# Quantum-dot-based Kitaev chains: Majorana quality measures and scaling with
increasing chain length

Viktor Svensson  Division of Solid State Physics and NanoLund, Lund
University, S-221 00 Lund, Sweden  Martin Leijnse  Division of Solid State
Physics and NanoLund, Lund University, S-221 00 Lund, Sweden

###### Abstract

Realizing Majorana bound states (MBSs) in short, well-controllable chains of
coupled quantum dots sidesteps the problem of disorder, but requires fine-
tuning and does not give the true topological protection inherent to long
chains. Here, we introduce a new quality measure that is applicable also in
the presence of strong electron-electron interactions and that quantifies the
closeness to topological protection of finetuned MBSs in short quantum-dot
chains. We call this measure local distinguishability because it puts a bound
to the degree an arbitrary local measurement can distinguish between two
states. We study the local distinguishability for quantum-dot chains of
different length. The three-dot chain is studied in detail, and we find that
it may not always be an improvement over the two-dot case, a fact that can be
understood within an effective model derived from perturbation theory. For
longer chains, the local distinguishability vanishes exponentially, signalling
a transition to a topological phase with two ground states that cannot be
distinguished by any local measurement.

##  I Introduction

The past 15 years have seen a surge of efforts aiming to realize the physics
of the so-called Kitaev chain [1], which features a topological
superconducting phase. This phase is associated with Majorana bound states
(MBSs), zero energy excitations with non-abelian statistics [2, 3, 4, 5, 6]
that could be used for topological quantum computation [7, 8, 9]. Many
different systems can be engineered to have a low-energy subspace that
realizes an effective Kitaev chain, including, for example, proximitized
nanowires [10, 11, 12, 13, 14, 15], magnetic adatoms [16, 17], and planar
Josephson junctions [18, 19, 20, 21]. However, the disorder present in all
these platforms complicates the realization and experimental detection of a
topological superconducting phase and MBSs [22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33].

Because of this, the proposal to implement a Kitaev chain in well-controllable
coupled quantum dots [34, 35, 36] has attracted increasing attention. Device
complexity increases with more dots and short chains are preferable, but
fortunately already the two-dot chain can host MBSs [35]. The price to pay in
short chains is that there is no true topological phase – the system must be
fine-tuned to a sweet spot to show signatures of MBSs and some protection
against perturbations. Because of the lack of formal topological protection,
the associated states have been called poor man’s MBSs. Various new ideas for
how to design and tune such two-dot chains have been explored theoretically
[37, 38, 39, 40, 41], and experimentally realized [42, 43, 44, 45].

Experimentally distinguishing MBSs from topologically trivial Andreev bound
states has proven to be a difficult problem. Experiments on the quantum-dot
platform [43, 44, 45] have already demonstrated correlated appearance and
splittings of zero-bias peaks at both ends of the chain, as well as signatures
in the nonlocal conductance that are consistent with (poor man’s) MBSs.
Conductance measurements in a setup including an additional probe quantum dot
could provide additional estimates of the nonlocal properties of the
wavefunction [46]. However, making conclusive statement about the closeness of
the observed states to true topological MBSs would require protocols aimed at
demonstrating non-abelian physics [47, 48, 49]. It is also an interesting
theoretical problem to define a MBS quality measure, that quantifies the
similarity of a given state to a true topological MBS. One such measure that
is based on the MBS wavefunction is the Majorana polarization (MP) [50, 51,
52, 38]. However, in the presence of electron-electron interactions, which are
typically very strong in the quantum-dot platform, the many-body nature of the
MBSs [53, 54, 55, 56, 57, 58, 59] complicates the interpretation of the MP.

In this article, we introduce a MBS quality measure based on the local
distinguishability of ground states in topological phases, which has no extra
difficulty in interacting systems and gives a rigorous bound on the protection
against perturbations. Increasing the length of the system should produce
better quality MBSs, and recent experimental results on a three-dot Kitaev
chain [60, 61] shows an increased stability of the energy spectrum under
certain perturbations. There is also recent theoretical work on scaling up to
longer length chains [62]. Here, we analyze the protection as a function of
the chain length, paying particular attention to the three-dot case. We find
that the protection in this case may be worse in some respects than the two-
dot case, and we explain these results with an effective Hamiltonian derived
by perturbation theory.

The paper is organized as follows: Section II describes the system and Section
III discusses how to characterize and quantify the quality of MBSs and how to
tune a system to a sweet spot in parameter space where such quality measures
are maximized. Section IV.1 then focuses on how to tune the three-dot chain
and shows that it can be worse than the two-dot chain in some respects, while
Section IV.2 contains a derivation of an effective model to explain these
results. Finally, Section IV.3 shows what happens when scaling to longer
chains.

##  II System

Figure 1: Sketch of setup where each quantum dot is coupled to the bulk
superconductor via a local proximity effect (with amplitude $\Delta$ and phase
$\phi$), and to its neighbours via both spin-preserving and -flipping
tunneling (with amplitudes $t$ and $t_{\text{so}}$). The dot levels
$\varepsilon$ and magnetic flux $\Phi$ (controlling the phase $\phi$) are used
to tune to a sweet spot.

The system we consider consists of a chain of quantum dots, each locally
coupled to a superconductor and coupled to each other via tunneling, see Fig.
1. This setup was proposed in Ref. [36] and the two-dot interacting version
was studied in detail in Ref. [40], where it was shown that for a wide range
of parameters it exhibits a sweet spot with localized MBSs, which can be
reached by tuning the dot levels $\varepsilon_{j}$ and the superconducting
phases $\phi_{j}$ (controlled by the magnetic fluxes $\Phi_{j}$ through the
loops). As $V_{Z},\Delta\rightarrow\infty$, the system reduces to a Kitaev
chain.

The system Hamiltonian consists of two parts,

$H=H^{\text{QD}}+H^{C},$ (1)

where $H^{\text{QD}}$ contains all the terms local to each of the $N$ dots and
$H^{C}$ contains the coupling between them. These parts take the form

$\displaystyle H^{\text{QD}}$
$\displaystyle=\sum_{j=1}^{N}\sum_{\sigma}(\varepsilon_{j}+\eta_{\sigma}V_{Z,j})n_{j\sigma}$
$\displaystyle+(\Delta_{j}e^{i\phi_{j}}d_{j\uparrow}^{\dagger}d_{j\downarrow}^{\dagger}+\text{h.c.})+U_{l,j}n_{j\uparrow}n_{j\downarrow}$
(2a) $\displaystyle H^{C}$
$\displaystyle=\sum_{j=1}^{N-1}\sum_{\sigma}(t_{j}d_{j\sigma}^{\dagger}d_{j+1\sigma}+\text{h.c.})$
$\displaystyle+(\eta_{\sigma}t_{\mathrm{so},j}d_{j\sigma}^{\dagger}d_{j+1\bar{\sigma}}+\text{h.c.})+U_{\mathrm{nl},j}N_{j}N_{j+1},$
(2b)

where $d^{\dagger}_{j\sigma}$ creates an electron with spin $\sigma$ on dot
$j$, $n_{j\sigma}=d^{\dagger}_{j\sigma}d_{j\sigma}$,
$N_{j}=n_{j\uparrow}+n_{j\downarrow}$ and $\eta_{\sigma}=\pm$. The local term
includes the dot levels $\varepsilon_{j}$, Zeeman splitting $V_{Z}$, induced
local pairing $\Delta_{j}$ and its phase $\phi_{j}$, and the local Coulomb
interaction $U_{l,j}$. The term coupling dots together includes spin-
conserving and spin-flip tunnelling $t_{j}$ and $t_{\mathrm{so},j}$, and the
non-local Coulomb interaction $U_{\mathrm{nl},j}$.

##  III Protection and sweet spot quality

In this section, we discuss how to characterize the kind of protection implied
by the existence of MBSs, the essential differences between short and long
chains, and define how we search for a sweet spot. We denote by $\ket{o}$ and
$\ket{e}$ the ground states in the odd and even fermionic parity sectors
respectively and define $\delta\rho=\outerproduct{o}{o}-\outerproduct{e}{e}$,
Their energy difference is $\delta E$ and $E_{\text{ex}}$ is the gap to
excited states.

The existence of non-overlapping Majorana operators implies that the two
ground states can’t be distinguished by any local measurement. Local means
here that the spatial size of the terms in the operator is smaller than the
separation between the MBSs. In particular, if the Hamiltonian is local, then
$\delta E=0$. This is the case for a long Kitaev chain in the topological
phase. This degeneracy is stable under any local perturbation $\mathcal{O}$ as

$\displaystyle\delta E$
$\displaystyle=\expectationvalue{H+\mathcal{O}}_{o}-\expectationvalue{H+\mathcal{O}}_{e}$
$\displaystyle=\Tr[\mathcal{O}\delta\rho]$ (3)

corresponds to the difference of a local measurement of the states.
Characterizing the quantity $\Tr[\mathcal{O}\delta\rho]$ will be one of the
main points of this article.

In short chains, even non-overlapping MBSs may be close enough to be coupled
together by the Hamiltonian. The Hamiltonian is in this sense non-local and
$\delta E$ may be non-zero even if the MBSs are perfectly localized to the
edges. This splitting introduces an undesirable dynamical phase in fusion and
braiding protocols. For a non-local Hamiltonian, $\delta E=0$ is an
independent condition from the separation of MBSs. Therefore, as we search for
the sweet spot, we will consider these independently.

###  III.1 Protection from fluctuating parameters

The stability of $\delta E$ is not necessarily related to the presence of
separated MBSs. Consider a Hamiltonian $H(\lambda)$ that is a function of some
parameter $\lambda$. Protection against fluctuations in $\lambda$ may arise in
several ways, some of which are

  1. 1.

$\norm{\frac{\partial H}{\partial\lambda}\delta\lambda}$ is small because the
Hamiltonian is naturally insensitive to variations in $\lambda$.

  2. 2.

$\Tr[\delta\rho\frac{\partial H}{\partial\lambda}]$ is small, because
$\delta\rho$ is orthogonal to this particular perturbation.

  3. 3.

There are non-overlapping MBSs that map between the ground states and the
Hamiltonian is local.

The first type has nothing to do with MBSs. The second type is implied by the
third, but can also occur when the MBSs overlap, see Sec. IV.1.1. The third
type is the strongest and implies protection based only on the locality of the
perturbations and the MBS wavefunction.

###  III.2 Majorana Polarization

In non-interacting systems the MBS wavefunction is simple to define and the
Majorana polarization (MP) can be used to characterize how separated the two
MBSs are [50, 51]. A generalization has been used in interacting systems[52,
38, 40] but in this case it cannot fully characterize MBSs with non-trivial
many-body content [54, 55, 56, 57, 58]. We use the definition from Ref. [40],

$\mathrm{MP}=\frac{\absolutevalue{\sum_{l,s}\matrixelement{e}{\gamma_{ls}}{o}^{2}}}{\sum_{l,s}\absolutevalue{\matrixelement{e}{\gamma_{ls}}{o}^{2}}},$
(4)

where $\ket{o}$ and $\ket{e}$ are the odd and even ground states, $l$ indexes
the fermions on the leftmost site 111In the full model, there are two fermions
on each site because there are two spins. In the effective model, see Sec.
IV.2, there is only one. and

$\displaystyle\gamma_{l+}$ $\displaystyle=d_{l}+d_{l}^{\dagger},$ (5a)
$\displaystyle\gamma_{l-}$ $\displaystyle=i(d_{l}-d_{l}^{\dagger}).$ (5b)

We have chosen to focus on the outer dots, and by reflection symmetry, we
don’t need to consider the rightmost dot. For a non-interacting system,
$\mathrm{MP}=1$ implies that there is no overlap of the MBSs on the outer dots
and the system is protected from perturbations at those dots. However, with
interactions, or imperfect $\mathrm{MP}$, there is no clear quantitative
relation between the $\mathrm{MP}$ and protection from perturbations.

###  III.3 Local distinguishability

Due to the difficulties with the MBS wavefunction, we introduce a measure
based on the local distinguishability of the ground states. This measure
provides a rigorous bound on the stability of $\delta E$ and is well-defined
even with interactions. Our construction is inspired by Ref. [56] where the
presence of MBSs is diagnosed by probing the ground states with local
measurements such as the charge on each dot. We extend this to cover all local
measurements.

Consider a perturbation $\mathcal{O}_{R}$ acting only in some subsystem $R$.
The rest of the system is denoted by the complement $R^{\complement}$. To
first order in this perturbation, the energy difference between the ground
states changes if the perturbation can distinguish the states, see Eq. III.
This quantity obeys

$|\Tr[\mathcal{O}_{R}\delta\rho]|=|\Tr[\mathcal{O}_{R}\delta\rho_{R}]|\leq\norm{\mathcal{O}_{R}}\norm{\delta\rho_{R}}$
(6)

where $\delta\rho_{R}=\Tr_{R^{\complement}}\delta\rho$ is the difference in
reduced density matrices and $\norm{\cdot}$ is the Frobenius norm 222Different
bounds result from using Hölder’s inequality
$|\Tr[\mathcal{O}_{R}\delta\rho]|\leq\norm{O_{R}}_{q}\norm{\delta\rho_{R}}_{p}$,
where $1/q+1/p=1$ and $\norm{\cdot}_{q}$ is the Schatten norm. Without
additional assumptions on the perturbation, we have no reason to prefer one
over another.. We see that $\norm{\delta\rho_{R}}$ bounds the effect of any
perturbation in the region $R$. In App. A we show that it is zero if there
exists a MBS outside of the region $R$.

In a topological phase, $\norm{\delta\rho_{R}}=0$ in all sufficiently local
regions. For our system, we take the subsystems to be each individual dot, and
define the local distinguishability ($\mathrm{LD}$) as

$\mathrm{LD}=\sqrt{\sum_{i}\norm{\delta\rho_{QD_{i}}}^{2}}$ (7)

which we minimize when optimizing for the sweet spot.

$\mathrm{LD}$ is simple to compute as it only relies on finding ground states.
It can be calculated in large interacting systems by using e.g. DMRG. In non-
interacting systems, the reduced density matrix can be reconstructed from the
single particle density matrix.

###  III.4 Optimizing for a sweet spot

In this section, we define how we optimize for sweet spots, both with and
without energy degeneracy and introduce notation to keep the different cases
apart. The term sweet spot is used interchangebly for all of them.

We denote by $\mathrm{mLD}_{0}$ the minimal LD point with degenerate ground
states, i.e. the solution to the minimization problem

$\displaystyle\begin{split}\min\quad&\mathrm{LD}\\\ \text{s.t.}\quad&\delta
E=0.\end{split}$ (8)

In practice, we impose the latter condition by solving the problem

$\displaystyle\min\quad$
$\displaystyle\mathrm{LD}+\lambda\absolutevalue{\delta E}$ (9)

which includes a penalty factor $\lambda$ that enforces the constraint. We
will compare this point to the point denoted by $\mathrm{mLD}$, defined by the
solution to

$\displaystyle\min\quad$ $\displaystyle\mathrm{LD},$

which does not require the states to be degenerate.

Another approach to defining a sweet spot, used in e.g. Refs. [41, 59], is to
optimize for a spot with degenerate ground states and where that degeneracy is
protected against fluctuations in the dot levels $\varepsilon$. In other
words, where $\frac{\partial\delta E}{\partial\varepsilon}=0$. We call this
the level-protected point. Since we also tune the phase difference
$\delta\phi$, we also consider a phase-protected point defined analogously.
These points are in general different from the $\mathrm{mLD}$ points.

##  IV Results

We start by analyzing the three-dot case in Sec. IV.1, comparing the different
sweet spots in detail. The observations will be explained with perturbation
theory in Sec. IV.2, and in Sec IV.3 we consider longer chain lengths. Our
results are produced with the code available at [65].

The constant parameters are set to

$\displaystyle t_{\text{so}}/t$ $\displaystyle=5$ (10a)
$\displaystyle\sqrt{t^{2}+t_{\text{so}}^{2}}$ $\displaystyle=\Delta/2$ (10b)
$\displaystyle V_{Z}$ $\displaystyle=3\Delta.$ (10c)

When $t>t_{\text{so}}$, it is advantageous to tune $\varepsilon$ in an
alternating manner to effectively enhance the spin-flipping process [40].
Picking $t<t_{\text{so}}$ is simpler, because the sweet spot is closer to
homogeneous. Similar results are expected in both cases as the effective
Hamiltonians (see Sec.IV.2 and App. D) are the same at first order in
perturbation theory.

For simplicity we also set

$U_{l}=U_{\mathrm{nl}}=0$ (11)

for the results in the main paper as we found that the essential features we
focus on in this work are qualitatively the same with and without Coulomb
interactions. The main thing to keep in mind is that there is a limit to how
large interaction strengths can be tolerated, see App. C for details.

###  IV.1 Three-dot system

In this section we consider a three-dot system and how to tune it to a sweet
spot. Assuming reflection symmetry, we have three independent parameters to
tune,
$\varepsilon_{1}=\varepsilon_{3}=\varepsilon,\,\varepsilon_{2},\,\delta\phi=\phi_{2}-\phi_{1}=\phi_{3}-\phi_{2}$.
In Sec. IV.1.1, we reduce it to two parameters by setting
$\varepsilon_{1}=\varepsilon_{2}$, making it easy to visualize. In Sec. IV.1.2
we study how the sweet spot can be improved by tuning $\varepsilon_{2}$
independently.

####  IV.1.1 Homogeneous tuning

Figure 2: Tuning $\varepsilon$ and $\delta\phi$ homogeneously in the three-dot
chain. Brighter color means a smaller $\mathrm{LD}$. The contours show the
energy splitting between ground states $\delta E$, with solid line signifying
degeneracy. The three points marked are defined in Sec III.4, and correspond
to points with minimal $\mathrm{LD}$, with and without energy degeneracy.

We start with a three-dot system where we tune the dot levels $\varepsilon$
and phase gradient $\delta\phi$ homogeneously. The heatmap in Fig. 2 shows
$\mathrm{LD}$ as these parameters are tuned, and the contour lines show the
energy splitting between the two ground states. Unfortunately, and differently
from the two-dot case, the solid lines where the ground states are degenerate
do not lie in the region with well separated MBSs and good protection.

In Fig. 2, the condition $\delta E=0$ is satisfied on the solid contour lines,
which has two disconnected branches. One branch features a point where it the
contour is tangent to the horizontal axis. At this point, the degeneracy is
protected from variations in $\varepsilon$. We call this the level branch. The
other branch is called the phase branch as it features a point tangent to the
vertical axis with protection against phase fluctuations. These tangential
points feature protection against a specific perturbation, an example of type
2 protection as defined in Sec. III.

On each branch, we can also define a point with minimal $\mathrm{LD}$ and
$\delta E=0$, denoted by $\mathrm{mLD}_{0}$ and marked in the figure. These
differ from the tangential points, but on the level branch they are very
close. The constraint of $\delta E=0$ severely limits the quality of the sweet
spot. The point marked $+$ and denoted by $\mathrm{mLD}$ has a non-zero
$\delta E$ but $\mathrm{LD}$ is much smaller. Here, there are well-separated
MBSs and the energy splitting is to some degree protected against any local
perturbation, but the ground states are not degenerate. We will see why in
Sec. IV.2.

####  IV.1.2 Inhomogeneous tuning

Figure 3: Sweet spot properties as a function of the middle dot detuning
$\delta\varepsilon_{2}$. The $\mathrm{mLD}_{0}$ points on the two branches
improve, but can’t clearly beat the two-dot sweet spot without reducing the
excitation gap. As the detuning increases, the phase branch becomes irrelevant
while the level branch should be thought of as an effective two-dot chain were
the middle dot acts as a tunnel barrier. At
$\delta\varepsilon_{2}\approx\Delta/5$, the location of the $\mathrm{mLD}$
point changes location discontinuously, see App. B.

Fig. 2 shows that there is a tension between energy degeneracy and good MBSs
at the sweet spot. Here, we check that this is still the case even if the
system is tuned inhomogeneously. We detune the middle dot so that
$\varepsilon_{2}=\varepsilon+\delta\varepsilon_{2}$.

In Fig. 3, we follow the phase branch and level branch of the sweet spot as a
function of the detuning $\delta\varepsilon_{2}$. The sweet spots improve,
especially the phase branch which reaches an $\mathrm{LD}$ and excitation gap
comparable to the two-dot case. When the detuning gets large enough, the phase
branch quickly gets worse, while the level branch features a small
$\mathrm{LD}$ but with a bad excitation gap. Effectively, the middle dot has
been detuned to act as a tunnel barrier and the three-dot chain can be
effectively reduced to a two-dot chain. We conclude that the three-dot chain
does not improve on the two-dot chain in this way of measuring the protection.
Note that our definition of $\mathrm{LD}$ does not cover all possible
perturbations, only those local to each dot. It also only quantifies first
order protection, and says nothing about higher order terms.

###  IV.2 Effective model

The three-dot chain seems to offer a worse protection against local
perturbations than the two-dot chain when we impose energy degeneracy. In this
section we show why, by deriving an effective model to second order in
$t/V_{Z}$. The second order terms give a Kitaev chain that includes next-
nearest neighbours hoppings and pairings. This makes the effective Hamiltonian
less local, and it can then more easily distinguish the ground states.

####  IV.2.1 Deriving the effective model

We can derive an effective Kitaev model using perturbation theory. We follow
[36, 40] and treat the hopping as a perturbation of $H^{\text{QD}}$. We first
transform $d_{n}\rightarrow d_{n}e^{i\phi_{n}/2}$ to put the phase winding on
the hopping. $H^{\text{QD}}$ is diagonalized by a Bogoliubov transformation

$\displaystyle d_{n\downarrow}^{\dagger}$
$\displaystyle=\frac{\sqrt{\beta_{n}-\epsilon_{n}}a_{n}^{\dagger}-\sqrt{\beta_{n}+\epsilon_{n}}b_{n}^{\dagger}}{\sqrt{2\beta_{n}}}$
(12a) $\displaystyle d_{n\uparrow}$
$\displaystyle=\frac{\sqrt{\beta_{n}+\epsilon_{n}}a_{n}^{\dagger}+\sqrt{\beta_{n}-\epsilon_{n}}b_{n}^{\dagger}}{\sqrt{2\beta_{n}}}$
(12b)

where $\beta_{j}=\sqrt{\varepsilon_{j}^{2}+\Delta_{j}^{2}}$. The $a$ and $b$
fermions have energies

$\displaystyle E_{a_{j}}$ $\displaystyle=\beta_{j}-V_{Z},$ $\displaystyle
E_{b_{j}}$ $\displaystyle=\beta_{j}+V_{Z},$

and we consider the regime where $E_{a}\ll E_{b}$ and integrate out the
$b$-fermions. Let $P$ be the projector on the subspace with no occupied
$b$-fermions, and $Q=I-P$. To second order in the hoppings, the effective
Hamiltonian in the subspace $P$ is

$\displaystyle H^{\text{eff}}$
$\displaystyle=PHP-\frac{1}{2E_{b}}PH^{\text{C}}QH^{\text{C}}P$
$\displaystyle=H^{\text{eff}}_{1}+H^{\text{eff}}_{2}.$ (13)

The $b$-fermions have been integrated out, but leave a trace in the form of
the second order term.

The first term is a standard Kitaev chain,

$H_{1}^{\text{eff}}=\sum_{n}\varepsilon_{a}a_{n}^{\dagger}a_{n}+(t_{aa}a_{n}^{\dagger}a_{n+1}+\Delta_{aa}a_{n}a_{n+1}+\mathrm{h.c})$
(14)

as derived previously for this model in [36, 40]. The explicit expressions for
the parameters for a homogeneous system are shown in App. D. The second term,

$H_{2}^{\text{eff}}=\sum_{n}\varepsilon_{2}a_{n}^{\dagger}a_{n}+t_{2}a_{n}^{\dagger}a_{n+2}+\Delta_{2}a^{\dagger}_{n}a^{\dagger}_{n+2}+\mathrm{h.c},$
(15)

includes longer range terms that couple next nearest neighbours. This is not
unique to this system, the same thing happens in other dot-based Kiteav chains
[66]. Kitaev chains with long range couplings have been studied in Refs. [67,
68, 69]. These terms are to blame for the complications of the three-dot sweet
spot, as they make the Hamiltonian more non-local.

Figure 4: The heatmap shows the $\mathrm{MP}$ (see Eq. 4) and the red contour
shows where the ground states are degenerate. The first order effective
Hamiltonian has a sweet spot where $\mathrm{MP}$ is maximized and the two
branches of $\delta E=0$ coincide. In the second order model (which closely
resembes the full model), the $\delta E=0$ lines does not coincide with the
maximum in $\mathrm{MP}$. This is due to the additional long range couplings
which can be significant if the Zeeman splitting is finite.

We confirm this with Fig. 4. It shows the tuning plot, now for the
$\mathrm{MP}$ for the first and second order effective Hamiltonian, and for
the full model. The first order effective model, which corresponds to a
standard Kitaev chain, has a definitive sweet spot where the ground states are
degenerate, $\mathrm{MP}=1$ and $\mathrm{LD}=0$. The two branches where
$\delta E=0$ cross at the sweet spot and the contours are tangential to both
axes. When the longer range terms of $H^{\text{eff}}_{2}$ are included, the
plot closely resembles the result of the full model, where the branches avoid
each other as well as the point with maximally separated MBSs. Comparing Figs.
2 and 4 we also see that the two measures of sweet spot quality, $\mathrm{MP}$
and $\mathrm{LD}$, are correlated.

###  IV.3 Scaling to longer chains

In this section, we optimize homogeneously over dot levels and phase
differences to find the sweet spot as a function of the length of the chain.
At large system sizes, exact diagonalization of the many body Hamiltonian is
prohibitively expensive, but since we consider a non-interacting theory at
this point, the reduced density matrices $\rho_{\text{QD}_{i}}$ on each dot
can be reconstructed from the correlators
$\expectationvalue*{c_{i}^{\dagger}c_{j}}$ and
$\expectationvalue*{c_{i}^{\dagger}c_{j}^{\dagger}}$[70, 71] which are
obtainable via standard Bogoliubov-de-Gennes techniques.

Figure 5: Sweet spot quality as a function of chain length with homogeneous
tuning. (a) $\mathrm{LD}$ has an exponential fall-off at large chain lengths,
implying exponentially better protection. At small lengths, the pattern is
less clear. (b) The energy splitting $\delta E$. (c) The excitation gap is
somewhat stable around $E_{\text{ex}}\sim\Delta/10$ for long chains. For $N=2$
and $4$, the excitation gap at $\mathrm{mLD}$ is very small, with only a very
small improvement in $\mathrm{LD}$ over $\mathrm{mLD}_{0}$.

As seen in Fig. 5(a), $\mathrm{LD}$ at the sweet spot falls off exponentially
as the chain length increases. This is expected in a topological phase. Fig.
5(b) shows the splitting of the ground state degeneracy $\delta E$, which is
constrained to be very small for $\mathrm{mLD}_{0}$. At the point
$\mathrm{mLD}$, there is no constraint on the energy splitting, but since
$\mathrm{LD}$ falls off exponentially, one would expect $\delta E$ to follow
the same pattern. Fig. 5(c) shows the excitation gap at the sweet spot, which
has an odd-even pattern but is otherwise quite stable at
$E_{\text{ex}}\sim\Delta/10$.

Fig. 6(a) shows the tuning diagram for a 40-dot chain, where a phase diagram
starts taking shape. For long enough chains, any point in the topological
phase has good protection and energy degeneracy and there is no need to fine-
tune to a sweet spot. The patterns seen in Fig. 6(a) can be matched with the
true phase diagram of the infinite system in 6(b). Since the second order
effective model captures the physics well, we use it to determine a
topological invariant $Q$ (see App. E) that signifies when the gap closes at
$k=0$ or $k=\pi$. It can take on the values $+1$ (trivial phase) or $-1$
(topological phase) and the boundary between these is plotted as the red
contour in Fig. 6(b). Due to the breaking of time reversal symmetry, there is
also a gapless phase where the gap closes at finite momentum [67], which can
be seen in the heatmap of Fig. 6(b) that shows the energy gap of the infinite,
periodic system.

Figure 6: Phase diagrams for finite and infinite systems. (a) In an open
system with 40 sites $\mathrm{LD}$ shows clear phase boundaries. (b) In an
infinite, periodic system, with the second order effective Hamiltonian, there
is a topological phase and a trivial phase separated by the red contour where
the topological invariant changes sign and the gap closes at $k=0$ or $k=\pi$.
The heatmap shows the energy gap which also reveals an extended region where
the gap closes at a finite momentum.

##  V Summary and conclusions

In this work, we have introduced the local distinguishability of ground states
as a quality measure of near-topological phases in finite and (possibly)
strongly interacting systems, and applied this measure to quantum-dot-based
Kitaev chains of varying length. As expected, the local distinguishability
decreases exponentially for long chains indicating a transition to a true
topological phase, but for short chains the pattern is different. In
particular, we have shown that the three-dot case can actually be worse than
the two-dot case. The explanation for this result is that the long-range
nature of the effective Hamiltonian introduces a trade-off between separated
MBSs and the energy degeneracy between the ground states. This has
consequences for protocols aiming to demonstrate the non-abelian nature of
MBSs, which rely on both energy degeneracy and the MBSs not overlapping [49].
It would be an interesting direction for future works to investigate if non-
abelian protocols can be designed that only require the ground state energy
splitting to be stable, not zero. On a more general level, much remains to be
understood concerning the topological-like protection and nonabelian
properties of strongly interacting, finite systems.

###### Acknowledgements.

We acknowledge stimulating discussions with Chun-Xiao Liu, William Samuelson,
Michael Wimmer and Mert Bozkurt and funding from the European Research Council
(ERC) under the European Unions Horizon 2020 research and innovation programme
under Grant Agreement No. 856526, the Swedish Research Council under Grant
Agreement No. 2020-03412, and NanoLund.

##  Appendix A Local Distinguishability and MBSs

Here, we show that $\norm{\delta\rho_{R}}=0$ if there exists a Majorana
operator in the complement. If there exists a Hermitian operator
$\gamma=I_{R}\otimes\gamma_{R^{\complement}}$ such that
$\ket{e}=\gamma\ket{o}$ and $\gamma_{R^{\complement}}^{2}=I$, then by the
cyclic property of the partial trace

$\Tr_{R^{\complement}}\outerproduct{e}{e}=\Tr_{R^{\complement}}\big{[}\gamma_{R^{\complement}}\outerproduct{o}{o}\gamma_{R^{\complement}}\big{]}=\Tr_{R^{\complement}}\outerproduct{o}{o}$
(16)

and therefore $\delta\rho_{R}=0$. This is a necessary condition for the
existence of MBSs outside of $R$, but not sufficient.

If there are several such operators, then $\delta\rho_{R}$ can only be non-
zero in regions that include all of them, as it vanishes if any one of them is
outside $R$.

##  Appendix B More on branches

In Fig. 7 some additional plots for the three-dot chain as a function of the
detuning of the middle dot, complementing Fig 3.

We plot the gradient of the ground state energy splitting
$\norm{\bm{\nabla}\delta E}$ in Fig. 7(a), where
$\bm{\nabla}\equiv\begin{bmatrix}\frac{\partial}{\partial\varepsilon}&\frac{\partial}{\partial\delta\phi}\end{bmatrix}$.
Note that we still optimize for a minimal $\mathrm{LD}$, not for minimal
gradient. The gradient differs somewhat from the $\mathrm{LD}$ since it is
sensitive to details such as how the Hamiltonian is parameterized. In other
words, it also includes the effect of type 1 and 2 protection as explained in
Sec. III. The location of the sweet spots is shown in Figures 7(b) and (c),
where we see that the $\mathrm{mLD}_{0}$ points on the level and phase branch
can be followed continuously while the $\mathrm{mLD}$ has a discontinuous
jump.

Figure 7: Similar to Fig. 3, we follow different sweet spots as a function of
the detuning $\delta\varepsilon_{2}$. (a) norm of the gradient of $\delta E$.
(b) $\delta\phi$ at the sweet spot. (c) $\varepsilon$ at the sweet spot.

##  Appendix C Interactions

Figure 8: $\mathrm{LD}$ at $\mathrm{mLD}_{0}$ (a) and at $\mathrm{mLD}$ (b) as
a function of the strength of local $U_{l}$ and non-local $U_{\mathrm{nl}}$
Coulomb interaction. This is similar to the two-dot chain studied in Ref.
[40]. The system can only tolerate a certain amount of interaction before
strongly degrading the quality of the sweet spot.

For the main conclusions of this article, we found that that interactions did
not play a central role, and excluded them for simplicity. In Ref. [40], the
role of interactions were studied in the two-dot system. There, the main
conclusion was that if they are sufficiently small, they only slightly affect
the quality of the sweet spot, while if they are too large, they may prohibit
the tuning to the sweet spot. For the three-dot chain, we find qualitatively
the same result, see Fig. 8. In Fig. 9 we show how the tuning plot changes
when interactions are included.

Figure 9: LD as a function of $\delta\phi$ and $\varepsilon$. Same as Fig. 2
but with finite interactions where $U_{l}=3\Delta,U_{\mathrm{nl}}=\Delta/100$.

##  Appendix D Effective Hamiltonian

In terms of the Bogoliubov quasiparticles where $H_{\mathrm{QD}}$ is diagonal,
the Hamiltonians takes the form

$\displaystyle H^{\text{QD}}=$
$\displaystyle\sum_{n}\varepsilon_{a}a_{n}^{\dagger}a_{n}+\varepsilon_{b}b_{n}^{\dagger}b_{n}$
(17)
$\displaystyle\begin{split}H^{C}=&\sum_{n}t_{aa}a_{n}^{\dagger}a_{n+1}+\Delta_{aa}a_{n}a_{n+1}+\\\
&+t_{ab}a_{n}^{\dagger}b_{n+1}+\Delta_{ab}a_{n}b_{n+1}-\Delta_{ba}a_{n+1}b_{n}\\\
&+t_{ba}b_{n}^{\dagger}a_{n+1}+\Delta_{bb}b_{n}b_{n+1}+t_{bb}b_{n}^{\dagger}b_{n}+\mathrm{h.c},\end{split}$
(18)

where

$\displaystyle\varepsilon_{a}$
$\displaystyle=\sqrt{\varepsilon^{2}+\Delta^{2}}-V_{Z}$ (19a)
$\displaystyle\varepsilon_{b}$
$\displaystyle=\sqrt{\varepsilon^{2}+\Delta^{2}}+V_{Z}$ (19b) $\displaystyle
t_{aa}$
$\displaystyle=-\frac{t\varepsilon\cos\left(\frac{\delta\phi}{2}\right)}{\sqrt{\Delta^{2}+\varepsilon^{2}}}+it\sin\left(\frac{\delta\phi}{2}\right)$
(19c) $\displaystyle t_{ab}$ $\displaystyle=-\frac{\Delta
t\cos\left(\frac{\delta\phi}{2}\right)}{\sqrt{\Delta^{2}+\varepsilon^{2}}}$
(19d) $\displaystyle\Delta_{aa}$ $\displaystyle=-\frac{\Delta
t_{\text{so}}\cos\left(\frac{\delta\phi}{2}\right)}{\sqrt{\Delta^{2}+\varepsilon^{2}}}$
(19e) $\displaystyle\Delta_{ab}$
$\displaystyle=\frac{t_{\text{so}}\varepsilon\cos\left(\frac{\delta\phi}{2}\right)}{\sqrt{\Delta^{2}+\varepsilon^{2}}}+it_{\text{so}}\sin\left(\frac{\delta\phi}{2}\right)$
(19f) $\displaystyle t_{ba}$ $\displaystyle=t_{ab}$ (19g) $\displaystyle
t_{bb}$ $\displaystyle=t_{aa}^{*}$ (19h) $\displaystyle\Delta_{bb}$
$\displaystyle=-\Delta_{aa}$ (19i) $\displaystyle\Delta_{ba}$
$\displaystyle=\Delta_{ab}^{*}$ (19j)

The first order term in the effective Hamiltonian is the projection onto
states with no $b$-fermions,

$H_{1}^{\text{eff}}=\sum_{n}\varepsilon_{a}a_{n}^{\dagger}a_{n}+(t_{aa}a_{n}^{\dagger}a_{n+1}+\Delta_{aa}a_{n}a_{n+1}+\mathrm{h.c}).$
(20)

The second order term is

$H_{2}^{\text{eff}}=\sum_{n}\varepsilon_{2}a_{n}^{\dagger}a_{n}+t_{2}a_{n}^{\dagger}a_{n+2}+\Delta_{2}a^{\dagger}_{n}a^{\dagger}_{n+2}+\mathrm{h.c}$
(21)

where

$\displaystyle\varepsilon_{2}$
$\displaystyle=2(|\Delta_{ab}|^{2}-|t_{ab}|^{2})/\varepsilon_{b}$ (22)
$\displaystyle t_{2}$ $\displaystyle=(-\Delta_{ab}^{\prime
2}-t_{ab}^{2})/\varepsilon_{b}$ (23) $\displaystyle\Delta_{2}$
$\displaystyle=2\Re{\Delta_{ab}t_{ab}}/\varepsilon_{b}$ (24)

It includes the effect of next nearest neighbour hoppings and pairings.

##  Appendix E Topological invariant

The topological invariant for the second order effective model (which is in
class D) signifies when the gap closes at $k=0$ or $k=\pi$ can be
characterized by

$Q\equiv\operatorname{sign}(\operatorname{Pf}(H_{0})\operatorname{Pf}(H_{\pi})).$
(25)

In the second order effective model, this is simply

$Q\equiv\operatorname{sign}((2\Re{t_{1}+t_{2}}+\varepsilon)(2\Re{-t_{1}+t_{2}}+\varepsilon).$
(26)

However, it should be noted that the gap can also close a finite momentum due
to the breaking of time-reversal symmetry [67].

## References

  * Kitaev [2001] A. Kitaev, Unpaired Majorana fermions in quantum wires, Physics-Uspekhi 44, 131 (2001).
  * Alicea  _et al._ [2011] J. Alicea, Y. Oreg, G. Refael, F. von Oppen, and M. P. A. Fisher, Non-Abelian statistics and topological quantum information processing in 1D wire networks, Nature Physics 7, 412 (2011).
  * Beenakker [2020] C. W. J. Beenakker, Search for non-Abelian Majorana braiding statistics in superconductors, SciPost Physics Lecture Notes , 15 (2020).
  * Alicea [2012] J. Alicea, New directions in the pursuit of Majorana fermions in solid state systems, Reports on Progress in Physics 75, 076501 (2012).
  * Leijnse and Flensberg [2012a] M. Leijnse and K. Flensberg, Introduction to topological superconductivity and Majorana fermions, Semiconductor Science and Technology 27, 124003 (2012a).
  * Aguado [2017] R. Aguado, Majorana quasiparticles in condensed matter, La Rivista del Nuovo Cimento 40, 523 (2017).
  * Nayak  _et al._ [2008] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. D. Sarma, Non-Abelian Anyons and Topological Quantum Computation, Reviews of Modern Physics 80, 1083 (2008).
  * Sarma  _et al._ [2015] S. D. Sarma, M. Freedman, and C. Nayak, Majorana Zero Modes and Topological Quantum Computation, npj Quantum Information 1, 15001 (2015).
  * Marra [2022] P. Marra, Majorana nanowires for topological quantum computation, Journal of Applied Physics 132, 231101 (2022).
  * Oreg  _et al._ [2010] Y. Oreg, G. Refael, and F. von Oppen, Helical Liquids and Majorana Bound States in Quantum Wires, Physical Review Letters 105, 177002 (2010).
  * Lutchyn  _et al._ [2010] R. M. Lutchyn, J. D. Sau, and S. D. Sarma, Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures, Physical Review Letters 105, 077001 (2010).
  * Vaitiekėnas  _et al._ [2020] S. Vaitiekėnas, G. W. Winkler, B. van Heck, T. Karzig, M.-T. Deng, K. Flensberg, L. I. Glazman, C. Nayak, P. Krogstrup, R. M. Lutchyn, and C. M. Marcus, Flux-induced topological superconductivity in full-shell nanowires, Science 367, eaav3392 (2020).
  * Mourik  _et al._ [2012] V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M. Bakkers, and L. P. Kouwenhoven, Signatures of Majorana Fermions in Hybrid Superconductor-Semiconductor Nanowire Devices, Science 336, 1003 (2012).
  * Das  _et al._ [2012] A. Das, Y. Ronen, Y. Most, Y. Oreg, M. Heiblum, and H. Shtrikman, Zero-bias peaks and splitting in an Al–InAs nanowire topological superconductor as a signature of Majorana fermions, Nature Physics 8, 887 (2012).
  * Deng  _et al._ [2016] M. T. Deng, S. Vaitiekėnas, E. B. Hansen, J. Danon, M. Leijnse, K. Flensberg, J. Nygård, P. Krogstrup, and C. M. Marcus, Majorana bound state in a coupled quantum-dot hybrid-nanowire system, Science 354, 1557 (2016).
  * Nadj-Perge  _et al._ [2013] S. Nadj-Perge, I. K. Drozdov, B. A. Bernevig, and A. Yazdani, Proposal for realizing Majorana fermions in chains of magnetic atoms on a superconductor, Physical Review B 88, 020407 (2013).
  * Nadj-Perge  _et al._ [2014] S. Nadj-Perge, I. K. Drozdov, J. Li, H. Chen, S. Jeon, J. Seo, A. H. MacDonald, B. A. Bernevig, and A. Yazdani, Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor, Science 346, 602 (2014).
  * Pientka  _et al._ [2017] F. Pientka, A. Keselman, E. Berg, A. Yacoby, A. Stern, and B. I. Halperin, Topological Superconductivity in a Planar Josephson Junction, Physical Review X 7, 021032 (2017).
  * Hell  _et al._ [2017] M. Hell, M. Leijnse, and K. Flensberg, Two-Dimensional Platform for Networks of Majorana Bound States, Physical Review Letters 118, 107701 (2017).
  * Fornieri  _et al._ [2019] A. Fornieri, A. M. Whiticar, F. Setiawan, E. Portolés, A. C. C. Drachmann, A. Keselman, S. Gronin, C. Thomas, T. Wang, R. Kallaher, G. C. Gardner, E. Berg, M. J. Manfra, A. Stern, C. M. Marcus, and F. Nichele, Evidence of topological superconductivity in planar Josephson junctions, Nature 569, 89 (2019).
  * Ren  _et al._ [2019] H. Ren, F. Pientka, S. Hart, A. T. Pierce, M. Kosowsky, L. Lunczer, R. Schlereth, B. Scharf, E. M. Hankiewicz, L. W. Molenkamp, B. I. Halperin, and A. Yacoby, Topological superconductivity in a phase-controlled Josephson junction, Nature 569, 93 (2019).
  * Kells  _et al._ [2012] G. Kells, D. Meidan, and P. W. Brouwer, Near-zero-energy end states in topologically trivial spin-orbit coupled superconducting nanowires with a smooth confinement, Physical Review B 86, 100503 (2012).
  * Prada  _et al._ [2012] E. Prada, P. San-Jose, and R. Aguado, Transport spectroscopy of NS nanowire junctions with Majorana fermions, Physical Review B 86, 180503 (2012).
  * Liu  _et al._ [2012] J. Liu, A. C. Potter, K. T. Law, and P. A. Lee, Zero-Bias Peaks in the Tunneling Conductance of Spin-Orbit-Coupled Superconducting Wires with and without Majorana End-States, Physical Review Letters 109, 267002 (2012).
  * Roy  _et al._ [2013] D. Roy, N. Bondyopadhaya, and S. Tewari, Topologically trivial zero-bias conductance peak in semiconductor Majorana wires from boundary effects, Physical Review B 88, 020502 (2013).
  * Liu  _et al._ [2017] C.-X. Liu, J. D. Sau, T. D. Stanescu, and S. Das Sarma, Andreev bound states versus Majorana bound states in quantum dot-nanowire-superconductor hybrid structures: Trivial versus topological zero-bias conductance peaks, Physical Review B 96, 075161 (2017).
  * Moore  _et al._ [2018] C. Moore, C. Zeng, T. D. Stanescu, and S. Tewari, Quantized zero-bias conductance plateau in semiconductor-superconductor heterostructures without topological Majorana zero modes, Physical Review B 98, 155314 (2018).
  * Reeg  _et al._ [2018] C. Reeg, O. Dmytruk, D. Chevallier, D. Loss, and J. Klinovaja, Zero-energy Andreev bound states from quantum dots in proximitized Rashba nanowires, Physical Review B 98, 245407 (2018).
  * Awoga  _et al._ [2019] O. A. Awoga, J. Cayao, and A. M. Black-Schaffer, Supercurrent Detection of Topologically Trivial Zero-Energy States in Nanowire Junctions, Physical Review Letters 123, 117001 (2019).
  * Vuik  _et al._ [2019] A. Vuik, B. Nijholt, A. R. Akhmerov, and M. Wimmer, Reproducing topological properties with quasi-Majorana states, SciPost Physics 7, 061 (2019).
  * Pan and Das Sarma [2020] H. Pan and S. Das Sarma, Physical mechanisms for zero-bias conductance peaks in Majorana nanowires, Physical Review Research 2, 013377 (2020).
  * Prada  _et al._ [2020] E. Prada, P. San-Jose, M. W. A. de Moor, A. Geresdi, E. J. H. Lee, J. Klinovaja, D. Loss, J. Nygård, R. Aguado, and L. P. Kouwenhoven, From Andreev to Majorana bound states in hybrid superconductor–semiconductor nanowires, Nature Reviews Physics 2, 575 (2020).
  * Hess  _et al._ [2021] R. Hess, H. F. Legg, D. Loss, and J. Klinovaja, Local and nonlocal quantum transport due to Andreev bound states in finite Rashba nanowires with superconducting and normal sections, Physical Review B 104, 075405 (2021).
  * Sau and Sarma [2012] J. D. Sau and S. D. Sarma, Realizing a robust practical Majorana chain in a quantum-dot-superconductor linear array, Nature Communications 3, 964 (2012).
  * Leijnse and Flensberg [2012b] M. Leijnse and K. Flensberg, Parity qubits and poor man’s Majorana bound states in double quantum dots, Physical Review B 86, 134528 (2012b).
  * Fulga  _et al._ [2013] I. C. Fulga, A. Haim, A. R. Akhmerov, and Y. Oreg, Adaptive tuning of Majorana fermions in a quantum dot chain, New Journal of Physics 15, 045020 (2013).
  * Liu  _et al._ [2022] C.-X. Liu, G. Wang, T. Dvir, and M. Wimmer, Tunable Superconducting Coupling of Quantum Dots via Andreev Bound States in Semiconductor-Superconductor Nanowires, Physical Review Letters 129, 267701 (2022).
  * Tsintzis  _et al._ [2022] A. Tsintzis, R. S. Souto, and M. Leijnse, Creating and detecting poor man’s Majorana bound states in interacting quantum dots, Physical Review B 106, L201404 (2022).
  * Liu  _et al._ [2023a] C.-X. Liu, A. M. Bozkurt, F. Zatelli, S. L. D. ten Haaf, T. Dvir, and M. Wimmer, Enhancing the excitation gap of a quantum-dot-based Kitaev chain (2023a), arXiv:2310.09106 [cond-mat] .
  * Samuelson  _et al._ [2024] W. Samuelson, V. Svensson, and M. Leijnse, Minimal quantum dot based Kitaev chain with only local superconducting proximity effect, Physical Review B 109, 035415 (2024).
  * Luna  _et al._ [2024] J. D. T. Luna, A. M. Bozkurt, M. Wimmer, and C.-X. Liu, Flux-tunable Kitaev chain in a quantum dot array (2024), arXiv:2402.07575 [cond-mat] .
  * Wang  _et al._ [2022] G. Wang, T. Dvir, G. P. Mazur, C.-X. Liu, N. van Loo, S. L. D. ten Haaf, A. Bordin, S. Gazibegovic, G. Badawy, E. P. A. M. Bakkers, M. Wimmer, and L. P. Kouwenhoven, Singlet and triplet Cooper pair splitting in hybrid superconducting nanowires, Nature 612, 448 (2022).
  * Dvir  _et al._ [2023] T. Dvir, G. Wang, N. van Loo, C.-X. Liu, G. P. Mazur, A. Bordin, S. L. D. ten Haaf, J.-Y. Wang, D. van Driel, F. Zatelli, X. Li, F. K. Malinowski, S. Gazibegovic, G. Badawy, E. P. A. M. Bakkers, M. Wimmer, and L. P. Kouwenhoven, Realization of a minimal Kitaev chain in coupled quantum dots, Nature 614, 445 (2023).
  * Zatelli  _et al._ [2023] F. Zatelli, D. van Driel, D. Xu, G. Wang, C.-X. Liu, A. Bordin, B. Roovers, G. P. Mazur, N. van Loo, J. C. Wolff, A. M. Bozkurt, G. Badawy, S. Gazibegovic, E. P. A. M. Bakkers, M. Wimmer, L. P. Kouwenhoven, and T. Dvir, Robust poor man’s Majorana zero modes using Yu-Shiba-Rusinov states (2023), arXiv:2311.03193 [cond-mat, physics:quant-ph] .
  * ten Haaf  _et al._ [2024] S. L. D. ten Haaf, Q. Wang, A. M. Bozkurt, C.-X. Liu, I. Kulesh, P. Kim, D. Xiao, C. Thomas, M. J. Manfra, T. Dvir, M. Wimmer, and S. Goswami, A two-site Kitaev chain in a two-dimensional electron gas, Nature 630, 329 (2024).
  * Souto  _et al._ [2023] R. S. Souto, A. Tsintzis, M. Leijnse, and J. Danon, Probing Majorana localization in minimal Kitaev chains through a quantum dot, Physical Review Research 5, 043182 (2023).
  * Liu  _et al._ [2023b] C.-X. Liu, H. Pan, F. Setiawan, M. Wimmer, and J. D. Sau, Fusion protocol for Majorana modes in coupled quantum dots, Physical Review B 108, 085437 (2023b).
  * Boross and Pályi [2024] P. Boross and A. Pályi, Braiding-based quantum control of a Majorana qubit built from quantum dots, Physical Review B 109, 125410 (2024).
  * Tsintzis  _et al._ [2024] A. Tsintzis, R. S. Souto, K. Flensberg, J. Danon, and M. Leijnse, Majorana Qubits and Non-Abelian Physics in Quantum Dot–Based Minimal Kitaev Chains, PRX Quantum 5, 010323 (2024).
  * Sedlmayr and Bena [2015] N. Sedlmayr and C. Bena, Visualising Majorana bound states in 1D and 2D using the generalized Majorana polarization, Physical Review B 92, 115115 (2015).
  * Sedlmayr  _et al._ [2016] N. Sedlmayr, J. M. Aguiar-Hualde, and C. Bena, Majorana bound states in open quasi-1D and 2D systems with transverse Rashba coupling, Physical Review B 93, 155425 (2016).
  * Aksenov  _et al._ [2020] S. V. Aksenov, A. O. Zlotnikov, and M. S. Shustin, Strong Coulomb interactions in the problem of Majorana modes in a wire of the nontrivial topological class BDI, Physical Review B 101, 125431 (2020).
  * Stoudenmire  _et al._ [2011] E. M. Stoudenmire, J. Alicea, O. A. Starykh, and M. P. A. Fisher, Interaction Effects in Topological Superconducting Wires Supporting Majorana Fermions, Physical Review B 84, 014503 (2011).
  * Wright and Veldhorst [2013] A. R. Wright and M. Veldhorst, Localized Many-Particle Majorana Modes with Vanishing Time-Reversal Symmetry Breaking in Double Quantum Dots, Physical Review Letters 111, 096801 (2013).
  * O’Brien  _et al._ [2015] T. E. O’Brien, A. R. Wright, and M. Veldhorst, Many-particle Majorana bound states: Derivation and signatures in superconducting double quantum dots, physica status solidi (b) 252, 1731 (2015).
  * O’Brien and Wright [2015] T. E. O’Brien and A. R. Wright, A many-body interpretation of Majorana bound states, and conditions for their localisation (2015), arXiv:1508.06638 [cond-mat] .
  * Kells [2015a] G. Kells, Many-body Majorana operators and the equivalence of parity sectors, Physical Review B 92, 081401 (2015a).
  * Kells [2015b] G. Kells, Multiparticle content of Majorana zero modes in the interacting p-wave wire, Physical Review B 92, 155434 (2015b).
  * Bozkurt  _et al._ [2024] A. M. Bozkurt, S. Miles, S. L. D. ten Haaf, C.-X. Liu, F. Hassler, and M. Wimmer, Interaction-induced strong zero modes in short quantum dot chains with time-reversal symmetry (2024), arXiv:2405.14940 [cond-mat] .
  * Bordin  _et al._ [2023] A. Bordin, X. Li, D. van Driel, J. C. Wolff, Q. Wang, S. L. D. ten Haaf, G. Wang, N. van Loo, L. P. Kouwenhoven, and T. Dvir, Crossed Andreev reflection and elastic co-tunneling in a three-site Kitaev chain nanowire device (2023), arXiv:2306.07696 [cond-mat] .
  * Bordin  _et al._ [2024] A. Bordin, C.-X. Liu, T. Dvir, F. Zatelli, S. L. D. ten Haaf, D. van Driel, G. Wang, N. van Loo, T. van Caekenberghe, J. C. Wolff, Y. Zhang, G. Badawy, S. Gazibegovic, E. P. A. M. Bakkers, M. Wimmer, L. P. Kouwenhoven, and G. P. Mazur, Signatures of Majorana protection in a three-site Kitaev chain (2024), arXiv:2402.19382 [cond-mat] .
  * Liu  _et al._ [2024] C.-X. Liu, S. Miles, A. Bordin, S. L. D. ten Haaf, A. M. Bozkurt, and M. Wimmer, Protocol for scaling up a sign-ordered Kitaev chain without magnetic flux control (2024), arXiv:2407.04630 [cond-mat] .
  * Note [1] In the full model, there are two fermions on each site because there are two spins. In the effective model, see Sec. IV.2, there is only one.
  * Note [2] Different bounds result from using Hölder’s inequality $|\Tr[\mathcal{O}_{R}\delta\rho]|\leq\norm{O_{R}}_{q}\norm{\delta\rho_{R}}_{p}$, where $1/q+1/p=1$ and $\norm{\cdot}_{q}$ is the Schatten norm. Without additional assumptions on the perturbation, we have no reason to prefer one over another.
  * Svensson [2024] V. Svensson, Cvsvensson/LongerPoorMansMajoranas (2024).
  * Miles  _et al._ [2023] S. Miles, D. van Driel, M. Wimmer, and C.-X. Liu, Kitaev chain in an alternating quantum dot-Andreev bound state array (2023), arXiv:2309.15777 [cond-mat] .
  * DeGottardi  _et al._ [2013] W. DeGottardi, M. Thakurathi, S. Vishveshwara, and D. Sen, Majorana fermions in superconducting wires: Effects of long-range hopping, broken time-reversal symmetry, and potential landscapes, Physical Review B 88, 165111 (2013).
  * Alecce and Dell’Anna [2017] A. Alecce and L. Dell’Anna, Extended Kitaev chain with longer-range hopping and pairing, Physical Review B 95, 195160 (2017).
  * Mahyaeh and Ardonne [2018] I. Mahyaeh and E. Ardonne, Zero modes of the Kitaev chain with phase-gradients and longer range couplings, Journal of Physics Communications 2, 045010 (2018).
  * Chung and Peschel [2001] M.-C. Chung and I. Peschel, Density-matrix spectra of solvable fermionic systems, Physical Review B 64, 064412 (2001).
  * Cheong and Henley [2004] S.-A. Cheong and C. L. Henley, Many-body density matrices for free fermions, Physical Review B 69, 075111 (2004).