# Recent progress on Majorana in semiconductor-superconductor heterostructures
— engineering and detection

Zhan Cao  Beijing Academy of Quantum Information Sciences, Beijing 100193,
China  Shumeng Chen  State Key Laboratory of Low Dimensional Quantum Physics,
Department of Physics, Tsinghua University, Beijing, 100084, China  Frontier
Science Center for Quantum Information, Beijing 100084, China  Gu Zhang
Beijing Academy of Quantum Information Sciences, Beijing 100193, China  Dong
E. Liu<EMAIL_ADDRESS>State Key Laboratory of Low Dimensional
Quantum Physics, Department of Physics, Tsinghua University, Beijing, 100084,
China  Beijing Academy of Quantum Information Sciences, Beijing 100193, China
Frontier Science Center for Quantum Information, Beijing 100084, China

###### Abstract

Majorana zero modes (MZMs) are exotic excitations (in condensed matter
systems) of fundamental scientific interest and hold great promise as the
basis of fault-tolerant topological quantum computation. Though MZMs have been
predicted in many platforms, their existence is still under debate. In this
paper, we review the recent progress in engineering and detecting MZMs in
semiconductor-superconductor heterostructures. We also briefly review the
protocols for implementing topological quantum computation by hybrid
semiconductor-superconductor nanowires.

topological superconductor, Majorana zero mode, semiconductor-superconductor
heterostructure, quantum transport, topological quantum computation

###### pacs:

73.63.–b, 03.65.Vf, 03.67.Lx, 74.45.+c

##  I Introduction

Majorana zero mode (MZM), also known as Majorana bound state, is an exotic
zero-energy excitation with equal electron and hole weights Read and Green
(2000); Kitaev (2001), in analog with Majorana fermion Majorana (1937) whose
antiparticle is itself. So far, no Majorana fermion as elementary particles
has been found in nature, however, MZMs are predicted to exist in topological
superconductors (SCs), please refer to reviews Alicea (2012); Leijnse and
Flensberg (2012); Stanescu and Tewari (2013a); Beenakker (2013, 2015); Elliott
and Franz (2015); Sato and Fujimoto (2016); Aguado (2017); Sato and Ando
(2017); Lutchyn  _et al._ (2018); Beenakker (2020); Flensberg  _et al._
(2021); Fu  _et al._ (2021); Marra (2022). In recent years, MZMs have
attracted broad interdisciplinary interest due to their unique exchange
statistics and potential for realizing fault-tolerant topological quantum
computation Kitaev (2003); Nayak  _et al._ (2008); Das Sarma  _et al._ (2015).

MZMs are predicted to exist at the interfaces of materials with different
topology, or in the vortices of a 2D topological SC (see, e.g., Ref. Alicea,
2012 for a pedagogical review). Motivated by its fundamental scientific
interest and promising application in quantum computation, MZMs and their
host, i.e., topological SCs, have received considerable experimental and
theoretical interest in the last decade. To date, topological SC candidates
can be classified into intrinsic and artificial ones. Since 2010, abundant
intrinsic topological SC candidates have been experimentally reported,
including, e.g., putative spinless chiral $p$-wave SC (Sr2RuO4 Maeno  _et al._
(2001); Das Sarma  _et al._ (2006)), metal-doped topological insulators
(CuxBi2Se3 Wray  _et al._ (2010); Sasaki  _et al._ (2011), SrxBi2Se3 Liu  _et
al._ (2015), NbxBi2Se3 Asaba  _et al._ (2017)), iron-based SCs (FeTe0.55Se0.45
Wang  _et al._ (2018), Li(Fe,Co)As Zhang  _et al._ (2019a), (Li,Fe)OHFeSe Liu
_et al._ (2018a)), transition metal dichalcogenides (2M-WS2 Yuan  _et al._
(2019)), non-centrosymmetric SCs (PbTaSe2 Bian  _et al._ (2016)), and
centrosymmetric SCs ($\beta$-Bi2Pd Lv  _et al._ (2017)). Meanwhile, a variety
of artificial topological SC candidates have been experimentally studied,
including, e.g., the topological insulator-$s$-wave SC heterostructure
(Bi2Te3-NbSe2 Xu  _et al._ (2015); Sun  _et al._ (2016), HgTe-Nb Wiedenmann
_et al._ (2016), HgTe-Al Bocquillon  _et al._ (2017); Deacon  _et al._
(2017)), ferromagnetic atomic chain on the surface of an $s$-wave SC (Fe-Pb
Nadj-Perge  _et al._ (2014)), semiconductor (SM) nanowire-$s$-wave SC
heterostructure (see Sec. II for a review), and ferromagnetic insulator-
metal-$s$-wave SC heterostructure (EuS-Au-V Manna  _et al._ (2020)).

Despite the ongoing search for MZMs in the last decade, the question of
whether we are witnessing proven evidence of MZMs is still being debated.
Currently, SM-SC heterostructures have become the most studied topological SC
candidates, as (i) they require only conventional materials and are compatible
with the well-developed SM growth techniques, and (ii) the roadmap for
implementing quantum computation with these SM techniques has been established
Aasen  _et al._ (2016); Plugge  _et al._ (2017); Karzig  _et al._ (2017). In
this paper, we review recent theoretical and experimental progress on
engineering and detecting MZMs in SM-SC heterostructures. We also briefly
review the related protocols for implementing quantum computation.

This paper is organized as follows. We review the available SM-SC
heterostructures for engineering MZMs in Sec. II and the experimental
protocols that have been performed to detect MZMs in Sec. III. In Sec. IV, we
review the recent progress of material choice and growth of SM-SC
heterostructures. In Sec. V, we provide a preliminary introduction to
topological quantum computation protocols implemented by hybrid SM-SC
nanowires. Finally, we give in Sec. VI a brief summary and the prospective
challenges in this field.

##  II Recent progress on engineering MZMs in SM-SC heterostructures

In 2010, R. M. Lutchyn et al. Lutchyn  _et al._ (2010) and Y. Oreg et al. Oreg
_et al._ (2010) independently proposed to engineer MZMs in a quasi-1D SM
nanowire proximitized to an $s$-wave SC, under the influence of a strong
Rashba spin-orbit coupling and a magnetic field induced Zeeman splitting. This
Majorana-nanowire proposal is rather attractive as it requires only
conventional SM, e.g., InAs or InSb, and $s$-wave SC, e.g., Al, to
equivalently realize the MZM-hosted Kitaev chain proposed by the pioneering
work Ref. Kitaev (2001), where instead an unconventional $p$-wave SC is
demanded.

In 2012, the device based on the Majorana-nanowire model was firstly
fabricated Mourik  _et al._ (2012) by placing part of an InSb nanowire on top
of a bulk SC NbTiN. Multiple metallic gates are placed next to the wire to
control both the chemical potential and the tunneling through a potential
barrier. An external magnetic field is also applied to generate the required
Zeeman splitting. Several transport features were observed in the tunneling
spectroscopy Mourik  _et al._ (2012), including (i) the appearance of a zero-
bias conductance peak (ZBCP) under a parallel magnetic field ranging from 0.07
to 1 T; (ii) the ZBCP remains when changing the applied voltages of different
gates over large ranges; (iii) the strong dependence of the zero-bias
conductance on the direction of the applied magnetic field; and (iv) the
missing of ZBCP after replacing the SC by a normal metal. These observations
coincide with the MZM signatures predicted by the Majorana-nanowire model.
Afterwards, similar results were reported in a few experiments using InSb or
InAs nanowires Deng  _et al._ (2012); Das  _et al._ (2012); Finck  _et al._
(2013); Churchill  _et al._ (2013). In these experiments, substantially
manifest subgap conductance coexists with the ZBCP. This phenomenon, known as
soft-gap, is detrimental to topological quantum computation as it degrades the
topological protection. Disorder impacts at the SM-SC interface are believed
to cause soft gap features Takei  _et al._ (2013). In 2015, with the technical
advances, these sub-gap states are dramatically suppressed in epitaxially
grown hybrid SM-SC nanowires Chang  _et al._ (2015); Krogstrup  _et al._
(2015), where a thin SC shell is directly grown on the pristine facets of the
SM nanowire [Fig. 1(a)] to greatly improve the cleanness of the SM-SC
interface. Note that the SM nanowire is partially covered by the SC shell
[Fig. 1(b)], so that the side gate can be used to tune the chemical potential
of the SM nanowire. To date, systems with partially-covered SM nanowires are
among the most preferred options in the realization of MZMs in hybrid SM-SC
structures Lutchyn  _et al._ (2018).

Figure 1: (a) Top view of the schematic of the tunneling spectroscopy
measurement of a hybrid SM-SC nanowire that is predicted to host a pair of
MZMs at the opposite wire ends. The tunnel gate controls the tunnel barrier
between the source lead and the hybrid nanowire, while the side gate tunes the
electron density in the SM nanowire. The hybrid nanowire can be either the
essential quasi-1D ones shown by (b)–(d) or the one etched on a 2D SM-SC
heterojunction. (b)–(e) Cross section of a hybrid SM-SC nanowire on a
dielectric material, in which a thin SC shell is epitaxially grown on pristine
facets of the SM nanowire. Note that the nanowire is partially covered by the
SC shell such that the chemical potential of the nanowire can be tuned by
voltage gates. A magnetic field $B$ in parallel to the nanowire is necessary
to induce a Zeeman splitting in the SM. (c) Side view of a full-shell hybrid
SM-SC nanowire. Instead of a Zeeman splitting induced by a magnetic field, the
magnetic flux threading through the cross-section of the SM core creates
fluxoids in the superconducting order parameter of the SC shell and drives the
system into the topological superconducting phase. (d) Cross section of an SM-
FMI-SC nanowire on a dielectric material. No magnetic field is needed, while
the magnetic proximity effects at the SM-FMI and SC-FMI interfaces are both
essential for achieving topological superconductivity. (e) Planar JJ formed in
a 2D SM covered by an SC shell. A pair of MZMs are predicted to localize at
two ends of the stripe region uncovered by the SC shell. By threading a
magnetic flux through the nearby superconducting loop, the phase difference
between the superconducting regions on the two sides of the stripe can be
tuned to drive the system into the topological superconducting phase with a
tiny magnetic field.

As mentioned above, the Majorana-nanowire model requires a large enough Zeeman
splitting in the SM nanowire to realize MZMs. This Zeeman splitting is
generated via an applied magnetic field, which might destroy the
superconductivity of the SC shell. Consequently, materials with large Landé g
factors are preferred, to produce a sufficient Zeeman splitting with a
comparatively small magnetic field. In 2020, it was proposed Vaitiekėnas  _et
al._ (2020) that a full-shell SM-SC nanowire, as shown in Fig. 1(c), requires
only a tiny parallel magnetic field to reach the topological phase. The
magnetic field aims to induce a magnetic flux $\Phi$ threading through the
full-shell geometry, rather than a Zeeman splitting, thus the requirement of a
large g factor is therefore removed, potentially increasing possible
candidates of SM materials. Specifically, the flux $\Phi$ creates fluxoids in
the superconducting paring of the SC shell, i.e., $\Delta=|\Delta|e^{in\phi}$,
with $\phi$ the angle around the nanowire axis and $n$ the integer closest to
the ratio $\Phi/\Phi_{0}$ with $\Phi_{0}=h/2e$ being the superconducting flux
quantum. It was predicted that MZMs can be engineered in the odd-$n$ cases.
Experimentally, for a full-shell nanowire with the SM core diameter of around
130 nm, and the SC shell thickness of 30 nm, the required magnetic field for
realizing the $n=1$ topological superconducting regime is about 0.1 T
Vaitiekėnas  _et al._ (2020). As another advantage, the full-shell structure
naturally protects the SM from impurities and random surface doping, thus
enabling a reproducible way of growing SM-SC nanowires with essentially
identical electrostatic environments. As one possible disadvantage, however,
full-shell SM-SC nanowires do not allow the control of their electron density
via metallic gates, due to the screening of the external electric fields by
the SC shell. Indeed, it is stated that the chemical potential of a full-shell
SM-SC nanowire is instead determined by the diameter of the SM core
Vaitiekėnas  _et al._ (2020). It is therefore expected to realize MZMs in
full-shell SM-SC nanowires with appropriate SM core diameter Vaitiekėnas  _et
al._ (2020). However, numerical studies Woods  _et al._ (2019) on the
electronic structure of full-shell InAs–Al nanowires indicate that this system
may be hard to be tuned to the topological phase. Specifically, it was shown
that the spin-orbit coupling, which is crucial in the realization of MZMs, is
generically weak for subbands considered in the Majorana-nanowire model, i.e.,
subbands near the Fermi surface. Consequently, the topological phase requires
a fine-tuned chemical potential, and thus a fine-tuning of the SM core
diameter. Though clear ZBCPs robust to the back-gate voltage have been
observed within a large scope in the anticipated $n=1$ topological
superconducting regime of full-shell hybrid InAs-Al nanowires Vaitiekėnas  _et
al._ (2020), their origin is still under debate. Indeed, a subsequent
tunneling spectroscopy experiment Valentini  _et al._ (2021) on full-shell
InAs-Al nanowires demonstrated that ZBCPs arising from topologically trivial
Andreev bound states (ABSs) could occur when the tunnel junction is longer
than 150 nm to accommodate an interacting quantum dot.

Based on the Majorana-nanowire model, to drive the SM nanowire into the
topological superconducting phase, the Zeeman field should be orthogonal to
the Rashba spin-orbit field in the nanowire. In devices discussed above, the
Rashba spin-orbit field is mainly determined by the electrostatic potential
across the wire section and thus perpendicular to the wire axis de Moor  _et
al._ (2018); Bommer  _et al._ (2019). Therefore, the applied magnetic field is
required to be parallel to the wire. This requirement is however unsatisfiable
for engineering topological nanowire networks, such as topological T-shaped
Alicea  _et al._ (2011) and X-shaped Zhou  _et al._ (2020) junctions in which
the nanowire segments have different axis directions. As a possible solution,
it was proposed Sau  _et al._ (2010) to generate a Zeeman splitting in a SM
from the exchange interaction between the ferromagnetic insulator (FMI) and
the SM. Following this proposal, in 2021, an FMI shell (EuS) was epitaxially
grown between the Al shell and the InAs nanowire with a layout shown in Fig.
1(d) Vaitiekėnas  _et al._ (2021). Without an applied magnetic field, in Ref.
Vaitiekėnas  _et al._ (2021) a ZBCP was observed and a remnant effective
Zeeman field exceeding 1 T was inferred. Numerical simulations Liu  _et al._
(2021a) on the electronic properties of this hybrid InAs/EuS/Al nanowire imply
that the magnetic proximity effects at the Al/EuS as well as the InAs/EuS
interfaces are both essential to achieve topological superconductivity,
without magnetic field applied.

MZMs are also predicted Hell  _et al._ (2017); Pientka  _et al._ (2017) to
exist in a planar Josephson junction (JJ) Fornieri  _et al._ (2019); Ren  _et
al._ (2019); Dartiailh  _et al._ (2021) that consists of a 2D SM electron gas
covered by two disconnected SC shells [Fig. 1(e)]. The narrow SM stripe
uncovered by SC forms the junction region. In the presence of a new tuning
knob, i.e., the phase difference of the JJ tuned by the magnetic flux induced
by a perpendicular magnetic field $B$, a zero or tiny parallel magnetic field
is required to induce Zeeman splitting. Theoretically, the topological
superconducting phase can be achieved (with two MZMs localized at the two ends
of the SM stripe) even without an applied magnetic field when the phase
difference is $\pi$ and the junction is transparent Pientka  _et al._ (2017).
In the presence of normal backscattering in the junction, a weak magnetic
field, in addition to the phase difference, is required to induce a Zeeman
splitting. The induced gap in planar JJs is relatively smaller than that of
other platforms mentioned above due to its 2D feature: now quasiparticles can
travel parallel to the junction, leading to a much longer flight time in the
non-superconducting SM region Laeven  _et al._ (2020). Fortunately, these
parallel quasiparticle trajectories, based on Ref. Laeven  _et al._ (2020),
can be greatly suppressed in a stripe with zigzag geometry, in which the
generated MZMs are expected to be more robust: they are protected by a much
larger (one-order larger in amplitude) induced gap and have much smaller
localization lengths Laeven  _et al._ (2020).

Basically, all the platforms shown in Figs. 1(b)–1(e) can be described by the
Majorana-nanowire model Lutchyn  _et al._ (2010); Oreg  _et al._ (2010) in
different dimensions with proper modifications. In most of the theoretical
works, the relevant parameters, e.g., effective electron mass, Landé g factor,
chemical potential, proximity induced gap, spin-orbit coupling strength, etc.,
are treated as independent variables in numerical simulations. In real
experiments, however, the tunable knobs are the magnetic field and gate
voltages, which together determine the values of all other parameters in a
complex way. In order to make a connection to realistic experiments, it is
suggested to study the hybrid SM-SC systems within the framework of
Schrödinger-Poisson method Mikkelsen  _et al._ (2018); Antipov  _et al._
(2018); Woods  _et al._ (2018), under the Thomas-Fermi approximation Wójcik
_et al._ (2018); Winkler  _et al._ (2019); Escribano  _et al._ (2019, 2020);
Liu  _et al._ (2021a). This is a systematic and close-to-experiment numerical
approach that takes into account all experimentally relevant effects, e.g.,
the orbital effect induced by a magnetic field, superconducting proximity
effect, SM-SC work function difference, and electrostatic environment.

##  III Recent progress on detecting MZMs

In addition to engineering MZMs, detecting MZMs in SM-SC heterostructures is
another challenging task. In this section, we review the latest progress on
the detection of MZM signatures. These detection methods include (i) the
tunneling spectroscopy detection of the existence of MZMs in Sec. III.1; (ii)
the detection of MZM non-locality in Sec. III.2; (iii) the detection of MZM
signature from ac Josephson effect in Sec. III.3; (iv) the detection of MZMs
electron-hole weights with Coulomb oscillation in Sec. III.4, and (v) nonlocal
tunneling spectroscopy measurement in Sec. III.5. In addition to the
experimental efforts above, other potential detection protocols (e.g.,
topological Kondo effect Béri and Cooper (2012); Altland  _et al._ (2014a);
Affleck and Giuliano (2013); Galpin  _et al._ (2014); Altland  _et al._
(2014b); Eriksson  _et al._ (2014a, b); Zazunov  _et al._ (2014); Liu  _et
al._ (2021b)) can be found in Ref. Zhang  _et al._ (2019b).

###  III.1 Tunneling spectroscopy

As one of the earliest introduced techniques, tunneling spectroscopy remains
among the most popular tools to detect MZMs in SM-SC heterostructures and
other topological SC candidates. Theoretically, MZMs are exotic zero-energy
excitations with equal electron and hole components. Because of this feature,
resonant tunneling into a MZM is predicted to arrive at a quantized ZBCP with
a height of $2e^{2}/h$ at zero temperature Sengupta  _et al._ (2001); Law  _et
al._ (2009); Flensberg (2010); Wimmer  _et al._ (2011). This quantized ZBCP,
being protected by topology, is predicted as robust against perturbations,
e.g., the tuning of magnetic field and gate voltages, as long as the topology
remains invariant.

However, since the first report of ZBCP in 2012, most of the observed ZBCPs
Deng  _et al._ (2012); Das  _et al._ (2012); Finck  _et al._ (2013); Churchill
_et al._ (2013); Deng  _et al._ (2016); Chen  _et al._ (2017); Suominen  _et
al._ (2017); Gül  _et al._ (2018); Sestoft  _et al._ (2018); Vaitiekėnas  _et
al._ (2018); Deng  _et al._ (2018); de Moor  _et al._ (2018); Bommer  _et al._
(2019); Grivnin  _et al._ (2019) are much smaller than the predicted quantized
value, except for some recently reported nearly-quantized ZBCPs Nichele  _et
al._ (2017); Pan  _et al._ (2022); Zhang  _et al._ (2021); Heedt  _et al._
(2021); Song  _et al._ (2021); Wang  _et al._ (2022a). Meanwhile, some other
predicted MZM signatures remain missing in most tunneling spectroscopy
measurements. These missing features include (i) the predicted correlation
between the emergence of ZBCPs and the close-and-reopen of the superconducting
gap gap Lutchyn  _et al._ (2010); Oreg  _et al._ (2010); and (ii) the
predicted oscillations of MZM splitting energy as a function of magnetic field
or chemical potential, dubbed as Majorana oscillations Das Sarma  _et al._
(2012); Rainis  _et al._ (2013). For issue (i), it is explained Stanescu  _et
al._ (2012) that the gap close-and-reopen is a bulk property of topological
phase transition that is not necessarily visible in the tunneling
spectroscopy. Specifically, in the topologically trivial phase, the lowest-
energy states are delocalized (localized) through the wire when the chemical
potential is lower (larger) than a critical value that is of the order of the
proximity-induced superconducting gap, resulting in an invisible (visible) gap
closure in the tunneling spectroscopy which is related to the local density of
states at the end of the wire. We mention that a recent experiment reported
the concomitant opening of a bulk gap with emerging ZBCPs. The Majorana
oscillation [issue (ii)], on the other hand, is explained as being sabotaged
by either the inverse proximity effect arising from drain-MZM coupling Danon
_et al._ (2017), or the Coulomb interaction between MZMs and the image charges
in the dielectric material around the SM nanowire Domínguez  _et al._ (2017).

Despite the explanations on the imperfect theory-experiment agreements, the
frequent observation of non-quantized ZBCPs in nanowire experiments since 2012
has stimulated many theoretical works on exploring the possible origins of
ZBCPs, see, e.g., Refs. Pientka  _et al._ (2012); Prada  _et al._ (2012); Liu
_et al._ (2012); Kells  _et al._ (2012); Pikulin  _et al._ (2012); Rainis  _et
al._ (2013); Roy  _et al._ (2013); Stanescu and Tewari (2013b); Liu  _et al._
(2017); Reeg  _et al._ (2018a); Moore  _et al._ (2018a); Vuik  _et al._
(2019); Pan  _et al._ (2021a); Cayao and Burset (2021). Most recently,
tunneling spectroscopy experiments of nanowires Chen  _et al._ (2019); Jünger
_et al._ (2020) have revealed non-Majorana ZBCPs owing to different
mechanisms. At present, these progress pose a serious doubt that all observed
ZBCPs in SM-SC heterostructures might originate from topologically trivial
zero-energy ABSs, see Ref. Prada  _et al._ (2020) for a review. Indeed,
following thorough numerical simulations, Ref. Pan and Das Sarma (2020)
further demonstrates that most of the experimental ZBCPs in InAs and InSb
nanowire devices might have been induced by strong disorders in the chemical
potential, proximity-induced superconducting gap, or effective Landé g factor.
It is worth mentioning that smooth-potential-induced topologically trivial
quasi-MZMs Vuik  _et al._ (2019); Moore  _et al._ (2018b); Pan and Das Sarma
(2020) can also lead to quantized ZBCP plateaus Moore  _et al._ (2018a). Both
MZMs and quasi-MZMs stand out from ABS interruptions, as the former ZBCPs
remain under the tuning of multiple experimental parameters. Notably, the
observation of ZBCP plateaus in real experiments is difficult as it depends on
experimental details, e.g., temperature, tunnel barrier, and contact
resistance, although the unknown contact resistance is expected as avoidable
in a four-terminal device Song  _et al._ (2021). Nevertheless, a recent
experiment Wang  _et al._ (2022a) observed plateau regions for ZBCPs within
$5\%$ of the quantized conductance value $2e^{2}/h$ when individually sweeping
back-gate voltage, tunnel-gate voltage, and parallel magnetic field. This
progress represents a further advance towards the detection of MZMs or quasi-
MZMs.

Figure 2: Schematics of the extended tunneling spectroscopy. (a) An
intentional quantum dot formed between the source lead and the hybrid SM-SC
nanowire enables the measurement of the nonlocality of MZMs. The quantum dot,
with an energy level $\varepsilon_{0}$ and a charging energy $U$, couples to
the left and right MZMs with tunneling amplitudes $t_{L}$ and $t_{R}$,
respectively, and the hybridization energy between the two MZMs is $\delta$.
(b) Tunneling spectroscopy measurement in the presence of a resistor in series
with the source lead. The resistor contains bosonic modes that dissipate
energy during transport processes. Transport is thus suppressed by coupling to
the resistor: this phenomenon is known as dynamical Coulomb blockade.

###  III.2 Extensions of tunneling spectroscopy

It was realized that tunneling spectroscopy protocol with minor modifications
can provide alternative insights (other than ZBCPs) to detect MZMs. For
instance, a quantum dot that locates at the end of the nanowire as shown in
Fig. 2(a) is predicted to detect the nonlocality of MZMs Prada  _et al._
(2017); Clarke (2017), while an Ohmic resistance in series with the electrode
Liu (2013); Liu  _et al._ (2022); Zhang  _et al._ (2022) as shown in Fig. 2(b)
could confirm the existence of MZMs via its unique universality class.

To start with, nonlocality is a unique property that can distinguish MZMs from
ABSs. Intuitively, tunneling spectroscopy obtained from a local probe is
unable to reflect the nonlocal property. Surprisingly, however, the
nonlocality is proposed to be measurable with the assistance of a quantum dot
Prada  _et al._ (2017); Clarke (2017), in a device schematically shown in Fig.
2(a). As indicated, there exists five energy scales, i.e., the dot level
$\varepsilon_{0}$, the Coulomb charging energy $U$, the dot-MZM couplings
$t_{L}$ and $t_{R}$, and the inter-MZM coupling $\delta$. In experiments, $U$
is inversely proportional to the dot size and $\varepsilon_{0}$ can be freely
tuned by the voltage gate near the dot. Note that $U=0$ also works for
measuring nonlocality as analyzed in Ref. Clarke (2017). The hierarchy of the
rest three energy scales characterizes a pair of well-separated MZMSs, an ABS,
or the cases in between. For a pair of MZMs each of which is localized at the
opposite ends of a finite-length nanowire [see Fig. 2(a)], both $\delta$ and
$t_{R}$ decay exponentially as a function of the wire length. In contrast, for
an ABS localized at the left wire end, so are its two Majorana components,
$t_{L}$ equals $t_{R}$ and $\varepsilon_{0}$ can be an arbitrary value
including zero. The ratio of $\sqrt{t_{R}/t_{L}}$ has been numerically
demonstrated to be an estimator that can quantify the degree of nonlocality of
a subgap bound state Prada  _et al._ (2017). Theoretically, different
amplitudes of $\delta$, $t_{L}$, and $t_{R}$ lead to different characteristic
tunneling spectroscopy patterns, from which the quantity $\sqrt{t_{R}/t_{L}}$
can be extracted, as predicted in Ref. Prada  _et al._ (2017). This protocol
has been implemented in an experiment Deng  _et al._ (2018) based on hybrid
InAs-Al nanowires, in which either a moderate degree of nonlocality,
suggesting a partially separated ABS, or a highly nonlocal subgap state
consistent with a pair of well-separated MZMs, were observed.

Previously, to observe quantized ZBCPs, noble metals (e.g., Au and Pt) with
negligible resistances are commonly chosen as lead materials (see e.g., Refs.
Deng  _et al._ (2016); Nichele  _et al._ (2017); Gül  _et al._ (2018);
Vaitiekėnas  _et al._ (2018); Zhang  _et al._ (2021); Song  _et al._ (2021);
Wang  _et al._ (2022a)). The authentication of the quantized ZBCP signature,
as has been discussed above, is however undermined by possible fake positive
signals arising from disorder-induced ABSs. Fortunately, these fake positive
signals can be suppressed by attaching a resistor (dissipation) to the leads
Liu (2013); Liu  _et al._ (2022), as schematically shown in Fig. 2(b). The
real MZM signal remains, however, for a resistor with a proper impedance.

Briefly, a resistor can be considered as a bath that contains bosonic modes.
During tunneling events, these modes can be excited to higher-energy levels,
during which energies are dissipated Grabert and Devoret (2013). This
phenomenon, known as dynamical Coulomb blockade, suppresses tunnelings at low
enough energies. Most importantly, dissipation influences the tunneling into a
MZM and an ABS in different ways Liu (2013); Liu  _et al._ (2022), leading to
distinct universal or non-universal features. One can thus distinguish the MZM
signal from the ABS ones by analyzing the transport features at low enough
energies, counter-intuitively, with dissipation intentionally included in the
lead. As the core idea, dissipation effects introduce interaction between
transported electrons and environmental bosons Grabert and Devoret (2013) and
cause renormalization effects of electron tunnelings as running the system
energy scale. Importantly, due to this renormalization effect, Majorana and
topologically trivial transports show completely different scaling behaviors
(in both temperature and voltage bias) that suffice to tell MZM from trivial
bound states. The idea to suppress ABS-induced ZBCPs has been recently
verified experimentally Zhang  _et al._ (2022) when the dissipation is strong
enough ($R\geq 0.21h/e^{2}$). With a smaller dissipation in the sample
($R\approx 0.1h/e^{2}$), the ABS-induced ZBCP is not fully suppressed at
experimentally accessible temperatures ($T>24$ mK) of Ref. Wang  _et al._
(2022b). Instead, the ABS-induced ZBCP now displays a clear and interesting
competition between dissipation and thermal fluctuations.

Figure 3: (Color online) Schematic JJ based on a hybrid SM-SC nanowire. The
tunnel gate tunes the junction transparency. If each nanowire segment on the
two sides of the junction hosts two well-separated MZMs, the fractional ac
Josephson effect is predicted to be measurable in such a device with a source-
drain bias voltage and low junction transparency.

###  III.3 Fractional Josephson effect

Josephson effect Josephson (1962, 1965) is a unique transport phenomenon
across a JJ. Experimental detection of MZMs also includes the probing of
fractional Josephson effect Kwon  _et al._ (2004) in SM-SC nanowire setups
Rokhinson  _et al._ (2012); Tiira  _et al._ (2017); Kamata  _et al._ (2018);
Laroche  _et al._ (2019) or HgTe-based topological insulator setups Wiedenmann
_et al._ (2016); Bocquillon  _et al._ (2017); Deacon  _et al._ (2017). A
hybrid SM-SC nanowire JJ is schematically shown in Fig. 3, where a fraction of
the SC shell in the middle is etched to create a JJ. A nearby tunnel gate is
used to tune the junction transparency.

When the hybrid SM-SC nanowire in Fig. 3 is driven to the topological
superconducting phase, the left (right) hybrid nanowire segment supports two
MZMs $\gamma_{1,2}$ ($\gamma_{3,4}$) at the ends. With a long-enough nanowire,
MZMs sufficiently decouple, and the two outer MZMs ($\gamma_{1,4}$) are
projected out of the low-energy subspace of the topological JJ. In this case,
the junction Hamiltonian effectively becomes
$H_{\textrm{eff}}(\phi)=t(\phi)(\hat{n}-1/2)$ where $t(\phi)$ is the inter-MZM
coupling as a function of the phase difference $\phi$ across the junction, and
$\hat{n}=(1+i\gamma_{2}\gamma_{3})/2$ is the number operator. Its eigenvalues
are conserved since $[\hat{n},H_{\textrm{eff}}]=0$. The eigenvalues $n=0$ and
$n=1$ correspond to the state of even and odd parities, respectively.
Following a simple symmetry analysis Marra (2022), $t(\phi)$ is
$4\pi$-periodic in $\phi$. Consequently, at zero temperature, the equilibrium
dc Josephson current mediated by a pair of MZMs becomes
$I(\phi)=-\frac{2e}{\hbar}(n-1/2)\partial_{\phi}t(\phi)$, which is also
$4\pi$-periodic in $\phi$ Kitaev (2001). This $4\pi$ periodicity is considered
as a MZM signature, as it is in strong contrast to the $2\pi$ periodicity in
conventional JJs. Physically, the doubling of periodicity reflects the
fractionalization of a MZM: it can be considered as ‘half’ of a regular
fermion.

In real experiments, this $4\pi$-periodic feature however encounters multiple
challenges Fu and Kane (2009); Lutchyn  _et al._ (2010); San-Jose  _et al._
(2012). To begin with, as the prerequisite of the $4\pi$-periodic feature,
parity conservation unfortunately can be violated by e.g., the hybridization
between the inner and outer MZMs or the inelastic particle transition that
involves the quasiparticle continuum above the gap (i.e., the quasiparticle
poisoning). Fortunately, these detrimental effects are less severe in ac
Josephson junctions San-Jose  _et al._ (2012); Houzet  _et al._ (2013);
Domínguez  _et al._ (2012); Sau and Setiawan (2017). Detection of the
fractional ac Josephson effect has been performed by measuring either the
Shapiro steps Shapiro (1963) or the Josephson frequency $f$. Specifically,
when influenced by a microwave with frequency $\Omega$, the dc component of
the current across a low-transparency topologically trivial JJ contains the
Shapiro steps at biases $V_{n}=nV_{0}$, for all positive $n$ and
$V_{0}=h\Omega/2e$ Shapiro (1963). To date, Shapiro steps at
$V=V_{0},~{}2V_{0},~{}3V_{0},~{}4V_{0}$ have been observed for JJ in the
trivial regime Rokhinson  _et al._ (2012). In the topological regime, by
contrast, only even numbers of $n$ are allowed theoretically Houzet  _et al._
(2013); Domínguez  _et al._ (2012); Sau and Setiawan (2017). This is, however,
not observed in a real experiment Rokhinson  _et al._ (2012), where the
Shapiro step at $V=V_{0}$ is absent while the one at $V=3V_{0}$ still remains.
This theory-experiment discrepancy has been visited by multiple works Pikulin
and Nazarov (2012); Domínguez  _et al._ (2012); Virtanen and Recher (2013); De
Cecco  _et al._ (2016); Sau and Setiawan (2017); Picó-Cortés  _et al._ (2017);
Le Calvez  _et al._ (2019) after taking more experimental details into
consideration. As another option, the fractional Josephson effect can be
detected by measuring the Josephson frequency. Indeed, in the topological
regime, the Josephson frequency is predicted San-Jose  _et al._ (2012) to be
half of that in the trivial regime. This phenomenon has been observed by Ref.
Kamata  _et al._ (2018). However, all Shapiro steps $V=nV_{0}$ are observed in
Ref. Kamata  _et al._ (2018), in contrast to the topological criteria
introduced above. The halved Josephson frequency was interpreted as trivial
changes in the superconducting circuit surrounding the device strongly
affected by the applied magnetic field Kamata  _et al._ (2018). The halved
Josephson frequency has also been reported in a JJ made of hybrid InAs-Al
nanowire Laroche  _et al._ (2019).

Figure 4: (Color online) (a) A schematic device for measuring Coulomb blockade
transport through a floating SM-SC nanowire island. Applying a parallel
magnetic field $B$ may induce a pair of MZMs at the two ends of the island.
The bias voltage $V_{SD}$ drives a charge current $I$ through the island, gate
voltages $V_{T1}$, $V_{T2}$ control the tunnel barriers between the metallic
leads and their nearest island ends, and $V_{G}$ controls the island electron
number through adjusting the electrostatic energy of the nanowire island. (b)
Typical Coulomb-blockade conductance peaks. Peak spacings for odd (o) and even
(e) Coulomb valleys are indicated as $S_{o}$ and $S_{e}$, respectively. Peak
heights at odd-to-even (even-to-odd) parity transitions occurring upon
increasing $V_{G}$ are denoted as $G_{o\rightarrow e}$ ($G_{e\rightarrow o}$).
In experiments, both the peak spacings and peak heights oscillate as $B$
various.

###  III.4 Coulomb blockade conductance oscillation

Coulomb blockade refers to the phenomenon where charge transport is suppressed
by the Coulomb interaction. As known, Coulomb blockade transport can provide
quantitative information about the charge and energy spectrum of a mesoscopic
island Von Delft and Ralph (2001); Aleiner  _et al._ (2002). It thus also
helps the detection of MZMs, especially their non-local features.

Figure 4(a) shows a common structure with Coulomb blockade involved. In
experiments Higginbotham  _et al._ (2015); Albrecht  _et al._ (2016); Sherman
_et al._ (2017); Albrecht  _et al._ (2017); Vaitiekėnas  _et al._ (2018);
O’Farrell  _et al._ (2018); Shen  _et al._ (2018, 2021); Bjergfelt  _et al._
(2021); Estrada Saldaña  _et al._ (2022); Valentini  _et al._ (2022) with
devices of the considered structure, a source-drain bias voltage is applied to
drive a charge current through a floating hybrid SM-SC nanowire. This finite-
size island has a charging energy $E_{C}$ and a proximity-induced
superconducting gap $\Delta$ (at $B=0$). The plunger gate voltage $V_{G}$,
applied beneath the island, controls the number of electrons on the island and
hence the number parity of the island. The couplings between the island and
two metallic leads are controlled by two tunnel gates. In addition to the
tunneling amplitudes tuned by the tunnel gates, the island-lead communications
also depend on the relative amplitudes of other relative energy scales
including $E_{C}$, $\Delta$, and the ABS (or MZM) energy.

Assuming that the superconducting island hosts an ABS with energy $E_{0}$ and
finite wavefunction distributions on both ends of the island. The most
interesting situation occurs when $E_{0}<E_{C}<\Delta$, where the parity of
the ground-state of the island can alternate between even and odd as the
voltage $V_{G}$ changes van Heck  _et al._ (2016). At a vanishingly small bias
voltage and low temperatures, generally, the transport of electrons through
the island is forbidden [the conductance valley shown in Fig. 4(b)] due to
blockade from Coulomb-induced charging energy (i.e., the Coulomb blockade). By
contrast, with a fine-tuned $V_{G}$, island states with different parities
share the same energy. This energy degeneracy of the island enables resonant
transport through the island, leading to the sharp ZBCP shown in Fig. 4(b).
The conductance peak heights associated with the even-to-odd ($G_{e\rightarrow
o}$) and odd-to-even ($G_{o\rightarrow e}$) ground-state parity transitions
are different, due to the fact that they are related to the electron-like and
hole-like components of the ABS (or MZM) of the island. Consequently, the
electron-hole weights of the ABS (or MZM) can be extracted from the positions
and heights of the consecutive Coulomb blockade conductance peaks Hansen  _et
al._ (2018).

By a rate-equation analysis, in the case where the tunnel couplings at both
ends of the nanowire are equal, it was found that Hansen  _et al._ (2018)

$\displaystyle\Lambda=\frac{G_{e\rightarrow o}}{G_{e\rightarrow
o}+G_{o\rightarrow
e}}=\frac{\frac{|u_{L}|^{2}|u_{R}|^{2}}{|u_{L}|^{2}+|u_{R}|^{2}}}{\frac{|u_{L}|^{2}|u_{R}|^{2}}{|u_{L}|^{2}+|u_{R}|^{2}}+\frac{|v_{L}|^{2}|v_{R}|^{2}}{|v_{L}|^{2}+|v_{R}|^{2}}},$
(1)

where $|u_{L/R}|^{2}=\sum_{\sigma}|u_{L/R,\sigma}|^{2}$,
$|v_{L/R}|^{2}=\sum_{\sigma}|v_{L/R,\sigma}|^{2}$, with $u_{L/R,\sigma}$,
$v_{L/R,\sigma}$ the spin-resolved local Bogoliubov-de Gennes amplitudes of
the ABS at the left/right end of the island. If the island has a space-
inversion symmetry, i.e., $|u_{L}|=|u_{R}|=|u|$ and $|v_{L}|=|v_{R}|=|v|$, Eq.
(1) simplifies to

$\Lambda=\frac{G_{e\rightarrow o}}{G_{e\rightarrow o}+G_{o\rightarrow
e}}=\frac{|u|^{2}}{|u|^{2}+|v|^{2}}.$ (2)

Clearly, in this case, the electron and hole weights can be obtained by
measuring the $\Lambda$ of Eq. (2) in experiments.

The ABS (or MZM) energy $E_{0}$, on the other hand, can be extracted from

$\pm E_{0}=\eta\langle S_{e(o)}\rangle-E_{C},$ (3)

that relates $E_{0}$ to the charging energy $E_{C}$ and the peak spacing
average $\langle S_{e(o)}\rangle$ [see Fig. 4(b)]. The charging energy and the
lever arm $\eta$ can be estimated from the Coulomb blockade diamonds features.

Equations (2) and (3) address how the information of an ABS (or a MZM) in an
SM-SC nanowire island can be obtained from Coulomb blockade transport
features. The obtained values of $\Lambda$ and $E_{0}$, which are magnetic-
field dependent in real experiments O’Farrell  _et al._ (2018); Shen  _et al._
(2018, 2021), can help to distinguish a MZM from ABSs, as the Coulomb blockade
transport mediated by a pair of MZMs is predicted to be distinct from that by
an ABS. More specifically, the MZMs mediated Coulomb blockade transport has
the following unique features: (i) $E_{0}$ decreases exponentially with
increasing the wire length Das Sarma  _et al._ (2012). (ii) $E_{0}$ and
$\Lambda$ are pined to 0 and $1/2$, respectively, for long wires in which
$|u|=|v|$ is guaranteed. For short wires with a finite MZM overlap, both
$E_{0}$ and $\Lambda$ oscillate with the magnetic field $B$. The oscillating
amplitude and period are both larger under a greater magnetic field, as long
as the system remains in the topological regime Hansen  _et al._ (2018). (iii)
The oscillations of $E_{0}$ and $\Lambda$ have a $\pi/2$ phase shift Hansen
_et al._ (2018). Features akin to theoretical predictions of feature (i) have
been claimed to be observed experimentally Albrecht  _et al._ (2016) by
measuring several devices with different wire lengths. In Ref. Albrecht  _et
al._ (2016), however, the observed oscillation of $E_{0}$ decays when $B$
increases, in strong conflict with the predicted feature (ii). In some other
experimental efforts O’Farrell  _et al._ (2018); Shen  _et al._ (2021), the
observed oscillation of $E_{0}$ also decays when $B$ increases, but the
$\pi/2$ phase shift mentioned in feature (iii) has been observed. Both
features (ii) and (iii) are however absent in Ref. Shen  _et al._ (2018).
Several theoretical works Chiu  _et al._ (2017); Dmytruk and Klinovaja (2018);
Fleckenstein  _et al._ (2018); Cao  _et al._ (2019) tried to explain these
discrepancies with different assumptions. However, the discrepancies can not
be fully explained by these theories, except for Ref. Cao  _et al._ (2019),
which assumes that there might exist steplike Rashba spin-orbit couplings
along these nanowire islands, due to the potentially nonuniform electrostatic
potential at the center and the two sides of the nanowires.

Before closure, we also mention that with the dissipation effect present as
shown in Sec. III.2, the tunneling signatures through a Coulomb blockade
island will be modified Liu  _et al._ (2020); and this proposal provides a
hallmark for the nonlocal coherent nature in the Fu-teleportation Fu (2010);
Semenoff and Sodano (2006); Hützen  _et al._ (2012), which are different in
comparison to the incoherent transports from other non-topological Coulomb-
blockade systems.

Figure 5: (Color online) (a) Schematic of a three-terminal transport device.
Two normal leads are connected to a central grounded superconducting region,
where the tunnel barriers at two ends can be controlled by tunnel gates. (b)
Schematics of a system comprising a spinless quantum dot coupled to one end of
a floating hybrid SM-SC nanowire. The conductance through the dot is measured
by adding two external leads.

###  III.5 Three-terminal transport

As a more direct option Rosdahl  _et al._ (2018); Pan  _et al._ (2021b); Hess
_et al._ (2021); Pikulin  _et al._ (2021), the nonlocal nature of MZMs can be
revealed by measuring both local and nonlocal tunneling spectroscopy in a
three-terminal device, as schematically shown in Fig. 5(a). Briefly, as MZMs
are predicted as correlated states localized at two ends of a hybrid SM-SC
nanowire, a strong correlation is anticipated between local tunneling
spectroscopy measured at each ends Das Sarma  _et al._ (2012), even when two
MZMs fully decouple. This criterion is preferred to be carried out in
experiments with long enough hybrid SM-SC wires. Indeed, correlation might as
well appear in topologically trivial systems, if the wire is shorter than the
localization length of regular fermionic states or ABSs Hess  _et al._ (2021).

The nonlocal conductance spectroscopy employs two spatially separated leads to
probe three global properties: the bulk superconducting gap, the induced gap,
and the induced coherence length Rosdahl  _et al._ (2018). Among them, the
bulk superconducting gap is barely a benchmark of the presence of
superconductivity. With the presence of the bulk superconductivity, the second
measured quantity, i.e., the induced gap conveys the information of possible
topological phase transition: near the transition point, the induced gap is
predicted to experience a gap closure and reopen. The observed topological
phase transition, if accompanied by a strong correlation of local conductance
at both ends of the nanowire, is currently believed as a strong evidence of
the presence of MZMs. Following this protocol, some three-terminal experiments
Anselmetti  _et al._ (2019); Puglia  _et al._ (2021); Ménard  _et al._ (2020);
Wang  _et al._ (2022c); Pöschl  _et al._ (2022); Banerjee  _et al._ (2022)
have recently been carried out in SM-SC heterostructures.

As the latest experimental progress Aghaee  _et al._ (2022), the Microsoft
group realizes the non-local tunnel measurement, which is believed as a more
trustworthy protocol than the standard tunneling spectroscopy method (shown in
Sec. III.1) for the MZM detection. More specifically, Ref. Aghaee  _et al._
(2022) claims to have detected the topological superconducting phase
transition following the three-terminal transport protocol. This protocol
starts with a fast-detection Pikulin  _et al._ (2021) of local tunneling
signal at both sides of the wire. By doing so, they narrow down the phase
space to a selected area with ZBCP observed at both ends of the wire. Finally,
careful non-local measurement is taken within the selected area, through which
a topological phase transition (in the bulk) and the corresponding topological
regime have been identified. Remarkably, three (out of five) devices have
passed this protocol Aghaee  _et al._ (2022). Although its data quality (e.g.,
the ZBCP height) requires further improvement, Ref. Aghaee  _et al._ (2022) is
highly inspiring and encouraging to the Majorana hunting community.

In addition to the protocol above, a recent theory Danon  _et al._ (2020) of
the three-terminal setup predicts that the ratio between the symmetric and
asymmetric components of the nonlocal conductance is directly related to the
local BCS charges (defined as $Q_{L/R}=|u_{L/R}|^{2}-|v_{L/R}|^{2}$) of the
bound state close to the left and right normal leads. Essentially, the
measurement of local BCS charges $Q_{L/R}$ (or equivalently, electron-hole
weights $|u_{L/R}|^{2}$ and $|v_{L/R}|^{2}$) can help detecting MZMs Hansen
_et al._ (2018); Cao  _et al._ (2022a): For a pair of well-separated MZMs
localized at the two ends of a hybrid nanowire, there exists
$|u_{L/R}|^{2}=|v_{L/R}|^{2}$ and $Q_{L/R}=0$. This protocol of detecting
local BCS charge in a three-terminal setup has been tested in a recent
experiment Ménard  _et al._ (2020), but no stable zero BCS charge has been
observed.

Given the ability of fabricating multi-terminal hybrid SM-SC nanowires and
quantum dots therein, another three-terminal protocol, as shown in Fig. 5(b),
has been proposed to detect MZM. As another related proposal, Ref. Liu and
Baranger (2011) studies the system where a resonant level couples to one
localized MZM of a floating hybrid SM-SC nanowire. In this case, the presence
of MZM blocks half of the tunneling channels through the resonant level,
leading to a ZBCP with height $e^{2}/2h$ at zero temperature. As a possible
extension of Ref. Liu and Baranger (2011), it is proposed to stabilize the MZM
signal by introducing a dissipative resonant level system that couples to the
partner of the MZM under detection Zhang and Baranger (2020); Zhang and
Spånslätt (2020). Briefly, the coupling between the MZM pairs, which is
predicted to sabotage the MZM signal Liu and Baranger (2011), will be
suppressed by the presence of dissipation, thus manifesting the $e^{2}/2h$
MZM-featured conductance.

##  IV Material aspects

###  IV.1 Material choice

The mostly studied epitaxial hybrid SM-SC nanowires thus far are InAs-Al or
InSb-Al Lutchyn  _et al._ (2018). A few works Sestoft  _et al._ (2018); Mayer
_et al._ (2020); Moehle  _et al._ (2021) also studied a similar combination,
InAsSb-Al, since it was predicted Winkler  _et al._ (2016) that InAsSb has a
much stronger intrinsic spin-orbit coupling than InAs and InSb. To enhance the
induced SM superconducting gap and its tolerance to magnetic field and thermal
fluctuation, $s$-wave SCs Sn and Pb have been tested in the fabrication of
hybrid InSb-Sn Pendharkar  _et al._ (2021), InSb-Pb Jung  _et al._ (2021), and
InAs-Pb Kanne  _et al._ (2021) nanowires. Tunneling spectroscopy measurements
show that such SM-SC nanowires based on Sn or Pb have larger induced
superconducting gaps, critical magnetic fields, and higher critical
temperatures than InAs-Al and InSb-Al nanowires, see Table 1.

Despite the above advantages from superior SCs, InAs and InSb nanowires were
predicted to commonly suffer from disorder interruptions that can potentially
sabotage the detection of MZMs (see, e.g., Refs. Pan and Das Sarma (2020); Das
Sarma and Pan (2021)). Indeed, these disorders, either long or short-ranged,
are known to induce topologically trivial ZBCPs Pientka  _et al._ (2012);
Prada  _et al._ (2012); Liu  _et al._ (2012); Kells  _et al._ (2012); Pikulin
_et al._ (2012); Rainis  _et al._ (2013); Roy  _et al._ (2013); Stanescu and
Tewari (2013b); Liu  _et al._ (2017); Reeg  _et al._ (2018a); Moore  _et al._
(2018a); Vuik  _et al._ (2019); Pan  _et al._ (2021a); Cayao and Burset (2021)
or unexpected properties missed by the analysis on pristine SM-SC nanowires
Hui  _et al._ (2015); Cole  _et al._ (2016); Liu  _et al._ (2018b);
Fleckenstein  _et al._ (2018); Cao  _et al._ (2019); Kiendl  _et al._ (2019).
In real devices, disorder may arise from the surface oxidation of the SC, the
imperfect substrates and gates, and more seriously, from the unintentional
charged impurities Pantelides (1978), the randomly distributed twin defects
Caroff  _et al._ (2009), and the stacking faults Shtrikman  _et al._ (2009) in
SM nanowires. Typically, charged impurities in InAs and InSb nanowires could
induce strong chemical potential fluctuations to the order of a few meV Woods
_et al._ (2021), which are much larger than the induced superconducting gap
(around 0.2 meV), making it impossible to access the topological
superconducting phase.

Table 1: Typical values of induced superconducting gap $\Delta_{\textrm{ind}}$, critical temperature $T_{c}$, and critical parallel magnetic field $B_{c,\parallel}$ extracted from tunneling spectroscopy measurements of hybrid SM-SC nanowires comprising InAs (or InSb) and different SC materials Al, Pb, and Sn. |  Al Deng  _et al._ (2016); Vaitiekėnas  _et al._ (2018); de Moor  _et al._ (2018) |  Sn Pendharkar  _et al._ (2021) |  Pb Kanne  _et al._ (2021)
---|---|---|---
$\Delta_{\textrm{ind}}$ (meV) | 0.22—0.27 | 0.7 | 1.25
$T_{c}$ (K) | $\sim 1$ | $\sim 3.7$ | $\sim 7$
$B_{c,\parallel}$ (T) | $\sim 2$ | $\sim 4$ | $\sim 8.5$

Most recently, the electrostatic and electronic properties of PbTe, a IV-VI SM
material, have been numerically studied Cao  _et al._ (2022b) as a possible
remedy for the interruption from disorders. In comparison to InAs and InSb,
PbTe has currently displayed several advantages. To start with, PbTe is known
to have a much larger dielectric constant ($\sim$ 1350) in comparison to that
($\sim$ 15) of InAs and InSb, leading to a higher tolerance of charged
impurity in PbTe. Meanwhile, as the sub-band of each valley can potentially
host MZMs, the multiple-valley structure of PbTe provides more options (in
comparison to the InAs and InSb situations) in the realization of the
topological superconducting phases.

Additional advantages of PbTe-Pb over the existing hybrid nanowires can be
expected: (i) Owing to the huge dielectric constant of PbTe, it has very high
electron mobilities (about $10^{6}~{}\textrm{cm}^{2}/\textrm{Vs}$) Springholz
_et al._ (1993); Ueta  _et al._ (1997), such that in submicron constrictions
lithographically patterned in PbTe quantum wells, sequential quantized
conductance steps were observed despite of a significant concentration of
charged defects near the constrictions Grabecki  _et al._ (1999, 2005, 2004,
2006). (ii) A high-resolution transmission electron microscopy revealed that
the epitaxially grown PbTe nanowires are free of stacking fault Dziawa  _et
al._ (2010). (iii) By virtue of the sizable parent gap $\Delta$ of Pb, a
superconducting gap comparable with those of InAs-Al and InSb-Al can be
induced in PbTe-Pb even by a weak coupling between Pb and PbTe, where the
effective Landé g factor and spin-orbit coupling in PbTe do not get reduced
much by the renormalization effects on SM from SC Stanescu  _et al._ (2011);
Cole  _et al._ (2015); Stanescu and Das Sarma (2017); Reeg  _et al._ (2018b).
Meanwhile, at weak SM-SC couplings, the induced superconductivity in the SM is
immune to any nonmagnetic disorder in the SC Hui  _et al._ (2015); Cole  _et
al._ (2016); Liu  _et al._ (2018b). Finally, (iv) Pb shell on PbTe nanowire
has less lattice mismatch than that of InAs-Al and InSb-Al. The lattice
matching can even be further improved by buffering a thin CdTe layer between
PbTe and Pb, as CdTe and PbTe share almost the same lattice constant. The
advantages mentioned above have inspired preliminary experimental efforts on
the epitaxial growth and transport characterization of PbTe nanowires
Schlatmann (2021); Schellingerhout  _et al._ (2022); Geng  _et al._ (2022);
Jiang  _et al._ (2022) and hybrid PbTe-Pb nanowires Jiang  _et al._ (2022).

Figure 6: Schematic and energy curves (as a function of the effective
dimensionless gate voltage $n_{g}$) of a charge qubit [(a) and (d)], transmon
qubit [(b) and (e)], and topological qubit [(c) and (f)]. Qubits of the first
two types are based on conventional SCs while the third ones require
topological SCs. States $|0\rangle$ and $|1\rangle$ [in (d) and (e)] of both
the charge and transmon qubits refer to the system ground state and the first
excited state, respectively. In a charge qubit (d), both state energies and
the energy difference $\Delta E$ between the ground state and the first
excited state oscillate with $n_{g}$. The oscillation of energies increases
the complexity of the qubit manipulation. Indeed, when $n_{g}$ is tuned to
around the odd values of (d), $\Delta E$ becomes much larger than the energy
difference between the first and second excited states, making it hard to
control the qubit state. For transmon qubit in (e), the problem with
oscillating energies is absent. However, in the transmon qubit setup, the
energy difference $E_{12}$ is very close to $E_{01}$. As a consequence, state
leakage may occur when controlling the qubit with fast pulses. A topological
qubit (f) contains four MZMs that combine into four degenerate states. Two of
them ($|00\rangle$ and $|11\rangle$) are in the even parity, and the other two
($|01\rangle$ and $|10\rangle$) are in the odd parity. States with the same
parity (either even or odd) have the same energy that is different from that
with the opposite parity [see curves of (f)]. Qubit manipulation is realized
between degenerate states with the same parity. From (f), it can be seen that
since the qubit is encoded within the same parity, charge noise then does not
influence the qubit.

###  IV.2 Material growth

As one recent progress in the hybrid SM-SC nanowire fabrication, the
integration of the selective-area growth technique with the shadow deposition
of SC Jung  _et al._ (2021); Jiang  _et al._ (2022) improves the quality of
fabricated nanowire networks. Previously, with the out-of-plane growth
technique, nanowire networks were constructed by merging multiple wires during
the growth Fadaly  _et al._ (2017). To guarantee the phase-coherent and
ballistic transport of the network, the out-of-plane growth technique has an
extremely high requirement (with accuracy to the order of nanometer) on the
positioning of the grown nanowires. The requirement of high precision of the
out-of-plane growth technique limits its application in the design of more
complex networks. The requirement of accuracy is however relaxed by the in-
plane selective area growth technique Krizek  _et al._ (2018); Aseev  _et al._
(2018); Op het Veld  _et al._ (2020). With this technique, an amorphous mask,
on which the trenches for SM network growth can be opened, is firstly
deposited on the substrate. The right growth conditions confine the subsequent
nanowire synthesis to the mask openings, allowing for scalable and highly
flexible growth of complex in-plane selective area networks. Subsequently, a
SC shell was epitaxially grown on the surface of the SM nanowire networks and
then selectively etched for covering metallic contacts or the tuning of
electrostatics by voltage gates. The etching process generally damages the SM
surface and thus prevents ballistic electron transport. To overcome this
shortcoming, several shadow walls, with designed heights and widths can be
fabricated before the SM nanowire network growth Jung  _et al._ (2021); Heedt
_et al._ (2021); Jiang  _et al._ (2022). The walls act as shadowing objects
during the subsequent SC deposition and allow for selective growth on the SM
network.

##  V Topological quantum computation based on hybrid SM-SC nanowires

Qubit is the fundamental building block in quantum computation. In theory, a
qubit can be constructed by any two-state system with states denoted as
$|0\rangle$ and $|1\rangle$. These states can be, e.g., the occupied and
unoccupied status of an energy level, the spin-up and spin-down of an
electron, or the left and right polarization of a photon. One of the most
formidable difficulties for almost all existing qubit platforms is the
decoherence problem caused by inevitable environmental noises Nielsen and
Chuang (2002). By contrast, MZMs can form topological qubits that store
information nonlocally. Specifically, $N$ pairs of spatially well-separated
MZMs in a topological material will form a $2^{N-1}-$ fold degenerate ground
state subspace, which is separated from the high-energy subspace by a
topological gap. If the temperature is much smaller than the topological gap,
the information encoded in the degenerate ground state subspace becomes
insensitive to quantum decoherence induced by local noise or fluctuations:
this is the so-call topological protection Kitaev (2003); Nayak  _et al._
(2008); Das Sarma  _et al._ (2015). In addition, due to the non-Abelian
exchange statistics obeyed by MZMs Moore and Read (1991); Nayak and Wilczek
(1996), braiding of MZMs along with the topological charge measurement (e.g.,
fermion parity measurement in topological SCs) can generate complete fault-
tolerant Clifford gate operations Kitaev (2003); Nayak  _et al._ (2008).
Certain non-Clifford gates, e.g., T-gate, can be realized by preparing noisy
magic states along with state distillation protocols Bravyi and Kitaev (2005).
Consequently, Majorana qubit system becomes an important candidate to realize
a universal fault-tolerant quantum computation. In this section, we show the
advantages of topological qubits over charge qubits Bouchiat  _et al._ (1998)
and transmon qubits Koch  _et al._ (2007), the latter two can be constructed
by conventional $s$-wave SCs.

Two adjacent SCs connected by a normal conductor can form a so-called JJ. As
shown in Fig. 6(a), two parallel JJs made of mesoscopic superconducting
islands can be considered as a charge qubit Jacobs (2014). Two lowest-energy
eigenstates of the system play the role of the qubit states $|0\rangle$ and
$|1\rangle$. In the charge qubit, there exist two relevant competing energy
scales: the Josephson coupling $E_{J}$ across the junction and the Coulomb
charging energy $E_{C}$. For $E_{J}=E_{C}$, as shown in Fig. 6(d), the low-
lying energies strongly depend on the gate voltage $V_{g}$ parameterized with
a dimensionless quantity $n_{g}=CV_{g}/e$, where $C$ is the capacitance of the
island. This remarkable charge dispersion amounts to that the charge qubit is
sensitive to charge noise Ithier  _et al._ (2005). Theoretically, the charge
dispersion can be reduced exponentially by increasing the ratio $E_{J}/E_{C}$.
It was proposed that enhanced $E_{J}/E_{C}$ is accessible in a special design
depicted in Fig. 6(b) Koch  _et al._ (2007), in which a charge qubit created
by sawtoothed superconducting islands is shunted by two transmission lines.
This setup is called transmon qubit Koch  _et al._ (2007). As shown in Fig.
6(e), when $E_{J}$ is an order larger than $E_{C}$, the charge dispersion is
almost completely eliminated. However, in the transmon qubit setup, the energy
difference between the second and the first excited states ($E_{12}$) is very
close to that between the first excited and the ground states ($E_{01}$),
similar to the spectrum of a 1D harmonic oscillator. When the transmon qubit
is controlled by fast pulses with wide frequency response, this low
anharmonicity of transmon qubit may lead to leakage out of the computational
subspace defined by the two lowest-lying energy eigenstates Steffen  _et al._
(2003); Motzoi  _et al._ (2009). In addition, although decoherence caused by
charge noise can be resolved, there are other types of decoherence in transmon
qubit, e.g., due to quasiparticle poisoning Catelani  _et al._ (2011); Sun
_et al._ (2012). We note that hybrid SM-SC nanowires have been employed to
create gate tunable transmon qubit, known as gatemon qubit Larsen  _et al._
(2015); de Lange  _et al._ (2015); Luthi  _et al._ (2018); Sabonis  _et al._
(2020); van Zanten  _et al._ (2020) and see Ref. Aguado (2020) for a review.
The combination between transmon qubits and Majorana-based qubits has been
proposed Hassler  _et al._ (2011); van Heck  _et al._ (2012); Hyart  _et al._
(2013); Ginossar and Grosfeld (2014); Yavilberg  _et al._ (2015); Keselman
_et al._ (2019); Ávila  _et al._ (2020a, b) to construct topological transmon
qubits with topological protection against local sources of noise. In a
Majorana box qubit (MBQ) Plugge  _et al._ (2017), also named as two-sided
tetron Karzig  _et al._ (2017), consisting of four MZMs at the ends of two
parallel topological SCs shown in Fig. 6(c), both the topological degeneracy
Fu (2010) shown in Fig. 6(f) and the charging energy can resolve both the
decoherence and quasiparticle poisoning problems at once.

Figure 7: Three Majorana parity readout protocols and their corresponding
parity-dependent curves. (a) A protocol to read out the Majorana parity by
measuring the occupation (via the charge sensor labeled by “CS”) of the
quantum dot that couples to two MZMs. The dot has the energy $\epsilon_{1}$.
(b) For MZMs with different parities, the dot occupation $\langle
n_{1}\rangle_{0}$ follows different $\epsilon_{1}$-dependent curves. One can
detect the MZM parity as long as the dot is off-resonance, i.e.,
$\epsilon_{1}\neq 0$. (c) The protocol that detects the MZM parity by coupling
the charge sensor to a quantum dot (QD2). (d) After tuning QD1 and QD2 to be
degenerate at time $t=0$, the dot state Rabi-oscillates between two states.
The time $\tau_{m}$ of a projective measurement indirectly conveys the
information of the MZM parity. (e) Majorana parity readout from interference
between two trajectories: the one through the reference arm R (yellow) and
that through the SC (gray). (f) The corresponding conductance curves
$G_{z}(\phi)$ as functions of the dimensionless magnetic flux.

As mentioned previously, MZMs can potentially provide a solution for realizing
fault-tolerant quantum computation. MBQ scheme Plugge  _et al._ (2017); Karzig
_et al._ (2017) is one of the most promising schemes among various MZM-based
topological qubit proposals. The qubits in this scheme are actually
topologically protected superconducting charge qubits, which include two or
more parallel hybrid SM-SC nanowires bridged by conventional SCs. One can
imagine this MBQ device as a mesoscopic topological superconducting island.
This island has a constant Coulomb charging energy and is connected to other
constituents, i.e., trivial SCs and other MBQs, via JJs. Most interestingly,
when the charging energy is much greater than the Josephson energy, the system
is in a topological Cooper pair box regime Plugge  _et al._ (2017); Karzig
_et al._ (2017), where the nonequilibrium quasiparticle tunneling (or
quasiparticle poisoning) outside the MBQ can be significantly reduced. This
MBQ scheme is thought to be better than the earlier topological transmon qubit
schemes Hassler  _et al._ (2011); van Heck  _et al._ (2012); Hyart  _et al._
(2013); Ginossar and Grosfeld (2014), which are more easily affected by
quasiparticle poisoning outside the qubit island Catelani  _et al._ (2011);
Sun  _et al._ (2012). Usually, each MBQ contains four MZMs (or six MZMs, where
two extra MZMs are ancillary ones to realize the Majorana braiding protocol
Clarke  _et al._ (2011)) located at the wire ends. These MZMs are described by
Majorana fermion operators $\gamma_{j}=\gamma_{j}^{\dagger}$, where
$j=1,2,3,4$. Due to the strong charging energy on the island, the fermion
parity of the island is a conserved quantity. Supposing that the four MZMs are
spatially well separated and the parity is fixed as even or odd, the ground
state of the MBQ is two-fold degenerate and behaves as a spin-1/2 freedom.
Consequently, due to the non-Abelian exchange statics obeyed by MZMs Moore and
Read (1991); Nayak and Wilczek (1996), Pauli operators for the MBQ can be
represented by Majorana fermion operators Plugge  _et al._ (2017):
$\sigma_{x}=i\gamma_{1}\gamma_{2}$, $\sigma_{y}=i\gamma_{3}\gamma_{1}$, and
$\sigma_{z}=i\gamma_{2}\gamma_{3}$.

Figure 8: Scalable designs of topological quantum computation networks Karzig
_et al._ (2017) based on the protocol of Fig. 7(a). With the dot-MZM couplings
switched on, the Fermion parity of two MZMs and the joint parity of four MZMs
can be obtained from either the charge sensing or the differential capacitance
measurement of the quantum dot (not shown in this figure). (a) The one-sided
hexon architecture. (b) The two-sided hexon architecture. For both cases, each
qubit (in each dashed box) consists of six MZMs connected by a conventional SC
(gray).

In quantum computation, qubit readout represents a critical issue. Below, we
briefly review three major protocols of MBQ readout, while a few other readout
protocols were reviewed in Ref. Beenakker (2020). The first MBQ readout
protocol Karzig  _et al._ (2017), as shown in Fig. 7(a), includes a quantum
dot that tunnel-couples to two MZMs $\gamma_{1}$ and $\gamma_{2}$ of the MBQ
(with amplitudes $t_{1}$ and $t_{2}$, respectively), and also weakly couples
to a charge sensor. When the gate voltage $n_{g}$ of the dot (not shown) is
tuned to the regimes where the dot-MBQ system is near on-resonance, the
average charge on the dot shows different $n_{g}$-dependence, for states with
different parities [Fig. 7(b)]. Thus, the detection of the dot occupation by a
nearby charge sensor at different gate voltages enables the readout of the MBQ
parity. In addition, the qubit readout can also be realized in this setup by
measuring the dot differential capacitance Karzig  _et al._ (2017). The setup
of the second MBQ readout protocol Plugge  _et al._ (2017) is shown in Fig.
7(c), in which two quantum dots are tunnel coupled to their nearest MZMs
$\gamma_{1}$ and $\gamma_{2}$ in the MBQ. Two dots are also inter-coupled
through a reference arm $R$. Physically, electrons can tunnel back and forth
between the two dots either through the box or via the arm. As a result, the
average charge on dot 2 oscillates in time with a period that depends on the
MBQ parity, as shown in Fig. 7(d). A charge sensor near dot 2 can perform a
projective measurement on the instantaneous average charge of dot 2, to obtain
the MBQ parity. We mention that at some special moments, e.g., $t=\tau_{m}$,
where the extremes of the curves in Fig. 7(d) arrive, the qubit readout is
ideal. Alternatively, the readout in frequency can be achieved with a
resonator that capacitively couples to dot 2 Plugge  _et al._ (2017).
Interestingly, in a similar two-dot setup, the qubit information can be
inferred from the charge states of two dots through a Landau-Zener transition
Zhang and Liu (2021). In this case, the charge sensing readout is
nondemolition and robust against the low-frequency fluctuations Zhang and Liu
(2021). The third MBQ readout protocol Plugge  _et al._ (2017) can be
implemented in a dot-free setup under a tunable magnetic flux $\phi$, as shown
in Fig. 7(e). With a small bias voltage applied between the two terminals of
such an Aharonov–Bohm ring island, the interplay between electron cotunneling
and interference dominates the transport. As shown in Fig. 7(f), the measured
$\phi$-dependent conductance can clearly reveal the even and odd MBQ parity.

In practice, the potential weakness of each readout protocol above may pose
challenge for experimental implementation. For the third protocol, phase
coherence in the reference arm $R$ requires small bias voltage, such that
cotunneling conductances are small, thus this current-based readout schemes
would be limited by the time needed for data accumulation Plugge  _et al._
(2017). For the second protocol, this single-shot projective measurement of
the charge of the quantum dot is ideal when the occupation or inoccupation
probability vanishes at particular times, i.e., $\tau=\tau_{m}$ in Fig. 7(d),
thus it requires fine tuning and rapid manipulation of system parameters
Plugge  _et al._ (2017). Simply re-measure multiple times can not improve the
readout fidelity Plugge  _et al._ (2017). For the first protocol, the charge
state of the quantum dot is only weakly correlated with the state of the MBQ
thus the readout requires longer measurement time Steiner and von Oppen
(2020). In addition, one has to repeatedly couple and decouple the MBQ and
quantum dot, which is also prone to errors. However, a recent theoretical work
proposed that a projective readout of the MBQ can indeed be robustly
implemented even though the MBQ is coupled to the quantum dot during the
entire readout procedure Steiner and von Oppen (2020). Recently, the effects
of noise Khindanov  _et al._ (2021) and Markovian dynamics Munk  _et al._
(2020) on the measurement of this readout protocol have been studied. There
exist crucial remarks on the feasibility of the first readout protocol: Charge
measurement is very fast and accurate, and represents a well-understood
technique in the SM community and is compatible with large magnetic fields
Karzig  _et al._ (2017). Additionally, the inclusion of charge sensors in the
qubit plane does not prevent from scaling the system to a two-dimensional
array of qubits Karzig  _et al._ (2017).

With the first readout protocol introduced above, one can detect the parity of
a single qubit, and the joint parity of two neighbouring qubits with designed
scalable structures Karzig  _et al._ (2017). Several scalable designs with the
Majorana qubits are proposed in Refs. Karzig  _et al._ (2017); Vijay and Fu
(2016), including the one-sided and two-sided hexon structures (see Fig. 8),
and the linear and two-sided tetron structures (not shown). In the hexon
architectures, one computational qubit consists of four MZMs and two ancillary
ones. The latter two MZMs are included to assist Majorana braiding Clarke  _et
al._ (2011). Figures 8(a) and 8(b) present the one-sided hexon and the two-
sided hexon architectures Karzig  _et al._ (2017). If the couplings between a
quantum dot and two MZMs are switched on, the parity of the corresponding two
MZMs can be read out from the quantum dot measurement as shown in previous
paragraphs. The joint parity of four MZMs can be obtained using two quantum
dots, where the couplings between the dot and two neighboring MZMs are
switched on for both dots [see Figs. 8(a) and 8(b)]. References Karzig  _et
al._ (2017); Vijay and Fu (2016) also demonstrate that braiding operations can
be realized by applying a sequence of Majorana parity measurements. In theory,
all single-qubit and two-qubit Clifford operations are then realizable in
hexon architectures using two-MZM parity measurements and four-MZM joint
parity measurements, without the need of performing braidings in real space.

##  VI Summary and perspectives

To summarize, we have reviewed the recent progress on (i) the engineering and
detection of MZMs in SM-SC heterostructures, and (ii) the MBQ-based
topological quantum computation. Briefly, after an extensive effort in the
last decade, the theory on the MZM realization has been widely accepted;
several challenges however remain on the experimental side, especially due to
the extremely high sample requirements. Among the reviewed SM-SC
heterostructures, the most promising one is perhaps the partially covered SM-
SC nanowire, due to its simple realization and easy tunability. It also has a
closer connection to our ultimate purpose: the realization of topological
quantum computation, following the architectures shown in Fig. 8.

On the MZM detection, conclusive experimental evidence of MZMs in the SM-SC
heterostructures remains missing. The major difficulty in the MZM detection
comes from two reasons. The first reason is that the topological phase and MZM
are significantly influenced by the control capability of the electrostatic
potential Mikkelsen  _et al._ (2018); Woods  _et al._ (2018); Antipov  _et
al._ (2018) and the disorder effects Das Sarma and Pan (2021). In fact, a low
charged impurity density or a high tolerance to charged impurity is required
for a hybrid SM-SC device to enter the topological regime. This requirement
might be satisfied by choosing new SM and SC materials. For instance, Refs.
Cao  _et al._ (2022b); Jiang  _et al._ (2022); Schlatmann (2021) has
investigated the possibility to engineer MZMs in a hybrid PbTe-Pb nanowire. As
for the fabrication of the complex nanowire network shown in Fig. 8, it is
within reach of the selective area growth technique Krizek  _et al._ (2018);
Aseev  _et al._ (2018); Op het Veld  _et al._ (2020); Jiang  _et al._ (2022);
Geng  _et al._ (2022) that employs a pre-designed mask to define a network
with an arbitrary pattern. The second reason is that Majorana signatures, e.g.
the most well-known zero bias conductance in the standard tunneling
spectroscopy, are easily contaminated or buried by false positive signatures
due to ABSs or other disorder effects. Those false positive signatures in the
detection can be resolved by introducing a dissipative environment in the
tunneling Liu (2013); Liu  _et al._ (2022); Zhang  _et al._ (2022), where the
non-Majorana signals will be suppressed and the Majorana signals remain. In
addition, the combination of local and nonlocal measurements in the same
device, e.g. the recent Microsoft experiment Aghaee  _et al._ (2022), could
significantly increase the reliability of the experimental data. Finally, we
want to emphasize that any experiments, including those listed in Sec. III,
can greatly aid the development of this area, once data with better precision
can be achieved.

##  VII Acknowledgements

This work was supported by the National Natural Science Foundation of China
(Grants No. 12004040 and No. 11974198), the Innovation Program for Quantum
Sci- ence and Technology (Grant No. 2021ZD0302400), and Tsinghua University
Initiative Scientific Research Program.

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