Method of creating majorana bound states in a quantum-dot chain

Abstract

The invention is about a method of creating a pair of Majorana bound states in a quantum-dot chain of N alternating first-type and second-type quantum dots, each second-type quantum dot being configured to host a respective Andreev bound state, ABS, wherein N is an integer greater than or equal to two The method comprises: tuning the quantum-dot chain to a Majorana sweet-spot for creating the pair of Majorana bound states on two outer quantum dots of the quantum-dot chain, the quantum-dot chain being configured to implement a Kitaev chain, and controlling at least one of the respective ABS as a fermion in the implementing of the Kitaev chain by the quantum-dot chain. The invention can have applications in quantum-computing methods and / or quantum-computing devices / chips that are implementing topological quantum computing based on Majorana bound states.

Description

[0001]Method of creating Majorana bound states in a quantum-dot chain Field of the invention The invention relates to a method of creating Majorana bound states in a quantum-dot chain, which may be used in topological quantum computing. The invention further relates to a quantum-dot chain and an apparatus for creating Majorana bound states. Background art Alexei Kitaev proposed an idealized one-dimensional chain of fermions with the goal of creating topologically protected quantum states that may be used in quantum computing [Kitaev]. An example of Kitaev’s Hamiltonian HKN of the proposed idealized one-dimension chain of N sites---also idealized N-site Kitaev chain---is [Eqn.1] ^ = ∑^ ( ^ ^ KN ^^^ε^^^^^+ (^ ^^^^^^+ Δ ^^^^^^+ ℎ. ^. )) wherein ^ and denote fermionic annihilation and creation operators, respectively, parameter ε^relates to an on-site energy of site k, the parameter t relates to an amplitude for single-electron coupling, the parameter Δ relates to Andreev-type coupling between neighboring sites, and h.c. denotes “Hermitian conjugate”; for notational convenience, fN+1=0. Kitaev found among other things that when tuning the chain to = 0 and ^ = Δ, chain hosts two Majorana bound states at the two boundary sites k=1 and k=N of the chain. Such a parameter regime for creating two Majorana bound states at the boundary of a Kitaev chain is also called a Majorana sweet-spot. Finding a suitable physical system for realizing a Kitaev chain and a physical system that may be controllably tuned to its Majorana sweet-spot in order to create Majorana bound states is a challenging task. However, Majorana bound states are regarded as a promising candidate for implementing topological quantum computing [Nayak, et al], [Sarma, et al] and may thus pave the way to realize and enable stable quantum computing. Therefore, there is a need for a method of creating Majorana bound states and / or apparatus for creating Majorana bound states. Summary of the invention A general object of the invention is to find a method of creating Majorana bound states and / or an apparatus capable of creating Majorana bound states. Therefore, according to a first aspect of the invention, a method as defined in appended claim 1 is provided. According to a second aspect of the invention, a quantum-dot chain as defined in appended claim 10 is provided. According to a third aspect of the invention, an apparatus as defined in appended claim 12 is provided. The above-referenced aspects of the invention enable to efficiently create Majorana bound states. The invention can have applications in quantum-computing methods and / or quantum-computing devices / chips that are implementing topological quantum computing based on Majorana bound states. Further more detailed advantages are discussed further below in the detailed description. Brief description of the drawings Embodiments of the present disclosure will be described herein below with reference to the accompanying drawings. However, the embodiments of the present disclosure are not limited to the specific embodiments and should be construed as including all modifications, changes, equivalent devices and methods, and / or alternative embodiments of the present disclosure. The terms “have,” “may have,” “include,” and “may include” as used herein indicate the presence of corresponding features (for example, elements such as numerical values, functions, operations, or parts), and do not preclude the presence of additional features. The terms “A or B,” “at least one of A or / and B,” or “one or more of A or / and B” as used herein include all possible combinations of items enumerated with them. For example, “A or B,” “at least one of A and B,” or “at least one of A or B” means (1) including at least one A, (2) including at least one B, or (3) including both at least one A and at least one B. The terms such as “first” and “second” as used herein may modify various elements regardless of an order and / or importance of the corresponding elements, and do not limit the corresponding elements. These terms may be used for the purpose of distinguishing one element from another element. For example, a first element may be referred to as a second element without departing from the scope the present invention, and similarly, a second element may be referred to as a first element. It will be understood that, when an element (for example, a first element) is “(operatively or communicatively) coupled with / to” or “connected to” another element (for example, a second element), the element may be directly coupled with / to another element, and there may be an intervening element (for example, a third element) between the element and another element. To the contrary, it will be understood that, when an element (for example, a first element) is “directly coupled with / to” or “directly connected to” another element (for example, a second element), there is no intervening element (for example, a third element) between the element and another element. The expression “configured to (or set to)” as used herein may be used interchangeably with “suitable for” “having the capacity to” “designed to” “adapted to” “made to,” or “capable of” according to a context. The term “configured to (set to)” does not necessarily mean “specifically designed to” in a hardware level. Instead, the expression “apparatus configured to…” may mean that the apparatus is “capable of…” along with other devices or parts in a certain context. The terms used in describing the various embodiments of the present disclosure are for the purpose of describing particular embodiments and are not intended to limit the present disclosure. As used herein, the singular forms are intended to include the plural forms as well, unless the context clearly indicates otherwise. All of the terms used herein including technical or scientific terms have the same meanings as those generally understood by an ordinary skilled person in the related art unless they are defined otherwise. The terms defined in a generally used dictionary should be interpreted as having the same or similar meanings as the contextual meanings of the relevant technology and should not be interpreted as having ideal or exaggerated meanings unless they are clearly defined herein. According to circumstances, even the terms defined in this disclosure should not be interpreted as excluding the embodiments of the present disclosure. The person skilled in the art will understand that the features described above and / or below may be combined in any way deemed useful. The drawings of the present disclosure show examples / embodiments of the invention, which will be described in detail hereinafter. It is to be understood that one or more of elements / components shown and / or described in one or more of these examples / embodiments and not in others may be used in those others too unless mechanical or other limitations prevent such an implementation. Moreover, describing features of different examples / embodiments in a single passage does not automatically mean that those features are inextricably linked. They may be applied separately from one another. The below description disclosing the present invention contains among other things concepts devised by the inventors to interpret and understand prior art. The present invention is discussed in more detail below, with reference to the attached drawings, in which: Fig.1a shows a known quantum-dot chain for implementing a 2-site Kitaev chain. Fig. 1b shows a further known quantum-dot chain for implementing a 3-site Kitaev chain. Figs. 2a and 2b show a minimal quantum-dot chain for implementing a 2-site Kitaev chain. Fig.3a shows a relationship involving coupling strengths and a Zeeman field. Fig. 3b shows a relationship involving a Zeeman field, spin-orbit strength and energy gap. Fig.3c shows a charge-stability diagram including a Majorana sweet-spot. Fig.3d shows Majorana wavefunctions and a relationship involving conductance, bias voltage and detuning. Fig. 4 shows a relationship between excitation energy and normal-state energy for different parameter values of the Zeeman field. Figs.5a and 5b show a relationship between effective couplings in dependence on an angle of the Zeeman field. Fig. 6a shows a quantum-dot chain with two first-type quantum dots surrounding one second-type quantum dot hosting an Andreev bound state and for implementing a 3-site Kitaev chain. Fig. 6b shows Majorana wavefunctions for the quantum-dot chain of Fig. 6a tuned to a Majorana sweet-spot. Fig. 6c shows a detuning spectrum of the quantum-dot chain of Fig. 6a including a comparison to an effective 3-site Kitaev chain with next-nearest neighbor coupling. Fig. 7a shows a quantum-dot chain with one first-type quantum dot centred in between two second-type quantum dots each hosting an Andreev bound state. Fig. 7b shows Majorana wavefunctions for the quantum-dot chain of Fig. 7a tuned to a Majorana sweet-spot. Fig. 7c shows a detuning spectrum of the quantum-dot chain of Fig. 7a including a comparison to an effective 3-site Kitaev chain. Fig.8 shows schematics of processes leading to next-nearest-neighbor couplings. Figs. 9a and 9b show quantum-dot chains with alternating first-type and second-type quantum dots. Fig.10 shows energy spectra of quantum-dot chains of different lengths. Fig.11 shows a method of the present invention in a flow-chart. Fig.12 shows an apparatus configured to implement the method. Figs.13a-13d show measurement results. Detailed description Majorana bound states (MBS) are quantum states that can appear in systems involving superconductors, and are defined as states that are particle-hole symmetric. They are predicted to exhibit non-Abelian anyon behavior and can be used to implement protected quantum bits. A physical system with a chain of N sites whose dynamics is mainly governed by a Kitaev-type Hamiltonian such as HKN is hereinafter also referred to as an N-site Kitaev chain. The dynamics of a real physical system typically comprises further effects in addition to those modelled by HKN; one may also say the dynamics corresponds to an effective Kitaev chain comprising the dynamics of the idealized Kitaev chain. For simplicity, a Kitaev chain is referred to hereinafter. Since Kitaev’s proposal of an idealized Kitaev chain, many efforts were made to realize an N-site Kitaev chain (^ ≥ 2) and to tune the chain to its Majorana sweet-spot in order to create Majorana bound states. Proposals that were made include, among others, 1D topological superconductivity involving semiconductor-superconductor nanowires, as well as planar Josephson junctions and magnetic adatom chain on top of superconductivity. Very recently, quantum-dot chains were manufactured that can realize the dynamics of Kitaev chains and that host a pair of Majorana bound states (MBSs) when being tuned to the Majorana sweet-spot [Dvir et al; Bordin et al, 2]. Fig. 1a illustrates an example of such a quantum-dot chain (1) for implementing a 2- site Kitaev chain. The quantum-dot chain (1) of this example comprises a semiconductor (40), quantum dots (10, 20, 11), a superconductor (202), tunnelling-control gates (30, 31, 32, 33), plunger control gates (101, 201, 111). The semiconductor (40) for hosting the quantum dots (10, 20, 11) may be based on for example a nanowire, a nanowire network (grown for example by selective area growth), two-dimensional electron gas, 2DEG, or two-dimensional hole-gas, 2DHG. The quantum dots (10, 20, 11) may refer to regions in the semiconductor (40) defined by confining potentials via the control gates and eventually further involving depletion gates (not shown in Fig.1a). The quantum-dot chain (1) in this example is based on the semiconductor (40) that hosts the three quantum dots (10, 20, 11). The quantum dots (10, 20, 11) may be defined and controlled by the tunnelling-control gates (30, 31, 32, 33) and the potential-control gates (101, 201, 111) (also: plunger gates). Such control gates are also referred to as electrostatic gates or finger gates, as they may be implemented as electrostatic control gates on which voltages can be applied. The potential-control gates (101, 201, 111) may be configured to provide control of chemical potentials / depths of potential wells in the respective quantum dots. The tunnelling-control gates (31, 32) may be configured to control tunnelling-amplitudes related to single-electron tunnelling between nearest-neighbor quantum dots of the respective quantum dots (10, 20, 11). The tunnelling-control gates (30, 33) may be configured to control tunnelling to leads / lead elements (not illustrated in Fig. 1a) for example for performing measurements. The quantum-dot chain (1) may comprise further elements such as leads (not shown in Fig.1) and layers in the semiconductor (also not shown in Fig.1) that may be used for performing measurements on the quantum-dot chain (1). However, such details regarding manufacturing are not outlined here, but are known, and may be found for example in [Bordin et al, 1; Bordin et al, 2; Dvir et al]. In Fig. 1a, the center quantum dot (20) is proximitized by the superconductor (202). The arrangement of the center quantum dot (20) involving the proximitized superconductor (202) distinguishes the center quantum dot (20) from its neighboring quantum dots (10, 11). The center quantum dot (20) in this example may also be considered to correspond to a semiconductor-superconductor hybrid segment (20, 202). The semiconductor-superconductor hybrid segment (20, 202) can host one or more Andreev bound states, ABSs (ψABS), for example in the center quantum dot (20). An ABS is an electron-hole superposition state and / or a quasi-particle state that can be formed in the center quantum dot (20). In this example quantum-dot chain, an ABS can be used to couple the two boundary quantum dots (10, 11). In particular, the ABSs formed in the center quantum dot (20) can be used as a mediator for realizing couplings needed to implement a 2- site Kitaev chain. The ABS (ψABS) may mediate a single-electron tunnelling process also referred to as elastic co-tunnelling, ECT. In ECT, a single electron may transition from one quantum dot (e.g., 10) via the center quantum dot (20) to the other quantum dot (e.g., 11). The ABS (ψABS) may further mediate a Cooper-pair coupling / splitting process referred to as Crossed-Andreev reflection, CAR. In CAR, two electrons in the superconductor (202) may form a Cooper pair in the semiconductor-superconductor hybrid segment (20, 202) (or: ABS region (20, 202)) and may separately tunnel into the neighboring boundary quantum dots (10, 11), respectively. Similarly, two electrons may tunnel from the neighboring boundary quantum dots (10, 11) into the superconductor (202) to form a Cooper pair. The ABS formed in the center quantum dot (20) may mediate ECT and CAR processes. Thereby, the quantum-dot chain (1) of three quantum dots (10, 20, 11) may realize a 2-site Kitaev chain model HK2 of Eqn. (1) with N=2. The quantum-dot (10) corresponds to a first site k=1 and the quantum-dot (11) corresponds to a second site k-2. In the realization, an ECT process can realize the tunnelling term with amplitude t in HK2, the term wherein an electron is transitioning between site 1 (quantum-dot 10) and site 2 (quantum-dot 11). A CAR process may realize the superconducting coupling term with amplitude Δ in HK2, the term Δ(^^^^+ ^ ^ ^ ^ ^ ^ ), wherein two electrons are paired in a Cooper-pair type process in the semiconductor-superconductor hybrid segment (20, 202) and are moving into the neighboring quantum dots (10, 11) (∼ Δ ^ ^ ^ ^ ^ ^ ) or two electrons are moving out of the neighboring quantum dots (10, 11) and into the semiconductor-superconductor hybrid segment (20, 202) (∼ Δ ^^^^). When being tuned to its sweet-spot via the control gates (101, 201, 111, 31, 32), the quantum-dot chain (1) exhibits a pair of Majorana bound states, at its boundary quantum dots (10, 11). A first Majorana bound state (MBS1) may be formed in one outer quantum dot (10), and a second Majorana bound state (MBS2) may be formed in the other outer quantum dot (11). MBSs may be understood as a special kind of Andreev bound states, i.e. bound quantum states under the involvement of a superconductor, that exhibit particle-hole symmetry. They are expected to be non-Abelian anyons and are expected to be used in fault- tolerant quantum computing. The outer quantum dots (10, 11) may be considered as examples of normal quantum dots. The normal quantum dots (10, 11) can be considered distinguished in their role and physical behaviour from the center quantum dot (20), because the center quantum dot (20) is configured to host at least one ABS by means of the proximitized superconductor (202). The center quantum dot (20) is hereinafter also called a ABS quantum dot and / or a superconducting quantum dot. In general, a normal quantum dot is hereinafter also referred to as a D-dot. An ABS quantum dot is hereinafter also referred to as an A-dot. The quantum-dot chain (1) shown in Fig. 1a is an example of an D-A-D quantum-dot chain, wherein a first (10) and second (11) D-dot correspond to two sites of a 2-site Kitaev chain. The 2-site Kitaev chain is implemented by the D-A-D quantum-dot chain (1), wherein the A-dot is configured to host an ABS for mediating the ECT and CAR processes realizing the coupling terms in the 2-site Kitaev chain. Fig. 1b shows another known example of a recently manufactured quantum-dot chain (1) comprising 5 quantum dots for implementing a 3-site Kiteav chain HK3 [Bordin et al, 2]. The quantum-dot chain (1) in this example includes 3 normal quantum dots (10, 11, 12) and 2 ABS quantum dots (20, 21), the ABS quantum dots (20, 21) corresponding to semiconductor-superconductor hybrid segments (20, 21; 202, 212) that are each configured to host a respective ABS. The ABSs can mediate respective ECT and CAR couplings for realizing the 3-site Kitaev chain, in which the 3 normal quantum dots (10, 11, 12) correspond to 3 sites k=1, k=2, k=3 of the 3-site Kitaev chain and the 2 ABS quantum dots (20, 21) realize the tunnelling and Andreev-type coupling terms between neighboring normal quantum dots (10, 11, 12). When tuning the quantum-dot chain (1) of Fig. 1b to its Majorana sweet spot, a pair of Majorana bound states (MBS1, MBS2) is formed in the two outer quantum dots (10, 12) of the quantum-dot chain (1). The inventors have identified a new method of creating a pair of Majorana bound states in a quantum-dot chain of N alternating first-type and second-type quantum dots, each second-type quantum dot being configured to host a respective Andreev bound state, ABS, wherein N is an integer greater than or equal to two. The method comprises: tuning the quantum-dot chain to a Majorana sweet-spot for creating the pair of Majorana bound states on two outer quantum dots of the quantum-dot chain, the quantum-dot chain being configured to implement a Kitaev chain, and controlling at least one of the respective ABS as a fermion in the implementing of the Kitaev chain by the quantum-dot chain. By controlled the at least one ABS as a fermion in implementing a Kitaev chain by the quantum-dot chain, the second-type quantum dots hosting ABSs can be used on equal footing as the first-type quantum dots. Thus, the inventors have found that it is possible to control the ABSs as taking the role of a fermion itself in implementing a Kitaev chain when tuning the quantum-dot chain of N alternating first-type and second-type quantum dots to its Majorana sweet-spot. The inventors have identified that semiconductor-superconductor hybrid segments for hosting ABSs can be controlled as quantum dots participating in the Kitaev chain themselves, instead of merely being controlled as interaction mediators. In other words, at least one ABS can be controlled so as to have a different functional role. The controlling of the at least one ABS so as to function as a fermion in implementing a Kitaev chain may include controlling parameters such as direction and strength of an applied magnetic field, on-site potentials, and / or tunnelling amplitudes. Details and example are discussed further below. In the above-recited examples with reference to Figs. 1a and 1b, the ABSs hosted in semiconductor-superconductor hybrid segments (20, 21) were controlled / used merely as mediators for ECT and CAR processes, i.e. merely mediate the desired couplings / interactions between quantum dots (10, 11, 12) in a quantum-dot chain (1). Thereby, an N-site Kitaev chain HKN may be realized by a quantum-dot chain (1) of length 2N-1 and having alternating first-type (10, 11, 12) and second-type (20, 21) quantum dots, wherein the second-type quantum dots (20, 21) are configured to host a respective ABS used as mediators for ECT and CAR couplings. In the above examples of Figs. 1a and 1b, a second-type quantum dot (20, 21) is implemented as a semiconductor-superconductor hybrid segment for hosting ABSs. However, according to the new method, an ABS is controlled as a fermion itself in a Kiteav chain. The ABS can be regarded for example as implementing a fermionic operator fi in HKN. Thereby, an ABS quantum dot participates itself as a fermionic site in implementing the Kitaev chain. In other words, a semiconductor-superconductor hybrid segment can be controlled in a way so as to effectively function as a quantum dot in the Kitaev chain as well, instead of merely mediating an interaction between neighbouring quantum dots. By controlling the ABS as a fermion in implementing a Kitaev chain, the effective length of the implemented Kitaev chain is increased as compared to controlling the ABS merely as a mediator. This aspect is discussed next with reference to Figs.1a and 1b. For example, in Fig.1a, when controlling the ABS (^^^^) formed in the A-dot (20) as a mediator for ECT and CAR couplings, the D-A-D quantum-dot chain (1) of Fig. 1a implements a 2-site Kitaev chain HK2with the D-dots (10, 11) corresponding to the fermionic sites k=1 and k=2, respectively, and the A-dot (20) hosting an ABS being used as a mediator. However, when using the new method the inventors have devised, the D-A-D quantum-dot chain (1) of Fig.1a can now implement a 3-site Kitaev chain in which a first D- dot (10) corresponds to a first fermion site k=1 with f1, the A-dot (20) corresponds to a second fermion site k=2 with f2, and a second D-dot (11) corresponds to a third fermion site k=3 with f3, of the 3-site Kitaev chain, respectively. That is, the ABS hosted in the A-dot (20) may be regarded as corresponding to a fermion state f2in the 3-site Kitaev chain. When thus tuning the 3-site quantum-dot chain (1) to its Majorana sweet-spot and controlling / using the ABS as a fermion in the 3-site Kitaev chain instead of as merely a mediator, the effective length of the implemented Kitaev chain increases from 2 to 3. In general, a longer Kitaev chain can provide topologically better protected Majorana bound states than a shorter Kitaev chain. For example, for longer chains, one may find a relatively wider range of parameter values supporting MBS than compared to shorter chains. Therefore, when using the new method to create Majorana bound states, the created Majorana bound states are better protected and hence may be more useful when implemented in topological quantum computing. As another example, in Fig. 1b, when controlling / using the ABSs formed in the second-type quantum dots (20, 21) as mediators for ECT and CAR couplings, the D-A-D-A- D quantum-dot chain (1) of Fig.1b implements a 3-site Kitaev chain HK3 with the D-dots (10, 11, 12) corresponding to the fermion sites k=1, k=2, k=3, respectively. However, when using the new method, the ABSs hosted in the A-dots / second-type quantum dots (20, 21) are controlled as fermions in the 5-site Kitaev chain, and thereby the 5- site quantum-dot chain (1) of alternating first-type (10, 11, 12) and second-type (20, 21) quantum dots implements a 5-site Kitaev chain instead of merely a 3-site Kitaev chain. The Majorana bound states MBS1 and MBS2 created in the outer quantum dots (10, 12) by using the new method are thus better topologically protected. The new method may thus provide one or more of the following technical advantages. An N-site Kitaev chain can be implemented by a quantum-dot chain of only N alternating first-type and second-type quantum dots, instead of 2N-1 quantum dots. Therefore, less quantum dots are needed for implementing an N-site Kitaev chain and the complexity and footprint in manufacturing and controlling quantum-dot chains for implementing an N-site Kitaev chain is reduced. For example, with the invention, a 2-site Kitaev chain can be implemented using only a two-site quantum-dot chain of one D-dot and one A-dot. This example is discussed further below for illustration purposes in more detail with reference to Figs.2-5. Furthermore, existing 2N-1 site quantum-dot chains can now be controlled to realize a 2N-1 Kitaev chain instead of only an N-site Kitaev chain. Thereby, the method enables the creation of Majorana bound states that are better protected than before due to the effective length of the implemented Kitaev chain then being 2N-1 instead of N. Furthermore, controlling an 2N-1 site quantum-dot chain of alternating (D-A-D-A- D…) quantum dots may be an involved task due to the control of corresponding 2N-1 on-site control gates (potential-control gates) and corresponding tunnelling-coupling gates. For example, when tuning to the Majorana sweet spot, all involved gates should be suitably controlled. The invention simplifies control of a quantum-dot chain for implementing an N- site Kitaev chain. Furthermore, when manufacturing a quantum-dot chain for implementing an N-site Kiteav chain, the quantum-dot chain according to known methods would require 2N-1 quantum dots together with respective potential-control gates and tunnelling-control gates. With the new method, the manufacturing of a quantum-dot chain for implementing an N-site Kitaev chain is simplified, because only N quantum dots are needed. Therefore, when scaling up, for example for building a computer chip for implementing topological quantum computing, the method may provide useful due to its reduced footprint. For example, when scaling up a prior-art device to add one more site in a Kitaev chain, one would need to add both a semiconductor-superconductor hybrid segment as well as another normal quantum dot, making the device fabrication a more challenging task. With the invention, scaling up is simplified. Furthermore, longer Kitaev chains can provide more stable / topologically better protected Majorana bound states as compared to shorter Kitaev chains. Therefore, the invention enables to more efficiently create Majorana bound states that show better protection. Furthermore, the inventors found that an excitation energy gap can be enhanced when using the new method. An enhanced excitation gap leads to a stronger resilience when the MBSs are used for quantum computation. Details and preferred aspects of the invention are discussed next with reference to examples which are shown for illustration purposes but not to limit the invention. In particular, the examples illustrate ways in which an ABS may be controlled to participate as a fermion in implementing a Kitaev chain. A model Hamiltonian useful for understanding the invention is discussed first in the context of a 2-dot system. Model Hamiltonian Figs. 2a and 2b show an example of the invention for N=2 in which a quantum-dot chain (1) of one first-type quantum dot (10) and one second-type quantum dot (20) is controlled. An order of the quantum dots (10, 20) is not essential but may be swapped, as is illustrated in Fig. 2b. The illustrated elements including control gates (30, 31, 32, 101, 201), semiconductor (40), and superconductor (202) are as described with reference to Figs.1a and 1b and are not repeated here. Furthermore, a magnetic-field generator may be provided. The magnetic-field generator may be configured to generate magnetic fields acting on the quantum dots (10, 20). The quantum-dot chain (1) may comprise further elements such as leads and layers in the semiconductor that are not shown in Figs. 2a and 2b but that may be used for performing measurements on the quantum-dot chain (1). Such details regarding manufacturing and measuring are not outlined here, but are known, and may be found for example in [Bordin et al, 1; Bordin et al, 2; Dvir et al]. The second-type quantum dot (20) is configured to host an ABS (^ABS). In this example, the second-type quantum dot (20) is proximitized by a superconductor (202) so as to realize a semiconductor-superconductor hybrid segment that can host the ABS. The pair of first-type and second-type quantum dots is hereinafter also referred to as a dot-ABS pair or D-A pair, wherein the “dot” or “D” refers to the first-type quantum dot (10) and the “ABS” or “A” refers to the second-type quantum dot (20) that corresponds to the semiconductor-superconductor hybrid quantum dot capable of hosting ABSs. The first-type quantum dot (10) is also referred to as a D-dot (10), and the second-type quantum dot (20) as an A-dot (20), respectively. In the following exemplary considerations, a global magnetic field is applied to the quantum dots (10, 20) so that a spin polarization axis of the quantum dots (10, 20) may coincide and Zeeman spin splittings EZD and EZA are induced in the D-dot (10) and A-dot (20), respectively. A model Hamiltonian HDA for the D-A pair (10, 20) may be formulated as HDA= HD+ HA+ HT, wherein HDis a Hamiltonian for the D-dot (10) with a single spinful orbital, HAis a Hamiltonian of the A-dot (20), and HT is a tunnel Hamiltonian between the D-dot (10) and the A-dot (20). The Hamiltonians HD, HA, and HT may be described as follows: HD= (ε^+ EZD) ^^↑+ (ε^- EZD) ^^↓+ UD^^↑^^↓, In HD, the operator ^^^= ^^ ^^^is related to an occupancy number of a spin-polarized dot orbital with spin σ, the parameter ε^is related to a normal-state energy and / or chemical potential of the D-dot (10), the parameter EZD is a strength of the Zeeman spin splitting induced in the D-dot (10) by the magnetic field (also: Zeeman energy EZD), and the parameter UDdenotes a Coulomb energy in the D-dot (10). In HA, the parameter EA= is an excitation energy with a parameter related to a normal-state energy or chemical potential of the A-dot (20), the parameter Δ^corresponds to an induced pairing gap, and the parameter EZA is a strength of the induced Zeeman spin splitting in the A-dot (20) (also: Zeeman energy or Zeeman field EZA). In HA, the operator γ ^ ^^= σ^^^+ ^^^is a Bogoliubov operator of the ABS with spin σ = ±1 corresponding to spin up and spin down, respectively, wherein ↑=↓ and ↓=↑. The operators ^^and ^ ^ ^ correspond to fermionic annihilation and creation operators, for example related to describing an electron entering or leaving the A-dot (20). The parameters u and v are the corresponding Bardeen-Cooper-Schrieffer, BCS, coherence factors related by ^^+ ^^= 1, wherein u is given by ^^=(1 / 2)(1 + ε^ / ^^). The Bogoliubov operators are related to quasi-particles created by linear combinations with electron components (u) and hole components (v). That is, a quasi-particle excitation is a coherent superposition of a particle / electron state (u) and a hole state (v). Further background on Bogoliubov operators and quasi-particle superposition states may be found in [Altland, Simons, e.g. Section 6.4]. The Hamiltonian HA models only one pair of sub-gap ABSs while above-gap continuum states are neglected, which may be justified by focusing on a low-energy regime. In HT, the parameter t0is the tunnelling amplitude relating to tunnelling between the quantum dots. The tunnelling amplitude t0 may also be referred to as a coupling strength, tunnelling coupling and / or tunnelling strength between the D-dot (10) and the A-dot (20). The matrix Uso (“so” for spin-orbit) is a unitary matrix given by , wherein diagonal elements describe spin-conserving tunnelling processes and off-diagonal elements describe spin-flipping tunnelling processes. For example, if α = 0, only spin- conserving tunnelling processes take place, while if α = π / 2, only spin-flipping tunnelling processes take place. The parameter α thus characterizes an amount of spin rotation accumulated in a tunnel process, due to spin-orbit interactions. The parameter σ^= ^^^(θ)σ^+ ^^^(θ)σ^corresponds to a spin-orbit field. The case θ = 0 corresponds to a spin-orbit field being perpendicular to the applied magnetic field. The parameter θ may characterize an angle between the spin-orbit field and the applied magnetic field. Without loss of generality, one may assume that the direction of the applied magnetic field is fixed along σ^and the spin-orbit field σ^can be rotated. The induced pairing gap Δ^may be measured, e.g. by varying a conductance G of a lead element (not illustrated in Fig. 2a / 2b) in a spinful regime with no Zeeman field applied, as shown in [Bordin et al, 1, Fig. 1e therein]. Similarly, the spin-orbit interaction strength α may be measured / determined, for example using a method as in [Wang et al.]. Its value is typically material dependent. For example, in Indium antimonide (InSb) nanowires, values in a range of about 0.2-0.4 have been found. With the invention, the quantum-dot chain (1) can be tuned to its Majorana sweet spot to create a pair of Majorana bound states (MBS1, MBS2), wherein the tuning comprises controlling an ABS in the A-dot (20) to participate as a fermion in implementing a Kitaev chain. This is discussed next. Low-energy effective theory and sweet-spot conditions In a low-energy regime and weak tunnelling and strong Zeeman spin splitting regime (t0 << EZD, EZA), high-energy spin states may be neglected and one may consider keeping only one spin-polarized state near the Fermi energy. The low-energy regime may be achieved in operation for example by cooling down the quantum-dot chain (1) to a milli-Kelvin range. Without loss of generality, one may restrict the discussion to spin-down states as the low-energy degrees of freedom on both quantum dots of the A-D pair (10, 20). This corresponds to the D-dot being near a degeneracy of being empty or singly occupied (possibly on a background of filled energy levels), and the A-dot being near a degeneracy of a singlet and doublet state. The A-D pair may then be modelled by an effective Hamiltonian given by wherein the spin-up orbitals in the D-dot (10) and γ^↑in the A-dot (20) are projected away, which is justified by the low-energy regime. The parameter t corresponds to a tunnel amplitude for spin-conserving processes and the parameter tso corresponds to a tunnel amplitude for spin-flipping processes, respectively, defined as Interestingly, since the ABS / Bogoliubov operator γ ^ ^ ^↓ = −^^↓ + ^^↑is a coherent superposition of both electron (u) and hole (v) components, single electron tunnelling from the D-dot (10) to the A-dot (20) simultaneously creates and annihilates an ABS excitation, giving both normal and Andreev-like effective couplings [Eqn.2] ^eff= −^^, Δeff= −^so^ . Thereby, the effective Hamiltonian is identical to a two-site Kitaev chain defined by the Hamiltonian HK2 as In the two-site Kitaev Hamiltonian HK2, the operators ^^ are annihilation and creation operators of a spinless fermion on site k, respectively, with k=1, k=2, the parameter ε^is related to an on-site energy of site k, the parameters t12 and Δ^^relate normal and Andreev-like tunnelling between adjacent sites, respectively. The h.c. denotes the Hermitian conjugate. The correspondence between parameters to identify and HK2is: The operator of the D-dot (10) takes the role of the spinless fermion operator f1.The Bogoliubov operator γ^↓corresponding to an ABS of the A-dot (20) takes the role of f2. The Majorana sweet-spot for the 2-site Kitaev chain corresponds to a parameter regime in which = ε^and |^^^| = |Δ^^|. That leads to the following three Majorana sweet- spot conditions for the present example: [Eqn.3]: ε^− ^^^= 0 [Eqn.4]:^ε^ ^ + Δ^ ^ − ^^^= 0 [Eqn.5]: |^^| = |^^^^| Eqn. 3 may mean that the dot orbital energy ε^− ^^^is zero. Eqn.4 may mean that the ABS energy EA-EZAvanishes and is adjusted to the Fermi energy of the superconductor (202). Eqn. 5 may mean that magnitudes of normal and Andreev-like couplings / tunnelings are fine-tuned into balance. Thereby, the inventors have identified that the ABS γ^↓may be controlled as a fermion in implementing a Kitaev chain. In this example, the ABS γ^↓is a spinless fermion in the sense that a magnetic field / Zeeman field is applied to the A-dot (20) to cause a Zeeman spin splitting, so that one ABS spin-state (γ^↑) may be projected away in a low-energy approximation while one ABS spin-state (w.l.o.g., spin-down) remains. In general, preferably, at least one ABS is controlled as a spinless fermion in the implementing of the Kitaev chain. That may correspond to controlling an ABS of a second- type quantum dot (20) by applying a magnetic field / Zeeman field. The Zeeman field may be sufficiently strong so as to reduce the effective dynamics to one spin type, thereby resulting in an ABS participating as a spinless fermion in the implemented Kitaev chain. A spinless state generally refers to a situation in which spin splitting is present due to a Zeeman field. For example, by applying a magnetic field to the quantum dots, a spin splitting may be observed in which spin-up and spin-down energy states are separated by a Zeeman splitting (EZ). In this case, a spin-up state may have an energy different than a spin-down state, for example a larger energy. A particle occupying an energy state that is energetically separated due to an applied Zeeman field may be referred to as a spinless particle state or spinless state. The Zeeman field thus causes a non-degeneracy / energy-separation of spin-up and spin-down. In contrast, a spinful state generally refers to a situation in which no Zeeman field is applied. For example, when no magnetic field is applied to the quantum dots, a spin degeneracy may be observed. An energy state in a quantum dot may be occupied by both spin-up and spin-down electrons / particles. In this case, both spin-up and spin-down states may have a same energy. For example, the induced pairing gap Δ^may be measured in a spinful state regime. One or more of the following parameters may be controlled (varied / tuned) to tune the quantum-dot chain (1) to its Majorana sweet spot and / or to control one or more ABSs: - the parameter t0 related to a tunnelling amplitude, by controlling a tunnel-gate voltage applied to the tunnelling-control gate (31); - the parameter ε^related to a normal-state energy and / or chemical potential in the D-dot (10), by controlling a voltage applied to the potential-control gate (101); - the parameter related to a normal-state energy and / or chemical potential in the A-dot (20), by controlling a voltage applied to the potential-control gate (201); - the parameters EZA and EZD related to Zeeman splitting strengths in the respective D-dot (10) and A-dot (20), by controlling a strength of the applied magnetic field / Zeeman field via a magnetic-field generator configured to generate magnetic fields; - the parameter θ related to an angle between the spin-orbit field and the applied magnetic field, by controlling the direction of the applied magnetic field via the magnetic-field generator. These parameters may be regarded as tuning knobs for tuning the quantum-dot chain (1) to its Majorana sweet spot. Next, examples are discussed of tuning the quantum-dot chain (1) to its Majorana sweet spot while controlling an ABS as a fermion in implementing a Kitaev chain by the quantum-dot chain (1). Tuning strength of Zeeman field As an example of a first tuning knob, one may search for the Majorana sweet-spot by primarily changing the strength of the Zeeman fields EZA and EZD by varying a strength of the applied magnetic field. A direction of the magnetic field may be fixed, for example to be perpendicular to the spin-orbit field (θ = 0). Among the three sweet-spot conditions Eqns. 3-5 introduced in the previous subsection, Eqn. 3 for the zero energy of the dot orbital can be readily satisfied because changes in the value of EZD can always be compensated by thereafter adjusting the dot chemical potential EDby controlling the dot control gate (101). By contrast, Eqns.4 and 5 are more subtle and mutually constrained. Specifically, by applying a sufficiently large Zeeman field, a zero-energy ABS (EA-EZA=0) can be obtained with Here a negative normal-state energy solution (ε^< 0) is chosen to have a hole- dominant ABS (v>u; i.e., an ABS is hole-dominated if the Bogoliubov component v is dominant over the Bogoliubov component u, that is if v>u.), such that a balance between normal and Andreev tunnelings indicated in Eqn. 5 is likely to reach in the weak spin-orbit interaction regime (tso < t). Now the normal and Andreev tunneling amplitudes defined in Eqn.2 become These relations indicate that by increasing the strength of the Zeeman field, one changes the electron and hole components of the zero-energy ABS, i.e., increasing v from 1 / √2 to close to 1 while decreasing u from 1 / √2 to nearly zero. Fig. 3a shows that in the large Zeeman limit, the Andreev tunnelling|Δeff|would increase from ^^^^^(α) / √2 to about ^^^^^(α)while the normal tunnelling|^eff|would decrease from ^^^^^(α) / √2 to zero. The solid lines in Fig. 3a and later figures relate to the analytical approximations as derived herein, and the dashed line with solid dots relates to a comparison to a full numerical model. The solid and dashes lines corresponding is a further confirmation of the validity of the analytical approximations. In the weak spin-orbit regime with α < π / 4, the two coupling strengths will be equal to each other at the following particular value of field strength: That is, the dot-ABS pair (10, 20) reaches its sweet-spot and hosts a pair of Majorana bound states (MBS1, MBS2) located on the D-dot (10) and the A-dot (20), respectively. One can tune the magnetic field / Zeeman field to induce a sufficiently strong Zeeman spin splitting in the D-dot (10) and the A-dot (20), thereby allowing to obtain a zero-ABS energy state with a hole dominant ABS (v>u). Eqn. 6 is an example showcasing that the Majorana sweet-spot conditions Eqns. 3-5 can be satisfied by tuning the strength of the Zeeman field, thereby varying the parameters EZAand EZD, and by further tuning the parameters ε^and ε^related to the D-dot (10) and A- dot (20) orbital / normal-state energies, respectively. Furthermore, an excitation energy gap Egap for exciting an ABS is twice the effective coupling strength. At the sweet-spot, the excitation gap is The excitation gap refers to the energy difference between the MBSs and a first excited state. A larger gap is expected to increase stability in MBS-based qubits. Fig. 3b shows an increase of the excitation gap with spin-orbit interaction strength α. The sweet-spot value ^∗ ZAof the Zeeman field strength decreases in this example with an increasing α. In particular, values of ^∗ ZAonly slightly larger than Δ^may constrain the possible detuning range of the ABS energy to be relatively narrow. Therefore, in choosing an optimal value of spin-orbit interaction α for the dot-ABS pair, there exists a trade-off between the size of Egap and the range of allowed ε^for the effective Kitaev model. Figs 3a-3c include numerical simulations of the D-A pair (10, 20) based on the Hamiltonian ^eff DA. The numerical simulations corroborate the above-described analytical results obtained from low-energy theory. The chosen values of t0 = 0.2 and α = 0.3 are an example of the weak tunnelling regime and weak spin-orbit interaction regime (α < π / 4). As shown in Fig. 3a, numerically extracted values of |^eff| and |Δeff| as a function of EZAare in excellent agreement with the analytical results. In Fig. 3b, the calculated ^∗ ZAand Egapalso match very well with the analytical predictions. Fig. 3c shows a charge-stability diagram involving an energy difference Δ^ between the odd- and even-parity ground states in the (ε^, ε^) plane. A charge-stability diagram may be obtained by measurements for different values of the parameters ε^, ε^. For example, the energy difference Δ^ being zero corresponds to observing conductance in the quantum-dot chain (1) at zero bias voltage. As illustrated in Fig. 3c, one can observe lines of parameters values along which conductance may be observed. The lines in Fig. 3c cross at two corners, both of which correspond to a possible Majorana sweet-spot. In particular, a Majorana sweet-spot appears in the right-bottom corner, in this example when ^ZA≈ 1.75 Δ^, which is consistent with an analytically predicted value of ^∗ ZA= 1.77Δ^. Here the right-bottom corner corresponds to a spin-down dot orbital and a hole-like ABS / hole-dominated ABS (v>u). This Majorana sweet-spot corresponding to a hole-dominated ABS is indicated by a cross in Fig.3c. The hole-dominated ABS corresponds to a parameter regime in which tso< t. In the strong spin-orbit regime (tso > t), a similar reasoning as shown above holds, but in this strong spin-orbit regime a sweet-spot can now be found for an electron-dominated ABS (u>v). Fig. 3d shows that indeed a pair of MBSs emerges at the Majorana sweet-spot, wherein one MBS, Majorana 1, is localized on the D-dot (10), while another MBS, Majorana 2, is localized on the A-dot (20). In Fig. 3d, the A↑ and A↓ indicate components of the fermion operators ^ that ^ ↑ / ↓occur in the ABS used as a fermion, = −^^↓ + ^^↑. As illustrated, the MBS on the A-dot (20) in this sweet-spot regime is hole dominated (v>u). Fig. 3e shows a calculated tunnel conductance spectroscopy in the(^bias, δε^)plane. The spectroscopy shows a stable zero-bias peak and a parabola-shaped gap peak, consistent with conductance features of MBSs in a two-site Kitaev chain. Fig. 4 shows the excitation energy EA-EZA of the ABS as a function of the normal- state energy ε^, for different strengths of the Zeeman field EZA. As shown in Fig.4, as soon as the Zeeman field EZA is sufficiently strong, a hole-dominated (v>u) or an electron-dominated (u>v) zero-energy ABS may be obtained by tuning ε^. The particular condition of EZA>Δ^for obtaining both cases is not essential but is specific to the modelling assumptions of this example. Similarly, the precise formula / relationship Eqn. 6 is merely an example showcasing that by varying the Zeeman field, a Majorana sweet spot may be found in which the ABS may be controlled to be used as a (e.g. spinless) fermion in implementing a Kitaev chain by the quantum-dot chain (1). For illustration, when one would additionally consider a charging energy U on the A- dot (20), one may obtain a different formula. For example, including the charging energy in the respective Hamiltonian description the A-dot (20) part in HDA, one may consider in the electron basis with c corresponding to the electron / fermion operator. In this case, one may find a sweet-spot value for the Zeeman field to be reduced by U / 2, i.e. However, the procedure of finding the Majorana sweet spot by controlling an ABS to participate as a fermion in implementing a Kitaev chain does not change. For example, one may just as outlined above control the ABS as the fermion by varying Zeeman fields EZA, EZD and the on-site potentials and ε^to a Majorana sweet-spot. The Majorana sweet-spot may correspond to a parameter regime with zero ABS energy and a hole-dominated ABS. Tuning direction of Zeeman field As a further example of another tuning knob, one may change the direction of the applied magnetic field for finding a Majorana sweet-spot. For illustration purposes, the following considerations are made under an assumption that the magnetic field strength is fixed. The assumption is not essential however, as both strength and direction may also be tuned together. Furthermore, also for illustration purposes, the field is assumed to be rotating in a plane where its direction can be parallel or perpendicular to the spin-orbit field. Physically, since the Zeeman and spin-orbit fields are the only two spinful fields, one can equivalently consider rotating the spin-orbit field direction while keeping the Zeeman field unchanged. Rotating the Zeeman field changes teff and Δeffby changing the spin- conserving and spin-flipping tunnelling amplitudes t and tso (see Eqn. 2), with u and v of the zero-energy ABS being fixed by the value of EZA(denoted as u(EZA) and v(EZA), respectively). One can then obtain One may focus on 0 ≤ θ ≤ π / 2, since the strength of the effective couplings are π- periodic and symmetric around θ = 0. Fig. 5a shows the|^eff|and|Δeff|curves as a function of the angle θ. When the magnetic field aligns with the spin-orbit field, i.e. when we have θ = π / 2, then|Δeff|(which is a triplet pairing term) would be suppressed completely due to spin conservation. Therefore, a sweet-spot of the dot-ABS pair may appear in the angle sweep when the Zeeman field strength is sufficiently large (EZA> ^∗ ZA), giving |Δeff| > |^eff| at θ = 0. Fig. 5a thus shows that reducing the angle θ between SOI and Zeeman field enables to recover a sweet-spot when slightly above the sweet-spot value. Thus, as illustrated above, by varying the direction of the applied magnetic field / Zeeman field, one may tune to a Majorana sweet-spot regime at which the ABS (γ^↓) is controlled as a fermion in implementing a Kitaev chain by the quantum-dot chain (1). The above-mentioned examples of varying the strength and direction of the applied magnetic field / Zeeman field and the on-site potentials / orbit energies and ε^are not the only tuning knobs for tuning to a Majorana sweet-spot. An example of tuning to the Majorana sweet-spot by tuning the tunnelling-amplitude t0 is discussed next. Tuning induced pairing gap in the quantum dot As a further example of yet another tuning knob, one may consider a superconducting pairing gap Δ^in the D-dot (10). In experiment, such a superconducting pairing gap Δ^may be induced by proximity effects from the ABS and / or continuum states in the adjacent A-dot (20). This superconducting pairing gap Δ^can be controlled (e.g. enhanced) by controlling the tunnel barrier height t0 / tunnelling amplitude t0 (e.g. lowering) via the tunnelling-control gate (31). While such a tuning process changes the tunnel amplitude t0, the sweet-spot condition Eqn.5 may still be valid in the weak coupling regime of t0< 2 EZA. The superconducting pairing gap Δ^may be accounted for in the model Hamiltonian by adding a pairing potential of the form Δ ^ ^ ^^↑^↓ + ℎ. ^. to the dot Hamiltonian HD in the Hamiltonian HDA. As such, the dot orbital in the D-dot (10) is now proximitized into an ABS itself with the electron and hole components being wherein is determined by the zero-energy condition for the dot ABS system. The approximations are made in the weak proximity effect regime, i.e., Δ^≪ ^^. Thereby, up to the leading order of Δ^ / ^^, uD = 1 becomes a constant and only ^^∼ Δ^grows linearly with Δ^. As a result, the effective couplings between the D-dot (10) and the A-dot (20) then become That is, teffdecreases with the magnitude of the induced pairing in the dot, while Δeffincreases with it. Fig. 5b shows the curves of |^eff| and |Δeff| as a function of Δ^, with a fixed Zeeman field strength and direction being perpendicular to the spin-orbit field. Therefore, when the applied Zeeman field strength is not large enough, we have |Δeff| < |^eff| at Δ^= 0, and enhancing the induced gap in the quantum dot may facilitate the search for the sweet-spot. Fig. 5b shows an increased pairing on the normal dot recovers a sweet-spot when δ^^^is below the sweet-spot value. In the discussion above, we have assumed that the D-dot is tuned close to a degeneracy of the dot states |↓^(single down spin) and |0^(empty level). If instead the D-dot is tuned close to the degeneracy |↑↓^(doubly occupied) and |↓^, the effective couplings instead become ^^^^= −^^^^; Δ^^^= ^^^. Also in this case, a sweet spot can be found, by tuning / controlling the ABS state appropriately. Table 1 summarizes conditions on t and tso to get a sweet spot for all possible combinations of dot configurations. Table 1: conditions on t and tsoin order to be able to tune into a sweet spot depending on the tuning of the different dots. D-dot |↑↓^ and |↓^ |↓^ and |0^ A-dot ABS with ^ > ^ ^ > ^^^^ < ^^^ABS with ^ > ^ ^ < ^^^^ > ^^^Next, examples are described for 3-site quantum-dot chains (1) of alternating first-type and second-type quantum dots. Fig. 6a shows a D-A-D quantum-dot chain (1) as in Fig. 1a. A magnetic-field generator configured to generate magnetic fields may be included. For example, the magnetic- field generator may be configured to generate Zeeman fields in the quantum-dot chain (1). Fig. 7a shows a A-D-A quantum-dot chain (1) and as well including a magnetic-field generator (50) as in Fig.6a. In both illustrations of Figs.6a and 7a, each occurring A-dot (20, 21) is proximitized by a respective superconductor (202, 212) which induces respective pairing gaps and ABSs. Tunnelling-control gates (30, 31, 32, 33) may be configured to control respective tunnelling amplitudes t0 between respective neighboring quantum dots. Potential-control gates (201, 101, 211) may be configured to control on-site potentials of the respective quantum dots they can act on. Such control gates are as described further above, and are not described here again. The examples of Figs. 6a and 7a go beyond the minimal setup of the D-A pair described above with reference to Figs. 2-5. Longer quantum-dot chains allow to explore additional interesting aspects of the Kitaev chain model. For example, the physical phenomenon of bulk-edge correspondence, which is absent in a minimal setup, starts to emerge in the topological phase of a three-site Kitaev chain. What’s more, the protection of the Majorana zero modes may be enhanced gradually from a quadratic protection in a minimal setup to an exponential protection in a sufficiently long chain. Without loss of generality, one may focus on the scenario as discussed above where the applied Zeeman field is perpendicular to the spin-orbit field and no pairing gap is induced on the normal dots. As such, the sweet-spots can be obtained by varying the Zeeman field strength alone. For simplicity of the description, the below example descriptions assume homogeneity in the quantum-dot chains of Figs. 6a and 6b, i.e., all the quantum dots are considered to have the same physical parameters, and so are the ABS and the tunneling terms. The invention is not limited to the homogeneity assumption however. The D-A-D quantum-dot chain shown in Fig.6a may be described by a Hamiltonian The Hamiltonians HDL and HDR (for the left D-dot (10) and the right D-dot (11), respectively) are as the Hamiltonian HDand the Hamiltonian HAis as HAfor an A-dot (20), as already described for a D-A pair Hamiltonian HDAin Figs. 2-4. The coupling Hamiltonians HTLA and HTRA incorporate single-electron tunnelling between the left D-dot (10) and the A- dot (20), and between the right D-dot (11) and the A-dot (20), respectively. Thus, even though additional labels are introduced relative to the description of Figs. 2-4 for taking more quantum dots into account, the specific forms of the individual Hamiltonians are still the same as in HDA. Assuming, for illustration purposes, homogeneity for the parameters in the Hamiltonian HDAD, one can reach a sweet-spot of the three-site D-A-D Kitaev chain by tuning the left and right quantum-dot pairs (D-A and A-D) into their own respective sweet-spots, using for example the parameters as shown above under Tuning strength of Zeeman field. Fig. 6b shows the resulting Majorana wavefunction profiles at ^ ∗ ∗ ZA= ^ZA, ε^= ε^and = ε∗ ^ found in Figs. 3a and 3c. The created two Majorana bound states are shown to be localized completely on the two outer quantum dots (10, 11), respectively, and separated in space further by the hybrid segment / the A-dot (20) in between. That indicates a Majorana sweet-spot of a three-site Kitaev chain. The energy of the two separated Majorana modes is split into a finite value of EMBS≈0.01, which is approximately one tenth of the excitation gap. Furthermore, as shown in Fig. 6c, the energy spectrum of the D-A-D chain as a function of detuning energies, i.e. δε = δεDL= δεDR= may be regarded as deviating from an expected cubic scaling behaviour of ^ ∼(δε)^. The physical mechanism underlying thus finite energy splitting may be understood in terms of a low-energy effective theory of the D-A-D chain. A low-energy effective Hamiltonian HDAD,eff, corresponding to a three-site Kitaev chain Hamiltonian, includes both first- and second-order terms as below, The first-order term is a corresponding generalization of the dot-ABS pair, and has the form wherein The Hamiltonian describes an idealized three-site Kitaev chain at the sweet- spot. The low-energy effective Hamiltonian HDAD,eff further comprises the second-order effective Hamiltonian wherein The parameters tDDand ΔDDare next-nearest-neighbor couplings between the two outer dots of the D-A-D quantum-dot chain (1). These next-nearest-neighbor couplings are mediated by a higher-energy spin-up ABS in the hybrid / A-dot (20) in a virtual tunnelling process. Such next-nearest-neighbor couplings are absent in the minimal dot-ABS pair, but show up when implementing a three-site Kitaev chain by a 3-site quantum-dot chain (1). The coupling strength scales as ^^ ^ / ^ZA, and the approximations in the equations for tDD and ΔDDare made based on ^so≪ ^ and ^ ≪ ^ in the system. Thereby, a realistic D-A- D chain is deviating from an idealized three-site Kitaev chain by a next-nearest-neighbor tunnelling effect. Fig.6c also shows a spectral comparison with an effective 3-site Kitaev chain model including next-nearest-neighbor tunnelling. Such a more realisitc 3-site Kitaev chain may be modelled as wherein fi is a (spinless) fermion on the i-th site, ε^is the on-site energy, and t and Δ are the normal and Andreev tunnelling between adjacent sites, and t’ and Δ’ are the next-nearest- neighbor tunnelings. Choosing ^ = Δ = and t’=tDD and Δ’=0, the energy spectrum of HK3 (dashed line in Fig.6c) shows an excellent agreement with that of the realistic D-A-D chain (1). This validates the low-energy analysis that a fine-tuned D-A-D quantum-dot chain (1) of Fig. 6a is implementing a realistic three-site Kitaev chain comprising the idealized 3- site Kitaev chain and second-order couplings via next-nearest-neighbor tunnelings. Fig. 7a shows a A-D-A quantum-dot chain (1). I.e., a second-type quantum dot (20) (A-dot (20)) is followed by a first-type quantum dot (10) (D-dot (10)) which is turn is followed by a second-type quantum dot (21) (A-dot (21)), each second-type quantum dot (20, 21) being configured to host an ABS. The Hamiltonian of the A-D-A quantum-dot chain may be described as wherein the Hamiltonians HDof the D-dot (10), HALof the left A-dot (20), HARof the right A- dot (21), and HTLDand HHRDof the respective tunnelings are defined as before for HDA. In the low-energy regime, the form of the effective Hamiltonian HADA,eff is wherein the first-order term is The parameters and operators are defined as before for ^^^, with the index “L” corresponding to the left A-dot (20), the index “R” corresponding to the right A-dot (21), and respective coupling parameters to the center D-dot (10) defined accordingly. The second-order effective Hamiltonian, which describes next-nearest-neighbor tunnelling processes between the two outer ABS quantum dots (20, 21), is with coupling parameters This derived effective Hamiltonian of the ABS-dot-ABS / A-D-A configuration may be regarded similar to that of a D-A-D configuration, except that now the roles of D-dots and A-dots are inverted. The next-nearest neighbor coupling strength is inversely proportional to Zeeman energy and is thus now weakened owing to an enhanced g factor in a D-dot, thus mitigating the finite energy splitting at the sweet-spot. I.e., the Zeeman splitting EZis typically smaller in an A-dot as compared to in a D-dot (EZA<EZD), because of the closer proximity of the superconductor. Fig. 7b shows that when tuning the A-D-A quantum dot chain (1) to its sweet-spot, Majorana bound states are created at the outer quantum dots (10, 11) of the A-D-A quantum dot chain (1). Furthermore, in this example, one ABS of the first A-dot (20), γ^↓, and another ABS of the second A-dot (21), γ^↓, are each controlled as a fermion in the implementation of the Kitaev chain by the A-D-A quantum-dot chain (1). Since the ABSs are controlled based on applying a Zeeman field, one may also say the ABSs are each controlled as a spinless fermion in the implementation of the Kitaev chain by the A-D-A quantum-dot chain (1). Fig. 7c shows a detuning spectrum for the A-D-A case, including comparison of the realistic 3-site Kitaev model and HADA. When compared to Fig.6c, one can observe a smaller energy splitting at the sweet-spot for the A-D-A configuration of Fig. 7a as compared to the D-A-D configuration of Fig.6a. Fig. 8 illustrates that a Zeeman splitting in A-dots may be relatively weaker as compared to D-dots, EZD ≈ 2 EZA. Due to the Zeeman splitting strength being smaller for A- dots, the second-order next-nearest neighbor coupling strength in an A-D-A quantum-dot chain is relatively weaker as compared to in a D-A-D quantum-dot chain. This leads to a smaller energy splitting of A-D-A (Fig.7c) as compared to D-A-D (Fig.6c). Figs 9a and 9b illustrate further longer chains. Fig. 9a shows a D-A-D-A configuration and Fig.9b a A-D-A-D-A configuration. The D-A-D-A quantum-chain may be for implementing a 4-site Kitaev chain by using the new method. The A-D-A-D-A- quantum- chain may be for implementing a 5-site Kitaev chain by using the new method. In both illustrations, each occurring A-dot (20, 21, 22) is proximitized by a respective superconductor (202, 212, 222) which induces respective pairing gaps and ABSs. Tunnelling- control gates (30, 31, 32, 33, 34, 35) may be configured to control respective tunnelling amplitudes t0between respective neighboring quantum dots. Potential-control gates (101, 111, 201, 211, 221) may be configured to control on-site potentials of the respective quantum dots they can act on. Such control gates are as described further above, and are not described here again. As also illustrated in Figs.9a and 9b, when tuning the quantum-dot chains (1) to their respective Majorana sweet-spot and controlling the ABSs of each second-type (20, 21, 22) quantum dot to participate in implementing a Kitaev chain, Majorana bound states (MBS1, MBS2) are created at the respective two outer quantum dots of the respective quantum-dot chain. At the example of Fig.9b, by using the new method including controlling the ABSs of A-dots (20, 21, 22) as fermions in implementing a 5-site Kitaev chain, the Majorana bound states (MBS1, MBS2) are created at the A-dots (20, 22) that are the outer two / the boundary quantum dots of the A-D-A-D-A chain (1). Further scaling up the D-A chains to a quantum-dot chain of N alternating first-type and second-type quantum dots, wherein each second-type quantum dot is configured to host a respective ABS, one may obtain a scaling behaviour for 1 ≤ ^ ≤ ^ − 1. Here Γ^denotes normal and / or Andreev coupling between any two sites separated by a distance of k. For example, Γ^denotes nearest-neighbor coupling. The parameter EZis the Zeeman splitting of either dot or ABS located between the two sites considered. Physically, Γ^originates from virtual tunnelling processes that includes k times single-electron tunnelling events via k-1 different high-energy states that are gapped from the Fermi energy due to Zeeman spin splitting. From the scaling behaviour, one may see that the strength of Γ^decays exponentially with the distance between the effectively coupled two sites, with the decay length being approximately The range of the effective couplings thus decreases with an increase of Zeeman energy in the quantum-dot chain of alternating first-type and second-type quantum dots. As a result, the low-energy effective Hamiltonian of the dot-ABS chain may, asymptotically in the large Zeeman energy limit (^^≫ ^^), be regarded as an idealized Kitaev chain. The decay length indicates that this effective two-site coupling is short-ranged in nature, and thus the topological property of the Kitaev chain should be manifested in the long-chain limit. Fig.10 shows energy spectra of N-site alternating D-A quantum-dot chains around the sweet-spot. For intermediate chain lengths (e.g. N=3), the next-nearest-neighbor coupling has an appreciable effect on the Majorana energy at the sweet-spot. As the chain is further scaled up (^ ≥ 4), this Majorana energy splitting is quickly suppressed due to the short-range nature of this effective coupling as shown in Eq. (30). Thus, as shown in Fig.10, the finite Majorana energy splitting starts to be suppressed to nearly zero as the chain length increases to for example four or five sites. Furthermore, the parameter range (δε) where this near-zero energy exists also increases with the chain length, indicating that the protection of the created Majorana bound states approaches topological. In general, the method of the present invention is not limited to a particular size of the quantum-dot chain of N alternating first-type and second-type quantum dots. The number N of total quantum dots may be any integer equal to or larger than 2. Fig. 11 shows the method in a flow-chart. Below, reference signs to Fig. 9a are used to illustrate the method, but not to limit the method to the chosen reference. Fig. 11 shows a method of creating a pair of Majorana bound states in a quantum-dot chain (1) of N alternating first-type (10, 11) and second-type (20, 21) quantum dots, each second-type quantum dot (20, 21) being configured to host a respective Andreev bound state, ABS (γ^↓), wherein N is an integer greater than or equal to two, the method comprising: tuning (action A1) the quantum-dot chain (1) to a Majorana sweet-spot for creating the pair of Majorana bound states on two outer quantum dots (10, 21) of the quantum-dot chain (1), the quantum-dot chain (1) being configured to implement a Kitaev chain, and controlling (action A2) at least one of the respective ABS (γ^↓) as a fermion in the implementing of the Kitaev chain by the quantum-dot chain (1). Preferably, the Kitaev chain is an N-site Kitaev chain, the quantum-dot chain (1) is configured to implement the N-site Kitaev chain, and each one of the respective ABS hosted in second-type quantum dots (20, 21) is controlled as a fermion in the implementing of the N- site Kitaev chain by the quantum-dot chain (1). By controlling each of the respective ABS hosted in the second-type quantum dots as a fermion, the N-site quantum-dot chain (1) of alternating first-type (10, 11) and second-type (20, 21) quantum dots can be controlled to implement an N-site Kitaev chain. Thereby, the quantum-dot chain (1) is put to more effective use by implementing a longer Kitaev chain, as compared to for example when controlling the ABSs merely as mediators for ECT and CAR couplings. Preferably, each of the controlled at least one ABSs is controlled as a spinless fermion by applying a Zeeman field to the quantum-dot chain (1) so as to induce Zeeman spin splitting (EZD, EZA) in the quantum dots (10, 11, 20, 21) of the quantum-dot chain (1). By applying the Zeeman field to the quantum-dot chain (1), the induced Zeeman spin spitting can be exploited in controlling the ABSs. For example, as illustrated with reference to the figures, the ABSs corresponding to Bogoliubov particles may be controlled as effectively spinless fermions in the implementing of the Kitaev chain based on the induced Zeeman spin splitting. Preferably, the quantum-dot chain (1) is cooled down to a milli-Kelvin range, such as 1mK-1000mK, so as to operate the quantum-dot chain in a low-energy regime. More preferably, the quantum-dot chain (1) is cooled down to in between 1mK-100mK. A concrete range may depend on the materials being used. In general, operating with even lower temperatures may be even more preferred and the invention is not limited to a minimal operating temperature. For example, preferably, the quantum-dot chain (1) is cooled down to below 1000mK or below 100mK. By operating in the low-energy regime, the dynamics of the quantum-dot chain (1) may be effectively controlled to a spinless regime in which ABS may be better controlled. The low-energy regime may facilitate the tuning to the Majorana sweet-spot. Preferably, the method further comprises, for the tuning to the Majorana sweet-spot and / or the controlling of the at least one ABS, one or more of: controlling, by a magnetic-field generator, a strength (EZA, EZD) of the applied Zeeman field; controlling, by the magnetic-field generator, a direction (θ) of the applied Zeeman field; controlling, by respective potential-control gates (101, 201, 111, 211), normal- state energies (^^, ^^) of the quantum dots (10, 11, 20, 21); and / or controlling, by respective tunnelling-control gates (31, 32, 33), one or more tunnelling amplitudes (t0) related to single-electron tunnelling between respective neighboring quantum dots. By controlling one or more of these parameters, one may controllably tune the quantum-dot chain (1) to satisfy one or more conditions (e.g., Eqns. 3-5) for the Majorana sweet-spot of the quantum-dot chain (1). For example, one may measure a charge-stability diagram for different values of the strength of the Zeeman field and the normal-state energies (ε^, ε^) until the Zeeman field is strong enough to lead to a charge-stability diagram as illustrated in Fig. 3c for a D-A configuration, which includes a Majorana sweet-spot where two lines are crossing. Controlling the normal-state energies may correspond to controlling chemical potentials by the respective potential-control gates (101, 201, 111, 211), for example by controlling voltages applied to the respective potential-control gates. More examples of tuning parameters are discussed above with reference to Figs.1-10 and further below. Preferably, the method further comprises, for the tuning to the Majorana sweet-spot and / or the controlling of the at least one ABS, controlling the ABS to be a hole-dominant (v>u) ABS or an electron-dominant (u>v) ABS. For example, as illustrated with respect to Figs.2-5, one may control a strength of the applied Zeeman field (EZA) and a normal-state energy (ε^) related to a second-type quantum dot to a regime correspond to < 0 and zero-ABS energy (Eqn. 4, EA=0). This parameter regime is illustrated in Fig. 3c by the cross. In this parameter regime, the controlled ABS is a hole-dominated (v>u) ABS, which may be preferred in a weak-orbit interaction regime in which tso < t as outlined above in the context of Eqn.5. As illustrated in Table 1 further above, a sweet spot can also be found for a strong-orbit interaction regime. Each second-type quantum dot (20, 21) corresponds to a respective semiconductor- superconductor hybrid segment that involves a proximitized superconductor (202, 212), each respective semiconductor-superconductor hybrid segment being suitable for hosting the respective ABS. Semiconductor-superconductor hybrid segments are particularly useful and effective for implementing second-type quantum dots that can host ABSs. In other words, a semiconductor-superconductor hybrid segment can be controlled in a way so as to effectively function as a quantum dot in the Kitaev chain as well, instead of merely mediating an interaction between neighbouring quantum dots. The normal quantum dots are referred to as first-type quantum dots and the semiconductor-superconductor hybrid segments are referred to as second-type quantum dots. Only the second-type quantum dots (20, 21) correspond to semiconductor- superconductor hybrid segments, but not the first-type quantum dots (10, 11). In other words, this means that the quantum-dot chain (1) corresponds to a chain of alternating quantum dots (referred to as first-type quantum dots) and semiconductor-superconductor hybrid segments (referred to as second-type quantum dots different from the first-type quantum dots). The quantum-dot chain (1) of N alternating first-type and second-type quantum dots is particularly effective in creating Majorana bound states and implementing an N-site Kitaev chain based on controlling ABSs as fermions, when precisely only the second-type quantum dots correspond to semiconductor-superconductor hybrids, but not the first-type. In this way, it is explicitly not necessary to control the strength of the proximity effect by e.g. changing the transparency to the superconducting contact. Also, a single superconducting contact per dot is sufficient. Also, the semiconductor-superconductor hybrids involving a single superconducting conduct may be relatively easy to manufacture. Furthermore, using the method based on an alternating configuration of such quantum dots may be resource-efficient with a reduced footprint when implementing an N-site Kitaev chain with N alternating first- type and second-type quantum dots. Preferably, the controlling of the respective potential-control gates (101, 201, 111, 211) and / or tunnelling-control gates (31, 32, 33) is based on controlling a voltage applied to the respective gates. For example, each gate may be implemented as an electrostatic control gate. Preferably, if ^ ≥ 3, the tuning of the quantum-dot chain (1) to the Majorana sweet- spot comprises tuning pairs of respective neighboring quantum dots to satisfy Majorana sweet-spot conditions. As illustrated in the context of the D-A-D and A-D-A configurations, longer chains may be tuned to the Majorana sweet-spot by considering pairs of neighboring quantum dots (i.e., nearest-neighboring) and tuning the respective pairs to their respective Majorana sweet- spot conditions. Preferably, the tuning of the quantum-dot chain (1) to the Majorana sweet-spot comprises measuring a charge-stability diagram for different values of at least one of the strength of the Zeeman field, the direction of the Zeeman field, the normal-state energies, and / or the tunnelling amplitudes, and finding the Majorana sweet-spot (e.g., a parameter regime corresponding to the Majorana sweet-spot) by considering the charge-stability diagram for which at least two crossings can be observed. Typically, when the applied Zeeman field is strong enough, the measured charge- stability diagram shows at least two crossings as illustrated in Fig.3b. A crossing may refer to a crossing region of conductance lines in the charge-stability diagram, as shown in Fig. 3b. Measuring the charge-stability diagram that way may be a particularly efficient way in finding the Majorana sweet-spot while controlling the one of more ABS as fermions in the implementing of the Kitaev chain. Preferably, a semiconductor (40) hosts the quantum dots (10, 11, 20, 21), the semiconductor being based on a two-dimensional electron gas, 2DEG, a two-dimensional hole gas, 2DHG, a nanowire, and / or nanowire networks. A quantum-dot chain for creating a pair of Majorana bound states may be provided, the quantum-dot chain comprising: N alternating first-type and second-type quantum dots, each second-type quantum dot being configured to host a respective Andreev bound state, ABS, wherein N is an integer greater than or equal to two; a plurality of control gates configured to control respective orbit energies; and one or more tunnelling control gates configured to control respective tunnelling couplings between two consecutive quantum dots of the quantum-dot chain, wherein at least one of two outer quantum dots of the quantum-dot chain is a second- type quantum dot. The method according to the present invention suggests the fabrication of such new quantum-dot chains (1) comprising N alternating first-type (10, 11) and second-type (20, 21) quantum dots, each second-type quantum dot (20, 21) being configured to host a respective Andreev bound state, ABS (γ^↓), wherein N is an integer greater than or equal to two; wherein at least one (21) of two outer quantum dots (10, 21) of the quantum-dot chain is a second-type quantum dot. Examples of such quantum-dot chains (1) are illustrated in Figs.2a, 2b, 7b, 9a, 9b. As mentioned in the context of Figs. 6-8, including more dots may further supress an energy splitting occurring between the created Majorana bound states. Preferably, the quantum-dot chain (1) is implemented based on a semiconductor (40) hosting the quantum dots (10, 11, 20, 21), the semiconductor being based on a two- dimensional electron gas, 2DEG, and / or a two-dimensional hole gas, 2DHG, or the quantum- dot chain (1) is implemented based on a nanowire or nanowire network. Examples of implementations based on 2DEGS / 2DHGs / nanowires may be found the list of references. However, none of these implementations shows an outer quantum dot corresponding to a second-type quantum dot configured to host an ABS. Preferably, each second-type quantum dot (20, 21) is proximitized by a respective superconductor (202, 212). Fig. 12 shows an apparatus (2) configured to implement the method and / or including preferred and / or optional features discussed herein. The apparatus (2) comprises a memory (3) and a processor (4). The apparatus may further comprise a control unit (5) for executing operations such as measurements, changing voltages, etc., e.g. for controlling the quantum- dot (1). For example, a magnetic-field generator (50) may be included in the apparatus (2). Thus, the method may be computer-implemented in the apparatus (2). The apparatus (2) is configured to control the quantum-dot chain (1), the apparatus (2) comprising: a processor (3); and a memory (4) storing instructions, which, when executed by the processor (3), cause the apparatus (2) to execute a method of any one of the preceding methods described above. Figs. 13a-d show measurement results for an effective two-site quantum-dot chain based on a nanowire and as illustrated in Figs.2a-2b. Fig. 13a shows measurement results from an experimental implementation of the setup shown in Figs. 2a and 2b, based on a semiconductor nanowire system. The different panels shown in Fig. 13a show experimental charge stability diagrams for different values of a tunnel gate voltage ^^. Fig. 13a thus demonstrates an example of tuning the tunnel gate voltage to tune the device into a sweet spot with two Majorana bound states. In other words, Fig. 13a shows measurement results relating to fine-tuning to a Majorana sweet-spot by tuning a tunnel-amplitude t0 via varying a voltage VT that is applied to the tunnelling-control gate (31) and obtaining a charge-stability diagram for different voltages applied to the gates (31, 101, 201). The voltages VHwere applied to potential-control gate (201) and the voltages VQDto potential-control gate (101). The concrete values are not essential, but may be chosen depending on circumstances and may depend on the manufactured quantum-dot chain (1) for example. The top left figure indicates two lines approaching in the charge-stability diagram, indicating that a Majorana sweet-spot can be reached at which an ABS in the A-dot (20) is controlled as a fermion in implementing the Kitaev chain. Therefore, reaching a Majorana sweet-spot at which an ABS in the A-dot (20) is controlled as a fermion in implementing the Kitaev chain can be facilitated by fine-tuning tunnelling-amplitudes via tunnelling-control gates. For example, as outlined further above under “Tuning induced pairing gap in the quantum dot”, tuning the tunnel-amplitude t0 via tunnelling-control gates may be used for fine-tuning to a Majorana sweet-spot. Fig. 13b shows measurement results from the same system as in Fig.13a, but now controlling the angle of the magnetic field ^ with the nanowire to change a respective effective coupling. In other words, Fig.13b shows measurement results relating to fine-tuning to a Majorana sweet-spot by fixing a strength of an applied magnetic field parallel to the nanowire implementing the quantum dots and varying the angle θ (ϕ in Fig. 13b) of the magnetic field / i.e. varying the direction of the magnetic field. Thus, the charge-stability diagram can be altered by varying the direction, confirming that finding the Majorana sweet spot while controlling an ABS as a fermion in the implementation of a Kitaev chain can be facilitated by including varying the direction of the magnetic field. Figs. 13c-d show measurement results relating to fine-tuning to obtain a charge- stability diagram at which two Majorana sweet-spots are observed. In Fig.13c, one Majorana sweet-spot can be recognized in the bottom right corner and another one in the top left corner, each corner corresponding to the crossing of lines. Fig. 13d shows measurement results from an experimental implementation of the setup shown in Figs. 2a and 2b, based on a 2DEG. The different panels show experimental charge stability diagrams for different values of the magnetic field strength ^, demonstrating that this allows to tune the device into a sweet spot with two Majorana bound states. Thus, Fig. 13d also shows measurement results relating to fine-tuning to a Majorana sweet-spot. The measurement results show the formation of a crossing in the charge-stability diagram when increasing the magnetic field strength, in this example from 100mT to 200mT. Clauses. Clause 1. A method of creating a pair of Majorana bound states in a quantum-dot chain (1) of N alternating first-type (10, 11) and second-type (20, 21) quantum dots, each second-type quantum dot (20, 21) being configured to host a respective Andreev bound state, ABS (γ^↓), wherein N is an integer greater than or equal to two, the method comprising: tuning the quantum-dot chain (1) to a Majorana sweet-spot for creating the pair of Majorana bound states on two outer quantum dots (10, 21) of the quantum-dot chain (1), the quantum-dot chain (1) being configured to implement a Kitaev chain, and controlling at least one of the respective ABS (ψABS, γ^↓) as a fermion in the implementing of the Kitaev chain by the quantum-dot chain (1). Clause 2. The method of clause 1, wherein the Kitaev chain is an N-site Kitaev chain, the quantum-dot chain (1) is configured to implement the N-site Kitaev chain, wherein the method further comprises controlling each one of the respective ABS hosted in second-type quantum dots (20, 21) as a fermion in the implementing of the N-site Kitaev chain by the quantum-dot chain (1). Clause 3. The method of any one of the preceding clauses, further comprising controlling each one of the controlled at least one ABSs as a spinless fermion by applying a Zeeman field to the quantum-dot chain (1) so as to induce Zeeman spin splitting (EZD, EZA) in the quantum dots (10, 11, 20, 21) of the quantum-dot chain (1). Clause 4. The method of any one of the preceding clauses, further comprising one or more of: controlling, by a magnetic-field generator, a strength (EZA, EZD) of the applied Zeeman field; controlling, by the magnetic-field generator, a direction (θ) of the applied Zeeman field; controlling, by respective potential-control gates (101, 201, 111, 211), normal-state energies (^^, ^^) of the quantum dots (10, 11, 20, 21); and / or controlling, by respective tunnelling-control gates (31, 32, 33), one or more tunnelling amplitudes (t0) related to single-electron tunnelling between respective neighboring quantum dots. Clause 5. The method of any one of the preceding clauses, further comprising controlling one or more of the respective ABS to be a hole-dominant (v>u) ABS or an electron-dominant (u>v) ABS. Clause 6. The method of any one of the preceding clauses, wherein each second-type quantum dot (20, 21) corresponds to a respective semiconductor-superconductor hybrid segment that involves a proximitized superconductor (202, 212), each respective semiconductor-superconductor hybrid segment being suitable for hosting the respective ABS. Clause 7. The method of clause 6, wherein only the second-type quantum dots (20, 21) correspond to semiconductor-superconductor hybrid segments, but not the first-type quantum dots (10, 11). Clause 8. The method of clause 4, wherein the controlling of the respective potential-control gates (101, 201, 111, 211) and / or tunnelling-control gates (31, 32, 33) is based on controlling a voltage applied to the respective gates. Clause 9. The method of any one of the preceding clauses, wherein if ^ ≥ 3, the tuning of the quantum-dot chain (1) to the Majorana sweet-spot further comprises tuning pairs of respective neighboring quantum dots to satisfy Majorana sweet-spot conditions. Clause 10. The method of any one of the preceding clauses, wherein the tuning of the quantum-dot chain (1) to the Majorana sweet-spot comprises measuring a charge-stability diagram for different values of at least one of the strength of the Zeeman field, the direction of the Zeeman field, the normal-state energies, and / or the tunnelling amplitudes, and finding the Majorana sweet-spot by considering the charge-stability diagram for which at least two crossings can be observed. Clause 11. The method of any one of the preceding clauses, further comprising cooling the quantum-dot chain (1) down to a milli-Kelvin range, such as in between 1mK and 100mK, so as to operate the quantum-dot chain in a low-energy regime. Clause 12. A quantum-dot chain (1) comprising: N alternating first-type (10, 11) and second-type (20, 21) quantum dots, each second- type quantum dot (20, 21) being configured to host a respective Andreev bound state, ABS (γ^↓), wherein N is an integer greater than or equal to two; wherein at least one (21) of two outer quantum dots (10, 21) of the quantum-dot chain is a second-type (20, 21) quantum dot. Clause 13. The quantum-dot chain (1) of clause 12, wherein the quantum-dot chain (1) is implemented based on a semiconductor (40) hosting the quantum dots (10, 11, 20, 21), the semiconductor being based on a two-dimensional electron gas, 2DEG, and / or a two- dimensional hole gas, 2DHG, and / or wherein the quantum-dot chain (1) is implemented based on a nanowire or nanowire network. Clause 14. The quantum-dot chain (1) of any of clause 12 or clause 13, wherein each second- type quantum dot (20, 21) is proximitized by a respective superconductor (202, 212). Clause 15. An apparatus (2) configured to control the quantum-dot chain (1) of claim 12, the apparatus (2) comprising: a processor (3); and a memory (4) storing instructions, which, when executed by the processor (3), cause the apparatus (2) to execute the method of any one of the preceding method clauses. The following list of references is incorporated herein by way of reference. List of references [Altland, Simons] Alexander Altland, Ben D. Simons, Condensed Matter Field Theory, Cambridge University Press, 2010. [Bordin et al, 1] Alberto Bordin, Guanzhong Wang, Chun-Xiao Liu, Sebastiaan LD ten Haaf, Grzegorz P Mazur, Nick van Loo, Di Xu, David van Driel, Francesco Zatelli, Sasa Gazibegovic, et al., Controlled crossed Andreev reflection and elastic co-tunneling mediated by Andreev bound states, arXiv:2212.02274 (2022). [Bordin et al, 2] Alberto Bordin, Xiang Li, David van Driel, Jan Cornelis Wolff, Qingzhen Wang, Sebastiaan L. D. ten Haaf, Guanzhong Wang, Nick van Loo, Leo P. Kouwenhoven, Tom Dvir, Crossed Andreev reflection and elastic co-tunneling in a three-site Kitaev chain nanowire device, arXiv:2306.07696 (2023). [Dvir et al] Tom Dvir, Guanzhong Wang, Nick van Loo, Chun-Xiao Liu, Grzegorz P. Mazur, Alberto Bordin, Sebastiaan L. D. ten Haaf, Ji-Yin Wang, David van Driel, Francesco Zatelli, Xiang Li, Filip K. Malinowski, Sasa Gazibegovic, Ghada Badawy, Erik P. A. M. Bakkers, Michael Wimmer, and Leo P. Kouwenhoven, Realization of a minimal Kitaev chain in coupled quantum dots, Nature 614, 445–450 (2023). [Kitaev] Alexei Kitaev, Unpaired Majorana fermions in quantum wires, arXiv, Physics- Uspekhi 44, 131 (2001). [Nayak, et al] Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar Das Sarma, Non-Abelian anyons and topological quantum computation, Rev. Mod. Phys. 80, 1083–1159 (2008). [Sarma, et al] Sankar Das Sarma, Michael Freedman, and Chetan Nayak, Majorana zero modes and topological quantum computation, Npj Quantum Information 1, 15001 EP – (2015). [Sau, Sarma] Jay D. Sau and S. Das Sarma, Realizing a robust practical Majorana chain in a quantum-dot-superconductor linear array, Nat. Commun.3, 964 (2012). [G. Wang et al] Guanzhong Wang, Tom Dvir, Grzegorz P. Mazur, Chun-Xiao Liu, Nick van Loo, Sebastiaan L. D. ten Haaf, Alberto Bordin, Sasa Gazibegovic, Ghada Badawy, Erik P. A. M. Bakkers, Michael Wimmer, and Leo P. Kouwenhoven, Singlet and triplet cooper pair splitting in hybrid superconducting nanowires, Nature 612, 448–453 (2022). [Q. Wang et al] Qingzhen Wang, Sebastiaan LD ten Haaf, Ivan Kulesh, Di Xiao, Candice Thomas, Michael J Manfra, and Srijit Goswami, Triplet cooper pair splitting in a two- dimensional electron gas, arXiv:2211.05763 (2022).

Claims

What is claimed is:

1. A method of creating a pair of Majorana bound states in a quantum-dot chain (1) of N alternating first-type (10, 11) and second-type (20, 21) quantum dots, each second-type quantum dot (20, 21) being configured to host a respective Andreev bound state, ABS (γ^↓), wherein N is an integer greater than or equal to two, the method comprising: tuning the quantum-dot chain (1) to a Majorana sweet-spot for creating the pair of Majorana bound states on two outer quantum dots (10, 21) of the quantum-dot chain (1), the quantum-dot chain (1) being configured to implement a Kitaev chain, and controlling at least one of the respective ABS (ψABS, γ^↓) as a spinless fermion in the implementing of the Kitaev chain by the quantum-dot chain (1) by applying a Zeeman field to the quantum-dot chain (1) so as to induce Zeeman spin splitting (EZD, EZA) in the quantum dots (10, 11, 20, 21) of the quantum-dot chain (1), wherein each second-type quantum dot (20, 21) corresponds to a respective semiconductor-superconductor hybrid segment that involves a proximitized superconductor (202, 212), each respective semiconductor-superconductor hybrid segment being suitable for hosting the respective ABS, wherein only the second-type quantum dots (20, 21) correspond to semiconductor- superconductor hybrid segments, but not the first-type quantum dots (10, 11).

2. The method of claim 1, wherein the Kitaev chain is an N-site Kitaev chain, the quantum- dot chain (1) is configured to implement the N-site Kitaev chain, wherein the method further comprises controlling each one of the respective ABS hosted in second-type quantum dots (20, 21) as a fermion in the implementing of the N-site Kitaev chain by the quantum-dot chain (1).

3. The method of any one of the preceding claims, further comprising controlling each one of the controlled at least one ABSs as a spinless fermion by applying the Zeeman field to the quantum-dot chain (1).

4. The method of any one of the preceding claims, further comprising one or more of: controlling, by a magnetic-field generator, a strength (EZA, EZD) of the applied Zeeman field;controlling, by the magnetic-field generator, a direction (θ) of the applied Zeeman field; controlling, by respective potential-control gates (101, 201, 111, 211), normal-state energies (^^, ^^) of the quantum dots (10, 11, 20, 21); and / or controlling, by respective tunnelling-control gates (31, 32, 33), one or more tunnelling amplitudes (t0) related to single-electron tunnelling between respective neighboring quantum dots.

5. The method of any one of the preceding claims, further comprising controlling one or more of the respective ABS to be a hole-dominant (v>u) ABS or an electron-dominant (u>v) ABS.

6. The method of claim 4, wherein the controlling of the respective potential-control gates (101, 201, 111, 211) and / or tunnelling-control gates (31, 32, 33) is based on controlling a voltage applied to the respective gates.

7. The method of any one of the preceding claims, wherein if ^ ≥ 3, the tuning of the quantum-dot chain (1) to the Majorana sweet-spot further comprises tuning pairs of respective neighboring quantum dots to satisfy Majorana sweet-spot conditions.

8. The method of any one of the preceding claims, wherein the tuning of the quantum-dot chain (1) to the Majorana sweet-spot comprises measuring a charge-stability diagram for different values of at least one of the strength of the Zeeman field, the direction of the Zeeman field, the normal-state energies, and / or the tunnelling amplitudes, and finding the Majorana sweet-spot by considering the charge-stability diagram for which at least two crossings can be observed.

9. The method of any one of the preceding claims, further comprising cooling the quantum- dot chain (1) down to a milli-Kelvin range, such as in between 1mK and 100mK, so as to operate the quantum-dot chain in a low-energy regime.

10. A quantum-dot chain (1) comprising: N alternating first-type (10, 11) and second-type (20, 21) quantum dots, each second- type quantum dot (20, 21) being configured to host a respective Andreev bound state, ABS (γ^↓), wherein N is an integer greater than or equal to two;wherein at least one (21) of two outer quantum dots (10, 21) of the quantum-dot chain is a second-type (20, 21) quantum dot, wherein each second-type quantum dot (20, 21) corresponds to a respective semiconductor-superconductor hybrid segment that involves a proximitized superconductor (202, 212), each respective semiconductor-superconductor hybrid segment being suitable for hosting the respective ABS, wherein only the second-type quantum dots (20, 21) correspond to semiconductor- superconductor hybrid segments, but not the first-type quantum dots (10, 11).

11. The quantum-dot chain (1) of claim 12, wherein the quantum-dot chain (1) is implemented based on a semiconductor (40) hosting the quantum dots (10, 11, 20, 21), the semiconductor being based on a two-dimensional electron gas, 2DEG, and / or a two- dimensional hole gas, 2DHG, and / or wherein the quantum-dot chain (1) is implemented based on a nanowire or nanowire network.

12. An apparatus (2) configured to control the quantum-dot chain (1) of claim 12, the apparatus (2) comprising: a processor (3); and a memory (4) storing instructions, which, when executed by the processor (3), cause the apparatus (2) to execute the method of any one of the preceding method claims.

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