theory without the restriction
$\Delta\gg|\epsilon|$ and accounting for an Aharonov-Bohm flux $f$ inclosed by
the two quantum dots and two superconductors. We note that in the limit
$\Delta\gg|\epsilon|$, those terms that depend on the Aharonov-Bohm flux are
higher-order contributions [19] and correspond to virtual tunneling paths as
depicted in Fig. 1(f) of the main text.

We adopt a more general description of the tunnel Hamiltonian, which is
$\boldsymbol{k}$ dependent and includes a Peierls phase in the presence of an
external magnetic vector potential $\boldsymbol{A}$. For this we replace
$t_{jn}$ by $t_{jn}\exp\left(\shortminus
i\boldsymbol{k}\cdot\boldsymbol{r}_{j}-{\textstyle\frac{i\pi}{\phi_{0}}}\int\limits_{\boldsymbol{r}_{n}}^{\boldsymbol{r}_{j}}\\!\text{d}\boldsymbol{l}\cdot\\!\boldsymbol{A}\right)$,
where $\boldsymbol{r}_{j}$ is the position on the superconductor that couples
to both quantum dots, each located at position $\boldsymbol{r}_{n}$. In
addition, $\phi_{0}=\frac{hc}{2e}$ is the superconducting flux quantum. (We
assume the associated Zeeman splitting on the dot to be small and neglect it.)
From the symmetry arguments presented in the main part, it still follows that
the total Hamiltonian has a similar form as Eq. (5) in the main text. The
generalized result then becomes (without the restriction
$\Delta\gg|\epsilon|$):

$\displaystyle H$
$\displaystyle=\textstyle\Gamma_{R}^{2}(\frac{C}{|\epsilon|}+\frac{C_{0}+C_{1}}{\Delta})\mathbf{S}_{1}\cdot\tilde{\mathbf{S}}_{2}+\Gamma_{L}^{2}(\frac{C}{|\epsilon|}+\frac{C_{0}+C_{1}}{\Delta})(U^{\dagger}\mathbf{S}_{1}U)\cdot\tilde{\mathbf{S}}_{2}$
(13)
$\displaystyle\textstyle+2\Gamma_{L}\Gamma_{R}\left[\frac{C_{1}}{\Delta}\cos(f)+(\frac{C}{|\epsilon|}+\frac{C_{0}}{\Delta})\cos(\varphi)\right]\left[\cos(\frac{\alpha}{2})\mathbf{S}_{1}\cdot\tilde{\mathbf{S}}_{2}-\sin(\frac{\alpha}{2})\mathbf{u}\cdot(\mathbf{S}_{1}\times\tilde{\mathbf{S}}_{2})\right]$
$\displaystyle\textstyle+\Gamma_{L}\Gamma_{R}\sin\left(\frac{\alpha}{2}\right)\left[\frac{C_{1}}{\Delta}\sin(f)\mathbf{u}\cdot(\mathbf{S}_{1}+\tilde{\mathbf{S}}_{2})+(\frac{C}{|\epsilon|}+\frac{C_{0}}{\Delta})\sin(\varphi)\mathbf{u}\cdot(\mathbf{S}_{1}-\tilde{\mathbf{S}}_{2})\right]+g(\varphi,f).$

In order to reveal the structure of the spin interaction, we rewrite Eq. (13)
in a form similar to Eq. (6) in the main text:

$H=h(S^{\|}_{1}-\tilde{S}^{\|}_{2})+h_{z}(S^{\|}_{1}+\tilde{S}^{\|}_{2})+J_{\|}S^{\|}_{1}\tilde{S}^{\|}_{2}+J_{\perp}\mathbf{S}^{\perp}_{1}\cdot\tilde{\mathbf{S}}^{\perp}_{2}+J_{\text{DM}}(\mathbf{S}_{1}\times\tilde{\mathbf{S}}_{2})^{\|}+g(\varphi,f).$
(14)

Here, the effective magnetic fields, spin-exchange constant, and a spin-
irrelevant constant $g(\varphi,f)$, respectively, are given by

$\displaystyle
h=\textstyle\Gamma_{L}\Gamma_{R}\left(\frac{C}{|\epsilon|}+\frac{C_{0}}{\Delta}\right)\sin\left(\frac{\alpha}{2}\right)\sin(\varphi),$
(15a) $\displaystyle
h_{z}=\textstyle\Gamma_{L}\Gamma_{R}\frac{C_{1}}{\Delta}\sin\left(\frac{\alpha}{2}\right)\sin(f),$
(15b) $\displaystyle
J_{\|}=\textstyle(\Gamma_{R}^{2}+\Gamma_{L}^{2})\left(\frac{C}{|\epsilon|}+\frac{C_{0}+C_{1}}{\Delta}\right)+2\Gamma_{L}\Gamma_{R}\left[\frac{C_{1}}{\Delta}\cos(f)+\left(\frac{C}{|\epsilon|}+\frac{C_{0}}{\Delta}\right)\cos(\varphi)\right]\cos(\frac{\alpha}{2}),$
(15c) $\displaystyle
J_{\perp}=\textstyle\left[\Gamma_{R}^{2}+\Gamma_{L}^{2}\cos\left(\alpha\right)\right]\left(\frac{C}{|\epsilon|}+\frac{C_{0}+C_{1}}{\Delta}\right)+2\Gamma_{L}\Gamma_{R}\left[\frac{C_{1}}{\Delta}\cos(f)+\left(\frac{C}{|\epsilon|}+\frac{C_{0}}{\Delta}\right)\cos(\varphi)\right]\cos(\frac{\alpha}{2}),$
(15d) $\displaystyle
J_{\text{DM}}=-\textstyle\Gamma_{L}^{2}\left(\frac{C}{|\epsilon|}+\frac{C_{0}+C_{1}}{\Delta}\right)\sin(\alpha)-2\Gamma_{L}\Gamma_{R}\left[\frac{C_{1}}{\Delta}\cos(f)+\left(\frac{C}{|\epsilon|}+\frac{C_{0}}{\Delta}\right)\cos(\varphi)\right]\sin(\frac{\alpha}{2}),$
(15e) $\displaystyle
g(\varphi,f)=\textstyle\frac{1}{2}\Gamma_{L}\Gamma_{R}\left[\frac{C_{1}}{\Delta}\cos(f)\cos(\frac{\alpha}{2})-(\frac{C_{0}}{\Delta}+\frac{C}{|\epsilon|})\cos(\varphi)\cos(\frac{\alpha}{2})\right]+(t_{L1}^{2}t_{R1}^{2}+t_{L2}^{2}t_{R2}^{2})\frac{\rho_{F}^{2}\pi^{2}C_{0}}{2\Delta}\cos(\varphi).$
(15f)

We have parameterized the tunneling coupling by $\Gamma_{j}=\rho_{F}\pi
t_{j1}t_{j2}$. The dimensionless parameters $C$, $C_{0}$, and $C_{1}$ are
given by

$\displaystyle C$
$\displaystyle=\left[\int\limits_{-\infty}\limits^{\infty}\frac{\text{d}x}{\pi
h_{0}(x)g_{0}(x)}\right]^{2}$ (16) $\displaystyle C_{0}$
$\displaystyle=\int\limits_{-\infty}\limits^{\infty}\int\limits_{-\infty}\limits^{\infty}\frac{\text{d}x\,\text{d}y}{\pi^{2}h_{0}(x)h_{0}(y)g_{0}(x)g_{0}(y)[h_{0}(x)+h_{0}(y)]}$
(17) $\displaystyle C_{1}$
$\displaystyle=\int\limits_{-\infty}\limits^{\infty}\int\limits_{-\infty}\limits^{\infty}\frac{\text{d}x\,\text{d}y\,[h_{0}(x)-\frac{|\epsilon|}{\Delta}]}{\pi^{2}[h_{0}(x)+h_{0}(y)][g_{0}(x)]^{2}g_{0}(y)},$
(18)

where $h_{0}(x)=\sqrt{x^{2}+1}$,
$g_{0}(x)=\sqrt{x^{2}+1}+\frac{|\epsilon|}{\Delta}$ [19]. The phase $\varphi$
and Aharonov-Bohm flux $f$, respectively, are given by

$\displaystyle\varphi=\varphi_{L}-\varphi_{R}-\frac{\pi}{\phi_{0}}\int_{\mathbf{r}_{R}}^{\mathbf{r}_{L}}(d\boldsymbol{\ell}_{1}+d\boldsymbol{\ell}_{2})\cdot\mathbf{A},$
(19) $\displaystyle
f=\frac{\pi}{\phi_{0}}\int_{\mathbf{r}_{R}}^{\mathbf{r}_{L}}(d\boldsymbol{\ell}_{1}-d\boldsymbol{\ell}_{2})\cdot\mathbf{A}.$
(20)

Here $\boldsymbol{\ell}_{n}$ corresponds to the path
$\boldsymbol{r}_{R}\rightarrow\boldsymbol{r}_{n}\rightarrow\boldsymbol{r}_{L}$
and $f$ is the dimensionless Aharonov-Bohm flux $f$ running through the area
between the dots and the superconductors [19]. In addition to the effective
staggered magnetic fields [first term in Eq. (14)] there exists a symmetry-
allowed Zeeman term [second term in Eq. (14)]. Generally, the external flux
$f$ offers one more experimentally tunable parameter, which increases the
tunability of our scalable architecture. Again, if $\Gamma_{L}=\Gamma_{R}$, we
can achieve the purely Ising coupling with $J_{\perp}=J_{DM}=0$:

$\displaystyle\left(\frac{C}{|\epsilon|}+\frac{C_{0}+C_{1}}{\Delta}\right)\cos\left(\frac{\alpha}{2}\right)=$
(21)
$\displaystyle-\left[\frac{C_{1}}{\Delta}\cos(f)+\left(\frac{C}{|\epsilon|}+\frac{C_{0}}{\Delta}\right)\cos(\varphi)\right].$

In the absence of flux, Eq. (21) reproduces the criterion derived in the main
part. The flux $f$ gives us one more parameter to be used to achieve the
optimal regime with high fidelities (see Fig. 3). We also note that, if
$\Gamma_{L}\neq\Gamma_{R}$ it is not possible to achieve the pure Ising
regime.

##  Appendix C Josephson supercurrent

The detection of spin interactions can be also realized by observing the
Josephson supercurrent $\hat{I}$, being defined by the derivative of $H$ with
respect to $\varphi$, i.e., $\hat{I}=\partial H/\partial\varphi$. Using Eq.
(14), we obtain for the supercurrent

$\displaystyle\hat{I}={\hat{I}}_{s}\sin\varphi+I_{0}\sin\left(\frac{\alpha}{2}\right)(S^{\|}_{1}-\tilde{S}^{\|}_{2})\cos\varphi,$
(22)

with

$\displaystyle\hat{I}_{s}=-2I_{0}\Big{[}$
$\displaystyle\cos\left(\frac{\alpha}{2}\right)\mathbf{S}_{1}\cdot\tilde{\mathbf{S}}_{2}-\sin\left(\frac{\alpha}{2}\right)(\mathbf{S}_{1}\times\tilde{\mathbf{S}}_{2})^{\|}$
(23)
$\displaystyle-\frac{1}{4}\cos\left(\frac{\alpha}{2}\right)\Big{]}-I_{1}.$

Note that the Josephson current $\hat{I}$ is still an operator in spin space,
indicated by the hat. This current involves only “two-lead” paths, as
manifested by the current amplitudes that contain simultaneously tunneling
amplitudes from right and left superconducting leads:

$\displaystyle
I_{0}=\Gamma_{L}\Gamma_{R}\left(\frac{C}{|\epsilon|}+\frac{C_{0}}{\Delta}\right),$
(24) $\displaystyle
I_{1}=(t_{L1}^{2}t_{R1}^{2}+t_{L2}^{2}t_{R2}^{2})\frac{\rho_{F}^{2}\pi^{2}C_{0}}{2\Delta}.$
(25)

In the presence of SOI ($\alpha\neq 0$), the supercurrent contains an
anomalous term proportional to $\cos\varphi$. This means that the supercurrent
$\hat{I}$ is finite even if $\varphi=0$. This phase shift in the supercurrent
can be exploited to detect the presence of the SOI, depending on the
orientations of the dot spins relative to each other. In turn, this spin
dependence of the phase shift can also be used as a readout of the spin states
of the dots: if one spin is aligned along its quantization axis, while the
other one is antialigned along its respective quantization axis, the amplitude
of the Josephson current at $\varphi=0$ will be maximal for finite SOI. In the
opposite extreme case, when the spins are both aligned along their respective
quantization axes, the current is minimal at $\varphi=0$, i.e., it vanishes.

For the special case considered in the main text,
$\Gamma_{L}=\Gamma_{R}=\Gamma$ and $|\epsilon|\ll\Delta$, we have $C_{0}\simeq
0$ and $C\simeq 1$ (and $C/|\epsilon|\gg C_{0}/\Delta)$. Then Eqs. (22) and
(23) reduce to Eqs. (8) and (9), respectively, of the main text.

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# Probing Script Knowledge from Pre-Trained Models

Zijia Jin¶, Xingyu Zhang♮, Mo Yu♣, Lifu Huang♠
¶New York University, ♮Xi’an Jiaotong University, ♣WeChat AI, ♠Virginia Tech
<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Script knowledge is critical for humans to understand the broad daily tasks
and routine activities in the world. Recently researchers have explored the
large-scale pre-trained language models (PLMs) to perform various script
related tasks, such as story generation, temporal ordering of event, future
event prediction and so on. However, it’s still not well studied in terms of
how well the PLMs capture the script knowledge. To answer this question, we
design three probing tasks: inclusive sub-event selection, starting sub-event
selection and temporal ordering to investigate the capabilities of PLMs with
and without fine-tuning. The three probing tasks can be further used to
automatically induce a script for each main event given all the possible sub-
events. Taking BERT as a case study, by analyzing its performance on script
induction as well as each individual probing task, we conclude that the
stereotypical temporal knowledge among the sub-events is well captured in
BERT, however the inclusive or starting sub-event knowledge is barely encoded.

##  1 Introduction

A script is a structure that describes a stereotyped sequence of events that
happen in a particular scenario Schank and Abelson (1975, 2013). It allows
human to keep track of the states and procedures that are necessary to
complete various tasks from daily lives to scientific processes. Taking the
task of _Eating in a Restaurant_ as an example. A classic example script for
this task may consist of a chain of subevents, such as _Enter_
$\rightarrow$_Order_ $\rightarrow$_Eat_ $\rightarrow$_Pay (and Tip)_
$\rightarrow$_Leave_. The script knowledge has shown benefit to many
downstream applications, such as story generation Li et al. (2013, 2018); Guan
et al. (2019); Zhai et al. (2019); Lin et al. (2022), machine reading
comprehension Tian et al. (2020); Ostermann et al. (2018); Sugawara et al.
(2018), commonsense reasoning Ding et al. (2019); Huang et al. (2019); Bauer
and Bansal (2021) and so on.

Recent large-scale pre-trained language models (PLMs) Devlin et al. (2019);
Liu et al. (2019); Radford et al. (2019); Raffel et al. (2019) have shown
competitive performance on many natural language processing tasks. Abundant
studies have demonstrated that these models either directly capture certain
types of syntactic Goldberg (2019); Clark et al. (2019); Htut et al. (2019);
Rosa and Mareček (2019), factual Petroni et al. (2019a, 2020); Bouraoui et al.
(2020); Wang et al. (2020) and commonsense knowledge Zhou et al. (2020);
Rajani et al. (2019); Lin et al. (2020) during the pre-training or acquire
inductive capability to more efficiently induce such knowledge from natural
language text Pandit and Hou (2021); Bosselut et al. (2019). However, as
another important type of cognitive and schematic knowledge describing human
routine activities, scripts are not yet well probed in the language models by
prior studies.

To investigate how well the pre-trained language models have captured the
script knowledge, in this work, we design three probing tasks and language
model prompting methods to probe the script knowledge from PLMs, and further
leverage the language model prompting methods to induce the scripts given the
main events. Specifically, we aim to answer the following two research
questions:

_Whether and what script knowledge is captured by the pre-trained language
models._ To answer this question, we design three sub-tasks to probe the
script knowledge, including inclusive sub-event selection (i.e., whether a
sub-event is included or excluded in a main event or task), starting sub-event
selection (i.e., which sub-event is the start of the script for a particular
main event), and sub-event temporal ordering (i.e., predicting a temporal
before or after relation between two sub-events). On these sub-tasks, we
explore both template-based and soft prompting methods to query the knowledge
from pre-trained language models. By investigating their performance gaps to
the fine-tuning results, we find that both the inclusive and starting sub-
event selection sub-tasks have relatively poorer performance than that of
temporal ordering, which is likely due to the lack of relevant objectives to
encourage the models to capture such knowledge during pre-training, and
further suggests future research directions to enhance the PLMs to better
capture the script knowledge.

_How to better generate the scripts from these pre-trained models._ With the
language model prompting methods, we can select the inclusive sub-events of a
particular script, the starting sub-event and subsequent events by predicting
the temporal order among all the inclusive sub-events, which can ultimately
generate a sequence of events as the script of a main event. Thus, we further
design a benchmark dataset to fine-tune the models for the three sub-tasks and
evaluate their performance on generating the whole scripts for various main
events from diverse domains and topics.

The contributions of this work can be summarized as follows:

  * •

We are the first to formulate the sub-tasks and set up benchmark datasets to
probe the script knowledge from pre-trained language models.

  * •

We are the first to research on the generation and evaluation of the whole
scripts from pre-trained language models.

##  2 Related Work

#### Script Knowledge

The definition of Script Knowledge was first proposed in 1981 Feigenbaum et
al. (1981), which aims to detect the relation between two events. Chambers and
Jurafsky (2008) created the first unsupervised data-driven method based on
point-wise mutual information (PMI) to automatically extract narrative event
chains. Recently, researchers explored deep neural networks, especially large-
scale pre-train language models to predict the temporal relation between two
events Pustejovsky et al. (2003); Chambers (2013); Ferraro and Durme (2016);
Reimers et al. (2016) or generate the future event Pichotta and Mooney (2014);
Jans et al. (2012); Zhang et al. (2020). Comparing with these studies, our
work focuses more on investigating how well the PLMs encode or capture the
script knowledge from pre-training and their bottleneck, suggesting possible
directions for future research.

#### Language Model Probing

Probing is a popular way to detect what knowledge is encoded in PLMs. At
first, probing method is designed for detect morphology knowledgeBelinkov et
al. (2017) ,syntactic knowledge Peters et al. (2018) and semantic
knowledgeTenney et al. (2019). Then researchers began to pay more attention to
more complex knowledge like commonsense knowledge. The two main standard
approaches in probing commonsense knowledge is building classifiersHewitt and
Liang (2019) or filling text in the gapPetroni et al. (2019b). In our study,
we extend the accuracy based methods and designed a series of downstream tasks
specific to Scripts Knowledge.

##  3 Method

Figure 1: Overview of the probing approaches for (1) Inclusive Sub-event
Selection, (2) Starting Sub-event Selection, and (3) Sub-event Temporal
Ordering. And an overall evaluation stage for generating scripts with main
events and subevents as input.

###  3.1 Script Knowledge Probing

Our first goal is to probe the script knowledge from pre-trained language
models. To do so, we divide the script knowledge into three categories: the
Inclusive and starting relation between each sub-event and main event,
indicating whether the sub-event should be included in or the start of the
script of a particular main event, and the temporal relation (i.e., Before or
After) among the sub-events. To probe these knowledge from PLMs, we design the
following tasks.

#### Task 1: Inclusive Sub-event Selection

As Figure 1 shows, given a main event, e.g., "Clean laundry", and a candidate
sub-event, e.g., "Gather dirty clothes.", we aim to have the language model to
determine whether the sub-event belongs to the script of the target main
event. To do so, we use [MASK] to connect them into a whole sequence and use a
PLM to encode the sequence into contextual representations. In order to
predict the Inclusive relation, we apply a linear function (i.e., a MLM head)
to project the [MASK] into a probability distribution over the whole
vocabulary of the PLM. By exploring many candidate tokens from the target
vocabulary to represent each relation, we finally select “include” to denote
the Inclusive relation and “except” for Exclusive.

#### Task 2: Starting Sub-event Selection

Given a main event and a set of sub-events that are predicted to belong to the
script of the main event, we aim to select the most probable sub-event as the
start of the script. We formulate it as a sequence classification problem. We
concatenate the main event and each sub-event candidate with a prompt "start
with", e.g., Taking bus start with finding bus stop, and use a MLP layer to
predict a score indicating how likely the sub-event is the start of the script
of the main event, based on the contextual representation of the [CLS]. As a
result, we use the sub-event with the highest score as the first sub-event. We
design a margin based loss function to encourage the score of the positive
start sub-event to be higher than others

$L(s^{*},s_{i})=\sum_{\tilde{s}_{i}\in{\tilde{S}}}max(score(\tilde{s}_{i})+m-score(s^{*}),0)$

where $s^{*}$ represents the positive start sub-event of a particular script
and $\tilde{S}$ denotes the set of other sub-events from the same script. The
margin $m$ is a hyper-parameter, which is set as 1.0 in our experiment. During
inference, given a set of candidate sub-events, we compare their scores and
select the one with the highest score as the starting sub-event.

#### Task 3: Sub-event Temporal Ordering

This probing task is to show the capability of the PLMs on correctly
organizing the sub-events into a temporally ordered event sequence. To do so,
we design a new language model probing approach following Petroni et al.
(2019c). As shown in Figure 1, given two subevents, e.g., "put clothes in
dryer." and "turn on dryer." , we use [MASK] to connect them into a sequence
and use a PLM to encode it. The temporal relation is predicted by comparing
the probability of tokens “before” and “after” based on the contextual
representation of [MASK].

###  3.2 Script Induction with PLMs

The second goal in this work is to design a simple yet effective approach to
automatically induce scripts based on PLMs. Given a particular main event and
a set of candidate sub-events, to induce the script for the target main event,
we design a pipeline approach consisting of three steps: (1) selecting a
subset of inclusive sub-events from all the candidates; (2) determining the
starting sub-event; and (3) ordering all the inclusive sub-events by
predicting the temporal relation between each pair of them. These three steps
correspond to the three approaches designed for script knowledge probing.

##  4 Experiment Setup

We take BERT-base-uncased Devlin et al. (2019) as the target PLM to
investigate how well it encodes the script language via the three probing
tasks. We combine three script datasets, including DeScript Wanzare et al.
(2016), OMICS Gupta and Kochenderfer (2004) and Stories Trinh and Le (2018),
where each main event is annotated with 7 to 122 scripts written by different
crowd-sourcing workers. We sample 60 main events as the evaluation set, 39
main events as the development set and use the remaining 98 main events for
training. For the main events in training and development sets, we keep all
the scripts, while for each main event in the evaluation set, we only keep the
longest script as the target. Table 1 shows the statistics of each dataset.

Datasets | # Main Events | # Scripts
---|---|---
Training | 98 | 4,685
Development | 39 | 1,791
Evaluation | 60 | 60
Table 1: Data statistics for training, development and evaluation Sets.

To create the training samples for the inclusive sub-event selection task, for
each script, we use all the ground truth subevents as positive samples and
randomly choose 100 times of negative samples from other main events’ scripts.
For evaluation, as the inclusive sub-event selection requires a pool of all
the possible candidate events, we combine the sub-events of all scripts in the
evaluation dataset. To create the training samples for the start sub-event
selection task, we use the first sub-event of each script as the positive
sample and all the remaining sub-events from the same script as the negative
samples. During the inference, we select the starting sub-event from the
inclusive sub-events predicted by the inclusive sub-event selection approach.
We use accuracy as the evaluation metric. Finally, for the temporal ordering
task, we create each training sample based on each sub-event together with one
of its following sub-events. We randomly shuffle the order of each pair of
sub-events and create its corresponding label: "before" or "after". To
evaluate the quality of the temporal ordering among all the sub-events, we
first generate a script based on the predicted temporal order and then use
ROUGE-L to evaluate the longest common subsequence between the generated
script and the gold script.

We compare the following approaches for each probing task as well as the
script induction:

#### BERT Pre-trained:

Directly use the pre-trained BERT model to make the predictions on the
evaluation set.

#### BERT Fine-tuning:

Fine-tune BERT with task-specific training data and evaluate those fine-tuned
models on the evaluation set.

#### BERT Ptuning:

Following the Ptuning framework Liu et al. (2021), fine-tune the parameters of
both BERT model and prompt tokens.

#### BERT Ptuning Freeze:

Only fine-tune the prompt tokens while freezing the parameters of BERT model.

##  5 Results and Analysis

###  5.1 Overall Script Induction

We first show the results of end-to-end script induction given each main event
and the pool of all candidate sub-events. As Table 2 shows, without any fine-
tuning, BERT-Pretrained can barely induce any reasonable scripts. The high
precision and low recall indicates that the bottleneck is likely in correctly
selecting the inclusive sub-events for each main event. However, with fine-
tuning either on the whole BERT parameters or a few prompt parameters, the
script induction performance can be improved significant, demonstrating that
the pre-trained BERT actually captures certain level of script knowledge but
requires external probes to induce such knowledge from it. Finally, by
analyzing of the performance of fine-tuning approaches, we notice a more
significant improvement on recall. We conjecture that with fine-tuning, the
inclusive sub-event selection is more likely to be improved.

Method | Rouge-L
---|---
Rec | Prec | F-score
BERT-Pretrained | 3.25 | 22.60 | 4.81
BERT-Finetuning | 37.19 | 28.07 | 28.73
BERT-Ptuning | 48.70 | 28.78 | 32.52
BERT-Ptuning-Freeze | 4.82 | 16.19 | 7.25

Table 2: Performance of script induction

Method | Inclusive Subevent Selection | Starting Subevent Selection | Temporal Ordering
---|---|---|---
Rec | Prec | F-score | Accuracy | Rouge-L F1
BERT-Pretrained | 7.44 | 0.64 | 1.17 | 18.33 | 63.79
BERT-Finetuning | 33.83 | 44.71 | 38.51 | 21.66 | 62.87
BERT-Ptuning | 31.16 | 56.24 | 40.10 | 20.00 | 63.62
BERT-Ptuning-Freeze | 98.69 | 0.52 | 1.03 | 28.33 | 66.02

Table 3: Performance on each individual task.

###  5.2 Probing on Individual Tasks

We further analyze the capability of BERT on encoding each type of script
knowledge based on the three probing tasks. To avoid error propagation, for
both starting sub-event selection and temporal ordering, we use the gold
inclusive sub-events of each main input as input.

As Table 3 shows, for inclusive sub-event selection, without fine-tuning, both
BERT-Pretrained and BERT-Ptuning-Freeze cannot correctly select any inclusive
sub-events. This is likely due to the discrepancy between the pre-training
objectives of BERT (i.e., MASK language modeling and next sentence prediction)
with the objective of inclusive sub-event selection. With fine-tuning, the
performance of both BERT-Finetuning and BERT-Ptuning is improved
significantly, which is aligned with our assumption in Section 5.1. Starting
sub-event selection is hard to all the approaches, which is likely due to two
reasons: one is the limited training samples, and the other is that though we
formulate each sub-task as mask prediction to better induce the knowledge from
BERT, the pattern “Main_Event starts with Sub_Event” is less likely to appear
in the unlabeled corpus than other patterns, such as “Main_Event includes
Sub_Event” and “Event_A before/after Event_B”. Finally, all the approaches
show consistently descent performance on temporal ordering, no matter whether
BERT is fine-tuned or not, demonstrating that BERT has well captured the
relations among the events with stereotypical temporal orders, possibly due to
the next sentence prediction objective during pre-training.

##  6 Conclusion

In this work, we investigate the capability of large-scale pre-trained
language models (PLMs) on capturing three aspects of script knowledge:
inclusive sub-event knowledge, starting sub-event knowledge and temporal
knowledge among the sub-events from the same script. These three types of
knowledge can be further leveraged to automatically induce a script for each
main event given all the possible sub-events. We use BERT as a target PLM. By
analyzing its performance on script induction as well as each individual
probing task, we achieve the conclusions that the stereotypical temporal
knowledge among the sub-events is well captured in BERT, however the inclusive
and starting sub-event knowledge are not well encoded.

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