2023

These authors contributed equally to this work.

These authors contributed equally to this work.

These authors contributed equally to this work.

These authors contributed equally to this work.

[1]Leo P. Kouwenhoven

1]QuTech and Kavli Institute of NanoScience, Delft University of Technology,
2600 GA Delft, The Netherlands

2]Department of Applied Physics, Eindhoven University of Technology, 5600 MB
Eindhoven, The Netherlands

# Robust poor man’s Majorana zero modes using Yu-Shiba-Rusinov states

Francesco Zatelli  David van Driel  Di Xu  Guanzhong Wang  Chun-Xiao Liu
Alberto Bordin  Bart Roovers  Grzegorz P. Mazur  Nick van Loo  Jan C. Wolff
A. Mert Bozkurt  Ghada Badawy  Sasa Gazibegovic  Erik P. A. M. Bakkers
Michael Wimmer<EMAIL_ADDRESS>Tom Dvir  [  [

###### Abstract

The recent realization of a two-site Kitaev chain featuring “poor man’s
Majorana” states demonstrates a path forward in the field of topological
superconductivity. Harnessing the potential of these states for quantum
information processing, however, requires increasing their robustness to
external perturbations. Here, we form a two-site Kitaev chain using
proximitized quantum dots hosting Yu-Shiba-Rusinov states. The strong
hybridization between such states and the superconductor enables the creation
of poor man’s Majorana states with a gap larger than
$70\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}$. It also greatly
reduces the charge dispersion compared to Kitaev chains made with non-
proximitized quantum dots. The large gap and reduced sensitivity to charge
fluctuations will benefit qubit manipulation and demonstration of non-abelian
physics using poor man’s Majorana states.

Kitaev chains based on quantum dots (QDs) coupled via a hybrid semiconductor-
superconductor heterostructure are a promising avenue for the creation of
Majorana bound states Kitaev.2001 ; Sau.2012 . Even a minimal chain,
consisting of only two QDs, supports fine-tuned Majorana zero modes known as
“poor man’s Majoranas” (PMMs) Leijnse.2012 . These PMM states do not benefit
from topological protection, but already exhibit robustness to local
perturbations and quadratic protection from global fluctuations in the
chemical potential Leijnse.2012 . Moreover, PMM states obey non-abelian
exchange statistics, thus providing a favourable platform for braiding and
Majorana-based qubit experiments in the near future Liu.2022.Fusion ;
Boross.2023 ; Tsintzis.2023 ; Pino.2023 . In Ref. Dvir.2023 , we have recently
realized the two-site Kitaev chain by coupling two QDs formed in an InSb/Al
hybrid nanowire. While providing the necessary proof of concept, the gap
separating PMMs from excited states is not much higher than the electron
temperature. Furthermore, appreciable noise in the measured spectra indicate
sensitivity to charge fluctuations, making practical use of such states
infeasible.

Insensitivity to charge noise has been successfully achieved in transmon
qubits by increasing the ratio between superconducting coupling (Josephson
energy) and charging energy, thereby suppressing the charge dispersion of the
spectrum Koch.2007 ; schreier2008Suppressing . We follow a similar approach by
strongly coupling our QDs to the superconductor. This induces superconducting
correlations on the QDs leading to energy eigenstates known as Yu-Shiba-
Rusinov (YSR) states Yu1965bound ; shiba1968classical ;
rusinov1969superconductivity ; Buitelaar.2002 ; Bauer.2007 ; meng2009self ;
GroveRasmussen.2009 ; Chang.2013 ; Jellinggaard.2016 ; Grove-Rasmussen.2018 .
Spin-polarized YSR states can also serve as the sites of a Kitaev chain, as
investigated theoretically Fulga.2013 ; Nadj-Perge.2013 ; Pientka.2015 ;
schecter2016self ; Liu.2022 ; Tsintzis.2022  and experimentally using scanning
tunneling microscopy NadjPerge2014 ; ruby2015end ; Jeon.2017 ;
schneider2021Topological ; schneider2022Precursors . In this work, we realize
a two-site Kitaev chain using YSR states in a hybrid semiconducting InSb
nanowire partly covered with a superconducting Al film. We show that the
sensitivity of the resulting PMMs to charge fluctuations affecting the QDs
decreases by two orders of magnitude compared to a QD-based Kitaev chain
Dvir.2023 . In addition, we measure a gap between ground and excited states of
$E_{\mathrm{gap}}=$76\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}$$,
a threefold increase compared to our previous report Dvir.2023 . Complementary
results are also reported in a parallel study on a 2D hybrid platform using an
InAsSb/Al two-dimensional electron gas tenHaaf.2023 , demonstrating the wider
applicability of our approach.

Figure 1: YSR states formed by hybridizing QDs with an ABS. a, False-colored
scanning electron micrograph and measurement circuit of the device. Scale bar
is $200\text{\,}\mathrm{n}\mathrm{m}$. b, Illustration of the QD-ABS-QD model.
Two QDs are coupled to the same ABS via spin-conserving and spin-flipping
tunneling. QD and YSR regimes are defined by the relative energy scales shown
below the sketch. c, Numerical CSD using zero-bias conductance of a QD coupled
to an ABS. “E”/“O” indicate even/odd occupation of the QD. d, e, Numerical
conductance spectrum of a QD coupled to an ABS as a function of QD chemical
potential when ABS is far (d) and close to its energy minimum (e). In e,
green/pink arrows indicate avoided crossings due to spin-conserving/spin-
flipping tunneling. f–h, Same as c–e for the left QD coupled to an ABS,
experimentally measured using local conductance. Gray/blue ticks in f mark
$V_{\mathrm{H}}$ values at which g/h are measured. The right QD is off
resonance. In h, we extract a lever arm $\alpha\approx 0.05e$ by fitting the
spectrum with the black dotted line at the zero-energy crossing. This
procedure overestimates the actual lever arm since it does not account for
capacitance to the normal lead. For comparison, we plot the profile of a
typical Coulomb diamond in our device with gray dotted lines (see Fig. ED3a).
The $V_{\mathrm{LP}}$ range in g is shifted because of a gate jump affecting
the left QD. i–k, Same as f–h, but for the right QD. The extracted lever arm
is $\alpha\approx 0.04e$.

##  1 Fabrication and setup

Fig. 1a shows a scanning electron microscope image of the reported device. It
consists of an InSb nanowire (green) placed on top of a series of bottom gates
Badawy.2019 . The middle part of the nanowire is covered by a thin Al shell
(blue), forming a superconductor-semiconductor hybrid whose electrochemical
potential is controlled by a plunger gate ($V_{\mathrm{H}}$). On both sides of
the hybrid segment, QDs are formed in the nanowire using three gates each. The
electrochemical potential of each QD is controlled by a plunger gate
($V_{\mathrm{LP}}$ and $V_{\mathrm{RP}}$ for the left and right QDs,
respectively), and the couplings between the QDs and the hybrid are controlled
by tunnel gates ($V_{\mathrm{LT}}$ and $V_{\mathrm{RT}}$ for the left and
right QDs, respectively). A normal lead is attached to each QD, separated by
another gate-defined tunnel barrier. The superconducting lead is kept grounded
at all times. In addition, the two normal leads are connected to off-chip
multiplexed resonators for fast RF reflectometry measurements Hornibrook.2014
, using the setup described in Ref. wang2022parametric . Each lead is voltage
biased independently with respect to the grounded Al, with voltages
$V_{\mathrm{L}}$ and $V_{\mathrm{R}}$ on the left and right leads,
respectively. The currents ($I_{\mathrm{L}}$ and $I_{\mathrm{R}}$ on the left
and right leads), the local conductances
($G_{\mathrm{LL}}=dI_{\mathrm{L}}/dV_{\mathrm{L}}$,
$G_{\mathrm{RR}}=dI_{\mathrm{R}}/dV_{\mathrm{R}}$), and the non-local
conductances ($G_{\mathrm{RL}}=dI_{\mathrm{R}}/dV_{\mathrm{L}}$,
$G_{\mathrm{LR}}=dI_{\mathrm{L}}/dV_{\mathrm{R}}$) are measured
simultaneously. When the full conductance matrix is measured, we correct for
line resistance as described in the Methods section. Further fabrication and
setup details can be found in our previous publications Heedt.2021 ;
Borsoi.2021 ; Mazur.2022 . The experiment is conducted in a dilution
refrigerator with a base temperature of $30\text{\,}\mathrm{m}\mathrm{K}$. A
magnetic field of $B=$150\text{\,}\mathrm{m}\mathrm{T}$$ is applied along the
nanowire axis, inducing a Zeeman splitting of approximately
$200\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}$ in the QDs (Fig.
ED1).

##  2 YSR states in quantum dots

We model our system using a three-site model in which the hybrid is considered
as a single Andreev bound state (ABS) in the atomic limit Bauer.2007 ;
meng2009self  tunnel-coupled to two QDs (see schematics in Fig. 1b)
Tsintzis.2022 . The QDs have charging energy $U$, Zeeman splitting
$E_{\mathrm{Z}}$, and chemical potentials $\mu_{\mathrm{L}},\mu_{\mathrm{R}}$
for the left and right QD, respectively. The ABS has an induced gap $\Gamma$,
which in the atomic limit Bauer.2007 ; meng2009self  can be identified with
its tunnel coupling to the bulk superconductor. Its charging energy is
negligible due to the screening of the grounded Al film. It also has Zeeman
splitting $E_{\mathrm{ZH}}$, which is smaller than that of the QDs due to
metallization of the ABS Reeg.2018 . We ensure $E_{\mathrm{ZH}}<\Gamma$ so
that the ground state of the ABS is always a BCS singlet. The electron-hole
composition of the ABS depends on its chemical potential $\mu_{\mathrm{H}}$.

In our model, the QDs are coupled to the ABS by spin-conserving and spin-
flipping tunneling due to spin-orbit interaction, with amplitudes $w$ and
$w_{\mathrm{SO}}$, respectively Tsintzis.2022 . The hybridization between QDs
and the ABS becomes significant when $w,w_{\mathrm{SO}}\sim\Gamma$. As a
consequence, the QDs become proximitized and form YSR states Grove-
Rasmussen.2018 ; Liu.2023 , which we distinguish from ABSs because of their
large charging energy Bauer.2007 ; meng2009self ; GroveRasmussen.2009 ;
Chang.2013 ; kirsanskas2015YuShibaRusinov ; Jellinggaard.2016 . We refer
readers to the Supplementary Information and the parallel work of Ref.
Liu.2023  for theory models of the strong coupling regime investigated in this
work.

To understand the nature of the YSR states formed in proximitized QDs, we
first examine the coupling between a single QD and an ABS in the hybrid
segment, following Ref. Grove-Rasmussen.2018 . Fig. 1c shows the theoretical
zero-bias conductance of a QD-ABS charge stability diagram (CSD), while the
second QD is off-resonance. The two vertical features indicate parity
transitions of the system, largely corresponding to the consecutive filling of
a single orbital of the QD by two electrons. The S-shaped conductance features
result from renormalization of the QD energy via hybridization with the ABS.
The ABS reaches its energy minimum at charge neutrality, i.e.,
$\mu_{\mathrm{H}}=0$, where its excitation is equal-parts electron and hole.
The hybridization with the QD is also maximal here due to their minimal energy
separation, evident in the enhanced zero-bias conductance as local Andreev
reflection becomes stronger Grove-Rasmussen.2018 ; Liu.2023 . The QD spectra
in panels d and e further reveal its hybridization with the ABS. When the
latter is away from charge neutrality, the QD spectrum as a function of
$\mu_{\mathrm{L}}$ (Fig. 1d) exhibits straight features reminiscent of Coulomb
diamonds. As the ABS approaches $\mu_{\mathrm{H}}=0$ (Fig. 1e), such features
evolve into an eye-shaped spectrum typical of YSR states Pillet.2010 ;
Lee.2014 ; Jellinggaard.2016 ; Scherbl2020 ; Scherbl2022 . The arrows indicate
avoided crossings in the excited states of the spectrum produced by spin-
conserving (green) and spin-flipping (pink) tunneling between the ABS and the
QD levels Liu.2023 .

The hybrid segment of our device features multiple discrete ABSs well-
separated from each other in $V_{\mathrm{H}}$ (Fig. ED2). We operate in a
$V_{\mathrm{H}}$ range containing a single ABS level. In Fig. 1f, we show the
zero-bias conductance measured with the left lead as a function of
$V_{\mathrm{LP}}$ and $V_{\mathrm{H}}$. $V_{\mathrm{RP}}$ is fixed to keep the
right QD off-resonance. We observe the QD-ABS charge stability diagram
features described in panel c, indicating the presence of an ABS reaching its
energy minimum at $V_{\mathrm{H}}\approx$337\text{\,}\mathrm{m}\mathrm{V}$$.
The QD-ABS hybridization is further confirmed by the QD spectrum being in
agreement with the model when the ABS is away from (Fig. 1g) and at its energy
minimum (Fig. 1h).

The YSR zero-energy excitations in Fig. 1h are our building blocks of a Kitaev
chain. Compared to a non-proximitized QD zero-energy crossing, these YSR
crossings have noticeably weaker energy dispersion as a function of gate (see
dashed lines in panel h). To quantify this observation, we can estimate the
lever arm of the YSR excitation at charge degeneracy using its gate-dispersion
slope: $\alpha\equiv\partial E/\partial V_{\mathrm{LP}}$. In contrast to the
above-gap QD lever arm of $\alpha\approx 0.4e$ in our devices (Fig. ED3a),
this subgap lever arm reduces to $\alpha\approx 0.2e$ (Fig. ED3b) when the ABS
is detuned from its charge neutrality. Tuning the ABS to its energy minimum
further reduces the lever arm to $\alpha\approx 0.05e$ (Fig. 1h). This signals
a strong reduction in the effective charge of the fermionic excitation,
attributable to charging energy renormalization, QD-ABS hybridization, and
electron-hole superposition Schindele.2014 ; gramich2017Andreev , as detailed
in Fig. ED4.

The right QD shows similar behavior when coupled to the same ABS (Fig. 1i–k).
However, when the ABS is away from charge neutrality, the subgap conductance
is not significantly suppressed (Fig. 1i) and the QD spectrum still shows
typical YSR features (Fig. 1h), albeit with a small superconducting coupling.
This could be due to residual proximity resulting from direct coupling between
the QD and the superconducting film since $V_{\mathrm{H}}$ does not affect it
appreciably (Fig. ED5).

##  3 Coupled YSR states

Figure 2: ECT and CAR coupling between YSR states. a, Illustration of the
effective two-site system of YSR states coupled via ECT ($t$) and CAR
($\Delta$). b–d, Conductance matrices of CSDs measured at different
$V_{\mathrm{H}}$. By tuning the electrochemical potential of the ABS in the
hybrid, it is possible to continuously vary $t$ and $\Delta$. In b, avoided
crossings along the anti-diagonal are observed, indicating $t>\Delta$. The
opposite regime is shown in d, with all the avoided crossings along the
diagonal, indicating $t<\Delta$. The crossover between these two regimes is
shown in c, where two crossings indicate $t\approx\Delta$.

Recent theoretical Liu.2022 ; Tsintzis.2022  and experimental Wang.2022 ;
Dvir.2023 ; Bordin.2022  works have shown that an ABS can mediate elastic co-
tunneling (ECT) and crossed Andreev reflection (CAR) between QDs. These two
processes implement the hopping and pairing terms of the original Kitaev chain
model Kitaev.2001 ; Sau.2012 . To form PMM states in a two-site chain, the
amplitudes of both terms must be equal Leijnse.2012 . Such control can be
achieved by tuning the electrochemical potential of the ABS in the hybrid
nanowire Liu.2022 ; Tsintzis.2022 ; Dvir.2023 ; Bordin.2022 .

Similar effective ECT and CAR couplings, with respective amplitudes $t$ and
$\Delta$, also emerge between YSR states (Fig. 2a) formed by strongly coupling
QDs to the same ABS Trocha2015 ; Scherbl2019 ; Liu.2023 . To observe them, we
turn to the CSD of two such YSR states, akin to those explored in Refs.
EstradaSaldaa2020 ; Krtssy2021 . In Fig. 2b, we show the zero-bias conductance
matrix measured as a function $V_{\mathrm{RP}}$ and $V_{\mathrm{LP}}$ when the
ABS is tuned away from its energy minimum. All elements of the conductance
matrix show prominent resonances arising from the two charge transitions of
each QD. The type of avoided crossings observed in the CSD serves as an
indication of the coupled-YSR system’s ground state Tsintzis.2022 ; Dvir.2023
. Avoided crossings along a negative diagonal, as seen in all four resonances
in Fig. 2b, show hybridization between states with the same total charge. This
is the ground state of the system when $t>\Delta$. This observation is further
confirmed by the negative non-local conductance that is characteristic of ECT
Dvir.2023 .

Increasing the value of $V_{\mathrm{H}}$ leads to a change in the CSD, shown
in Fig. 2c. Here, the bottom-left and the top-right resonances show avoided
crossings along a positive diagonal associated with $\Delta>t$. In the top-
left and bottom-right, the resonance lines cross each other, indicating a PMM
sweet spot, with $t\approx\Delta$. Finally, bringing the ABS close to its
energy minimum by a further increase of $V_{\mathrm{H}}$ tunes all of the
quadrants of the CSD to the $\Delta>t$ regime (Fig. 2d). The theoretical model
reproduces the observed evolution of the CSDs (Fig. ED6a–c).

##  4 Gate control of CAR and ECT

Figure 3: ECT and CAR coupling as a function of the ABS chemical potential. a,
CSD measured using RF reflectometry. The center of the avoided crossing,
marked with a gray cross, is identified as
$\mu_{\mathrm{L}}=\mu_{\mathrm{R}}=0$. b, Spectrum measured on both sides of
the device at the center of the avoided crossing shown in a. The inner peaks
correspond to excited states at $\absolutevalue{t-\Delta}$, while the outer
ones correspond to excited states at $t+\Delta$. By summing and subtracting
these energies, it is possible to extract $t$ and $\Delta$ for each
$V_{\mathrm{H}}$ value. c, Spectrum as a function of $V_{\mathrm{H}}$ while
$\mu_{\mathrm{L}}=\mu_{\mathrm{R}}=0$. Each line of the spectrum was measured
as described above. The blue tick indicates the $V_{\mathrm{H}}$ value at
which a and b were measured. d, ECT and CAR couplings extracted from c. e,
Zoom-in on the zero-energy crossings highlighted in c. The white dotted lines
are fits to extract the slope of the linear splitting. The data reported here
was collected for the bottom-right charge transition shown in Fig. 2. More
details about data processing and data for all charge degeneracies are
reported in Fig. ED7.

ABS-mediated ECT and CAR couplings between QDs are controlled by the chemical
potential of ABS Liu.2022 ; Bordin.2022 . At the energy minimum of the ABS,
its excitation is equally electron- and hole-like, both parts interfering
constructively to enhance CAR and destructively to quench ECT. Finite values
of $\mu_{\mathrm{H}}$ lead to an imbalance between the electron and hole parts
of the ABS, decreasing the value of the CAR coupling while enhancing ECT. This
control over the ECT and CAR amplitudes guarantees a PMM sweet spot when the
two QDs are coupled via a single ABS Liu.2022 ; Bordin.2022 ; Tsintzis.2022 .
To demonstrate that this description can be extended to YSR states coupled via
an ABS, we study how the couplings $t$ and $\Delta$ vary as the
electrochemical potential of the ABS changes.

The magnitudes of $t$ and $\Delta$ can be extracted by measuring the
excitation spectrum of the system at $\mu_{\mathrm{L}}=\mu_{\mathrm{R}}=0$,
when spectral splitting is determined by the couplings alone. We limit the
discussion here to the bottom-right crossing of Fig. 2, noting that the other
crossings exhibit qualitatively similar behavior (Fig. ED7). At each fixed
value of $V_{\mathrm{H}}$, we measure a CSD (Fig. 3a) and set the QD gates to
the center of an avoided crossing. There, the subgap spectrum exhibits two
sets of electron-hole symmetric peaks at energies $\absolutevalue{t-\Delta}$
and $t+\Delta$ (Fig. 3b), as detailed in Methods. By fitting the measured
spectrum with two pairs of Gaussians symmetric around $V_{\mathrm{L,R}}=0$, we
extract the energy of the excited states and calculate $t$ and $\Delta$. We
repeat this procedure for different values of $V_{\mathrm{H}}$ and collect the
spectra in Fig. 3c, where each line was measured as described above.

Fig. 3d shows the extracted values of ECT and CAR amplitudes. At
$V_{\mathrm{H}}\approx$322\text{\,}\mathrm{m}\mathrm{V}$$ and
$V_{\mathrm{H}}\approx$355\text{\,}\mathrm{m}\mathrm{V}$$, the outer peaks
almost merge with the inner ones, indicating $\absolutevalue{t-\Delta}\approx
t+\Delta$. Inspecting the corresponding CSDs (see Fig. 2b for an example, all
the CSDs are available in the online repository), we can see that ECT is
dominant and, therefore, $t\gg\Delta$. Upon varying $V_{\mathrm{H}}$, the two
peaks split into four well-separated peaks, signaling that CAR is increasing
although ECT still prevails. At
$V_{\mathrm{H}}\approx$333\text{\,}\mathrm{m}\mathrm{V}$$ and
$345\text{\,}\mathrm{m}\mathrm{V}$, the two inner peaks merge into a single
zero-bias peak. These are two sweet spots where $t\approx\Delta$. In between
them, CAR dominates, as the CSDs can confirm (see Fig. 2d). Around
$V_{\mathrm{H}}\approx$338\text{\,}\mathrm{m}\mathrm{V}$$, the outer peaks
merge again, this time indicating $\Delta\gg t$. This feature is the peak in
$\Delta$ and dip in $t$ seen in Fig. 3d, because of the aforementioned
interference effects Liu.2022 ; Bordin.2022 . Finally, Fig. ED6d shows that
the same spectral features are reproduced by the theoretical model.

The zero-energy crossings in Fig. 3e allow us to characterize the robustness
of the PMM sweet spots to charge fluctuations affecting the ABS and,
consequently, $t$ and $\Delta$, causing a splitting of the zero-energy states.
Indeed, we observe that each zero energy state splits with a linear dependence
on $V_{\mathrm{H}}$ as predicted by theory, indicating the lack of protection
against deviations from the condition $t=\Delta$ known for a two-site Kitaev
chain Leijnse.2012 . We extract a slope of $\frac{\partial E}{\partial
V_{\mathrm{H}}}\approx$7\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{m}\mathrm{V}$$
and
$3\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{m}\mathrm{V}$
for the two crossings, comparable to what we measured in Ref. Dvir.2023 .

##  5 Majorana sweet spot

Figure 4: Poor man’s Majorana sweet spot for coupled YSR states. a,
Conductance matrix of a crossing in the CSD, when $t\approx\Delta$. b,
Spectrum of both sides measured at the sweet spot in the center of the
crossing of panel a. c, Conductance matrix of the spectrum as a function of
$V_{\mathrm{RP}}$ (along the gray dotted line in a). The right QD is detuned
across the sweet spot, while the left one is kept on resonance. d, Conductance
matrix of the spectrum measured detuning both QDs simultaneously along the
antidiagonal along the pink dashed line in a. e, f, $G_{\mathrm{RR}}$ as a
function of the simultaneous detunings of both sites, $\delta V_{\mathrm{LP}}$
and $\delta V_{\mathrm{RP}}$, away from the sweet spot, using QDs (e,
replotted from Ref. Dvir.2023 ) and YSR states (f, from the same dataset
presented in d). For comparison, we have plotted the two gate dispersions with
the same scale in both plots. The green dotted lines correspond to the
expected energy splitting of a PMM realized with QDs, the blue dashed lines to
that of a PMM realized with YSR states, as detailed in the Methods section.

Finally, we measure the spectrum and stability of the PMM states against
perturbation of the QDs. We tune our device to the sweet spot at
$V_{\mathrm{H}}=$333.5\text{\,}\mathrm{m}\mathrm{V}$$, where $t\approx\Delta$.
In Fig. 4a, we show the full conductance matrix of the CSD at the sweet spot.
As expected, the resonance lines cross each other and non-local conductance
alternates between positive and negative values. The spectrum measured at the
sweet spot in the center of the CSD (Fig. 4b) shows, on both sides of the
device, a zero-bias conductance peak clearly separated from the excited
states. As a result of the much stronger coupling between YSR states, the
first excited states reside at energies of
$E_{\mathrm{gap}}\approx$76\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}$$,
three times larger than in our previous report Dvir.2023  and significantly
above the electron temperature.

One of the hallmarks of PMM states is their stability against local
perturbations. In Fig. 4c, we measure the spectrum of the QDs varying
$V_{\mathrm{RP}}$ while keeping the left QD on resonance. The observed zero-
bias conductance peak persists within the range of the investigated charge
degeneracy. Tuning $V_{\mathrm{LP}}$ while keeping the right QD on resonance
(Fig. ED8b) shows the same qualitative features. The same behavior can be
reproducibly observed with other QD and ABS orbitals, as shown in Fig. ED9.

If the electrochemical potentials of both QDs are detuned from the sweet spot,
PMM states are expected to split quadratically. This is verified in Fig. 4d,
where we measure the spectrum while detuning both QDs along the antidiagonal
path shown in Fig. 4a. The spectrum taken along the diagonal path is shown in
Fig. ED8c. Numerical simulations reproducing these measurements are reported
in Fig. ED6e–i. For comparison, we plot the quadratic dispersion measurements
of the QD-based vs YSR-based PMMs side by side in panels e and f. The
curvature of the energy-gate dispersion close to the sweet spot is directly
proportional to the dephasing rate resulting from charge noise affecting both
QDs Aasen.2016 ; Knapp.2018 ; Mishmash.2020 ; Boross.2022  and is therefore a
measure of the Majorana states’ robustness against it. Comparing the overlaid
curves in Fig. 4e and f, we find the gate dispersion curvature reported in
this work to be a factor of $\sim 150$ lower than the non-proximitized case
(details in Materials and Methods). This striking reduction can be fully
explained by the decreased lever arm of the YSR states, $\alpha$ and the
increased $E_{\mathrm{gap}}$, since the curvature expected from the theory
model is $\alpha^{2}/E_{\mathrm{gap}}$ (see Materials and Methods).

To illustrate the effect of reduced charge dispersion on the coherence of a
potential poor man’s Majorana qubit, we calculate the dephasing rates using
realistic charge noise estimations and data presented above (see Supplementary
Information). While perturbation of QDs’ potentials is expected to be the
dominant mechanism of energy splitting and thus dephasing for non-proximitized
PMMs, the drastic reduction in gate dispersion of YSR-based PMMs makes it
negligible compared to dephasing caused by deviations from the $t=\Delta$
condition. The estimated $T_{2}^{*}\sim$10\text{\,}\mathrm{n}\mathrm{s}$$ is
now limited by noise affecting the couplings $t$ and $\Delta$, nearly an order
of magnitude higher than the charge-noise-limited $T_{2}^{*}$ extracted from
data in our previous report Dvir.2023 . Importantly, the expected dephasing
time is now also much longer than the adiabatic limit
$\hbar/E_{\mathrm{gap}}\sim$10\text{\,}\mathrm{p}\mathrm{s}$$, which sets an
upper bound on how fast Majorana states can be manipulated without populating
the excited states Aasen.2016 ; Boross.2023 .

##  6 Conclusion

In conclusion, we have demonstrated the formation of YSR states by hybridizing
QDs with a common ABS. The coupling between these YSR states can be controlled
by varying the electrochemical potential of the ABS, thus realizing a fully
tunable two-site Kitaev chain. The resulting PMM states have two significant
improvements over those in a non-proximitized QD chain. First, the stronger
coupling between the YSR states triples the gap between the PMM and excited
states, protecting the former from finite temperature excitations Knapp.2018
and enabling faster adiabatic operations Aasen.2016 ; Boross.2023 . Second,
the reduced charge dispersion of the YSR states enhances the robustness of the
PMMs against charge noise affecting both QDs by more than 100. Thanks to
these, even a two-site Kitaev chain realized using YSR states should suffice
for a prototypical Majorana qubit and verification of non-abelian properties
with fusion and braiding experiments. Despite the present lack of protection
against tunnel-coupling noise, the expected coherence of Majorana qubits made
from PMM states is increased by close to an order of magnitude compared to the
first report Dvir.2023 . In the future, increasing the number of sites can
mitigate noise affecting the tunnel-coupling rates. Estimations using
parameters of the YSR-based PMMs suggest that a Majorana qubit realized with
Kitaev chains as short as 3 to 5 sites could already achieve dephasing times
comparable to those predicted for continuous nanowires Sau.2012 ; Knapp.2018 .

##  7 Data availability and code availability

Raw data presented in this work, the data processing/plotting code, and code
used for the theory calculations are available at
https://zenodo.org/records/10013728.

##  8 Acknowledgements

This work has been supported by the Dutch Organization for Scientific Research
(NWO) and Microsoft Corporation Station Q. We thank John M. Hornibrook and
David J. Reilly for providing the frequency multiplexing chips. We thank
Martin Leijnse, Ville Maisi, Pasquale Scarlino, Athanasios Tsintzis, Rubén
Seoane Souto, Karsten Flensberg, Srijit Goswami, Sebastiaan L. D. ten Haaf,
Qingzhen Wang, Ivan Kulesh, and Yining Zhang for helpful discussions.

##  9 Author contributions

DvD, FZ, AB, GPM, NvL, and JCW fabricated the device. FZ, DvD, DX, BR, and TD
performed the electrical measurements. TD and GW designed the experiment. FZ,
DvD, GW, and TD analyzed the data. FZ, DvD, TD, GW, and LPK prepared the
manuscript with input from all authors. TD and LPK supervised the project. CXL
and AMB developed the theoretical model with input from MW. G.B., S.G., and
E.P.A.M.B. performed InSb nanowire growth.

##  10 Competing interests

The authors declare no competing interests.

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##  11 Materials and Methods

###  11.1 Device fabrication

The InSb/Al hybrid nanowire device presented in this work was fabricated using
the shadow-wall lithography technique Heedt.2021 ; Borsoi.2021 .

A substrate is patterned with Ti/Pd gates. $10\text{\,}\mathrm{n}\mathrm{m}$
of AlOx and $10\text{\,}\mathrm{n}\mathrm{m}$ of HfOx are deposited by ALD as
gate dielectric. HSQ shadow-walls are then patterned. Nanowires are deposited
and pushed next to the shadow-walls using an optical micro-manipulator. An
$8\text{\,}\mathrm{n}\mathrm{m}$ Al shell is deposited at alternating angles
of $15\text{\,}\mathrm{\SIUnitSymbolDegree}$ and
$45\text{\,}\mathrm{\SIUnitSymbolDegree}$ with respect to the substrate,
followed by a capping layer of $20\text{\,}\mathrm{n}\mathrm{m}$ of AlOx.
Normal leads in ohmic contact with the nanowire are fabricated by Ar milling
and evaporation of Cr/Au.

Additional details about the substrate fabrication and the Al deposition are
described in Ref. Mazur.2022 .

###  11.2 Transport measurements and data processing

The measurements are done in a dilution refrigerator with a base temperature
of $30\text{\,}\mathrm{m}\mathrm{K}$. A magnetic field of
$150\text{\,}\mathrm{m}\mathrm{T}$ is applied along the nanowire axis except
in Fig. ED1c,d.

The three-terminal setup used to measure the device is illustrated in Fig. 1a.
The SC lead is always kept grounded. The two normal leads can be voltage-
biased independently. When a bias is applied to one side, the other one is
kept grounded. The currents $I_{L}$ and $I_{R}$ are measured separately.
Digital multimeters and lock-in amplifiers are used to read the voltage
outputs of the current meters. AC excitations of
$5\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{V}$ RMS are applied on each side
with different frequencies ($39\text{\,}\mathrm{H}\mathrm{z}$ on the left and
$29\text{\,}\mathrm{H}\mathrm{z}$ on the right), except for Fig. ED2 and Fig.
ED5d–f, where excitations of
$10\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{V}$ RMS were used.

Off-chip multiplexed resonators Hornibrook.2014  connected to the two normal
leads are used for fast RF reflectometry measurements. This measurement scheme
was employed to speed up the tune-up of the device, for Fig. 3, and for Fig.
ED7, as explained in Fig. 3a. Additional details about the reflectometry setup
are described in Ref. wang2022parametric .

###  11.3 Spinless PMM spectrum

The low-energy spectrum of the coupled YSR system arising from the model
described in Supplementary Information can be reduced to that of a spinless
PMM model Leijnse.2012 ; Liu.2022 ; Tsintzis.2022 ; Liu.2023 ; Tsintzis.2023 .

In this simplified model, the non negative energy eigenvalues are Leijnse.2012

$E_{\pm}=\sqrt{\mu_{+}^{2}+\mu_{-}^{2}+t^{2}+\Delta^{2}\pm
2\sqrt{(\mu_{+}^{2}+\Delta^{2})(\mu_{-}^{2}+t^{2})}},$

where $\mu_{\pm}=\frac{\mu_{\mathrm{L}}\pm\mu_{\mathrm{R}}}{2}$,
$\mu_{\mathrm{L}}$ and $\mu_{\mathrm{R}}$ are the chemical potentials of the
two sites, and $t$, $\Delta$ are the couplings. The spectrum is symmetric
around zero energy because of particle-hole symmetry. If
$\mu_{\mathrm{L}}=\mu_{\mathrm{R}}=0$, the eigenvalues reduce to
$E_{+}=t+\Delta$ and $E_{-}=\absolutevalue{t-\Delta}$. These are the
excitation energies that we use to extract $t$ and $\Delta$ from the measured
spectra in Fig. 3.

The gate dispersion lines plotted in Fig. 4e,f are calculated using the lowest
excitation energy. When $t=\Delta$ we obtain

$E_{-}=\sqrt{\alpha^{2}\delta V^{2}+2t^{2}-2\sqrt{t^{2}(t^{2}+\alpha^{2}\delta
V^{2})}},$

where $\alpha$ is the lever arm converting voltage to chemical potential and
$\delta V$ is the simultaneous detuning of each gate away from the sweet spot.
For the PMM realized with QDs, we have used $\alpha=0.33e$ and
$t=$12\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}$$ Dvir.2023 .
For the PMM realized with YSR states we have used $\alpha=0.05e$ (Fig. ED1)
and $t=$38\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}$$ (Fig.
4b). Finally, the quadratic splitting of PMM states can be derived by
expanding the expression above for small $\delta V$, which gives

$E_{-}\approx\frac{\alpha^{2}}{2t}\delta V^{2}.$

###  11.4 Series resistance correction

The effect of series resistance in the fridge line and other parts of the
circuit on transport measurements of a three-terminal device of our type is
described in Ref. Martinez2021 , as well as how to correct for it. In our
setup, the resistance of the voltage source is $100\text{\,}\Omega$ and that
of the current meter is $200\text{\,}\Omega$. Additional series resistance
comes from the fridge lines and the ohmic contacts. Correcting for the voltage
fall over these resistors in series to the device requires knowledge of the
exact resistance values, which we presently cannot obtain before the next
sample exchange. Therefore, we make use of the fact that the bulk Al
superconducting gap in local conductance measurements should stay constant
across all gate values to arrive at an estimation of the total series
resistance in each fridge line.

In Fig. ED10a,c it is possible to observe that the superconducting gap appears
enlarged when there is finite subgap conductance, indicating unaccounted-for
series resistance. To estimate it, we correct the $G_{\mathrm{LL}}$
measurement shown in Fig. ED10a for different trial values of the series
resistance. We extract the energies of the coherence peaks and calculate their
variance. Finally, we choose the value that minimizes the variance as the
optimal series resistance. We find an optimal value of
$3.65\text{\,}\mathrm{k}\Omega$, in addition to the resistance of the voltage
source and current meter. The corrected measurement is shown in Fig. ED10b.
All the data processing steps are available in the online repository. In Fig.
ED10c,d we show that the same series resistance value also corrects an
analogous measurement on the right side. In the regimes relevant to this work,
the voltage divider effect caused by the series resistance does not strongly
affect local conductance. On the other hand, it can be more pronounced for
non-local conductance, as discussed in Fig. ED10e,f. Using the extracted
resistance value, we apply the correction method described in Ref.
Martinez2021  to the data presented in Fig. 2, Fig. 3, Fig. 4, Fig. ED2, Fig.
ED5d, Fig. ED7, Fig. ED8, Fig. ED9d–k, and Fig. ED10b,d,f, where the full
conductance matrix is measured.

###  11.5 Device tune-up

The device is controlled with seven bottom gates (Fig. 1a). To perform tunnel
spectroscopy of the hybrid segment in Fig. ED2 and Fig. ED5, we use
$V_{\mathrm{LT}}$ and $V_{\mathrm{RT}}$ to form a tunnel barrier and apply a
large positive voltage to $V_{\mathrm{LP}}$, $V_{\mathrm{RP}}$, and the outer
gates. $V_{\mathrm{H}}$ is used to control the electrochemical potential in
the hybrid segment, where discrete states are confined because of the tunnel
barriers. QDs are formed by reducing the voltage on the outer gates. This
creates additional tunnel barriers next to the normal leads, thus shaping a
confining potential. $V_{\mathrm{LP}}$ and $V_{\mathrm{RP}}$ control the
electrochemical potential of the QDs. Finally, the coupling between the QDs
and the ABS can be controlled using $V_{\mathrm{LT}}$ and $V_{\mathrm{RT}}$.
By increasing their voltage, we can achieve the strong coupling regime between
the QDs and the ABS in the hybrid as explained in Fig. 1. For all the
measurements presented in the main text, the gates defining the tunnel
barriers were maintained at the same value.

##  12 Supplementary Information

###  12.1 Theoretical model

The physical system of double QDs connected by a hybrid segment in the middle
can be well described by a three-site model as below

$\displaystyle H=\sum_{i=L,H,R}H_{i}+H_{T},$ $\displaystyle
H_{i}=(\mu_{i}+E_{Zi})n_{i\uparrow}+(\mu_{i}-E_{Zi})n_{i\downarrow}+U_{i}n_{i\uparrow}n_{i\downarrow}+\Gamma_{i}(c^{\dagger}_{i\uparrow}c^{\dagger}_{i\downarrow}+c_{i\downarrow}c_{i\uparrow}),$
$\displaystyle
H_{T}=w_{L}(c^{\dagger}_{H\uparrow}c_{L\uparrow}+c^{\dagger}_{H\downarrow}c_{L\downarrow})+w_{SOL}(c^{\dagger}_{H\downarrow}c_{L\uparrow}-c^{\dagger}_{H\uparrow}c_{L\downarrow})$
$\displaystyle\quad+w_{R}(c^{\dagger}_{R\uparrow}c_{H\uparrow}+c^{\dagger}_{R\downarrow}c_{H\downarrow})+w_{SOR}(c^{\dagger}_{R\downarrow}c_{H\uparrow}-c^{\dagger}_{R\uparrow}c_{H\downarrow})+h.c..$
(1)

Here $H_{i}$ is the Hamiltonian for the QDs ($i=L,R$) or the ABS ($i=H$) in
the hybrid, $n_{i\sigma}=c^{\dagger}_{i\sigma}c_{i\sigma}$ is the occupancy
number of the QD orbital or normal state in the hybrid with spin $\sigma$,
$\mu_{i}$ is the chemical potential energy, $E_{Zi}$ is the induced Zeeman
energy, $U_{i}$ is the charging energy, and $\Gamma_{i}$ is the induced
superconducting gap. $H_{T}$ is the tunnel Hamiltonian between the QDs and the
ABS, where $w_{L/R}$ and $w_{SOL/SOR}$ are the amplitudes for the spin-
conserving and spin-flipping single electron tunneling processes. The
parameters used for the numerical simulations performed in this work, in unit
of $\Gamma_{H}\equiv\Gamma$, the induced gap on the ABS, are:
$E_{ZL}=E_{ZR}=1.5\Gamma$, $E_{ZH}=0.5\Gamma$ for Zeeman energy,
$U_{L}=U_{R}=5\Gamma$, $U_{H}=0$ for charging energy, $w_{L}=\Gamma$,
$w_{SOL}=w_{L}/2$, $w_{R}=0.75\Gamma$, $w_{SOR}=w_{R}/2$ for tunneling
amplitudes, and $\Gamma_{L}=0,\Gamma_{R}=\Gamma$ for induced gaps on the QDs.
Here the magnitude of the Zeeman energy on the ABS is weaker than that in the
QDs owing to the $g$ factor renormalization in the hybrid. Furthermore, an
extra pairing gap is added on the right QD to explain the residual Andreev
conductance when the ABS is far from its energy minimum, as shown in the
measurement in Fig. 1. The sweet spot investigated in Fig. ED6e–i corresponds
to $\mu_{H}=0.617\Gamma$, $\mu_{L}=2.290\Gamma$, and $\mu_{R}=-5.879\Gamma$.
Finally, in Fig. ED4, we calculate the electron and hole character of an
excitation between two states of different parity as

$u_{i}^{2}=\sum_{\sigma=\uparrow,\downarrow}\absolutevalue{\bra{o}c_{i\sigma}^{\dagger}\ket{e}}^{2},\quad
v_{i}^{2}=\sum_{\sigma=\uparrow,\downarrow}\absolutevalue{\bra{o}c_{i\sigma}\ket{e}}^{2},$
(2)

where $\ket{e}$ is the even state and $\ket{o}$ the odd one.

In the experimental device, conductance spectroscopy of the hybrid system is
measured in a three-terminal junction with two normal leads and one grounded
superconducting lead. To numerically simulate the transport measurement, we
attach two additional normal leads to the two QDs, respectively. The lead
Hamiltonians are

$\displaystyle
H_{\text{lead}}=\sum_{a=L,R}\sum_{\sigma=\uparrow,\downarrow}\sum_{k}(\varepsilon_{ka}-\mu_{\text{lead},a})f^{\dagger}_{ak\sigma}f_{ak\sigma},$
(3)

where $f_{ak\sigma}$ is the annihilation operator of a free electron with
momentum $k$ and spin $\sigma$ in lead-$a$, $\varepsilon_{ka}$ is the kinetic
energy, and $\mu_{\text{lead},a}$ is the lead chemical potential. The tunnel
Hamiltonian between leads and the QDs is

$\displaystyle
H_{T,\text{lead}}=\sum_{a=L,R}\sum_{\sigma=\uparrow,\downarrow}\sum_{k}w^{\prime}_{a}f^{\dagger}_{ak\sigma}c_{a\sigma}+h.c.,$
(4)

where $w^{\prime}_{a}$ is the tunneling amplitude between QD-$a$ and lead-$a$,
and is assumed to be spin-diagonal and independent of $k$. A more useful
physical quantity to characterize the lead coupling strength is
$\gamma_{a}=2\pi\rho w^{\prime 2}_{a}$ with $\rho$ being the density of states
at the Fermi energy of the normal lead. Based on the model Hamiltonians
introduced above, current and conductance matrices can be numerically
calculated by solving a rate equation for the reduced density matrix of the
hybrid system. We set the temperature $k_{B}T=0.04\Gamma$ and tunneling
strength $\gamma_{a}=0.03\Gamma$ to be the same for both normal leads.
Finally, when simulating the local Andreev conductance of a QD-ABS system, we
are actually considering a two-terminal junction by removing the quantum dot
and the normal lead on the other side.

###  12.2 Estimation of dephasing rates

A minimal Majorana qubit consists of two subsystems, each hosting one pair of
Majoranas: $\gamma_{A1},\gamma_{A2}$, and $\gamma_{B1},\gamma_{B2}$. It is
possible to encode a qubit in the subspace of either even or odd global
fermion parity. Without loss of generality, we focus on the even subspace. We
define qubit states $\ket{0}\equiv\ket{o_{A},o_{B}}$, with the fermion parity
of each subsystem being odd, and $\ket{1}\equiv\ket{e_{A},e_{B}}$, each being
even. The qubit Pauli operator $\sigma_{z}$ is then identified as
$i\gamma_{A1}\gamma_{A2}=i\gamma_{B1}\gamma_{B2}$ and the effective
Hamiltonian is Aasen.2016 ; Boross.2022 ; Tsintzis.2023

$H=\frac{i}{2}\left(E_{A}\gamma_{A1}\gamma_{A2}+E_{B}\gamma_{B1}\gamma_{B2}\right)=E\sigma_{z},$
(5)

where $E_{A,B}=E_{o_{A,B}}-E_{e_{A,B}}$ is the excitation energy of each
subsystem and $E\equiv(E_{A}+E_{B})/2$.

An ideal Majorana qubit has $E=E_{A}=E_{B}=0$. Uncontrolled fluctuations of
$E$ lead to qubit dephasing Aasen.2016 ; Knapp.2018 ; Mishmash.2020 ;
Boross.2022 . If the typical amplitude of $E$ fluctuation is $\delta E$, the
dephasing rate is $1/T_{2}^{*}\sim\delta E/\hbar$ Koch.2007 ; Aasen.2016 . In
the following estimations, we assume the two subsystems are similar and omit
details of the statistical distribution of the noise, so that $\delta
E_{A}\sim\delta E_{B}\sim\delta E$. Thus, we use the typical energy
fluctuations of one pair of PMM to estimate the dephasing rates of a qubit.

Energy splitting of a pair of PMM can occur due to two reasons: the degeneracy
splits quadratically when the QDs are detuned simultaneously
($\mu_{\mathrm{L}}=\pm\mu_{\mathrm{R}}\neq 0$) and linearly when the ECT and
CAR couplings are perturbed from the sweet spot ($t\neq\Delta$) Leijnse.2012 .
Other types of noise, e.g., magnetic fluctuations, enter the Hamiltonian
parametrically via these two mechanisms. In the main text, we have
characterized the sensitivity of the degeneracy to each of them by
intentionally tuning gate voltages away from the sweet spots. These gate
dispersion results allow us to estimate the dephasing rate given typical noise
amplitudes found in QD qubit devices. Concretely, we consider random
fluctuations of the gate voltages controlling the electrochemical potential of
the QDs ($V_{\mathrm{LP}}$, $V_{\mathrm{RP}}$) and ABS ($V_{\mathrm{H}}$), the
former causing on-site potential fluctuation and the latter that of the
tunneling amplitudes.

We first focus on charge noise affecting the QDs. In Fig. 4f, we have measured
the quadratic splitting of the PMM when the electrochemical potential of both
QDs is detuned from the sweet spot. Given the vanishing first derivative of
the gate dipersion, we extracted a second derivative of
$\partial^{2}E/\partial
V_{\mathrm{LP}}^{2}\approx$30\text{\,}\mathrm{e}\mathrm{/}\mathrm{V}$$, two
orders of magnitude smaller than the one measured in our previous report,
$\partial^{2}E/\partial
V_{\mathrm{LP}}^{2}\approx$4500\text{\,}\mathrm{e}\mathrm{/}\mathrm{V}$$
Dvir.2023 . In previous measurements of charge fluctuations of a QD
Petersson.2010 ; Dial.2013 ; Scarlino.2022 ; Connors.2022 ; Burkard.2023 ,
equivalent voltage noise on the gates is typically found to be $\delta
V\sim$10\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{V}$$ (after accounting for
the gate lever arm). A worst-case estimation of maximally (anti)correlated
gate noise on both QDs predicts a resulting energy fluctuation of $\delta
E\approx\frac{\partial^{2}E}{\partial V_{\mathrm{LP}}^{2}}\delta V^{2}$,
yielding $\sim$3\text{\,}\mathrm{n}\mathrm{e}\mathrm{V}$$ for the YSR-based
PMM states in this work and $\sim$450\text{\,}\mathrm{n}\mathrm{e}\mathrm{V}$$
for our previous work. These correspond to dephasing rates of $\delta
E/\hbar\sim$5\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$ and
$\sim$700\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$ for the two cases, a direct
consequence of the two orders of magnitude of gate dispersion suppression.

We now turn to charge noise affecting the ABS and thereby the couplings
between QDs, $t$ and $\Delta$. Given the charge sensitivity of the hybrid ABS
and its key effect on modulating the tunneling amplitudes, we assume the
primary source of $t-\Delta$ variation to be charge fluctuations of the ABS.
In Fig. 3e, we have estimated the noise sensitivity of two PMM sweet spots
against perturbation on $V_{\mathrm{H}}$. We found $\partial E/\partial
V_{\mathrm{H}}\approx 3$ to
$7\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{m}\mathrm{V}$,
comparable to our previous work Dvir.2023 . Although the exact sensitivity
depends on microscopic details, we note that the superconducting lead, in
general, strongly screens charge fluctuations. This explains the small lever
arm for an ABS in the hybrid segment, which is $\sim 1/20$ that of a normal QD
(see Fig. ED2 and Fig. ED3a). By considering the worst sensitivity and the
same fluctuations as before, we estimate typical splittings of $\delta
E\sim\frac{\partial E}{\partial V_{\mathrm{H}}}\delta
V=$70\text{\,}\mathrm{n}\mathrm{e}\mathrm{V}$$ and a dephasing rate of
$\sim$100\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$.

Comparing the dephasing rates, we find that on-site QD charge fluctuations
likely dominated dephasing of PMM qubits made from non-proximitzed QD states.
With the introduction of YSR-based Kitaev chain and its strongly suppressed
gate dispersion, the QD charge fluctuation will likely no longer be the main
source of dephasing. Instead, tunneling amplitude variations, not suppressed
by the formation of YSR states, becomes the more probable primary dephasing
mechanism. While reducing sensitivity to tunneling amplitude fluctuations is
beyond the scope of this paper, we remark here that a possible solution is to
increase the number of sites of the Kitaev chain to at least 3 Sau.2012 .

Finally, we comment on a few additional details not addressed by the
discussions above. We estimated energy fluctuations only using equivalent gate
fluctuations converted from previous charge noise measurements. This does not
describe other sources of decoherence including that due to quasiparticles
Rainis.2012 ; Knapp.2018  and electron-phonon coupling Knapp.2018 . We do not
expect our estimations to be drastically altered if these effects are
included, given the low electron temperature and that electron-phonon
dephasing rates measured in charge qubits realized in III/V semiconductors are
on the order of $100\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ Hofmann.2020 ;
Ranni.2023 , comparable to our estimated dephasing rates due to tunneling-rate
variations. Ultimately, a realistic quantitative noise model would require the
measurement of the noise spectrum and dephasing times. Despite all these
simplifications, the dephasing rate estimations indicate that there should be
a reasonable time window where it is possible to adiabatically manipulate PMM
states before they decohere significantly. This is a necessary condition to
demonstrate qubit, fusion, and braiding protocols based on PMM states
Liu.2022.Fusion ; Boross.2023 ; Tsintzis.2023 . In the future, longer Kitaev
chains are expected to increase the dephasing time, since the energy splitting
is predicted to decrease exponentially with the number of sites Sau.2012 .

##  13 Extended Data

Figure ED1: Characterization of the YSR states at $t\approx\Delta$. a, Local
conductance measurement of the spectrum of the left YSR state when the right
one is off-resonance. $V_{\mathrm{H}}=$333.5\text{\,}\mathrm{m}\mathrm{V}$$ is
tuned such that $t\approx\Delta$ for the crossing showed in Fig. 4a. The lever
arm at charge degeneracy is $\approx 0.05e$, extracted using the black dotted
line. b, Same as panel a, but for the right YSR state. We extract a lever arm
of $\approx 0.04e$. c, Local conductance measurement of the Zeeman splitting
of the left YSR state, at the same value of $V_{\mathrm{H}}$ as above. The
field is applied along the nanowire axis. The gray dotted line indicates the
field at which all the other measurements were conducted. The left dot was
positioned at the charge degeneracy point at
$B=$150\text{\,}\mathrm{m}\mathrm{T}$$ and the right one off-resonance. We
extract a g-factor of $\approx 24$, consistent with previously measured values
for strongly proximitized states in similar devices Mazur.2022 . The
corresponding Zeeman splitting at $150\text{\,}\mathrm{m}\mathrm{T}$ is
$\approx$200\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}$$. d,
Same as panel c, but for the right YSR state. We extract the same g-factor as
for the left YSR state.  Figure ED2: Spectroscopy of the hybrid segment.
Conductance matrix of the hybrid spectrum as a function of $V_{\mathrm{H}}$.
The hybrid segment hosts separated ABS that can mediate the coupling between
the QDs. The gray dashed lines indicate the range of $V_{\mathrm{H}}$ used for
the measurements presented in the main text. The lever arm is $\alpha\approx
0.02e$ as extracted from the black dotted lines superimposed on the ABS in
panel d, indicating a strong screening effect of the superconducting lead. For
this measurement, the two sides are tuned as tunnel barriers as described in
the Methods section. When QDs are formed, $V_{\mathrm{LT}}$ and
$V_{\mathrm{RT}}$ are increased to reach the strong coupling regime between
them and the ABS. This affects the ABS spectrum because of cross-coupling and
modification to the confining potential. Therefore, the $V_{\mathrm{H}}$
values and the energy of the ABS in this measurement are slightly different
from those presented in the main text.  Figure ED3: High-bias spectroscopy of
the QDs. a, Coulomb diamonds for the left QD when it is not coupled to any ABS
by closing $V_{\mathrm{LT}}$ and keeping ABSs off-resonance. From the overlaid
black dotted lines, we extract typical charging energies of
$\approx$3.5\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$ and lever arms of
$\alpha\approx 0.4e$. b, Local conductance spectroscopy of the left QD as a
function of $V_{\mathrm{LP}}$ after $V_{\mathrm{LT}}$ is increased to strongly
couple to an ABS. The measurement was taken in the same configuration as Fig.
1g, with the ABS off-resonance. We extract a charging energy of
$\approx$2\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$ and a lever arm of
$\alpha\approx 0.2e$, which are renormalized because of the lowered tunnel
barriers. c, Same as panel b, but for the right QD. Typical QD features are
not as clearly visible since the state is significantly proximitized by the
SC, as shown in Fig. 1j and in Fig. ED5a–c. By overlaying the visible
features, we extract a charging energy of
$\approx$2\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$ and a lever arm of
$\alpha\approx 0.2e$. These measurements were taken applying a field of
$150\text{\,}\mathrm{m}\mathrm{T}$ along the nanowire axis.  Figure ED4:
Properties of the YSR zero-energy excitation. a, Charge of the YSR excitation
residing on the left QD (blue line) and on the ABS (orange line). The charge
is calculated as $u_{i}^{2}-v_{i}^{2}$, with $i=L$ for the QD and $i=M$ for
the ABS, as defined in Supplementary Information. The gray dotted line
corresponds to the $\mu_{\mathrm{H}}$ value used for the sweet spot studied in
Fig. ED6e–i. The decrease of the charge residing on the QD explains part of
the lever arm reduction observed for YSR states (Fig. 1h). A further reduction
comes from charging energy renormalization. b, Weight of the YSR excitation
wavefunction on the left QD (blue line) and on the ABS (orange line). The
weight of the wavefunction on each site is defined as $u_{i}^{2}+v_{i}^{2}$.
When the ABS is close to its energy minimum at $\mu_{\mathrm{H}}=0$, the
hybridization between the QD and the ABS becomes stronger and the excitation
is delocalized on the two sites, thus reducing the charge residing on the QD.
c, Charge of the YSR excitation normalized for the wavefunction weight
residing on the left QD (blue line) and on the ABS (orange line). The
normalized charge is defined as $(u_{i}^{2}-v_{i}^{2})/(u_{i}^{2}+v_{i}^{2})$
and it quantifies the mixed electron and hole character. It ranges from $1$ to
$-1$ when the excitation is purely electron- and hole-like, respectively.
Although the charge of a YSR state is not a good quantum number while ECT/CAR
is defined using charge states, the predominance of a charge character
justifies our use of ECT and CAR in the main text, as investigated in more
detail in Ref. Liu.2023 .  Figure ED5: QDs directly proximitized by the
superconducting film. a–c, Same as Fig. 1c–e, but with an additional gate-
independent source of proximity for the QD, as discussed in Supplementary
Information. d, Conductance matrix for tunnel spectroscopy of the hybrid
section as a function of $V_{\mathrm{H}}$. The green tick marks the value of
$V_{\mathrm{H}}$ at which the measurements in e and f were taken. In this
regime, there are no subgap states. e, f, Local tunnel spectroscopy of the
left (b) and right (c) QDs at low $V_{\mathrm{H}}$. Although there are no
subgap states in the hybrid, it is possible to tune the QDs in a regime where
a gap and subgap states appear. We interpret this as arising from the QDs
being proximitized by the Al, indicating that the proximity of the QDs can be
not exclusively due to subgap states in the hybrid. These measurements were
taken applying a field of $150\text{\,}\mathrm{m}\mathrm{T}$ along the
nanowire axis.  Figure ED6: Numerical simulations of coupled YSR states a-c
Numerical conductance matrix of the CSDs of coupled YSR states varying
$\mu_{\mathrm{H}}$, reproducing Fig. 2. d, Numerical conductance spectrum of
coupled YSR states as a function of the ABS chemical potential. The spectrum
is calculated following the same approach discussed in Fig. 3 to maintain the
chemical potentials of the QDs at the center of the crossing. The theoretical
model reproduces the experimental findings presented in Fig. 3. e, Numerical
conductance matrix of a crossing in the CSD, when $t\approx\Delta$, at
$\mu_{\mathrm{H}}=0.617\Gamma$. f, g, Numerical conductance matrix of the
spectrum as a function of the right (f)/left (g) QD detuning along the
green/blue dotted line in panel a. The other QD is kept on resonance. The
numerical results reproduce the measurement shown in Fig. 4c/Fig. ED8b. h, i,
Numerical conductance matrix of the spectrum calculated detuning both QDs
simultaneously, along the pink/gray dashed line in panel e. The numerical
results reproduce the measurements shown in Fig. 4d/Fig. ED8c.  Figure ED7:
ECT and CAR coupling for all the resonances. a–d, (i) Conductance matrix of
the spectrum of two coupled YSR states as a function of $V_{\mathrm{H}}$,
measured as described in Fig. 3. The extracted peaks are marked with gray
dots. (ii) Energy of the excited states, calculated by averaging the peaks at
positive and negative bias of both the left and the right spectrum. From the
outer peaks we extract $t+\Delta$, from the inner ones
$\absolutevalue{t-\Delta}$. The gray dashed lines indicate a change of sign of
$t-\Delta$. In particular, $t<\Delta$ within the lines and $t>\Delta$ outside.
In c a gate jump affected the measurement of the spectrum for two values of
$V_{\mathrm{H}}$. These are highlighted with red dotted lines in c(ii) and
c(iii). Parts of d were presented in Fig. 3.  Figure ED8: Additional PMM
spectra. a, CSD for a PMM sweet spot. Same as Fig. 4a, but with different
paths highlighted. b, PMM spectrum measured along the vertical path of panel
a. The left QD is detuned, while the right one is kept on resonance. The
asymmetry with respect to $\mu_{\mathrm{L}}=0$ can be attributed to the
asymmetric energy dispersion of the left YSR state (see Fig. ED1a). c, PMM
spectrum measured detuning both QDs along the diagonal path of panel a. The
asymmetry with respect to $\mu_{\mathrm{L}}=\mu_{\mathrm{R}}=0$ can be
explained as discussed in panel b.  Figure ED9: PMM with different YSR and
ABSs. a–c Similarly to Fig. 1, QD-ABS CSDs (a), spectra of the YSR states when
the ABS is at its minimum energy (b), and for a $V_{\mathrm{H}}$ corresponding
to a $t=\Delta$ sweet spot (c), as described in the next panels. The colors of
the frames correspond to the $V_{\mathrm{H}}$ values indicated in panel a.
d–e, Conductance matrix for YSR states CSDs when the ABS is at its minimum
energy (d) and slightly detuned (e), as discussed in Fig. 3. The color of the
frames correspond to the $V_{\mathrm{H}}$ values indicated in panel a. f,
Zoom-in on the bottom right crossing of panel e, corresponding to a PMM sweet
spot. g, Left and right conductance spectra measured at the PMM sweet spot,
corresponding to the black dot at the middle of the crossing in f. The gap
between the ground and the first excited states is
$E_{\mathrm{gap}}\approx$71\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}$$.
h–k, Conductance matrix of the PMM spectra along the paths in panel f,
analogously to Fig. 4 and Fig. ED8.  Figure ED10: Estimation of series
resistance. a, Tunnel spectroscopy of the hybrid spectrum from both sides
varying $V_{\mathrm{LT}}$. Whenever there is finite subgap conductance, the
parent gap expands because of the voltage divider effect. b, Same measurement
as in panel a, after correcting the voltage divider effect. We find that a
series resistance of $R_{\mathrm{series}}=$3.65\text{\,}\mathrm{k}\Omega$$
corrects the gap increase discussed above. In addition, we also considered a
total resistance for the voltage source and current meter of
$300\text{\,}\Omega$. c, d, Same as panels a and b, but for the right side. e,
f, CSD presented in Fig. 4a, before (e) and after (f) correcting the voltage
divider effect. The local signal is not significantly affected. On the other
hand, the alternating positive and negative non-local signal becomes clearer
after the correction. In particular, it approaches zero in the middle of the
crossing, indicated by the black arrow. Without series resistance correction,
the finite non-local signal at the center of the crossing in panel e could be
misinterpreted as a signature of low Majorana polarization Tsintzis.2022 .