# A superconvergence result in the RBF-FD method

Andrej Kolar-Požun1,2 Mitja Jančič1 and Gregor Kosec1 1 Parallel and
Distributed Systems Laboratory, Jožef Stefan Institute, Ljubljana, Slovenia  2
Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana,
Slovenia<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Radial Basis Function-generated Finite Differences (RBF-FD) is a meshless
method that can be used to numerically solve partial differential equations.
The solution procedure consists of two steps. First, the differential operator
is discretised on given scattered nodes and afterwards, a global sparse matrix
is assembled and inverted to obtain an approximate solution. Focusing on
Polyharmonic Splines as our Radial Basis Functions (RBFs) of choice,
appropriately augmented with monomials, it is well known that the truncation
error of the differential operator approximation is determined by the degree
of monomial augmentation. Naively, one might think that the solution error
will have the same order of convergence. We present a superconvergence result
that shows otherwise - for some augmentation degrees, order of convergence is
higher than expected.

##  1 Introduction

Some of the most popular topics in numerical analysis are arguably numerical
solution procedures for Partial Differential Equations (PDEs). Both because
this field has numerous applications, as PDEs describe a variety of real-life
phenomena, and because of its many challenges - PDEs are typically much more
complicated than ordinary differential equations with only very simple cases
being solvable analytically.

Without delving too much into the history, let us mention that as far as real-
life applications go, the most commonly used methods for numerical solutions
of PDEs consider computational domains with some additional structure. For
example, in the widespread Finite Element (FEM) [11] and Finite Difference
(FDM) [13] Methods, this structure is a (typically triangular) mesh and a
regular grid, respectively.

The method that we focus on in this paper instead falls into the scope of
meshless methods that work directly with scattered nodes without any
additional structure and with very mild requirements on their geometry. The
main advantage of this approach is its greater flexibility (in particular when
it comes to adaptive solution procedures [7]), as we eliminate a possibly
cumbersome step of meshing, while maintaining an accurate description of
irregular geometries.

Over the last few decades, several different meshless methods have been
proposed. We refer an interested reader to [10] for a more in-depth survey. We
will focus on the Radial Basis Function-generated Finite Difference (RBF-FD)
method, proposed in the early 2000s [14] that is particularly attractive due
to its simplicity. It is a strong form method that directly generalises the
FDM by approximating a linear differential operator locally on a given
neighbourhood of a computational node. As its name implies, it is based on the
Radial Basis Functions (RBFs), which have historically been particularly
attractive for scattered data interpolation due to their positive definiteness
properties [5].

In our studies, we opt for the Polyharmonic Splines (PHS) as our RBF of
choice. Concretely, the radial cubics $\phi(r)=r^{3}$ will be our RBF basic
function. For clarity, we will refer to the employed solution procedure as PHS
RBF-FD. Compared to smooth RBFs, PHSs posses a weaker notion of conditional
positive definiteness, which requires us to add some number of monomials to
the approximation, a procedure known as monomial augmentation. Nevertheless
the advantage gained from using such RBFs is that they do not posses a shape
parameter, which makes them easier to use. For more information, we once again
refer the reader to [5].

Specifically for PHS RBF-FD, various studies of the method’s properties have
already been performed. In a paper by Flyer et al. [6] it is shown that the
PHS order is irrelevant as far as the convergence order of the approximation
is concerned. In fact, the latter is instead determined by the degree of the
augmenting monomials. This result has been further developed by Bayona [2],
who analysed monomials’ effect in greater detail and derived a formula
expressing the PHS RBF-FD truncation error.

PHS RBF-FD solution procedure is reminiscent of the standard FDM - The linear
differential operator appearing in the PDE is discretised locally at each
computation point. This allows us to transform the PDE into a set of algebraic
equations, which is then solved to obtain a numerical solution. Usually, the
order of convergence of such a solution procedure matches the order of
operation approximation error. However, in some special cases, the so called
superconvergence can occur, where the solution converges at a higher order
than expected. Superconvergence has been well researched in the context of
finite element methods [15], but has also been observed in methods that are
variants of the FDM, such as in the Shortley-Weller approximation, where
differing truncation orders at the boundary and the interior of the domain
cause a superconverge result [1]. More recently, another superconvergence
result was presented [9], where finite difference schemes, derived from weak
form with specially chosen quadrature points cause certain error cancellations
leading to the convergence order elevation.

In this paper we start by applying PHS RBF-FD to solve a common test problem -
The Poisson equation on a unit disc, albeit irregularly discretised. We first
show that the operator truncation error behaves as expected. However, for even
augmentation degrees, the solution convergence order is of approximately one
higher than expected.

In the following section our problem setup and the methods used are further
described. In section 3 our main result is shown and briefly analysed. We
conclude with section 4, repeating the main remarks and providing possible
ideas for further research.

##  2 Problem setup

We consider the Poisson problem:

$\nabla^{2}u(x,y)=f(x,y),$ (1)

for $x^{2}+y^{2}\leq 1$, where $f(x,y)$ is chosen such that the analytical
solution is

$u(x,y)=1+\sin(4x)+\cos(3x)+\sin(2y).$ (2)

$u(x,y)$, restricted to $x^{2}+y^{2}=1$ also gives us the Dirichlet boundary
conditions.

Figure 1: Analytical solution $u(x,y)$, shown on an example discretisation
generated during our solution procedure.

As mentioned, the problem will be solved with the PHS RBF-FD method. First,
the domain is discretised with the algorithm described in [4], providing a
locally regular discretisation, with a desired internodal distance $h$. An
example discretisation can be seen on Figure 1. To each discretisation point
$(x_{i},y_{i})$ we then associate the set of its $n$ closest nodes, known as
its stencil $S_{i}$. We set $n=(m+1)(m+2)$, where $m$ is the PHS RBF-FD
monomial augmentation degree. This is a common choice and can have a positive
impact on the method stability, as explained in [3].

A local approximation of $\nabla^{2}$ over the stencil is then obtained by the
standard RBF-FD procedure and involves solving an appropriate linear system of
equations. As this paper deals with a specific property of the RBF-FD method,
we will assume the reader already has some familiarity with it and not delve
into the details. If that is not the case, [8] should fill in the missing
details. We then obtain an approximation of $\nabla^{2}$ in the following
form:

$\nabla^{2}u(x_{i},y_{i})\approx\sum_{j\in S_{i}}w_{j}u(x_{j},y_{j}),$ (3)

where $w_{j}$ are the RBF-FD weights, expressing $\nabla^{2}$ as a linear
combination of function values in the the stencil $S_{i}$. Writing Equation 3
for each node in the domain interior, and an equation expressing the Dirichlet
boundary conditions for each node on the boundary, we obtain a global (and,
due to the locality of the approximation, sparse) system, which we then solve
to obtain a numerical solution. The BiCGSTAB iterative solver has been used
for that matter. All of the source code can be found on our public git
repository111https://gitlab.com/e62Lab/public/2024_cp_eurotherm_superconvergence.
Note that it relies heavily on our in-house (open source) C++ library Medusa
[12].

In what follows, we will use the just described procedure to numerically solve
our model problem and study the error convergence.

##  3 The main result

Two different errors will be considered. First, the operator approximation
error: For each point we compute the difference between the analytical and
RBF-FD $\nabla^{2}$ applied to $u(x,y)$ (Equation 2). For convenience we
introduce the vector $\textbf{e}_{\mathrm{op}}$, its $i$-th component being
the error at $(x_{i},y_{i})$. We will consider the mean error, which by a
slight abuse of notation will be denoted by $||\textbf{e}_{\mathrm{op}}||_{1}$
(despite it actually being the $\ell_{1}$ norm, divided by the number of
discretisation nodes $N$). All of the coming results have been observed also
in $\ell_{\infty}$ error norm, but will not be displayed for brevity reasons.

The error convergence of the $\nabla^{2}$ RBF-FD approximation is shown on the
left side of Figure 2. As our discretisation is irregular, we have computed
these errors over 30 different discretisation sets, varied by changing the
random seed in our previously mentioned discretisation algorithm. We can see
that for each monomial augmentation degree $m$, the errors scale as
approximately $\propto h^{m-1}$, which is in agreement with the theory -
operator discretisation error should scale as $\propto h^{m+1-l}$, where $l$
is the derivative order, in our case $l=2$ [8].

Figure 2: Convergence of mean errors under decreasing $h$. Operator and
solution errors are shown on the left and right, respectively for different
monomial augmentation degrees $m$. For each $m$, the errors have been computed
for 30 different discretisation sets. The results lie in the corresponding
shaded regions.

Next we consider the solution error, expressing the error of the whole
solution procedure when solving our chosen Poisson problem. Analogously to the
previous notation we introduce the vector $\textbf{e}_{\mathrm{sol}}$.
Solution error convergence can be seen on the right side of Figure 2, which
shows our main observation of the paper - convergence orders match those of
the operator approximation for odd augmentation degrees $m$, where the error
scales as $\propto h^{m-1}$, but we get an order higher for even $m$ \-
approximately $\propto h^{m}$. Additionally, the error spread for even $m$ is
noticeably larger, hinting at a different underlying mechanism.

For our analysis, we first consider an alternative procedure of studying error
convergence. Keeping $h$ fixed at $0.05$, we instead make a transformation
$u(x,y)\to u(Rx,Ry)$ and study the error scaling as we increase $R$,
effectively flattening the chosen solution. Error convergence with respect to
$R$ can seen on Figure 3. Both the operator and solution errors now scale as
$R^{m+1}$, which is expected.

Figure 3: Solution and operator error dependences on scaling factor $R$ shown
for different monomial augmentation degrees $m$. For each $m$, the errors have
been computed for 30 different discretisation sets. The results lie in the
corresponding shaded regions.

The lack of order elevation when studying the error dependence on $R$ implies
that for even augmentation degrees $m$ we do not actually gain any extra
approximation power (as in, method suddenly becoming exact for higher degree
monomials). Instead, an extra factor of $h$ must appear somewhere in the final
step of the solution procedure. Since that step involves solving a sparse
system with the global matrix $A$, it makes sense to look at it more closely.

We now attempt to study the error convergence more systematically by applying
Bayona’s explicit formula [2] for the RBF interpolation error over a stencil
around the point $(x_{0},y_{0})$:

$\begin{split}\hat{u}(x,y)-u(x,y)=\sum_{k=s+1}^{\infty}&L_{k}[u(x_{0},y_{0})]\big{(}\textbf{p}_{k}^{T}\boldsymbol{\hat{\psi}}(x,y)-p_{k}(x-x_{0},y-y_{0})-\\\
&-\textbf{p}_{k}^{T}W(P^{T}\boldsymbol{\hat{\psi}}(x,y)-\textbf{p}(x-x_{0},y-y_{0}))\big{)},\end{split}$
(4)

where the notation is similar to Bayona’s and we do not repeat it here.

Applying $\nabla^{2}$ over the above and evaluating at some discretisation
point $(x_{i},y_{i})$, we obtain a simplified formula for the operator
approximation error:

$\nabla^{2}\hat{u}(x_{i},y_{i})-\nabla^{2}u(x_{i},y_{i})=\sum_{k=s+1}^{\infty}L_{k}[u(x_{i},y_{i})](\textbf{p}_{k}^{T}\cdot\textbf{w}),$
(5)

where we have expressed the vector of $\nabla^{2}$ RBF-FD weights w as
$\textbf{w}=\nabla^{2}\boldsymbol{\psi}(x,y)$ and used the facts that
$\boldsymbol{\psi}(x,y)=(I-WP^{T})\boldsymbol{\hat{\psi}}(x,y)+W\textbf{p}(x,y)$
and $p_{k}(0,0)=0$ [2]. The weights w scale as $h^{-2}$, while
$\textbf{p}_{k}^{T}$ is a vector of monomials scaling as, for example
$h^{m+1}$ for the first terms in the error, and with higher exponents for
later terms. We can use this formula to analyse the operator error term by
term, grouped by powers of $h$. The results are seen on the left side of
Figure 4 and match the previously observed operator error scalings.

Next, we use this formula to also analyse the behaviour of the solution error.
Consider the RBF-FD approximation of $\nabla^{2}$, applied to the solution
error:
$\nabla^{2}\textbf{e}_{\mathrm{sol}}(i)=\nabla^{2}(\hat{u}(\textbf{x}_{i})-u(\textbf{x}_{i}))=f_{i}-\nabla^{2}u(\textbf{x}_{i})=\textbf{e}_{\mathrm{op}}(i)$.
This, together with the fact that the solution error on the boundary is equal
to $0$ (due to the Dirichlet boundary conditions) implies

$A\textbf{e}_{\mathrm{sol}}=\textbf{e}_{\mathrm{op}}.$ (6)

Therefore, we can analyse how each of the terms in Bayona’s error formula
behave also after the global system is inverted - we simply plug the
corresponding term as the RHS. Right side of Figure 4 shows that terms, that
originally contained an odd power of $h$, obtain an extra factor of $h$ after
the system is inverted, while the ones with even powers are unaffected. This
is why we observe superconvergence only for even augmentation degrees, as in
those cases the dominant error term contains odd powers of $h$, while for odd
augmentation degrees, the opposite applies.

Summarising, we have demonstrated that the observed order elevation can
already be seen by considering Bayona’s error formula, as written in Equation
5. Further study of the latter could be easier than of the original problem
(Equation 1), as we have essentially reduced it to the study of the solution
to a linear system, where the functional forms of the matrix $A$ and the RHS
are known (Equation 6). Further investigation is difficult, as it would
require an understanding of what precisely happens during the process of
inverting the matrix $A$, which is a challenge due to its size. Some ideas are
mentioned in the conclusion below.

Figure 4: Average error for each term in Bayona’s formula for the operator
approximation on the left, and the same terms after solving the global system
on the right. Once again, the results over 30 different discretisations are
shown. In both cases $m=2$ was fixed and first few leading terms in the
formula were considered.

##  4 Conclusion

This paper considered the error analysis of the RBF-FD method, applied to the
Poisson problem on a unit disc. We have opted for radial cubics as our RBF
with monomial augmentation, for which it is well established on how the
operator error scales with the internodal distance $h$.

Despite the simplicity of the model problem, we have noticed unexpected
behaviour of the solution error convergence: for even augmentation degrees,
the convergence order is of approximately one higher than predicted by the
theory.

We have managed to simplify the problem, showing that we can use Bayona’s
error formula to study the error behaviour systematically term by term.

There are multiple ways of proceeding in this research that we are actively
attempting in our further studies. First, one can make use of the known theory
mentioned in the introduction, which other authors studying similar phenomena
have successfuly applied. More tests of purely experimental nature are also
being performed to verify which properties of the problem or its solution
procedure affect the observed orders of convergence.

## Acknowledgments

The authors acknowledge the financial support from the Slovenian Research and
Innovation Agency (ARIS) in the framework of the research core funding No.
P2-0095 and the Young Researcher program PR-12347.

## References

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# High-fidelity two-qubit gates of hybrid superconducting-semiconducting
singlet-triplet qubits

Maria Spethmann<EMAIL_ADDRESS>Department of Physics, University
of Basel, 4056 Basel, Switzerland  RIKEN, Center for Emergent Matter Science
(CEMS), Wako-shi, Saitama 351-0198, Japan  Stefano Bosco  Department of
Physics, University of Basel, 4056 Basel, Switzerland  Andrea Hofmann
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  RIKEN, Center for Emergent Matter Science (CEMS), Wako-shi,
Saitama 351-0198, Japan

###### Abstract

Hybrid systems comprising superconducting and semiconducting materials are
promising architectures for quantum computing. Superconductors induce long-
range interactions between the spin degrees of freedom of semiconducting
quantum dots. These interactions are widely anisotropic when the semiconductor
material has strong spin-orbit interactions. We show that this anisotropy is
tunable and enables fast and high-fidelity two-qubit gates between singlet-
triplet (ST) spin qubits. Our design is immune to leakage of the quantum
information into non-computational states and removes always-on interactions
between the qubits, thus resolving key open challenges for these
architectures. Our ST qubits do not require additional technologically-
demanding components nor fine-tuning of parameters. They operate at low
magnetic fields of a few milli Tesla and are fully compatible with
superconductors. In realistic devices, we estimate infidelities below
$10^{-3}$, that could pave the way toward large-scale hybrid superconducting-
semiconducting quantum processors.

Figure 1:  Schematics of coupled hybrid ST qubits. Two ST qubits, each
comprising a double quantum dot, interact (a) via one superconductor or (b)
via a Josephson junction. (c) These setups effectively coincide with two
exchange-coupled double quantum dots, with fully tunable interactions. Strong
SOI induces spin rotations of an angle $\Phi_{\text{so}}$ around an axis
$\boldsymbol{n}_{\text{so}}$, and yield a large asymmetry of exchange
depending on the angle $\theta$ between $\boldsymbol{n}_{\text{so}}$ and the
Zeeman vector $\boldsymbol{n}_{B}$. This angle is controlled by the direction
of the applied magnetic field $\boldsymbol{B}$.

### Introduction.

Hybrid systems comprising superconductors and semiconductors are the workhorse
of modern quantum technology, with applications in low-power electronics [1]
and in neuromorphic [2] and quantum computing [3, 4]. Front-runner quantum
bits (qubits) are encoded in the spin of particles confined in semiconducting
quantum dots [5, 6, 7, 8] or in collective modes of superconducting devices
[9, 10]. Spin qubits are compact but challenging to address, while
superconducting qubits are bulky but easy to couple. By combining the best
properties of each architecture, hybrid qubits [11, 12, 13, 14] could
outperform the state-of-the-art and pave the way toward large-scale quantum
processors.

A key ingredient for effective hybrid systems is spin-orbit interaction (SOI).
In Josephson junctions, SOI induces a spin-dependent supercurrent that is
critical to manipulate and read out Andreev spin qubits [12, 11, 13, 14].
Topological encoding of quantum information in Majorana bound states [15, 16,
17, 18, 19, 20] and long-range entanglement of distant spins [21, 22, 23, 24,
25] also crucially require strong effective SOIs. SOIs are exceptionally large
in narrow-gap semiconducting nanowires [26, 27, 28, 29, 30] and in
nanostructures where the charge carriers are holes rather than electrons [31,
32, 33, 34, 35, 36, 37, 38, 39]. Hole gases in planar germanium (Ge)
heterostructures [40, 41, 42, 43, 44, 45] are particularly appealing because
of their compatibility with superconducting materials and the possibility of
engineered proximitized superconductivity [46, 47, 48, 49, 50, 51]. Recent
experiments with Ge [52, 53] demonstrated operations of singlet-triplet (ST)
spin qubits, encoding quantum information in the zero-spin subspace of two
coupled quantum dots [54], at milli-Tesla magnetic fields [52]. These fields
are compatible with current superconducting devices, opening various
opportunities for hybrid systems in a potentially nuclear spin free material
such as Ge.

In this work, we discuss a robust implementation of high-fidelity two-qubit
gates between distant ST qubits [55, 52, 53, 56, 57] in hybrid systems, see
Fig. 1. By taking full advantage of both the large SOI in the material and the
long-range spin-spin correlations induced by the superconductor [58, 59, 60],
our design overcomes the fundamental limitations of current two-qubit gates
between ST qubits, namely leakage to non-computational states [61, 62] and
crosstalk [63, 64] caused by always-on residual interactions between dots.

We find that leakage is naturally suppressed by the large SOI in hole systems,
yielding a tunable anisotropy in the exchange interactions of quantum dots
[65], and crosstalk vanishes by utilizing the phase response of the
supercurrent in a Josephson junction. We estimate infidelities below the
surface code threshold $\sim 10^{-3}$ [66] in realistic devices, fully
compatible with current technology [52], and without requiring additional
technologically-demanding tuning of the individual Zeeman energies [67, 68,
69, 70], nor fine-tuning of parameters [54, 71, 72]. Our two-qubit gate design
could push ST qubits towards higher coherence standards and boost the growing
field of hybrid superconducting-semiconducting quantum systems.

Figure 2:  Leakage to non-computational basis states. (a) Energy spectrum of
our setup, described by $H_{\text{spin}}$ [Eq. (Superconductor-mediated
exchange interactions.)], as a function of the global Zeeman splitting
$\bar{h}$ for $\mathcal{J}=0.4\,\mu$eV, $J_{1}=J_{2}=0$ and
$\Phi_{\text{so}}=0$ (no SOI). States with total spin $z-$component $S^{z}=1$
and $S^{z}=2$ (grey lines) are separated from the six $S^{z}=0$ states by the
large energy $\mathcal{O}(\bar{h})$. In the inset, we highlight the two
$S^{z}=0$ states (red lines) not belonging to the computational space. These
states have energies $\mathcal{O}(\delta h)$, comparable to the computational
states and cause large leakage in current devices. We use here $(\delta
h,\delta h_{1},\delta h_{2})/\bar{h}=(1/10,\leavevmode\nobreak\
1/20,\leavevmode\nobreak\ 1/20)$. (b) The coupling
$\mathcal{J}\gamma_{\perp}(\Phi)$ to the leading leakage states, and (c) the
effective qubit-qubit interaction $\mathcal{J}\gamma_{\parallel}(\Phi)$
defined in Eq. (ST qubits coupling.). For large SOIs, the SOI rotation angle
can become $\Phi_{\text{so}}=\pi/2$ and the leakage coupling
$\mathcal{J}\gamma_{\perp}(2\Phi_{\text{so}})$ vanishes when the Zeeman field
$\boldsymbol{n}_{B}$ is perpendicular to the SOI-vector
$\boldsymbol{n}_{\text{so}}$ ($\theta=\pi/2$). At this operational sweet spot,
the absolute qubit-qubit interaction
($|\mathcal{J}\gamma_{\parallel}(2\Phi_{\text{so}})|$) is also maximal,
yielding the fastest possible two-qubit gates. (d) Leakage as a function of
time $t$ in systems with large ($\Phi_{\text{so}}\approx\pi/2$) and without
($\Phi_{\text{so}}=0$) SOI. Without SOI, leakage is large and becomes
negligible only at specific system-dependent times (black line) or when a
large Zeeman energy difference between dot 2 and 3 is engineered (blue line).
In our setup, leakage is orders of magnitude smaller (green line) at the sweet
spot, and remains small also without fine-tuning the device (orange line). We
use here the same parameters as (a) with $\bar{h}=20\,\mu$eV.

### Setup.

We consider two ST qubits [54, 55, 53, 52, 56, 57] tunnel coupled via a
superconducting lead, see Fig. 1(a). An ST qubit comprises two exchange-
coupled quantum dots, each containing a single spin, and is accurately modeled
[73] by the Fermi-Hubbard Hamiltonian

$\displaystyle H$ ${}_{\text{DQD}}$ (1) $\displaystyle=$
$\displaystyle\sum_{\alpha\sigma\sigma^{\prime}}\\!\left(\\!\epsilon_{\alpha}\delta_{\sigma\sigma^{\prime}}\\!+\\!\frac{1}{2}\left(\boldsymbol{h}_{\alpha}\cdot\boldsymbol{\sigma}\right)_{\sigma\sigma^{\prime}}\\!\right)\\!d_{\alpha\sigma}^{\dagger}d_{\alpha\sigma^{\prime}}+\\!\mathcal{U}\sum_{\alpha}n_{\alpha\uparrow}n_{\alpha\downarrow}$
$\displaystyle+t_{0}\sum_{\begin{subarray}{c}\sigma\sigma^{\prime}\\\
\alpha\in\\{1,3\\}\end{subarray}}\left(U_{\text{so}}^{\sigma\sigma^{\prime}}(\Phi_{\text{so}})\,d_{\alpha+1,\sigma}^{\dagger}d_{\alpha\sigma^{\prime}}+\text{H.c.}\right).$

Here, $d_{\alpha\sigma}^{\dagger}$ creates a particle with spin
$\sigma\in\\{\uparrow,\downarrow\\}$ on the dot $\alpha=\\{1,2\\}$
($\alpha=\\{3,4\\}$) for the first (second) qubit with energy
$\epsilon_{\alpha}<0$ and Kronecker delta $\delta_{\sigma\sigma^{\prime}}$.
The spin states are split by the Zeeman field
$\boldsymbol{h}_{\alpha}=\mu_{B}\hat{g}_{\alpha}\boldsymbol{B}$, with $g$
tensor $\hat{g}_{\alpha}$, produced by an applied magnetic field
$\boldsymbol{B}$. Double-occupation of each dot, specified by the occupation
numbers $n_{\alpha\sigma}=d_{\alpha\sigma}^{\dagger}d_{\alpha\sigma}$, costs
the on-site Coulomb energy $\mathcal{U}$. Crucially, $H_{\text{DQD}}$ includes
tunneling events between the dots. These are parameterized by a real-valued
tunneling amplitude $t_{0}>0$ and by an SOI-induced spin-flip operator
$U_{\text{so}}(\Phi_{\text{so}})=\exp\left(i\Phi_{\text{so}}\boldsymbol{n}_{\text{so}}\cdot\boldsymbol{\sigma}/2\right)$
that rotates the spins around the SOI vector $\boldsymbol{n}_{\text{so}}$ by
the angle $\Phi_{\text{so}}\approx 2L/l_{so}$ [65, 74, 21], with dot-dot
distance $L$, SOI length $l_{so}$ and Pauli vector $\boldsymbol{\sigma}$. We
emphasize that $\Phi_{\text{so}}$ is widely tunable by electrically
controlling the position of the dots or the amplitude of the SOI; large values
of $\Phi_{\text{so}}\sim\pi$, corresponding to complete spin flips, were
recently measured in hole systems [65].

Superconductors are modeled by the mean-field BCS Hamiltonian

$H_{\text{S}}=\sum_{\boldsymbol{k}\sigma}\epsilon_{k}c_{\boldsymbol{k}\sigma}^{\dagger}c_{\boldsymbol{k}\sigma}-\sum_{\boldsymbol{k}}\Delta
c_{\boldsymbol{k}\uparrow}^{\dagger}c_{-\boldsymbol{k}\downarrow}^{\dagger}+\text{H.c.}\
,$ (2)

where $c_{\boldsymbol{k}\sigma}^{\dagger}$ creates an electron with wave
vector $\boldsymbol{k}$, spin $\sigma$, in a superconductor with
superconducting gap $\Delta>0$ and normal-state energy $\epsilon_{k}$. In the
following, $\epsilon_{k}$ and $\epsilon_{\alpha}$ are measured with respect to
the chemical potential.

Each ST qubit is tunnel-coupled to the superconductor as shown in Fig. 1(a),
described by the Hamiltonian

$\displaystyle H_{\text{T}}$
$\displaystyle=t_{S}\sum_{\boldsymbol{k}\sigma\sigma^{\prime}}U_{\text{so}}^{\sigma\sigma^{\prime}}(\Phi_{\text{so}})\left(c_{\boldsymbol{k}\sigma}^{\dagger}d_{2\sigma^{\prime}}+d_{3\sigma}^{\dagger}c_{\boldsymbol{k}\sigma^{\prime}}\right)+\text{H.c.}$
(3)

In analogy to Eq. (1), we account for the SOI-induced spin-flip by the
rotation $U_{\text{so}}(\Phi_{\text{so}})$. For illustration purposes, we
assume that all spin-flips occur with the same angle $\Phi_{\text{so}}$ and
direction $\boldsymbol{n}_{\text{so}}$. In the Supplemental Material (SM)
[75], we analyze the general case, which will leave the main results
unchanged, and we extend our model to Josephson junctions [Fig. 1(b)].

### Superconductor-mediated exchange interactions.

The tunnel coupling to the superconductors affects the spins confined in the
quantum dots [58, 76, 77, 78, 79]. To the lowest order in the tunneling
amplitude, the relevant mechanisms affecting the dots are elastic cotunneling
[80], where one particle tunnels from one dot to the next through an
electronic excitation of the superconductor, and local and crossed Andreev
processes [81, kürtössy2022parallel], where Cooper pairs are split and
recombined in the same and different dots, respectively [82]. In the regime
where $\Delta$ and $\mathcal{U}$ are large, the elastic cotunneling and local
Andreev processes are suppressed, and we can focus on crossed Andreev
processes only [82].

Crossed Andreev processes cause effective spin-spin correlations between
distant dots,

$H_{\text{CA}}=\Gamma_{\text{CA}}\left(\begin{array}[]{c}-d_{3\downarrow}\\\
d_{3\uparrow}\end{array}\right)^{T}U_{\text{so}}(2\Phi_{\text{so}})\left(\begin{array}[]{c}d_{2\uparrow}\\\
d_{2\downarrow}\end{array}\right)+\text{H.c.}$ (4)

The spin rotation $U_{\text{so}}(2\Phi_{\text{so}})$ combines two of the SOI
rotations in $H_{T}$ [Eq. (3)]. The coupling strength is
$\Gamma_{\text{CA}}=\pi t_{S}^{2}\rho_{F}$, where $\rho_{F}$ is the normal
density of states per spin of the superconductors. We consider the width of
the superconductors $w$ to be smaller than the superconducting coherence
length $\xi$, in the micrometer range [83, 84]. Beyond that, the resulting
interaction decreases exponentially $\propto e^{-2w/\xi}$.

When all quantum dots are occupied by a single particle in their ground state,
we derive the effective four-spin Hamiltonian for small $t_{0},\
\Gamma_{\text{CA}}\ll|\epsilon_{\alpha}|,\ \mathcal{U},\ h_{\alpha}$,

$\displaystyle H_{\text{spin}}=$
$\displaystyle\frac{1}{2}\sum_{\alpha}\boldsymbol{h}_{\alpha}\cdot\boldsymbol{\sigma}^{\alpha}+\frac{\mathcal{J}}{4}\boldsymbol{\sigma}^{2}\cdot\hat{R}_{\text{so}}(2\Phi_{\text{so}})\boldsymbol{\sigma}^{3}$
$\displaystyle+\frac{J_{1}}{4}\boldsymbol{\sigma}^{1}\cdot\hat{R}_{\text{so}}(\Phi_{\text{so}})\boldsymbol{\sigma}^{2}+\frac{J_{2}}{4}\boldsymbol{\sigma}^{3}\cdot\hat{R}_{\text{so}}(\Phi_{\text{so}})\boldsymbol{\sigma}^{4},$
(5)

with spin operators $\boldsymbol{\sigma}^{\alpha}$ of dot $\alpha$, energy
detunings $\tilde{\epsilon}_{1}=\epsilon_{1}-\epsilon_{2}$ and
$\tilde{\epsilon}_{2}=\epsilon_{3}-\epsilon_{4}$, and coupling constants

$J_{i}=\frac{4t_{0}^{2}\mathcal{U}}{\mathcal{U}^{2}-\tilde{\epsilon}_{i}^{2}}\
\text{and}\
\mathcal{J}=\frac{-4\Gamma_{\text{CA}}^{2}\mathcal{U}}{(\epsilon_{2}+\epsilon_{3})(2\mathcal{U}+\epsilon_{2}+\epsilon_{3})}.$
(6)

Importantly, the exchange interactions are anisotropic and are given by the
rotation matrices $\hat{R}_{\text{so}}(\Phi)$, which describe right-handed
rotations around the vector $\boldsymbol{n}_{\text{so}}$ of an angle $\Phi$
[65]. The energy spectrum of $H_{\text{spin}}$, highlighting the relevant
computational states, is shown in Fig. 2(a), and depends on the global,
averaged Zeeman splitting $\bar{h}$ as well as the Zeeman energy differences
$\delta h_{1}=h_{1}-h_{2}$, $\delta h=h_{2}-h_{3}$ and $\delta
h_{2}=h_{3}-h_{4}$.

We anticipate that in the setup sketched in Fig. 1(b), where a single
superconductor is substituted by a Josephson junction, the effective exchange
$\mathcal{J}$ becomes externally controllable by the superconducting phase
difference $\varphi$, enabling on-demand switching on and off of these
interactions, see SM [75]. We also emphasize that this effective model [Eq.
(Superconductor-mediated exchange interactions.)] is equivalent to a chain of
four QDs that are directly coupled by exchange interactions, as shown in Fig.
1(c). Consequently, our approach to reduce leakage and to achieve high-
fidelity two-qubit gates is valid also in these systems.

### ST qubits coupling.

We now derive the effective coupling between our two ST qubits. To simplify
the discussion, here we assume that the Zeeman fields of each spin $\alpha$
are aligned along the direction $\boldsymbol{n}_{B}$ but can change in
magnitude, i.e. $\boldsymbol{h}_{\alpha}=h_{\alpha}\boldsymbol{n}_{B}$; the
general case is discussed in the SM [75] and only the rotation matrices
$\hat{R}_{\text{so}}$ are renormalized.

When the global Zeeman field is much larger than the exchange couplings, we
can project $H_{\text{spin}}$ onto the computational subspace of the ST qubits
yielding the two-qubit Hamiltonian

$H_{\text{ST}}=\frac{1}{2}\boldsymbol{\mathcal{B}}_{1}\cdot\boldsymbol{\tau}^{1}+\frac{1}{2}\boldsymbol{\mathcal{B}}_{2}\cdot\boldsymbol{\tau}^{2}+\frac{\mathcal{J}\gamma_{\parallel}(2\Phi_{\text{so}})}{4}\tau_{z}^{1}\tau_{z}^{2}\
,$ (7)

where $\boldsymbol{\tau}^{1}$ ($\boldsymbol{\tau}^{2}$) is the Pauli vector
acting on the computational space of the first (second) ST qubit, spanned by
the states
{$|\\!\\!\uparrow_{1}\downarrow_{2}\rangle,|\\!\\!\downarrow_{1}\uparrow_{2}\rangle$}
({$|\\!\\!\uparrow_{3}\downarrow_{4}\rangle$,
$|\\!\\!\downarrow_{3}\uparrow_{4}\rangle$}), with
$|\\!\uparrow_{\alpha}\rangle$ pointing along $\boldsymbol{n}_{B}$. The
single-qubit terms
$\boldsymbol{\mathcal{B}}_{i}=\left(J_{i}\text{Re}\left[\gamma_{\perp}(\Phi_{\text{so}})\right],J_{i}\text{Im}\left[\gamma_{\perp}(\Phi_{\text{so}})\right],\delta
h_{i}\right)$ enable single qubit rotations. The superconductor mediates
effective Ising qubit-qubit interactions
$\propto\mathcal{J}\gamma_{\parallel}(2\Phi_{\text{so}})$ via crossed Andreev
reflection. Single- and two-qubit gates are enabled by controlling $J_{i}$ and
$\mathcal{J}$, respectively. Importantly, the SOI induces an anisotropic
factor described by the dimensionless functions

$\displaystyle\gamma_{\parallel}(\Phi)$
$\displaystyle=2\sin^{2}(\theta)\sin^{2}(\Phi/2)-1\ ,$
$\displaystyle\gamma_{\perp}(\Phi)$
$\displaystyle=\left[\cos\left(\Phi/2\right)+i\cos(\theta)\sin\left(\Phi/2\right)\right]^{2}.$
(8)

These functions are shown in Fig. 2(b)-(c), respectively. We emphasize that
they are fully controllable by the SOI angle $\Phi_{\text{so}}$ and by the
direction $\theta$ of the Zeeman vector $\boldsymbol{n}_{B}$ relative to the
SOI vector $\boldsymbol{n}_{\text{so}}$ (Fig. 1). In the SM [75] we calculate
higher-order corrections to the qubit-qubit interaction and the leakage
coupling.

Figure 3:  Two-qubit gates. (a) Control of qubit-qubit interactions
$\mathcal{J}$ by tuning the superconducting phase difference $\varphi$, see
Fig. 1(b). The interaction can be precisely switched on and off, removing
residual interactions and crosstalk that hinder scalability. In current
devices $\mathcal{J}\approx 0.4$ $\mu$eV, so short gate times of $T_{g}\sim 5$
ns are within reach. (b)-(c) Two-qubit gate infidelity $1-F$. (b) The large
SOI enables high-fidelity gates for a wide range of parameters close to the
sweet spot at $\Phi_{\text{so}}=\theta=\pi/2$. Here, $\delta h/\mathcal{J}=5$.
(c) Comparing the performance of different approaches, we observe that our
setup consistently outperforms other devices, yielding fidelities orders of
magnitude larger even without precisely fine-tuning the system to the sweet
spot. We used here $\bar{h}/\mathcal{J}=50$, $J_{1}=J_{2}=0$, and $(\delta
h_{1},\leavevmode\nobreak\ \delta h_{2})/\delta h=(1/2,\leavevmode\nobreak\
1/2)$.

### Leakage suppression by SOI.

The projection onto the ST qubit subspace leading to $H_{\text{ST}}$ is valid
when the non-computational states are decoupled from the computational space.
In a single ST qubit and at typical values of $\boldsymbol{B}$, the non-
computational subspace is well-separated by the large total Zeeman energy
$h_{\alpha}\sim 5$GHz, orders of magnitude larger than the characteristic
energy of $H_{\text{ST}}$, in the $10$ to $100$ MHz range. However, in two ST
qubits, there are two non-computational states in the computational energy
window, resulting in large leakage. These states
($|\\!\uparrow_{1}\uparrow_{2}\downarrow_{3}\downarrow_{4}\rangle$ and
$|\\!\downarrow_{1}\downarrow_{2}\uparrow_{3}\uparrow_{4}\leavevmode\nobreak\
\rangle$) are not affected by the large total Zeeman energy because they have
zero total spin, $S^{z}=0$, see Fig. 2(a).

This critical flaw of ST architectures is often addressed by fine-tuning
isotropic exchange interaction and magnetic field such that the leakage
vanishes at the target evolution time [54, 71, 72], see black and blue curves
in Fig. 2(d). Alternatively, leakage can be partially reduced by engineering a
large Zeeman energy difference $\delta h$ [67, 68, 69, 70]. These approaches,
however, require extremely precise control over $g$ factors and over exchange,
which is challenging to achieve in current experiments. Capacitive and
resonator-mediated couplings [85, 86, 87, 88] only yield weak interaction
strengths and result in slow gates.

In striking contrast, in our system the strong SOI offers a compelling way to
remove leakage, fully compatible with the state-of-the-art [65]. While the
qubit-qubit interactions are determined by
$\mathcal{J}\gamma_{\parallel}(2\Phi_{\text{so}})$, leakage is determined by
the matrix elements that couple the computational states
$|\\!\uparrow_{1}\downarrow_{2}\uparrow_{3}\downarrow_{4}\rangle$ and
$|\\!\downarrow_{1}\uparrow_{2}\downarrow_{3}\uparrow_{4}\rangle$ to the non-
computational states
$|\\!\uparrow_{1}\uparrow_{2}\downarrow_{3}\downarrow_{4}\leavevmode\nobreak\
\rangle$ and
$|\\!\downarrow_{1}\downarrow_{2}\uparrow_{3}\uparrow_{4}\rangle$, and are
given by $\mathcal{J}\gamma_{\perp}(2\Phi_{\text{so}})$ [75]. By operating the
systems at $\Phi_{\text{so}}=\pi/2$ and at
$\boldsymbol{n}_{B}\perp\boldsymbol{n}_{\text{so}}$ ($\theta=\pi/2$), one
maximizes the two-qubit interaction
[$\mathcal{J}\gamma_{\parallel}(\pi)=\mathcal{J}$] and suppresses leakage
[$\mathcal{J}\gamma_{\perp}(\pi)=0$]. At this operational sweet spot,
particles from dot 2 and 3 make a rotation of $\pi/2$ when tunneling to the
superconductor, but in opposite directions. Since Cooper pairs can only be
created from particles in the superconductor with opposite spin,
$|\\!\\!\downarrow_{2}\uparrow_{3}\rangle$ and
$|\\!\\!\uparrow_{2}\downarrow_{3}\rangle$ will not be affected by crossed
Andreev processes. In contrast, $|\\!\\!\downarrow_{2}\downarrow_{3}\rangle$
and $|\\!\\!\uparrow_{2}\uparrow_{3}\rangle$ will couple to Cooper pairs and
experience a shift in energy, resulting in the Ising qubit-qubit interaction
required for two-qubit gates. Although the anisotropic exchange also causes
leakage to states with $S^{z}\neq 0$ (e.g.
$|\\!\\!\uparrow_{1}\uparrow_{2}\uparrow_{3}\uparrow_{4}\rangle$), this
contribution is small because those states are at high energies, determined by
the large global Zeeman field, see Fig. 2(a).

As shown in Fig. 2(d), our SOI-induced leakage suppression can significantly
outperform current alternative approaches, removing also technologically
demanding constraints on the engineering of the devices. We also emphasize
that while leakage is minimal at $\Phi_{\text{so}}=\pi/2$ and
$\boldsymbol{n}_{B}\perp\boldsymbol{n}_{\text{so}}$, conditions achieved in
recent experiments [65], our system does not require precise fine-tuning and
leakage is significantly lower than the state-of-the-art for a wide range of
$\Phi_{\text{so}}$.

##  I Controlling exchange by Josephson junctions

The other critical obstacle to scaling up current ST architectures is the
residual exchange interaction between two qubits, which yields a dangerous
always-on coupling between ST qubits. Our hybrid device sketched in Fig. 1(b)
removes this issue by taking full advantage of the phase-tunability of
Josephson junctions. In this case, the effective exchange interaction
$\mathcal{J}$ becomes dependent on the superconducting phase difference
$\varphi$, yielding
$\mathcal{J}\to\mathcal{J}(\varphi)=4\mathcal{J}\cos^{2}\left(\frac{\varphi}{2}\right)$,
see the SM [75] for a detailed derivation. This interaction is thus maximal
when $\varphi=0$, and vanishes at $\varphi=\pi$, as shown in Fig. 3(a).
Because $\varphi$ is accurately controllable in experiments, our setup offers
a long-sought way to on-demand switch interactions on and off, removing the
main source of crosstalk in future large-scale ST qubit architectures.

### Two-qubit gates.

The SOI-induced anisotropy and the phase-tunability of the effective exchange
interactions in our setup enable fast and high-fidelity two-qubit gates, see
Fig. 3(b)-(c). By using realistic parameters $\epsilon_{\alpha}\approx-20$
$\mu$eV, $\Gamma_{\text{CA}}\approx 2$ $\mu$eV and assuming
$\mathcal{U},\leavevmode\nobreak\ \Delta\gg\epsilon_{\alpha}$, we estimate
that $\mathcal{J}$ in Eq. (6) can reach values up to $\mathcal{J}\approx 0.4$
$\mu$eV. Fast controlled-Z (cZ) entangling gates are then enabled by turning
the interactions on for a time
$T_{g}=\hbar\pi/|\mathcal{J}\gamma_{\parallel}(2\Phi_{\text{so}})|$, which may
take around $T_{g}\approx 5$ ns only. We estimate the fidelity by

$\displaystyle
F=\left|\frac{1}{4}\text{tr}(U_{\text{cZ}}^{\dagger}U_{\text{spin}})\right|^{2},$
(9)

where $U_{\text{cZ}}=\exp(-i\pi\tau_{z}^{1}\tau_{z}^{2}/4)$ is the ideal cZ
gate (up to single-qubit operations [5]) and
$U_{\text{spin}}=P_{\text{comp}}\exp(-iH_{\text{spin}}T_{g}/\hbar)P_{\text{comp}}$
is the time evolution generated by $H_{\text{spin}}$ in Eq. (Superconductor-
mediated exchange interactions.) projected onto the computational subspace by
the projection operator $P_{\text{comp}}$. Our approach accurately captures
leakage because $H_{\text{spin}}$ includes all $2^{4}$ spin states.

As shown in Fig. 3(c), our ST qubits substantially outperform current state-
of-the-art, reaching two orders of magnitude smaller values of infidelities,
below $10^{-4}$ at the optimal parameter spot. This value is limited by our
conservative choice of the global Zeeman energy $\bar{h}=50\mathcal{J}\sim
20\,\mu$eV, in contrast to current implementations where the fidelity is
limited by $\delta h\ll\bar{h}$. The infidelity remains below $10^{-3}$ for
small Zeeman energy differences $\delta h$ at values of $\Phi_{\text{so}}$ and
$\theta$ deviating up to $10\%$ from the sweet spot, demonstrating that our
approach does not require precise fine-tuning of the device, and that
fidelities larger by more than two orders of magnitude are within reach in
current experiments.

### Conclusion

In conclusion, hybrid ST qubit architectures comprising semiconducting quantum
dots with large SOI and superconductors can substantially outperform current
devices. In particular, the superconductor mediates correlations of distant
qubits via crossed Andreev processes. These processes are externally
controllable by the phase difference in Josephson junctions, removing
dangerous crosstalk caused by always-on residual qubit-qubit interactions. We
also show that large SOIs induce tunable anisotropies in these interactions
that strongly suppress leakage. When combined, these effects result in fast
and high-fidelity two-qubit gates, orders of magnitude more efficient than the
state-of-the-art, and could provide a significant step forward toward
implementing large-scale ST qubit quantum processors.

##  II Acknowledgements

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 (grant number
51NF40-180604). Finally, this work was financially supported by the JSPS
Kakenhi Grant No. 19H05610.

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