# Rabi and Ramsey oscillations of a Majorana qubit in a quantum dot-
superconductor array

Haining Pan  Department of Physics and Astronomy, Center for Materials Theory,
Rutgers University, Piscataway, NJ 08854 USA  Sankar Das Sarma  Condensed
Matter Theory Center and Joint Quantum Institute, Department of Physics,
University of Maryland, College Park, Maryland 20742, USA  Chun-Xiao Liu
<EMAIL_ADDRESS>QuTech and Kavli Institute of NanoScience, Delft
University of Technology, Delft, The Netherlands

###### Abstract

The Kitaev chain can be engineered within a quantum dot-superconductor array,
hosting Majorana zero modes at fine-tuned sweet spots. In this work, we
propose and simulate the occurrence of Rabi and Ramsey oscillations to
feasibly construct a minimal Majorana qubit in the quantum dot setup. Our
real-time results incorporate realistic effects, e.g., charge noise and
leakage, reflecting the latest experimental progress. We demonstrate that
Majorana qubits with larger energy gaps exhibit significantly enhanced
performance—longer dephasing times, higher quality factors, reduced leakage
probabilities, and improved visibilities—compared to those with smaller gaps
and with conventional quantum dot-based charge qubits. We introduce a method
for reading out Majorana qubits via quantum capacitance measurements. Our work
paves the way for future experiments on realizing Majorana qubits in quantum
dot-superconductor arrays.

_Introduction._ —Majorana zero modes are non-Abelian anyonic excitations
localized at the defects or edges of a topological superconductor Read and
Green (2000); Kitaev (2001); Nayak  _et al._ (2008); Alicea (2012); Leijnse
and Flensberg (2012a); Beenakker (2013); Stanescu and Tewari (2013); Jiang and
Wu (2013); Sarma  _et al._ (2015); Elliott and Franz (2015); Sato and Fujimoto
(2016); Sato and Ando (2017); Lutchyn  _et al._ (2018); Zhang  _et al._
(2019); Flensberg  _et al._ (2021); Das Sarma (2023). Qubits constructed from
the Majorana excitations are immune to local noise and are fault-tolerant
without active error corrections, offering a pathway to implementing error-
resilient topological quantum computing Nayak  _et al._ (2008); Sarma  _et
al._ (2015). Recently the quantum dot-superconductor array has become a
promising candidate for realizing topological Kitaev chains Kitaev (2001) in
solid-state physics using a concrete idea proposed a while ago Sau and Sarma
(2012). An advantage of this quantum-dot-based approach is the intrinsic
robustness against the effect of disorder that is ubiquitous in semiconductor-
superconductor Majorana platforms Pan and Das Sarma (2020); Ahn  _et al._
(2021); Das Sarma  _et al._ (2023); Das Sarma and Pan (2023); Taylor  _et al._
(2024). In addition, utilizing Andreev bound states in a hybrid region as
coupler enables precise control over the relative amplitudes of normal and
superconducting interactions between quantum dots Liu  _et al._ (2022); Bordin
_et al._ (2023); Wang  _et al._ (2022, 2023); Bordin  _et al._ (2024a), thus
allowing for fine-tuning of a quantum dot-superconductor array into a sweet
spot with optimally protected Majorana zero modes Kitaev (2001); Sau and Sarma
(2012); Leijnse and Flensberg (2012b). Tunnel spectroscopic signatures of
Majoranas have been observed in recent experiments on quantum dots using both
nanowires Dvir  _et al._ (2023); Zatelli  _et al._ (2023); Bordin  _et al._
(2024b) and two-dimensional electrons ten Haaf  _et al._ (2024).

To decisively establish a Majorana qubit and demonstrate its topologically
enhanced coherence, Rabi oscillation experiments on quantum-dot-based Kitaev
chains are necessary Sau and Sarma (2024). Additionally, understanding the
topological coherence and obtaining a sufficiently long coherence time is
crucial for detecting the non-Abelian statistics of Majorana anyons in fusion
Liu  _et al._ (2023) or braiding experiments Boross and Pályi (2024); Tsintzis
_et al._ (2024). Most importantly (and we demonstrate in the current work),
such a Rabi oscillation experiment is already feasible in currently available
platforms Dvir  _et al._ (2023); Zatelli  _et al._ (2023); ten Haaf  _et al._
(2024); Bordin  _et al._ (2024b), provided that two such minimal Kitaev chains
are interconnected via a common superconducting lead, and normal-tunnel-
coupled at their ends [see Fig. 1(a)].

In the current work, we propose Rabi and Ramsey oscillation experiments in a
minimal Majorana qubit composed of double two-site Kitaev chains [see Fig.
1(a)]. Our real-time simulations incorporate realistic effects such as charge
noise and leakage to the non-computational bases. We find that Majorana qubits
constructed from large-gap Kitaev chains significantly outperform those with
smaller gaps and conventional quantum dot-based charge qubits in terms of
dephasing time, quality factor, leakage probability, and visibility. In
addition, we propose a Majorana qubit readout method based on quantum
capacitance. Our work demonstrates the optimal route to the first step of
establishing Majorana qubit as a viable experimental entity, which has not
been achieved in the fifteen years of experiments Mourik  _et al._ (2012);
Nichele  _et al._ (2017); Zhang  _et al._ (2021); Aghaee  _et al._ (2023) and
twenty-five years of theory Read and Green (2000); Kitaev (2001); Nayak  _et
al._ (2008); Sau  _et al._ (2010); Lutchyn  _et al._ (2010); Oreg  _et al._
(2010) on topological quantum computing.

Figure 1:  (a) Schematic of a Majorana qubit composed of double two-site
Kitaev chains. (b) Bloch sphere. $\ket{\pm z}$ are defined as $\ket{ee}$ and
$\ket{oo}$, respectively. The dots represent the trajectory of the state
vector in a Rabi experiment.

_Setup and Hamiltonian._ —A minimal Majorana qubit consists of double two-site
Kitaev chains, as shown in Fig. 1(a). The Hamiltonian is

$\begin{split}&\mathcal{H}_{\text{tot}}=\mathcal{H}_{L}+\mathcal{H}_{R}+\mathcal{H}_{\text{tunn}},\\\
&\mathcal{H}_{a}=\sum_{i=1}^{2}\mu_{ai}n_{ai}+\left(t_{a}c^{\dagger}_{a2}c_{a1}+\Delta_{a}c_{a2}c_{a1}+\rm{H.c.}\right),\\\
&\mathcal{H}_{\text{tunn}}=\Gamma c^{\dagger}_{R1}c_{L2}+\rm{H.c.}\end{split}$
(1)

Here $\mathcal{H}_{a}$ with $a\in\left\\{L,R\right\\}$ is the Hamiltonian for
the left and right chain, respectively, with $\mu_{ai}$ ($i=1,2$) the onsite
energy of a spin-polarized dot orbital, $n_{ai}=c_{ai}^{\dagger}c_{ai}=0,1$
the occupancy number, and $t_{a}$ and $\Delta_{a}$ the strengths of the normal
and Andreev tunnelings. $H_{\text{tunn}}$ is the tunnel Hamiltonian, with
$\Gamma$ being the strength of single-electron transfer between dots from
different chains. In the current work, we are particularly interested in the
sweet spot of the system, which is defined as $\mu_{ai}=0$ and
$t_{a}=\Delta_{a}$. At that point, the even-parity ground state
$\ket{e}_{a}=(\ket{00}_{a}-\ket{11}_{a})/\sqrt{2}$ is degenerate with the odd-
parity one $\ket{o}_{a}=(\ket{10}_{a}-\ket{01}_{a})/\sqrt{2}$ within each
Kitaev chain, hosting a pair of Majorana zero modes at two separate quantum
dots. Here
$\ket{n_{1}n_{2}}_{a}=\left(c_{a1}^{\dagger}\right)^{n_{1}}\left(c_{a2}^{\dagger}\right)^{n_{2}}\ket{0}_{a}$,
and $\ket{0}_{a}$ is the vacuum state of chain-$a$. Since total Fermion parity
is conserved in the Hamiltonian of Eq. (1), we can focus on the subspace with
total parity even without loss of generality. As such, the ground-state
degeneracy is two-fold:

$\displaystyle\ket{ee}\equiv\ket{e}_{L}\otimes\ket{e}_{R},\quad\ket{oo}\equiv\ket{o}_{L}\otimes\ket{o}_{R},$
(2)

which form the basis states of a Majorana qubit.

_Rabi oscillations._ —In the qubit subspace spanned by $\ket{ee}$ and
$\ket{oo}$, the low-energy effective Hamiltonian is

$H_{\text{eff}}=\frac{\varepsilon}{2}\sigma_{z}+\frac{\Gamma}{2}\sigma_{x},$
(3)

where $\varepsilon\equiv E_{oo}-E_{ee}$ and $\sigma_{x/z}$ are Pauli X/Z
matrices. Here $\sigma_{z}$ rotation is proportional to the ground-state
energy splitting, which we choose to be $\varepsilon=t_{L}-\Delta_{L}$ by
detuning the hybrid region in the left chain Zatelli  _et al._ (2023).
$\sigma_{x}$ rotation is realized by single electron tunneling between the two
chains that can be controlled by a tunnel barrier. Motivated by the form of
$H_{\text{eff}}$ in Eq. (3), we perform a numerical simulation of the Rabi and
Ramsey experiments using the total Hamiltonian $\mathcal{H}_{\text{tot}}$ in
Eq. (1). Here we implement the qubit rotations by applying sequences of pulses
of $\varepsilon$ or $\Gamma$ instead of microwave driving because of the basis
state degeneracy. In particular, in the Rabi experiment the system is
initialized in $\ket{ee}$ of two decoupled Kitaev chains at their sweet spots.
This corresponds to the north pole of the Bloch sphere. We then turn on the
inter-chain tunneling $\Gamma$ and let the system evolve for a time $\tau$
before performing a readout in the $\sigma_{z}$ basis [see pulse profiles in
Fig. 2(a)]. Figure 2(b) shows the numerically calculated
$P_{ee}(\tau)\equiv\absolutevalue{\innerproduct{ee}{\psi(\tau)}}^{2}$ in the
$(\Gamma,\tau)$ plane. Indeed, the fringe pattern of Rabi oscillations
confirms that single-electron tunneling $H_{\text{tunn}}$ in Eq. (1) works as
a $\sigma_{x}$ rotation in the qubit subspace, with the oscillation frequency
being proportional to $\Gamma$. However, surprisingly, we also find that the
state wavefunction can leak out of the qubit subspace with a probability
$P_{\text{leak}}(\tau)\equiv 1-P_{ee}(\tau)-P_{oo}(\tau)$, which oscillates
periodically in time and increases with the tunneling strength $\Gamma$ [see
Fig. 2(c)]. Using time-dependent perturbation theory sup , we show that a
finite inter-chain tunneling $\Gamma$ inevitably induces a leakage to the
excited states of $\ket{e^{\prime}e^{\prime}}$ and
$\ket{o^{\prime}o^{\prime}}$, i.e.

$\displaystyle
P_{\text{leak}}(\tau)=P_{e^{\prime}e^{\prime}}(\tau)+P_{o^{\prime}o^{\prime}}(\tau)\approx\frac{\Gamma^{2}}{16\Delta^{2}}\sin^{2}\left(2\Delta\tau/\hbar\right),$
(4)

where $\ket{e^{\prime}}_{a}=(\ket{00}_{a}+\ket{11}_{a})/\sqrt{2}$ and
$\ket{o^{\prime}}_{a}=(\ket{10}_{a}+\ket{01}_{a})/\sqrt{2}$ are excited states
in each chain and $\Delta_{L}=\Delta_{R}=\Delta$. Here the oscillation
frequency of the leakage probability is $4\Delta/\hbar$ and the magnitude
scales with $\Gamma^{2}/\Delta^{2}$. On the other hand, in a Ramsey experiment
we first apply a pulse of $H_{\text{tunn}}$ to rotate the initial state
$\ket{ee}$ to the equator of the Bloch sphere, then let it evolve for a time
duration $\tau_{\text{wait}}$ in the presence of a finite $\varepsilon$, and
apply the same $H_{\text{tunn}}$ pulse again before the final readout [see
pulse profiles in Fig. 2(d)]. The simulated $P_{ee}(\tau)$ in the
$(\varepsilon,\tau_{\text{wait}})$ plane is shown in Fig. 2(e). Here the small
$P_{\text{leak}}$ in Fig. 2(f) is due to the $\sigma_{x}$ pulses, while
detuning the coupling $t_{L}-\Delta_{L}$ has a negligible impact on the
leakage probability. Both experiments are doable in the currently available
devices and provide complementary information about Majorana coherence.

Figure 2:  Numerical simulations in the clean limit. Upper panels: Numerical
simulation of a Rabi experiment. (a) Pulse profiles. (b) and (c) $P_{ee}$ and
$P_{\text{leak}}$ in Eq. (4) in the $(\Gamma,\tau)$ plane. Lower panels:
Numerical simulation of a Ramsey experiment. (d) Pulse profiles. (e) and (f)
$P_{ee}$ and $P_{\text{leak}}$ in the $(\varepsilon,\tau_{\text{wait}})$
plane. Here $\Delta_{L}=\Delta_{R}=\Delta$.

_Qubit dephasing._ —Charge noise is one of the primary sources of decoherence
in semiconductor-based qubits Hu and Das Sarma (2006); Petersson  _et al._
(2010); Dial  _et al._ (2013); Scarlino  _et al._ (2022); Connors  _et al._
(2022); Throckmorton and Das Sarma (2022); Burkard  _et al._ (2023); Paladino
_et al._ (2014). It can be induced by charge impurities in the environment or
fluctuations in the gate voltages nearby. As a $1/f$ noise, the fluctuations
are dominated by the low-frequency components, which can be modeled by the
quasi-static disorder approximation, since the zero-frequency part of the
noise dominates Ithier  _et al._ (2005); Boross and Pályi (2022). That is, in
each run of the Rabi or Ramsey experiment the Hamiltonian parameters in Eq.
(1) are subject to a static disorder that obeys normal distribution, and the
final readout measurement is averaged over 500 different disorder
realizations, giving $\expectationvalue{P_{ee}(\tau)}$. In particular, we
simulate and compare three different types of qubits: 1) semiconductor charge
qubit with one electron in double quantum dots Hayashi  _et al._ (2003);
Paladino  _et al._ (2014), 2) small-gap Majorana qubit Dvir  _et al._ (2023),
and 3) large-gap Majorana qubit Zatelli  _et al._ (2023); ten Haaf  _et al._
(2024). Here a small (large) gap in the Kitaev chain corresponds to the
scenario where the dot-hybrid coupling strength is smaller than (comparable
to) the induced gap in the hybrid region Liu  _et al._ (2024a). The mean
values and standard deviations of Hamiltonian parameters that are subject to
charge noises are chosen according to the values reported in relevant
experimental works, which are summarized in the supplemental materials sup .
Figure 3 shows the calculated Rabi and Ramsey oscillations of
$\expectationvalue{P_{ee}(\tau)}$ with dephasing for all three types of
qubits. The curves with decaying envelopes are further fitted using the
following formula

$\displaystyle\expectationvalue{P_{ee}(\tau)}=P_{0}+A\cos(2\pi
f\tau+\phi_{0})\exp{-(\tau/T_{2})^{\beta}},$ (5)

where $2A$ is the visibility, $T_{2}$ is the dephasing time, and $\beta$ is
the decaying exponent. Their values are summarized in Table 1, and in addition
we define the quality factor as $Q=2\pi fT_{2}$ and the leakage probability as
${P_{\text{leak}}}=\lim_{\tau_{0}\rightarrow\infty}\int_{0}^{\tau_{0}}\expectationvalue{P_{\text{leak}}(\tau)}d\tau/\tau_{0}$
in the long time limit where
$\expectationvalue{P_{\text{leak}}(\tau)}=1-\expectationvalue{P_{ee}(\tau)}-\expectationvalue{P_{oo}(\tau)}$
is the instantaneous value.

The Hamiltonian for a semiconductor charge qubit is

$\displaystyle H_{c}=\begin{pmatrix}\varepsilon_{L}&\Gamma\\\
\Gamma&\varepsilon_{R}\end{pmatrix},$ (6)

where the basis states are $\ket{10}$ and $\ket{01}$ with one electron in the
left or right quantum dot, $\varepsilon_{L}$ ($\varepsilon_{R}$) is the
corresponding orbital energy in the left (right) dot, and $\Gamma$ is the
interdot coupling strength. Here, the fluctuations of the dot energies
$\sigma_{\varepsilon}$ dominate the dephasing effect, compared to the
fluctuations of the interdot coupling strength $\sigma_{\Gamma}$, due to the
small magnitude of $\Gamma$. In the Rabi experiment, the dot energies are
tuned into a sweet spot of $\varepsilon_{L}=\varepsilon_{R}=0$, which is
insensitive to dot-energy detuning up to the first order, i.e., $\partial
E/\partial\varepsilon_{a}=0$. However, since the dot-energy fluctuations are
large and comparable to the interdot coupling strength, e.g.,
$\sigma_{\varepsilon}=3~{}\mu\text{eV}\simeq\Gamma=5~{}\mu$eV, the higher-
order contributions (e.g., $\delta E\sim\sigma^{2}_{\varepsilon}/\Gamma$) lead
to a short dephasing time $T_{2}\approx 2.50(7)~{}$ns for $x$ rotations, see
Fig. 3(a). In the Ramsey experiment, to implement the $z$ rotation, we choose
$\varepsilon_{L}=-\varepsilon_{R}=20~{}\mu$eV $\gg\Gamma$, which is much more
susceptible to charge noise as $\partial E/\partial\varepsilon_{a}\approx 1$.
Thus, the dephasing time is even shorter $T_{2}\approx 0.096(2)~{}$ns and the
visibility is reduced, see Fig. 3(d). The consistency between our $T_{2}$
estimates and the experimental measurements reported in Ref. Hayashi  _et al._
(2003) validates our modeling of the quantum dot devices.

In a minimal two-site Kitaev chain that is in the vicinity of the sweet spot,
the energy splitting between the even- and odd-parity ground states is
approximately $E\equiv E_{o}-E_{e}\approx\mu_{1}\mu_{2}/2t+(t-\Delta)$, where
the first term is due to the simultaneous detuning of onsite dot energies,
while the second term is the detuning of the hybrid region. In a small-gap
Majorana qubit (i.e., small $t\equiv\Delta$ limit), the dot-energy
fluctuations are comparatively dominant, giving a characteristic energy
splitting between the basis states $\delta E\sim\sigma^{2}_{\mu}/t$. For a
Majorana qubit defined in Fig. 1(a), such a $\delta E$ leads to noises in the
$\sigma_{z}$ basis. In the Rabi experiment, since the dot-energy noise
($\propto\sigma_{z}$) is orthogonal to the $\sigma_{x}$ rotation, the
dephasing effect of the dot energy fluctuations is strongly mitigated sup . As
such, $T_{2}\approx 9.25(5)~{}$ns is jointly determined by the fluctuations in
the dot energies ($\propto\sigma_{z}$) as well as in the interchain coupling
strengths ($\propto\sigma_{x}$), see Fig. 3(b). In the Ramsey experiment on
$\sigma_{z}$ rotations, the large dot-energy fluctuations
($\propto\sigma_{z}$) cause a more detrimental effect on qubit dephasing,
giving a much shorter dephasing time $T_{2}\approx 1.84(7)~{}$ns and a reduced
visibility $2A\approx 0.3716(8)$, see Fig. 3(e) and Table 1. Note that here
the dephasing effect of charge noise in $t_{a}$ and $\Delta_{a}$ is negligible
because of the weak dot-superconductor hybridization.

On the contrary, the performance of a large-gap Majorana qubit is much
improved in almost all aspects, e.g., dephasing time, quality factor,
visibility, and leakage probability. The strong dot-superconductor
hybridization not only strongly enhances the excitation gap of a Kitaev chain,
but also transforms the dot orbitals into Yu-Shiba-Rusinov states Luh (1965);
Shiba (1968); Rusinov (1969), thus significantly screening the electric charge
in the quantum dots Liu  _et al._ (2024a); Zatelli  _et al._ (2023); ten Haaf
_et al._ (2024). As a result, the energy splitting due to $\mu_{ai}$
fluctuations in the effective Kitaev chain is strongly suppressed, i.e.,
$\sigma^{2}_{\mu}/t$ is reduced by a factor of $\sim 300$ compared to the
small-gap Majorana qubit. Now the dominant source of dephasing in the Rabi
experiment is the charge noise in $\Gamma$, giving $T_{2}\approx
19.064(8)~{}$ns, see Fig. 3(c). In the Ramsey experiment, the fluctuations of
$t_{a}-\Delta_{a}$ begin to dominate the dephasing, giving $T_{2}\approx
11.23(1)~{}$ns, see Fig. 3(f). In addition, a larger excitation gap in the
Majorana qubit also greatly suppresses the leakage probabilities (see Table
1), consistent with the analytic estimates shown in Eq. (4).

Figure 3:  Numerical simulations including charge noises. Upper panels: Rabi oscillations of disorder-averaged $\expectationvalue{P_{ee}}$. Lower panels: Ramsey oscillations of disorder-averaged $\expectationvalue{P_{ee}}$. (a) and (d) semiconductor charge qubits. (b) and (e) small-gap Majorana qubits. (c) and (f) large-gap Majorana qubits. The blue dots are data from numerical simulations, while the red lines are fitting curves using Eq. (5). Here, the size of the disorder ensemble is 500.  Figure 4:  Quantum capacitance readout in Eq. (7) of the low-energy states in a Majorana qubit. $\alpha$ is the magnitude of the lever arm of quantum dots, assumed to be identical for all dots. $\Delta_{L}$ ($\Delta_{R}$) are the superconducting coupling strengths in the left (right) chains. Table 1: Comparison of qubit performances. Protocol |  Qubit properties |  Charge qubit |  Small-gap Majorana qubit |  Large-gap Majorana qubit
---|---|---|---|---
Rabi |  $T_{2}$ (ns) | 2.50(7) | 9.25(5) | 19.064(8)
$Q$ | 38(1) | 69.4(4) | 144.7(5)
$2A$ | 1.01(2) | 0.958(1) | 1.001(3)
$P_{\text{leak}}$ | - | 0.035 | $5.7(8)\times 10^{-4}$
| $\beta$ | 0.66(2) | 1.87(2) | 1.941(3)
Ramsey |  $T_{2}$ (ns) | 0.096(2) | 1.84(7) | 11.23(1)
$Q$ | 5.3(2) | 5.6(2) | 34.31(3)
$2A$ | 0.77(2) | 0.3716(8) | 0.950(5)
$P_{\text{leak}}$ | - | 0.0275(1) | $3.31(6)\times 10^{-5}$
| $\beta$ | 1.01(4) | 0.57(2) | 1.557(4)

_Qubit readout._ —To read out the Majorana qubits, we consider quantum
capacitance measurement, which is defined as

$\displaystyle C_{q}=-\frac{\partial^{2}E}{\partial V^{2}_{g}}$ (7)

in the zero-temperature limit Liu  _et al._ (2023). Here $E$ is the
eigenenergy, $V_{g}$ is the gate voltage that controls the dot energy via
$\mu_{ai}=\alpha_{ai}V_{g}$ with $\alpha_{ai}$ being the lever arm. Since the
measurement is performed when the two chains are decoupled, the result would
simply be a sum of the values in each chain, i.e., $C_{q}=C_{qL}+C_{qR}$.
Furthermore, in the equal-lever-arm regime
($\alpha_{a1}=\alpha_{a2}\equiv\alpha$), the quantum capacitance comes only
from the even-parity state within each chain, while that of the odd-parity one
is strongly suppressed Liu  _et al._ (2023). Thus the quantum capacitances of
$\ket{ee}$ and $\ket{oo}$ are

$\displaystyle
C^{ee}_{q}=\frac{\alpha^{2}}{\Delta_{L}}+\frac{\alpha^{2}}{\Delta_{R}},\quad
C^{oo}_{q}=0,$ (8)

which are distinct from each other and therefore can be used for qubit readout
sup . Following the argument, we further obtain that
$C^{eo}_{q}=\alpha^{2}/\Delta_{L}$ and $C^{oe}_{q}=\alpha^{2}/\Delta_{R}$,
which are different from both $C^{ee}_{q}$ and $C^{oo}_{q}$. Therefore in
addition to qubit readout, $C_{q}$ measurement can simultaneously reveal the
quasiparticle poisoning effect that transitions a Majorana qubit between
states in different total parity space.

_Discussion._ —In the numerical simulations, we regard $1/f$ charge noise as
the dominant source of decoherence in the proposed devices, neglecting the
quasiparticle poisoning effect because this is the prevailing situation in
semiconductor platforms. For example, a poisoning time around $\sim 1$ ms, as
reported in a similar semiconductor-superconductor hybrid device Aghaee  _et
al._ (2024), is much longer than the dephasing time considered here $\sim 10$
ns, making poisoning insignificant for the current consideration where $1/f$
charge noise dominates decoherence. In addition, here both Rabi and Ramsey
experiments are simulated using the most basic protocols for $x$ and $z$
rotations in order to demonstrate the working principles and to provide a fair
comparison between semiconductor charge qubits and Majorana qubits. We
emphasize that the system we consider Dvir  _et al._ (2023); Zatelli  _et al._
(2023); ten Haaf  _et al._ (2024); Bordin  _et al._ (2024b) is equivalent to
semiconductor charge qubits if all superconductivity is removed from
considerations. It is, therefore, possible to further improve the dephasing
time, e.g., by optimizing the pulse profiles, by designing a form of interdot
coupling that is more resilient against charge noises, or by further scaling
up the Kitaev chain Bordin  _et al._ (2024a, b); Liu  _et al._ (2024b). Such
considerations should be relevant once the basic Rabi and Ramsey oscillations
proposed by us are observed so that the elementary concept of a Majorana qubit
is established beyond the simplest transport measurements prevalent so far in
this subject. We emphasize that our work establishes the feasibility of Rabi
oscillations in the already existing experimental platforms of Refs. Dvir  _et
al._ (2023); Zatelli  _et al._ (2023); ten Haaf  _et al._ (2024); Bordin  _et
al._ (2024b).

_Summary._ —We propose and simulate Rabi and Ramsey oscillation experiments
for a minimal Majorana qubit defined in coupled quantum dot-superconductor
arrays. Our realistic calculations demonstrate that the performance of large-
gap Majorana qubits significantly surpasses that of small-gap counterparts and
traditional conventional charge qubits, although some enhancement over
semiconductor charge qubits should already manifest in the small-gap
platforms. Consequently, conducting such experiments is both feasible and
promising on currently available Kitaev chain devices, utilizing existing
control and measurement technologies. This would provide a crucial step toward
the realization of the first Majorana qubit in solid-state systems. In fact,
the observation of stable Rabi oscillations is synonymous with having a qubit,
and our work establishes that such a qubit experiment should be successful in
the existing Majorana platforms.

_Acknowledgements._ —We are particularly grateful to Xin Zhang, Francesco
Zatelli, F. Setiawan, Michael Wimmer, and Jay D. Sau for useful discussions.
H.P. is supported by US-ONR grant No. N00014-23-1-2357. C-X.L is supported by
a subsidy for top consortia for knowledge and innovation (TKI toeslag). S.D.S.
is supported by the Laboratory for Physical Sciences through the Condensed
Matter Theory Center at the University of Maryland.

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Supplemental Materials for “Rabi oscillation and coherence time of Majorana
qubit in quantum dot-superconductor array”

##  Appendix I Leakage probability in a Rabi experiment

The system of the double two-site Kitaev chain is described by the following
Hamiltonian:

$\mathcal{H}_{\text{tot}}=\mathcal{H}_{L}+\mathcal{H}_{R}+\mathcal{H}_{T}+\mathcal{H}_{\mu}$
(S-1)

where intra-chain coupling in the left chain (Site index 1 and 2) and right
chain (Site index 3 and 4) are

$\begin{split}\mathcal{H}_{L}=t\sum\left(c_{L2}^{\dagger}c_{L1}+c_{L1}^{\dagger}c_{L2}\right)+\Delta\left(c_{L2}c_{L1}+c_{L1}^{\dagger}c_{L2}^{\dagger}\right)\\\
\mathcal{H}_{R}=t\sum\left(c_{R2}^{\dagger}c_{R1}+c_{R1}^{\dagger}c_{R2}\right)+\Delta\left(c_{R2}c_{R1}+c_{R1}^{\dagger}c_{R2}^{\dagger}\right),\end{split}$
(S-2)

the inter-chain hopping is

$\mathcal{H}_{T}=\Gamma\left(c_{R1}^{\dagger}c_{L2}+c_{L2}^{\dagger}c_{R1}\right),$
(S-3)

and the onsite chemical potential is

$\mathcal{H}_{\mu}=\sum_{a\in\left\\{L,R\right\\}}\sum_{i=1}^{2}\mu_{a,i}c_{a,i}^{\dagger}c_{a,i}.$
(S-4)

Up to a particle-hole transformation, we can choose $t>0$, $\Delta>0$, and
$\Gamma>0$.

Here, without the tunneling term $\Gamma=0$, the two chains are decoupled,
where the sweet spot is achieved when $t=\Delta$, and $\mu_{a,i}=0$, leading
to the ground state manifold spanned by

$\ket{e}_{L}=\frac{1}{\sqrt{2}}\left(1-c_{L1}^{\dagger}c_{L2}^{\dagger}\right)\ket{0},\ket{o}_{L}=\frac{1}{\sqrt{2}}\left(c_{L1}^{\dagger}-c_{L2}^{\dagger}\right)\ket{0},$
(S-5)
$\ket{e}_{R}=\frac{1}{\sqrt{2}}\left(1-c_{R1}^{\dagger}c_{R2}^{\dagger}\right)\ket{0},\ket{o}_{R}=\frac{1}{\sqrt{2}}\left(c_{R1}^{\dagger}-c_{R2}^{\dagger}\right)\ket{0},$
(S-6)

for the left and right systems, respectively.

With the tunneling term, the ground state of the two chains can be spanned by
the two other single-chain excited states denoted as

$\ket{e^{\prime}}_{L}=\frac{1}{\sqrt{2}}\left(1+c_{L1}^{\dagger}c_{L2}^{\dagger}\right)\ket{0},\ket{o^{\prime}}_{L}=\frac{1}{\sqrt{2}}\left(c_{L1}^{\dagger}+c_{L2}^{\dagger}\right)\ket{0},$
(S-7)
$\ket{e^{\prime}}_{R}=\frac{1}{\sqrt{2}}\left(1+c_{R1}^{\dagger}c_{R2}^{\dagger}\right)\ket{0},\ket{o^{\prime}}_{R}=\frac{1}{\sqrt{2}}\left(c_{R1}^{\dagger}+c_{R2}^{\dagger}\right)\ket{0}.$
(S-8)

Therefore, $\ket{e}_{L}$, $\ket{e^{\prime}}_{L}$, $\ket{o}_{L}$, and
$\ket{o^{\prime}}_{L}$ ($\ket{e}_{R}$, $\ket{e^{\prime}}_{R}$, $\ket{o}_{R}$,
and $\ket{o^{\prime}}_{R}$) form the complete basis for the left (right)
chain.

Without the loss of generality, we choose to work in the even-total-parity,
leading to a complete basis of $\ket{ee}\equiv\ket{e}_{L}\ket{e}_{R}$,
$\ket{oo}$, $\ket{e^{\prime}e^{\prime}}$,$\ket{o^{\prime}o^{\prime}}$,
$\ket{ee^{\prime}}$, $\ket{oo^{\prime}}$, $\ket{e^{\prime}e}$, and
$\ket{o^{\prime}o}$. With this set of bases, the matrix representation of the
sum of Hamiltonian Eq. (S-2) and Eq. (S-3) is

$H_{L}+H_{R}+H_{T}=\begin{pmatrix}h_{+}&0\\\ 0&h_{-}\end{pmatrix},$ (S-9)

where $h_{+}$ and $h_{-}$ are

$h_{+}=\begin{pmatrix}-2\Delta&-\Gamma/2&0&-\Gamma/2\\\
-\Gamma/2&-2t&\Gamma/2&0\\\ 0&\Gamma/2&2\Delta&\Gamma/2\\\
-\Gamma/2&0&\Gamma/2&2t\end{pmatrix},$ (S-10)
$h_{-}=\begin{pmatrix}0&\Gamma/2&0&\Gamma/2\\\ \Gamma/2&0&-\Gamma/2&0\\\
0&-\Gamma/2&0&-\Gamma/2\\\ \Gamma/2&0&-\Gamma/2&0\end{pmatrix}.$ (S-11)

Similarly, the matrix representation of the onsite chemical potential Eq.
(S-4) is

$H_{\mu}=\frac{\mu_{LR}}{2}\mathds{1}+\frac{1}{2}\begin{pmatrix}0&h_{\mu}\\\
h_{\mu}^{\dagger}&0\end{pmatrix},$ (S-12)

where

$h_{\mu}=\begin{pmatrix}-\mu_{R}&0&-\mu_{L}&0\\\
0&\delta\mu_{R}&0&\delta\mu_{L}\\\ -\mu_{L}&0&-\mu_{R}&0\\\
0&\delta\mu_{L}&0&\delta\mu_{R}\end{pmatrix},$ (S-13)

with the shorthand notions of
$\mu_{LR}\equiv\sum_{a\in\left\\{L,R\right\\}}\mu_{a}$ ,
$\mu_{a}\equiv\mu_{a1}+\mu_{a2}$, and $\delta\mu_{a}\equiv\mu_{a1}-\mu_{a2}$.

###  I.1 Leakage due to $\Delta$, $t$, and $\Gamma$

In Eq. (2) in the main text, we considered the disorder effect in $\epsilon$
and $\Gamma$ before $\sigma_{z}$ and $\sigma_{x}$. Here, we will consider
their leakage effect separately to understand the leakage that is effectively
on $\epsilon$ and $\Gamma$.

We first consider the effect of the disorder only in $\Delta$, $t$, and
$\Gamma$, i.e., $\mu_{a,i}=0$, because Eq. (S-9) is block diagonal, and given
the initial state being $\ket{ee}$, we only need to consider the subspace of
$h_{+}$, where the Rabi oscillation is between $\ket{ee}$ and $\ket{oo}$, and
the leakage states are $\ket{e^{\prime}e^{\prime}}$ and
$\ket{o^{\prime}o^{\prime}}$.

We use the time-dependent perturbation theory, where Eq. (S-9) is decomposed
into the noninteracting part $H_{0}$

$H_{0}=\begin{pmatrix}-2\Delta&-\Gamma/2&0&0\\\ -\Gamma/2&-2t&0&0\\\
0&0&2\Delta&\Gamma/2\\\ 0&0&\Gamma/2&2t\end{pmatrix},$ (S-14)

and the perturbation $H_{1}$ as

$H_{1}=\begin{pmatrix}0&0&0&-\Gamma/2\\\ 0&0&\Gamma/2&0\\\ 0&\Gamma/2&0&0\\\
-\Gamma/2&0&0&0\end{pmatrix}.$ (S-15)

Conceptually, the first term $H_{0}$ in Eq. (S-14) accounts for the Rabi
oscillation between $\ket{ee}$ and $\ket{oo}$ (given the initial state being
$\ket{ee}$), and the second term $H_{1}$ in Eq. (S-15) leads to
$\ket{e^{\prime}e^{\prime}}$ and $\ket{o^{\prime}o^{\prime}}$.

The time evolution operator (in the Schrödinger picture) is expanded in the
Dyson series (truncated at the first order) as

$U(\tau)=e^{-iH_{0}\tau}\left(1-i\int_{0}^{\tau}d\tau_{1}H_{1,I}(\tau_{1})\right)e^{iH_{0}\tau},$
(S-16)

where $H_{1,I}(\tau_{1})$ is $H_{1}$ in the interacting picture

$H_{1,I}(\tau_{1})=e^{iH_{0}\tau}H_{1}e^{-iH_{0}\tau}=\sum_{i,j}\matrixelement{i}{H_{1}}{j}e^{i\left(E_{i}-E_{j}\right)\tau_{1}},$
(S-17)

and $\ket{i}$ is the eigenvector of $H_{0}$ with the eigenvalues of $E_{i}$.

The spectrum of $H_{0}$ in Eq. (S-14) is

$\begin{split}-E_{+}:&\quad\begin{pmatrix}\cos(\frac{\theta}{2})&\sin(\frac{\theta}{2})&0&0\end{pmatrix}^{\intercal}\\\
-E_{-}:&\quad\begin{pmatrix}-\sin(\frac{\theta}{2})&\cos(\frac{\theta}{2})&0&0\end{pmatrix}^{\intercal}\\\
E_{-}:&\quad\begin{pmatrix}0&0&-\sin(\frac{\theta}{2})&\cos(\frac{\theta}{2})\end{pmatrix}^{\intercal}\\\
E_{+}:&\quad\begin{pmatrix}0&0&\cos(\frac{\theta}{2})&\sin(\frac{\theta}{2})\end{pmatrix}^{\intercal}\\\
\end{split}$ (S-18)

where $E_{\pm}=\Delta+t\pm\sqrt{(\Delta-t)^{2}+(\Gamma/2)^{2}}$, and
$\tan\theta=\frac{\Gamma/2}{\Delta-t}$, We substitute the eigenvectors and
eigenvalues of $H_{0}$ into Eq. (S-17), with the initial state

$\ket{\psi(\tau=0)}=\ket{ee}=\begin{pmatrix}1&0&0&0\end{pmatrix}^{\intercal},$
(S-19)

we have the (unnormalized) final state in the Schrödinger picture as

$\ket{\psi(\tau)}=U(\tau)\ket{\psi(\tau=0)}=\ket{\psi(\tau)}=\begin{pmatrix}e^{i\omega\tau}\left[\cos(\Omega\tau)+i\sin(\Omega\tau)\cos\theta\right]\\\
e^{i\omega\tau}i\sin(\Omega\tau)\sin\theta\\\
\frac{\Gamma}{2\omega}\sin(\omega\tau)\sin(\Omega\tau)\sin\theta\\\
\frac{\Gamma}{2\omega}\left[-\sin(\Omega\tau)\cos\theta+i\cos(\Omega\tau)\right]\sin(\omega\tau)\end{pmatrix},$
(S-20)

where $\Omega=\sqrt{(\Delta-t)^{2}+(\Gamma/2)^{2}}$ and $\omega=\Delta+t$.

Therefore, the Rabi oscillation between $\ket{ee}$ and $\ket{oo}$ have the
probability densities of

$\begin{split}P_{ee}(\tau)&=\absolutevalue{\innerproduct{ee}{\psi(\tau)}}^{2}=\sin^{2}{\left(\Omega\tau\right)}\cos^{2}{\left(\theta\right)}+\cos^{2}{\left(\Omega\tau\right)}\\\
P_{oo}(\tau)&=\absolutevalue{\innerproduct{oo}{\psi(\tau)}}^{2}=\sin^{2}{\left(\theta\right)}\sin^{2}{\left(\Omega\tau\right)},\end{split}$
(S-21)

and the leakage is

$P_{\text{leak}}(\tau)=P_{e^{\prime}e^{\prime}}(\tau)+P_{o^{\prime}o^{\prime}}(\tau)=\absolutevalue{\innerproduct{e^{\prime}e^{\prime}}{\psi(\tau)}}^{2}+\absolutevalue{\innerproduct{o^{\prime}o^{\prime}}{\psi(\tau)}}^{2}=\frac{\Gamma^{2}\sin^{2}{\left(\omega\tau\right)}}{4\omega^{2}},$
(S-22)

indicating that the leakage frequency is $2\omega=2(\Delta+t)$, which is
independent of the $\Gamma$, and consistent with Fig. 2(b) in the main text.

Specifically, at the sweep spot $\Delta=t$, we have $\theta=\pi/2$ and the
probability densities are

$\begin{split}P_{ee}(\tau)&=\cos^{2}\left(\Omega\tau\right)\\\
P_{oo}(\tau)&=\sin^{2}\left(\Omega\tau\right)\\\
P_{\text{leak}}(\tau)&=\frac{\Gamma^{2}}{16\Delta^{2}}\sin^{2}\left(2\Delta\tau\right),\end{split}$
(S-23)

recovering the Rabi frequency $2\Omega=\Gamma$ in Frg. 2(a).

###  I.2 Leakage due to $\mu_{L1}$

To consider the leakage effect in $\mu_{L1}$ (or, equivalently for $\mu_{R2}$
due to the inversion symmetry), we set all other parameters to the sweet spot,
including $\Delta=t_{0}$ and $\mu_{L2}=\mu_{R1}=\mu_{R2}=0$. Following the
same time-dependent perturbation theory, the noninteracting part
$H^{\mu_{L1}}_{0}$ is

$H^{\mu_{L1}}_{0}=\frac{\mu_{L1}}{2}\mathds{1}+\begin{pmatrix}H_{0}&0\\\
0&h_{-}\end{pmatrix}$ (S-24)

where $H_{0}$ and $h_{-}$ are in Eq. (S-14) and in Eq. (S-11) (with
$\Delta=t$), and the perturbation term $H^{\mu_{L1}}_{1}$ is

$H^{\mu_{L1}}_{1}=\begin{pmatrix}H_{1}&h_{\mu_{L1}}\\\
h_{\mu_{L1}}^{\dagger}&0\end{pmatrix}$ (S-25)

where $H_{1}$ is in Eq. (S-15) and $H_{\mu_{L1}}$ is

$h_{\mu_{L1}}=\frac{1}{2}\begin{pmatrix}0&0&-\mu_{L1}&0\\\ 0&0&0&\mu_{L1}\\\
-\mu_{L1}&0&0&0\\\ 0&\mu_{L1}&0&0\end{pmatrix}.$ (S-26)

from Eq. (S-13).

Here, the noninteracting term $H^{\mu_{L1}}_{0}$ has the eigenvalues and
eigenvectors as

$\begin{split}-2\Delta-\frac{\Gamma}{2}+\frac{\mu_{1}}{2}:&\quad\frac{1}{\sqrt{2}}\begin{pmatrix}1&1&0&0&0&0&0&0\end{pmatrix}^{\intercal}\\\
-2\Delta+\frac{\Gamma}{2}+\frac{\mu_{1}}{2}:&\quad\frac{1}{\sqrt{2}}\begin{pmatrix}-1&1&0&0&0&0&0&0\end{pmatrix}^{\intercal}\\\
2\Delta-\frac{\Gamma}{2}+\frac{\mu_{1}}{2}:&\quad\frac{1}{\sqrt{2}}\begin{pmatrix}0&0&-1&1&0&0&0&0\end{pmatrix}^{\intercal}\\\
2\Delta+\frac{\Gamma}{2}+\frac{\mu_{1}}{2}:&\quad\frac{1}{\sqrt{2}}\begin{pmatrix}0&0&1&1&0&0&0&0\end{pmatrix}^{\intercal}\\\
\frac{\mu_{1}}{2}:&\quad\frac{1}{\sqrt{2}}\begin{pmatrix}0&0&0&0&1&0&1&0\end{pmatrix}^{\intercal}\\\
\frac{\mu_{1}}{2}:&\quad\frac{1}{\sqrt{2}}\begin{pmatrix}0&0&0&0&0&-1&0&1\end{pmatrix}^{\intercal}\\\
-\Gamma+\frac{\mu_{1}}{2}:&\quad\frac{1}{2}\begin{pmatrix}0&0&0&0&1&1&-1&1\end{pmatrix}^{\intercal}\\\
\Gamma+\frac{\mu_{1}}{2}:&\quad\frac{1}{2}\begin{pmatrix}0&0&0&0&-1&1&1&1\end{pmatrix}^{\intercal}\\\
\end{split}$ (S-27)

With the initial state being
$\ket{ee}=\begin{pmatrix}1&0&0&0&0&0&0&0\end{pmatrix}^{\intercal}$, the
(unnormalized) final state in the Schrödinger picture is

$\psi^{\mu_{L1}}(\tau)=\begin{pmatrix}e^{i\tau\left(2\Delta-\frac{\mu_{1}}{2}\right)}\cos{\left(\frac{\tau\Gamma}{2}\right)}\\\
ie^{i\tau\left(2\Delta-\frac{\mu_{1}}{2}\right)}\sin{\left(\frac{\tau\Gamma}{2}\right)}\\\
\frac{\Gamma}{4\Delta}e^{-\frac{i\tau\mu_{1}}{2}}\sin{\left(2\tau\Delta\right)}\sin{\left(\frac{\tau\Gamma}{2}\right)}\\\
\frac{i\Gamma}{4\Delta}e^{-\frac{i\tau\mu_{1}}{2}}\sin{\left(2\tau\Delta\right)}\cos{\left(\frac{\tau\Gamma}{2}\right)}\\\
\frac{1}{2}\mu_{1}e^{i\tau\left(\Delta-\frac{\mu_{1}}{2}\right)}\sin{\left(\frac{\tau\Gamma}{2}\right)}\left(\frac{1}{E_{-}}e^{\frac{i\tau\Gamma}{4}}\sin{\left(\frac{E_{-}\tau}{2}\right)}-\frac{1}{E_{+}}e^{-\frac{i\tau\Gamma}{4}}\sin{\left(\frac{E_{+}\tau}{2}\right)}\right)\\\
\frac{1}{2}\mu_{1}e^{i\tau\left(\Delta-\frac{\mu_{1}}{2}\right)}\cos{\left(\frac{\tau\Gamma}{2}\right)}\left(-\frac{1}{E_{-}}ie^{\frac{i\tau\Gamma}{4}}\sin{\left(\frac{E_{-}\tau}{2}\right)}+\frac{1}{E_{+}}ie^{-\frac{i\tau\Gamma}{4}}\sin{\left(\frac{E_{+}\tau}{2}\right)}\right)\\\
\frac{1}{2}\mu_{1}e^{i\tau\left(\Delta-\frac{\mu_{1}}{2}\right)}\cos{\left(\frac{\tau\Gamma}{2}\right)}\left(\frac{1}{E_{-}}ie^{\frac{i\tau\Gamma}{4}}\sin{\left(\frac{E_{-}\tau}{2}\right)}+\frac{1}{E_{+}}ie^{-\frac{i\tau\Gamma}{4}}\sin{\left(\frac{E_{+}\tau}{2}\right)}\right)\\\
\frac{1}{2}\mu_{1}e^{i\tau\left(\Delta-\frac{\mu_{1}}{2}\right)}\sin{\left(\frac{\tau\Gamma}{2}\right)}\left(\frac{1}{E_{-}}e^{\frac{i\tau\Gamma}{4}}\sin{\left(\frac{E_{-}\tau}{2}\right)}+\frac{1}{E_{+}}e^{-\frac{i\tau\Gamma}{4}}\sin{\left(\frac{E_{+}\tau}{2}\right)}\right)\end{pmatrix},$
(S-28)

where $E_{\pm}=2\Delta\pm\Gamma/2$.

Therefore, the leakage to the $\ket{e^{\prime}e^{\prime}}$ and
$\ket{o^{\prime}o^{\prime}}$ is the same

$P_{e^{\prime}e^{\prime}}(\tau)+P_{o^{\prime}o^{\prime}}(\tau)=\absolutevalue{\innerproduct{e^{\prime}e^{\prime}}{\psi^{\mu_{L1}}(\tau)}}^{2}+\absolutevalue{\innerproduct{o^{\prime}o^{\prime}}{\psi^{\mu_{L1}}(\tau)}}^{2}=\frac{\Gamma^{2}\sin^{2}{\left(2\tau\Delta\right)}}{16\Delta^{2}},$
(S-29)

and the leakage to $\ket{ee^{\prime}}$, $\ket{oo^{\prime}}$,
$\ket{e^{\prime}e}$, and $\ket{o^{\prime}o}$ is

$\begin{split}P_{ee^{\prime}}(\tau)+P_{oo^{\prime}}(\tau)+P_{e^{\prime}e}(\tau)+P_{o^{\prime}o}(\tau)&=\absolutevalue{\innerproduct{ee^{\prime}}{\psi^{\mu_{L1}}(\tau)}}^{2}+\absolutevalue{\innerproduct{oo^{\prime}}{\psi^{\mu_{L1}}(\tau)}}^{2}+\absolutevalue{\innerproduct{e^{\prime}e}{\psi^{\mu_{L1}}(\tau)}}^{2}+\absolutevalue{\innerproduct{o^{\prime}o}{\psi^{\mu_{L1}}(\tau)}}^{2}\\\
&=\frac{\mu_{L1}^{2}}{2}\left(\frac{\sin^{2}\left(\frac{E_{-}\tau}{2}\right)}{E_{-}^{2}}+\frac{\sin^{2}\left(\frac{E_{+}\tau}{2}\right)}{E_{+}^{2}}\right),\end{split}$
(S-30)

This introduces a superposition of two frequencies $E_{-}$ and $E_{+}$ in the
leakage frequency, where the envelope frequency is $E_{-}+E_{+}=4\Delta$ and
the carrier frequency is $E_{+}-E_{-}=\Gamma$.

###  I.3 Leakage due to $\mu_{L2}$

The leakage effect in the other onsite chemical potential is $\mu_{L2}$ (or
$\mu_{R1}$). Namely, we set $\mu_{L1}=\mu_{R1}=\mu_{R2}=0$, and $t=\Delta$.
This leads to the same noninteracting part $H^{\mu_{L2}}_{0}=H^{\mu_{L1}}_{0}$
as in Eq. (S-24), while the perturbation term $H^{\mu_{L2}}_{1}$ is

$H^{\mu_{L2}}_{1}=\begin{pmatrix}H_{1}&h_{\mu_{L2}}\\\
h_{\mu_{L2}}^{\dagger}&0\end{pmatrix}$ (S-31)

where $H_{1}$ is in Eq. (S-15) and $h_{\mu_{L2}}$ is

$h_{\mu_{L2}}=\frac{1}{2}\begin{pmatrix}0&0&-\mu_{L2}&0\\\ 0&0&0&-\mu_{L2}\\\
-\mu_{L2}&0&0&0\\\ 0&-\mu_{L2}&0&0\end{pmatrix}$ (S-32)

Here, the noninteracting term $H^{\mu_{L2}}_{0}$ has the same eigenvalues
(with $\mu_{L1}$ replaced with $\mu_{L2}$) and eigenvectors as in Eq. (S-27),
and therefore, the final state in the Schrödinger picture starting from
$\ket{ee}$ is

$\psi^{\mu_{L2}}(\tau)=\begin{pmatrix}e^{i\tau\left(2\Delta-\frac{\mu_{2}}{2}\right)}\cos{\left(\frac{\tau\Gamma}{2}\right)}\\\
ie^{i\tau\left(2\Delta-\frac{\mu_{2}}{2}\right)}\sin{\left(\frac{\tau\Gamma}{2}\right)}\\\
\frac{\Gamma}{4\Delta}e^{-\frac{i\tau\mu_{2}}{2}}\sin{\left(2\tau\Delta\right)}\sin{\left(\frac{\tau\Gamma}{2}\right)}\\\
\frac{i\Gamma}{4\Delta}e^{-\frac{i\tau\mu_{2}}{2}}\sin{\left(2\tau\Delta\right)}\cos{\left(\frac{\tau\Gamma}{2}\right)}\\\
\frac{\mu_{2}}{4}e^{i\tau\left(\Delta-\frac{\mu_{2}}{2}\right)}\left[\left(\frac{i\sin{\left(\frac{E_{-}\tau}{2}\right)}}{E_{-}}-\frac{i\sin{\left(\frac{E^{{}^{\prime}}_{+}\tau}{2}\right)}}{E^{{}^{\prime}}_{+}}\right)e^{-\frac{i\tau\Gamma}{4}}+\left(-\frac{i\sin{\left(\frac{E^{{}^{\prime}}_{-}\tau}{2}\right)}}{E^{{}^{\prime}}_{-}}+\frac{i\sin{\left(\frac{E_{+}\tau}{2}\right)}}{E_{+}}\right)e^{\frac{i\tau\Gamma}{4}}\right]\\\
\frac{\mu_{2}}{4}e^{i\tau\left(\Delta-\frac{\mu_{2}}{2}\right)}\left[\left(\frac{i\sin{\left(\frac{E_{-}\tau}{2}\right)}}{E_{-}}+\frac{i\sin{\left(\frac{E^{{}^{\prime}}_{+}\tau}{2}\right)}}{E^{{}^{\prime}}_{+}}\right)e^{-\frac{i\tau\Gamma}{4}}+\left(-\frac{i\sin{\left(\frac{E^{{}^{\prime}}_{-}\tau}{2}\right)}}{E^{{}^{\prime}}_{-}}-\frac{i\sin{\left(\frac{E_{+}\tau}{2}\right)}}{E_{+}}\right)e^{\frac{i\tau\Gamma}{4}}\right]\\\
\frac{\mu_{2}}{4}e^{i\tau\left(\Delta-\frac{\mu_{2}}{2}\right)}\left[\left(\frac{i\sin{\left(\frac{E_{-}\tau}{2}\right)}}{E_{-}}+\frac{i\sin{\left(\frac{E^{{}^{\prime}}_{+}\tau}{2}\right)}}{E^{{}^{\prime}}_{+}}\right)e^{-\frac{i\tau\Gamma}{4}}+\left(\frac{i\sin{\left(\frac{E^{{}^{\prime}}_{-}\tau}{2}\right)}}{E^{{}^{\prime}}_{-}}+\frac{i\sin{\left(\frac{E_{+}\tau}{2}\right)}}{E_{+}}\right)e^{\frac{i\tau\Gamma}{4}}\right]\\\
\frac{\mu_{2}}{4}e^{i\tau\left(\Delta-\frac{\mu_{2}}{2}\right)}\left[\left(-\frac{i\sin{\left(\frac{E_{-}\tau}{2}\right)}}{E_{-}}+\frac{i\sin{\left(\frac{E^{{}^{\prime}}_{+}\tau}{2}\right)}}{E^{{}^{\prime}}_{+}}\right)e^{-\frac{i\tau\Gamma}{4}}+\left(-\frac{i\sin{\left(\frac{E^{{}^{\prime}}_{-}\tau}{2}\right)}}{E^{{}^{\prime}}_{-}}+\frac{i\sin{\left(\frac{E_{+}\tau}{2}\right)}}{E_{+}}\right)e^{\frac{i\tau\Gamma}{4}}\right]\end{pmatrix},$
(S-33)

where $E_{\pm}=2\Delta\pm\Gamma/2$ and $E_{\pm}^{{}^{\prime}}=2\Delta\pm
3\Gamma/2$.

Therefore, the leakage to the $\ket{e^{\prime}e^{\prime}}$ and
$\ket{o^{\prime}o^{\prime}}$ is the same as Eq. (S-30), and the leakage to
$\ket{ee^{\prime}}$, $\ket{oo^{\prime}}$, $\ket{e^{\prime}e}$, and
$\ket{o^{\prime}o}$ is

$\begin{split}P_{ee^{\prime}}(\tau)+P_{oo^{\prime}}(\tau)+P_{e^{\prime}e}(\tau)+P_{o^{\prime}o}(\tau)&=\absolutevalue{\innerproduct{ee^{\prime}}{\psi^{\mu_{L2}}(\tau)}}^{2}+\absolutevalue{\innerproduct{oo^{\prime}}{\psi^{\mu_{L2}}(\tau)}}^{2}+\absolutevalue{\innerproduct{e^{\prime}e}{\psi^{\mu_{L2}}(\tau)}}^{2}+\absolutevalue{\innerproduct{o^{\prime}o}{\psi^{\mu_{L2}}(\tau)}}^{2}\\\
&=\frac{\mu_{L2}^{2}}{4}\left(\frac{\sin^{2}\left(\frac{E_{-}\tau}{2}\right)}{E_{-}^{2}}+\frac{\sin^{2}\left(\frac{E_{+}\tau}{2}\right)}{E_{+}^{2}}+\frac{\sin^{2}\left(\frac{E_{-}^{{}^{\prime}}\tau}{2}\right)}{{E_{-}^{{}^{\prime}}}^{2}}+\frac{\sin^{2}\left(\frac{E_{+}^{{}^{\prime}}\tau}{2}\right)}{{E_{+}^{{}^{\prime}}}^{2}}\right).\end{split}$
(S-34)

##  Appendix II Dephasing time $T_{2}$ estimation

###  II.1 Dephasing due to disorder in $\Gamma$

In this section, we consider the dephasing effect which is used to estimate
$T_{2}$. To focus only on the low energy sector, we work in the minimal
2-level system where the Hilbert space only includes $\ket{ee}$ and
$\ket{oo}$. The effect of $\Gamma$ acts like the magnetic field along the $x$
direction, and therefore, the effective two-level Hamiltonian is

$H_{x}=B_{x}\sigma_{x},$ (S-35)

where $B_{x}$ is quasi-static disorder following the Gaussian distribution
with the variance of $\sigma_{B_{x}}^{2}$ and mean of $\bar{B}_{x}$, i.e.,
$B_{x}\sim\mathcal{N}(\bar{B}_{x},\sigma_{B_{x}}^{2})$, $\sigma_{x}$ is the
Pauli X matrix, and the initial state is $\psi_{x}(\tau=0)=\ket{ee}$.

Under the evolution of $H_{x}$, the final state $\psi_{x}(\tau)$ is

$\psi_{x}(\tau)=e^{-iH_{x}\tau}\psi_{x}(\tau=0)=\begin{pmatrix}\cos(B_{x}\tau)\\\
-i\sin(B_{x}\tau)\end{pmatrix},$ (S-36)

Therefore, the probability of finding the state in $\ket{oo}$ is

$P_{oo}(\tau)=\absolutevalue{\innerproduct{oo}{\psi_{x}(\tau)}}^{2}=\sin^{2}(B_{x}\tau).$
(S-37)

Thus, the disorder-averaged probability is

$\expectationvalue{P_{oo}(\tau)}_{B_{x}}=\frac{1}{\sqrt{2\pi}\sigma_{B_{x}}}\int_{-\infty}^{\infty}dB_{x}e^{-\frac{(B_{x}-\bar{B}_{x})^{2}}{2\sigma_{B_{x}}^{2}}}\sin^{2}(B_{x}\tau)=\frac{1}{2}\left[1+e^{-{2\tau^{2}\sigma_{B_{x}}^{2}}}\cos(2\bar{B}_{x}\tau)\right].$
(S-38)

Therefore, the decay of the envelope of
$\expectationvalue{P_{oo}(\tau)}_{B_{x}}$ follows the Gaussian decay with a
prefactor of $e^{-{2\tau^{2}\sigma_{B_{x}}^{2}}}$, namely, $\beta=2$ in the
ansatz in Eq. (5) in the main text. This provides a fundamental understanding
of the ansatz in Eq. (5) in the main text.

###  II.2 Dephasing due to disorder in $\mu_{a,i}$, $\Delta$, and $t$

Besides the dephasing effect due to the disorder in $\Gamma$, we also consider
the dephasing effect due to the disorder in $\mu_{a,i}$, $\Delta$, and $t$.

In practice, the disorder of these quantities acts like the magnetic field
along the $z$ direction, and therefore, the effective two-level Hamiltonian is

$H_{xz}=B_{x}\sigma_{x}+B_{z}\sigma_{z},$ (S-39)

where $B_{x}$ here is constant, and $B_{z}$ is quasi-static disorder following
the Gaussian distribution with the variance of $\sigma_{B_{z}}^{2}$ and mean
of 0, i.e., $B_{z}\sim\mathcal{N}(0,\sigma_{B_{z}}^{2})$, $\sigma_{z}$ is the
Pauli Z matrix, and the initial state is again $\psi_{xz}(\tau=0)=\ket{ee}$.

Under the evolution of $H_{xz}$, the final state $\psi_{xz}(\tau)$ is

$\psi_{xz}(\tau)=e^{-iH_{xz}\tau}\psi_{xz}(\tau=0)=\begin{pmatrix}\cos(\sqrt{B_{x}^{2}+B_{z}^{2}}\tau)-i\sin(\sqrt{B_{x}^{2}+B_{z}^{2}}\tau)\cos(\theta)\\\
-i\sin(\sqrt{B_{x}^{2}+B_{z}^{2}}\tau)\sin(\theta)\end{pmatrix},$ (S-40)

where $\tan\theta=B_{x}/B_{z}$, and the probability of finding the state in
$\ket{oo}$ is

$P_{oo}(\tau)=\absolutevalue{\innerproduct{oo}{\psi_{xz}(\tau)}}^{2}=\sin^{2}\left(\sqrt{B_{x}^{2}+B_{z}^{2}}\tau\right)\frac{B_{x}^{2}}{B_{x}^{2}+B_{z}^{2}}.$
(S-41)

Therefore, the disorder-averaged probability is

$\begin{split}\expectationvalue{P_{oo}(\tau)}_{B_{z}}&=\frac{1}{\sqrt{2\pi}\sigma_{B_{z}}}\int_{-\infty}^{\infty}dB_{z}e^{-\frac{B_{z}^{2}}{2\sigma_{B_{z}}^{2}}}\sin^{2}\left(\sqrt{B_{x}^{2}+B_{z}^{2}}\tau\right)\frac{B_{x}^{2}}{B_{x}^{2}+B_{z}^{2}}\\\
&\approx\frac{1}{\sqrt{2\pi}\sigma_{B_{z}}}\int_{-\infty}^{\infty}dB_{z}e^{-\frac{B_{z}^{2}}{2\sigma_{B_{z}}^{2}}}\sin^{2}\left(\left(B_{x}^{2}+\frac{B_{z}^{2}}{2B_{x}}\right)\tau\right)\\\
&=\frac{1}{4}\left(2-\frac{B_{x}e^{2iB_{x}\tau}}{\sqrt{B_{x}(B_{x}-2i\sigma_{B_{z}}^{2}\tau)}}-\frac{B_{x}e^{-2iB_{x}\tau}}{\sqrt{B_{x}(B_{x}+2i\sigma_{B_{z}}^{2}\tau)}}\right)\end{split}$
(S-42)

where the second line assumes the $B_{z}\ll B_{x}$ such that
$\sqrt{B_{x}^{2}+B_{z}^{2}}\approx B_{x}+\frac{B_{z}^{2}}{2B_{x}}$, and
$\frac{B_{x}^{2}}{B_{x}^{2}+B_{z}^{2}}\approx 1$. Therefore, it shows a power-
law decay with
$\expectationvalue{P_{oo}(\tau)}_{B_{z}}\sim\tau^{-\frac{1}{2}}$.

##  Appendix III Model Hamiltonian and parameters in the numerical simulations

###  III.1 Model parameters for charge noises

In the numerical simulations of the Rabi and Ramsey oscillations shown in the
main text, we include charge noises assuming quasi-static disorder. That is,
the fluctuations of a model parameter $\lambda$ follows an uncorrelated
Gaussian distribution
$\lambda\sim\mathcal{N}(\overline{\lambda},\sigma_{\lambda}^{2})$. In
particular, in a charge qubit, the fluctuation amplitude of a dot level
$\varepsilon_{L,R}$ is chosen to be $\sigma_{\varepsilon}=3~{}\mu$eV and for
tunnel couplings we choose $\sigma_{\Gamma}/\Gamma=0.01$ corresponding to the
devices studied in Refs. Hayashi  _et al._ (2003); Paladino  _et al._ (2014).
Other parameters for the charge qubit are summarized in Table 2.

Supplementary Table 2: Parameters for charge qubits in $\mu$eV. Qubit | Protocol | $\overline{\Gamma}$ | $\overline{\varepsilon_{L}}$ | $\overline{\varepsilon_{R}}$ | $\sigma_{\varepsilon}$ | $\sigma_{\Gamma}$ | Figure
---|---|---|---|---|---|---|---
Charge | Rabi | 5 | 0 | 0 | 3 | 0.05 | Fig. 3(a)
Charge | Ramsey $x$ pulse | 5 | 0 | 0 | 3 | 0.05 | Fig. 3(d)
Ramsey $z$ pulse | 5 | 20 | -20 | 3 | 0.05

The parameters of a Majorana qubit are chosen according to Refs. Dvir  _et
al._ (2023); Zatelli  _et al._ (2023); ten Haaf  _et al._ (2024). In a small-
gap Majorana qubit, the fluctuation amplitudes of the dot orbitals are
identical to those in double quantum dots, i.e. $\sigma_{\mu}=3~{}\mu$eV. The
intra-chain couplings are $t=\Delta=12~{}\mu$eV with a fluctuation amplitude
$\sigma_{t}=0.016~{}\mu$eV. The inter-chain couplings are chosen to be
$\sigma_{\Gamma}/\Gamma=0.01$, similar to charge qubits. In a large-gap
Majorana qubit, since the dot charges are now significantly screened due to
strong dot-superconductor hybridization Zatelli  _et al._ (2023); Liu  _et
al._ (2024a), the fluctuation amplitude reduces to only one-tenth of the
normal dot, i.e. $\sigma_{\mu}=0.3~{}\mu$eV. The intra-chain coupling
strengths are now $t=\Delta=38~{}\mu$eV with fluctuations of
$\sigma_{t}=0.05~{}\mu$eV. The inter-chain coupling fluctuations are still
chosen to be $\sigma_{\Gamma}/\Gamma=0.01$. The other parameters for Majorana
qubit parameters are summarized in Table 3.

Supplementary Table 3: Parameters for Majorana qubits in $\mu$eV. Qubit | Protocol | $\overline{t_{a}}$ | $\Delta_{a}$ | $\overline{\Gamma}$ | $\overline{\mu_{ai}}$ | $\sigma_{t}$ | $\sigma_{\mu}$ | $\sigma_{\Gamma}$ | Figure
---|---|---|---|---|---|---|---|---|---
Small-gap Majorana | Rabi | 12 | 12 | 5 | 0 | 0.016 | 3 | 0.05 | Fig. 3(b)
Large-gap Majorana | Rabi | 38 | 38 | 5 | 0 | 0.05 | 0.3 | 0.05 | Fig. 3(c)
Small-gap Majorana | Ramsey $x$ pulse | 12 | 12 | 0.5 | 0 | 0.016 | 3 | 0 | Fig. 3(e)
Ramsey $z$ pulse | 14 | 12 | 0 | 0 | 0.016 | 3 | 0
Large-gap Majorana | Ramsey $x$ pulse | 38 | 38 | 0.5 | 0 | 0.05 | 0.3 | 0 | Fig. 3(f)
Ramsey $z$ pulse | 40 | 38 | 0 | 0 | 0.05 | 0.3 | 0

##  Appendix IV Quantum capacitance measurement of a Majorana qubit

In this section, we show that quantum capacitance measurement is capable of
reading out $\ket{ee}$ and $\ket{oo}$ in a Majorana qubit. We first review the
calculations for a single Kitaev chain before generalizing it to a Majorana
qubit composed of double Kitaev chains. The zero-temperature quantum
capacitance of a state is defined as

$\displaystyle C_{q}=-\frac{\partial^{2}E}{\partial V^{2}_{g}},$ (S-43)

where $E$ is the eigenenergy of the state, and $V_{g}$ is the gate voltage.
The Hamiltonian of a minimal Kitaev chain can be decomposed into even- and
odd-parity sectors due to fermi parity conservation. The even-parity
Hamiltonian is

$\displaystyle H_{\text{even}}=\begin{pmatrix}\ket{00}\\\
\ket{11}\end{pmatrix}^{T}\begin{pmatrix}0&\Delta\\\
\Delta&\mu_{L1}+\mu_{L2}\end{pmatrix}\begin{pmatrix}\bra{00}\\\
\bra{11}\end{pmatrix},$ (S-44)

where $\mu_{1,2}$ are the onsite energies of the two dots and $t,\Delta$ are
the normal and Andreev couplings. The ground state energy $E_{e}$ of $\ket{e}$
is

$\displaystyle
E_{e}=\frac{\mu_{L1}+\mu_{L2}}{2}-\sqrt{\Delta^{2}+\left(\frac{\mu_{L1}+\mu_{L2}}{2}\right)^{2}}.$
(S-45)

On the other hand, the derivative with respect to gate voltage is

$\displaystyle\frac{\partial}{\partial
V_{g}}=\alpha_{1}\frac{\partial}{\partial\mu_{L1}}+\alpha_{2}\frac{\partial}{\partial\mu_{L2}}\approx\alpha\left(\frac{\partial}{\partial\mu_{L1}}+\frac{\partial}{\partial\mu_{L2}}\right),$
(S-46)

where $\alpha_{i}\equiv d\mu_{i}/dV_{g}$ is the lever arm, and here we assume
that all dots share a similar value of lever arm $\alpha$. Therefore, it is
straightforward to obtain the quantum capacitance of the even-parity ground
state as below

$\displaystyle C^{e}_{q}=-\frac{\partial^{2}E_{e}}{\partial
V^{2}_{g}}=\frac{\alpha^{2}}{\sqrt{\Delta^{2}+\left(\frac{\mu_{L1}+\mu_{L2}}{2}\right)^{2}}}-\frac{\alpha^{2}(\mu_{L1}+\mu_{L2})^{2}}{4\left(\Delta^{2}+\left(\frac{\mu_{L1}+\mu_{L2}}{2}\right)^{2}\right)^{3/2}}.$
(S-47)

At the sweet spot of $\mu_{L1}=\mu_{L2}=0$, we have

$\displaystyle C^{e}_{q}=\frac{\alpha^{2}}{\Delta}.$ (S-48)

On the other hand, the Hamiltonian in the odd-parity sector is

$\displaystyle H_{\text{odd}}=\begin{pmatrix}\ket{10}\\\
\ket{01}\end{pmatrix}^{T}\begin{pmatrix}\mu_{L1}&t\\\
t&\mu_{L2}\end{pmatrix}\begin{pmatrix}\bra{10}\\\ \bra{01}\end{pmatrix},$
(S-49)

with ground-state energy being

$\displaystyle
E_{o}=\frac{\mu_{L1}+\mu_{L2}}{2}-\sqrt{t^{2}+\left(\frac{\mu_{L1}-\mu_{L2}}{2}\right)^{2}}.$
(S-50)

The corresponding quantum capacitance is

$\displaystyle C^{o}_{q}=0,$ (S-51)

due to the opposite signs of coefficients in front of $\mu_{L1}$ and
$\mu_{L2}$ in the term of $(\frac{\mu_{L1}-\mu_{L2}}{2})^{2}$.

We now generalize our calculations to the quantum capacitance of the double
Kitaev chain system. The eigenenergies of the ground states are simply the sum
of the left and right chains, i.e.,

$\displaystyle E_{ab}=E_{aL}(\mu_{L1},\mu_{L2})+E_{bR}(\mu_{R1},\mu_{R2}).$
(S-52)

where $a,b$ denotes the parity $e,o$. We note that the $\mu$ dependence of $E$
is separable between the left and right chains, which indicates that
$\partial/\partial
V_{g}\to\alpha\left(\frac{\partial}{\partial\mu_{L1}}+\frac{\partial}{\partial\mu_{L2}}\right)$
for the left chain energy, while $\partial/\partial
V_{g}\to\alpha\left(\frac{\partial}{\partial\mu_{R1}}+\frac{\partial}{\partial\mu_{R2}}\right)$
for the right one. Therefore, we have

$\displaystyle C^{ab}_{q}=-\frac{\partial^{2}E_{ab}}{\partial
V^{2}_{g}}=-\frac{\partial^{2}E_{aL}}{\partial
V^{2}_{g}}-\frac{\partial^{2}E_{bR}}{\partial
V^{2}_{g}}=C^{a}_{qL}+C^{b}_{qR},$ (S-53)

that is, the quantum capacitance of the state in the whole system is a sum of
the value in each chain separately. We therefore have

$\displaystyle
C^{ee}_{q}=\frac{\alpha^{2}}{\Delta_{L}}+\frac{\alpha^{2}}{\Delta_{R}},$
$\displaystyle C^{oo}_{q}=0,$ $\displaystyle
C^{eo}_{q}=\frac{\alpha^{2}}{\Delta_{L}},$ $\displaystyle
C^{oe}_{q}=\frac{\alpha^{2}}{\Delta_{R}}.$ (S-54)

Therefore, one can distinguish between $\ket{ee}$ and $\ket{oo}$ states using
quantum capacitance measurement. Furthermore, the values of $C_{q}$ for
$\ket{eo}$ and $\ket{oe}$ are generally very different from the qubit states.
Thus our method also provides a possible way to investigate the quasiparticle
poisoning effect by analyzing the readout results of $\ket{eo}$ and
$\ket{oe}$.