Qubits based on majorana zero modes and quantum-dot chains for hosting majorana zero modes

Abstract

The disclosure relates to qubits based on quantum-dot chains for hosting Majorana zero modes and has applications in topological quantum computing. The qubit is based on quantum-dot chains configured to host Majorana zero modes and comprises: a first quantum-dot chain comprising a first pair of two outer quantum dots and configured to host a first pair of Majorana zero modes in the first pair of two outer quantum dots; a second quantum-dot chain comprising a second pair of two outer quantum dots and configured to host a second pair of Majorana zero modes in the second pair of two outer quantum dots; and a readout system comprising one or more auxiliary quantum dots and configured to provide a coupling between the Majorana zero modes. The disclosure further relates to a two-qubit system and to quantum-dot chains for hosting Majorana zero modes and for providing a qubit based on Majorana zero modes.

Description

[0001] Qubits based on Majorana zero modes and quantum-dot chains for hosting Majorana zero modes

[0002] Field of the invention

[0003] The invention relates to qubits based on Majorana zero modes and quantum-dot chains for hosting Majorana zero modes.

[0004] Background art

[0005] Majorana zero modes are regarded as a promising candidate for implementing topological quantum computing [Nayak et al; Sarma et al] and may pave the way to realize and enable stable quantum computing.

[0006] For example, a qubit may be based on four Majorana zero modes and qubit gates may be based on suitably coupling the four Majorana zero modes [Plugge et al, Karzig et al], Majorana zero modes can encode a spin-1 / 2 degree of freedom and the suitable couplings can enable the implementation of qubit gates via realizing Pauli operators.

[0007] Recent developments to attempt realizing qubits and / or topological quantum computing based on Majorana zero modes include the use of topological superconductors for creating and controlling the Majorana zero modes. However, as of this moment, such topological superconductors are merely a theoretical concept and have not yet been demonstrated in experiments to exhibit Majorana zero modes. The realization and control of qubits based on Majorana zero modes therefore remains an open challenge.

[0008] There is thus a need to efficiently realize and control qubits based on Majorana zero modes.

[0009] Summary of the invention

[0010] A general object of the invention is to efficiently realize and control qubits based on Majorana zero modes.

[0011] Therefore, according to a first aspect of the invention, a qubit based on quantum-dot chains configured to host Majorana zero modes, as defined in appended claim 1, is provided.

[0012] According to further aspects of the invention, a two-qubit system as defined in appended claim 10, a quantum-dot chain as defined in appended claim 14, and a curved quantum-dot chain as defined in appended claim 15, are provided.

[0013] The above-referenced aspects of the invention enable to efficiently realize and control qubits based on Majorana zero modes. The invention can have applications in quantumcomputing methods and / or quantum-computing devices / chips that are implementing topological quantum computing based on Majorana zero modes. The disclosure further relates to a two-qubit system and to quantum-dot chains for hosting Majorana zero modes and for providing a qubit based on Majorana zero modes. Further more detailed advantages are discussed further below in the detailed description.

[0014] Brief description of the drawings

[0015] The present invention is discussed in more detail below, with reference to the attached drawings, in which:

[0016] Figs, la-lc show prior art quantum-dot chains for implementing a Kitaev chain and for hosting Majorana zero modes.

[0017] Figs. 2a and 2b show quantum-dot chains for implementing a Kitaev chain, wherein superconducting elements are part of a single connected superconductor.

[0018] Fig- 3 shows schematically a quantum-dot chain with a pair of two outer quantum dots hosting a pair of Majorana zero modes.

[0019] Fig. 4a shows a qubit based on quantum-dot chains hosting Majorana zero modes, with two quantum-dot chains coupled via a readout system.

[0020] Fig. 4b also shows a qubit based on quantum-dot chains hosting Majorana zero modes, with two quantum-dot chains coupled via a readout system as in Fig. 4b, but wherein additionally the involved quantum-dot chains share a common superconductor.

[0021] Figs. 5a-5c show examples of curved quantum-dot chains hosting Majorana zero modes.

[0022] Fig. 6 shows schematically a curved quantum-dot chain with a pair of two outer quantum dots hosting a pair of Majorana zero modes.

[0023] Fig. 7a shows a qubit based on quantum-dot chains hosting Majorana zero modes, with two curved quantum-dot chains coupled via a readout system that provides a coupling between the Majorana zero modes.

[0024] Fig. 7b also shows a qubit based on quantum-dot chains hosting Majorana zero modes, with two curved quantum-dot chains coupled via a readout system that provides a coupling between the Majorana zero modes, but wherein additionally the involved quantumdot chains share a common superconductor.

[0025] Figs. 8a and 8b show schematically a respective generalization of the examples of Figs. 7a and 7b, respectively.

[0026] Figs. 9a-9c show two-qubit systems based on quantum-dot chains hosting Majorana zero modes.

[0027] Figs. lOa-lOb show multi-qubit systems involving qubits and two-qubits based on quantum-dot chains hosting Majorana zero modes. Figs. 1 la-1 Id show examples of coupling a superconductor to a grounded superconducting lead via a gate-tuneable Josephson junction.

[0028] Detailed description

[0029] 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.

[0030] 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.

[0031] 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.

[0032] 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.

[0033] 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.

[0034] 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.

[0035] 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.

[0036] 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.

[0037] The inventors have identified that a qubit based on Majorana zero modes can be efficiently realized based on quantum-dot chains for hosting the Majorana zero modes.

[0038] Figs, la-lc show examples of prior art quantum-dot chains 100A for hosting Majorana zero modes. In each example, the quantum-dot chain 100A comprises a pair of two outer quantum dots 1(1) and 1(2) and is suitable for hosting a pair of Majorana zero modes (Yi, y2) in the two outer quantum dots 1(1), 1(2).

[0039] In Fig. la, the quantum-dot chain 100A further comprises a superconductor 2(1), a number of control gates 3(k), k=l, 3, 5, ..., 13, and a semiconductor 4(1). Fig. la also illustrates two Majorana zero modes Yi and Y2 that are formed in the quantum dots 1(1), 1(2). The quantum-dot chain 100A shown in Fig. la may be understood as a minimal quantum-dot chain for implementing for example a 2-site Kitaev chain as demonstrated in [Dvir et al], In Fig. lb, the quantum-dot chain 100 A further comprises a quantum dot 5(1), superconductors 2(1) and 2(3), a number of control gates 3(k), k=l, 3, 5, ..., 21, and a semiconductor 4(1).

[0040] As compared to the illustration of Fig. la, the example of Fig. lb shows an additional superconductor 2(3), a further quantum dot 5(1) and further control gates 3(k), k=15, 17, 19 21.

[0041] The two quantum dots 1(1) and 1(2) in this example are referred to as outer quantumdots 1(1), 1(2), as they form a boundary of the quantum-dot chain 100 A. In other words, the two outer quantum dots 1(1) and 1(2) are configured to be coupled to each other via a link / via bulk elements so as to enable hosting the pair of Majorana zero modes (y13y2).

[0042] In Fig. 1c, the quantum-dot chain 100A further comprises quantum dots 5(1) and 5(3), superconductors 2(1), 2(3), 2(5), a number of control gates 3(k), k=l, 3, 5, ..., 29, and a semiconductor 4(1).

[0043] In each of the examples, the superconductors 2(1), 2(3), 2(5) may also be referred to as superconducting couplers or superconducting elements. For instance, the superconducting coupler 2(1) may be configured to provide a coupling between the outer quantum dot 1(1) and the quantum dot 5(1). The coupling may be controllable via the involved control gates 3(3), 3(5), 3(7), 3(9), 3(11).

[0044] The semiconductor 4(1) may be configured to host the quantum dots 1(1), 1(3), 5(1), 5(3) and may be based on for example a nanowire, two-dimensional electron gas, 2DEG, a two-dimensional hole-gas, 2DHG, and / or nanowire networks [e.g., Aseev et al, Jason Jung et al, Op het Veld et al].

[0045] The quantum dots 1(1), 1(3), 5(1), 5(3) may refer to regions in the semiconductor 4(1) defined by confining potentials via the control gates 3(k) and eventually further involving depletion gates (not shown in Figs, la-lc).

[0046] The quantum-dot chain 100A in Figs, la-lc is based on the semiconductor 4(1) configured to host the respectively involved quantum dots 1(1), 1(3), 5(1), 5(3). The quantum dots 1(1), 1(3), 5(1), 5(3) may be defined and controlled by the control gates 3(k), wherein some of the control gates may correspond to tunnelling-control gates (3(1), 3(5), 3(9), ..., 3(29)), controlling tunnelling between regions, and potential-control gates (3(3), 3(7), 3(11), ..., 3(27); also: plunger gates or on-site control gates') controlling on-site potentials / chemical potentials. The control gates may also be referred to as electrostatic gates or finger gates, as they may be implemented as electrostatic control gates on which voltages can be applied. The on-site control gates may be configured to provide control of chemical potentials / depths of potential wells in the respective quantum dots. The tunnelling-control gates (3(1), 3(5), 3(9), ..., 3(29)) may be configured to control tunnelling-amplitudes related to single-electron tunnelling between nearest-neighbour quantum dots of the respective quantum dots. Outer tunnelling-control gates (3(1), 3(13) in Fig. la, 3(1), 3(21) in Fig. lb, 3(1), 3(29) in Fig. 1c) may be configured to control tunnelling to leads / lead elements (not illustrated in Fig. la) for example for performing measurements.

[0047] The quantum-dot chain 100A may comprise further elements such as leads (not shown in Figs, la-lc) and layers (e.g. dielectric layers) of and / or on top of the semiconductor (also not shown in Figs, la-lc) that may be used for performing measurements on the quantum-dot chain 100 A and for defining regions. For example, depletion gates may be used to define a region in which tunnelling amplitudes and potentials are controlled. However, such details regarding manufacturing and design details are known [Bordin et al, 1; Bordin et al, 2; Dvir et al; G. Wang et al; Q. Wang et al] and are therefore not outlined and repeated here.

[0048] Preferably, the quantum-dot chains described in the present document are configured to implement a Kitaev chain. For example, the quantum-dot chain 100 A, illustrated by example in Figs, la-lc, may be configured to implement a Kitaev chain [Dvir et al], Kitaev chains are briefly described next.

[0049] Alexei Kitaev proposed an idealized one-dimensional chain of interacting fermions with the goal of creating topologically protected quantum states that may be used in quantum computing [Kitaev], A simplified example of Kitaev’ s Hamiltonian HKN of the proposed idealized one-dimension chain of N sites is

[0050] [Eqn. 1] HKN= i ( fk + (t fk+ k + A fk+ifk + h. c. )) wherein fkand denote fermionic annihilation and creation operators, respectively, the parameter skrelates to an on-site energy of site k, the parameter t relates to an amplitude for single-electron coupling, the parameter A relates to Andreev-type coupling between neighbouring sites, h.c. denotes “Hermitian conjugate”, and fx+i=0 by convention.

[0051] Kitaev found that when tuning the chain to = 0 and t = A, the chain hosts two Majorana zero modes — certain quasi -particle states — at the two boundary sites k=l and k=N of the chain. Such a parameter regime for creating two Majorana zero modes at the boundary of a Kitaev chain is also called Majorana sweet-spot. 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 or a generalized Kitaev chain comprising the dynamics of the idealized Kitaev chain. For simplicity, a Kitaev chain is referred to hereinafter.

[0052] Since Kitaev’ s proposal of an idealized Kitaev chain, many efforts were made to realize an N-site Kitaev chain (IV > 2) and to tune the chain to its Majorana sweet-spot in order to create Majorana zero modes. Proposals that were made include, among others, ID topological superconductivity involving semiconductor-superconductor nanowires, as well as planar Josephson junctions and magnetic adatom chain on top of superconductivity.

[0053] Quantum-dot chains realizing the dynamics of Kitaev chains and hosting a pair of Majorana zero modes when being tuned to the Majorana sweet-spot have recently been realized [Dvir et al; Bordin et al, 2],

[0054] Fig. la illustrates an example of such a quantum-dot chain 100A for implementing a Kitaev chain. In Fig. la, the arrangement involving the proximitized superconductor 2(1) distinguishes the center region which may be regarded as corresponding to a semiconductorsuperconductor hybrid segment 2(1).

[0055] The semiconductor-superconductor hybrid segment involving the superconducting coupler 2(1) can host one or more Andreev bound states, ABSs. An ABS is an electron-hole superposition state and / or a quasi-particle state that can be formed in the region close to the superconducting coupler 2(1). In this example quantum-dot chain, an ABS can be used to couple the two boundary quantum dots 1(1), 1(2). The ABSs can for example be used as a mediator for realizing couplings needed to implement a 2-site Kitaev chain.

[0056] The ABS may mediate a single-electron tunnelling process also referred to as elastic co-tunnelling, ECT. In ECT, a single electron may transition from one outer quantum dot (e.g., 1(1)) via the semiconductor-superconductor hybrid segment 2(1) to the other quantum dot (e.g., 1(2)).

[0057] The ABS may further mediate a Cooper-pair coupling / splitting process referred to as Crossed-Andreev reflection, CAR. In CAR, two electrons in the superconducting coupler 2(1) may form a Cooper pair and may separately tunnel into the neighbouring boundary quantum dots 1(1), 1(2), respectively. Similarly, two electrons may tunnel from the neighbouring boundary quantum dots 1(1), 1(2) into the superconductor 2(1) to form a Cooper pair.

[0058] The ABS may mediate ECT and CAR processes. Thereby, the quantum-dot chain 100A shown in Fig. la may realize a 2-site Kitaev chain model HK2 of Eqn. (1) with N=2. The quantum-dot 1(1) corresponds to a first site k=l and the quantum-dot 1(2) 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 t f2fz), wherein an electron is transitioning between site 1 (quantum-dot 1(1)) and site 2 (quantum-dot 1(2)). An CAR process may realize the superconducting coupling term with amplitude A in HK2, the term A( / 2I + fi fz , wherein two electrons are paired in a Cooper-pair type process in the semiconductor-superconductor hybrid segment involving the superconducting coupler 2(1) and are moving into the neighbouring quantum dots 1(1), 1(2) (~ A / ^) or two electrons are moving out of the neighbouring quantum dots 1(1), 1(2) and into the semiconductor-superconductor hybrid segment (~ A f2fr

[0059] When being tuned to its sweet-spot via the control gates, the quantum-dot chain 100 A exhibits a pair of Majorana zero modes at its boundary quantum dots 1(1), 1(2) / two outer quantum dots 1(1), 1(2). A first Majorana zero mode yxmay be formed in one outer quantum dot 1(1), and a second Majorana zero mode in the other outer quantum dot 1(2).

[0060] Fig. lb shows another known example of a recently manufactured quantum-dot chain for implementing for example a 3 -site Kiteav chain HK3 [Bordin et al, 2],

[0061] Similarly as for Fig. la, when tuning the quantum-dot chain 100A of Fig. lb to its Majorana sweet spot, a pair of Majorana zero modes (y13y2) is formed in the two outer quantum dots 1(1), 1(2) of the quantum-dot chain 100 A. Longer Kitaev chains may be implemented, as illustrated by example in Fig. 1c. Furthermore, different ways of and parameter regimes for implementing Kitaev chains are known (e.g., [Bordin et al, 2]) and are therefore not described here.

[0062] The inventors have identified a particularly advantageous type of quantum-dot chain that is distinguished from the known quantum-dot chains 100A illustrated in Figs, la-lc. In these above-described prior-art quantum-dot chains 100A, the superconducting couplers 2(1), 2(3), 2(5) are separated from one another and are each grounded.

[0063] Figs. 2a-2b show examples of the preferred type of quantum-dot chains 100B that the inventors have identified as having technical advantages. In these examples, the involved superconductors / superconducting couplers 2(1), 2(3), 2(5) are not separated / separately grounded but are instead connected via a respective superconducting link 2(2) (also: superconducting connection 2(2) or superconducting bridge 2(2)) so as to form a superconductor 2. The superconductor 2 may also be referred to as one single connected superconductor 2 comprising the respective superconductor elements / parts 2(1), 2(2), 2(3), 2(5). Examples Figs. 2a-2b with reference to are discussed next. In Fig. 2a, as compared to the example of Fig. lb, the superconductors 2(1) and 2(3) are connected via the superconducting link 2(2) so that a superconductor 2 is formed. In other words, the superconductor 2 corresponds to a single connected superconductor 2 comprising the superconductor elements 2(1), 2(2), 2(3).

[0064] Similarly, in Fig. 2b, as compared to the example of Fig. 1c, the superconductors 2(1), 2(3), 2(5) are connected via the superconducting link 2(2) so that a superconductor 2 is formed. In other words, the superconductor 2 corresponds to a single connected superconductor 2 comprising the superconductor parts 2(1), 2(2), 2(3), 2(5).

[0065] In general, the quantum-dot chain 100B comprising two or more superconducting couplers (such as 2(1), 2(3), 2(5)) preferably comprises a superconductor 2 comprising the respective two or more superconducting couplers and a superconducting link 2(2) so as to form a single connected superconductor (2).

[0066] Providing a quantum-dot chain (100B) with a single connected superconductor 2 comprising all superconducting couplers (e.g., 2(1), 2(3) in Fig. 2a; 2(1), 2(3), 2(5) in Fig. 2b) has for example the technical advantage of facilitating the tuning of the quantum-dot chain 100B to its Majorana sweet spot. The single connected superconductor 2 can namely enable tuning involved superconducting coupling terms A of different sites (which may have different phases unlike in the simplified example of HKN) to have a same or substantially same phase. For example, by tuning a magnetic flux that goes through a loop defined by an involved pair of superconducting couplers and involved quantum dot, the corresponding superconducting coupling terms can be tuned, in particular to having an equal phase. Thereby, a scenario in which for example a phase difference between different superconducting couplers occurs can be more easily avoided by tuning the magnetic flux. Therefore, tuning the chain to its Majorana sweet spot can be facilitated.

[0067] In general, a quantum-dot chain 100B comprises: a pair of two outer quantum dots 1(1), 1(2) and is configured to host a pair of Majorana zero modes (y13y2) in the first pair of two outer quantum dots 1(1), 1(2); a chain of alternating quantum-dots (1(1), 1(2), 5(1)) and superconducting couplers (2(1), 2(3)); and a superconductor (2) comprising a superconducting link (2(2)) and the superconducting couplers (2(1), 2(3)) so as to form a single connected superconductor. In the example of Fig. 2a, quantum dots 1(1), 5(1), 1(2) and superconducting couplers 2(1), 2(3) form the chain of alternating quantum-dots and superconducting couplers. The superconductor 2 forms a single connected superconductor and comprises the superconducting link 2(2) and the superconducting couplers 2(1), 2(3).

[0068] In the example of Fig. 2b, quantum dots 1(1), 5(1), 5(3), 1(2) and superconducting couplers 2(1), 2(3), 2(5) form the chain of alternating quantum-dots and superconducting couplers. The superconductor 2 forms a single connected superconductor and comprises the superconducting link 2(2) and the superconducting couplers 2(1), 2(3), 2(5).

[0069] The quantum-dot chain 200B is not limited to the illustrations of Figs. 2a-2b, but may for instance comprise a longer chain of accordingly more quantum-dots and more superconducting couplers.

[0070] As outlined above, the new quantum-dot chains 100B can facilitate tuning to the Majorana sweet spot at which the two outer quantum dots 1(1), 1(2) host the pair of Majorana zero modes (y13y2) in the first pair of two outer quantum dots 1(1), 1(2).

[0071] Fig- 3 shows schematically a quantum-dot chain 100 with a pair of two outer quantum dots 1(1) and 1(2) hosting a pair of Majorana zero modes (y15y2). The shown schematics is used in other figures as well for illustration purposes.

[0072] In the present document, a quantum-dot chain with reference sign 100 or 100(j), j an integer, may also refer to a prior-art quantum-dot chain 100A or a quantum-dot chain 100B as conceptualized by the inventors.

[0073] In Fig. 3, a linking element 6(1) (or: coupling element 6(1)) is shown. The linking element 6(1) schematically represents control gates, further quantum dots and superconducting couplers as illustrated in Figs, la-lc, 2a, 2b. In the example of Fig. la, the linking element 6(1) comprises the superconductor 2(1) and the control gates 3(k), k=l, 3, . . ., 3(13). In the example of Fig. lb, the linking element 6(1) comprises the quantum dot 5(1), the superconductors 2(1) and 2(3), and the control gates 3(k), k=l, 3, 5, . . ., 21. In the example of Fig. 1c, the linking element 6(1) comprises the quantum dots 5(1) and 5(3), the superconductors 2(1), 2(3), 2(5), and the control gates 3(k), k=l, 3, 5, . . ., 29. In the examples of Figs. 2a-2b, the linking element 6(1) comprises the superconducting link 2(2) so that the superconductor 2 is formed. The outer quantum dots 1(1) and 1(2) and one or more of the elements of the linking element 6(1) may be based on / formed in a semiconductor 4(1) (not illustrated in Fig. 3) as described above and is thus not repeated here.

[0074] The linking element 6(1) is not limited to one of the illustrations of Figs, la-lc, 2a, 2b, but further combinations may be considered so that the quantum-dot chain 100 is configured to host the pair of Majorana zero modes (y15y2) in its two outer quantum dots 1(1) and 1(2); i.e., so that the quantum-dot chain 100 can be controlled to a parameter regime at which the pair of Majorana zero modes (y15y2) can be formed in its two outer quantum dots 1(1) and 1(2). For example, further quantum dots 5(i) and further superconducting couplers 2(j) together with corresponding further control gates 3(k) (i, j, k integers) may be considered to provide longer quantum-dot chains 100 in which the two outer quantum dots 1(1) and 1(2) are further separated from one another. For example, other combinations and control regimes for tuning the quantum-dot chain 100 into its Majorana sweet-spot for hosting the pair of Majorana zero modes (y15y2) may be considered.

[0075] Fig. 4a shows a qubit 300 comprising quantum-dot chains 100(1) and 100(3) each hosting a respective pair Majorana zero modes, y15y2, and y3, y4, and a readout system 7(1). The read-out system 7(1) in this example comprises tunnelling-control gates 8(1), 8(3) and one or more auxiliary quantum dots 9(1), 9(3).

[0076] Fig. 4a illustrates two auxiliary quantum dots 9(1) and 9(3), but the readout system 7(1) is not limited thereto. For example, the readout system 7(1) may be configured to have one auxiliary quantum dot, or may be configured to have more than two auxiliary quantum dots. The one or more auxiliary quantum dots may be controlled for example via corresponding one or more on-site control gates 10(1), 10(3), 10(5). The illustration shows three on-site control gates 10(1), 10(3), 10(5), which may be suitable for controlling the two illustrated auxiliary quantum dots 9(1), 9(3). In general, the number of on-site control gates may be fabricated according to a preferred number of auxiliary quantum dots one wishes to implement. A position or region in which an auxiliary quantum dot is formed may be determined by the configuration / position of the respective control gate / auxiliary quantum dot relative to the tunnelling-control gates 8(1), 8(3).

[0077] Fig. 4a shows two tunnelling-control gates 8(1) and 8(3). Each tunnelling-control gate may be implemented for example as an electrostatic gate and may be controllable / tuneable via applying a voltage and / or modifying the voltage. The tunnelling-control gate 8(1) may be configured to control single-electron tunnelling between the outer quantum dot 1(2) of the first quantum-dot chain 100(1) and the one or more auxiliary quantum dots 9(1), 9(3). Similarly, the tunnelling-control gate 8(3) may be configured to control single-electron tunnelling between the outer quantum dot 1(3) of the second quantum-dot chain 100(3) and the one or more auxiliary quantum dots 9(1), 9(3).

[0078] The two quantum-dot chains 100(1) and 100(3) are coupled via the readout system 7(1). More specifically, due to the geometric configuration of this example, the read-out system 7(1) can couple the Majorana zero modes y2and y3via the one or more auxiliary quantum dots 9(1), 9(3). For example, via suitably controlling the on-site control gates 10(1), 10(3), 10(5) and the tunnelling-control gates 8(1) and 8(3), the parity operator p23■= IY2Y3 corresponding to coupling the Majorana zero modes y2and Y3canbe implemented.

[0079] The indirect coupling of the Majorana zero modes by means of the one or more auxiliary quantum dots 9(1), 9(3) is preferred as it enables coupling the Majorana zero modes while preserving parity. Thus, the readout system 7(1) provides an efficient way of implementing a single-qubit gate such as the parity operator p23:=iY2Y3 - Theoretical details and background regarding the qubit states and qubit gates operating thereon based on coupling Majorana zero modes may be found for example in [Plugge et al, Karzig et al].

[0080] Fig. 4b shows a qubit 300 similar to the one shown in Fig. 4a, but in which instead a superconducting connection 2(2) / superconducting link 2(2) is provided so as to provide a superconducting connection between superconducting couplers involved in the respective links 6(1), 6(3) of the respective quantum-dot chains. In other words, a superconductor 2 is formed that corresponds to a single connected superconductor 2 and that comprises superconducting couplers 2(j), j=l, 3, 5, ..., involved in the first and second linking elements 6(1) and 6(3) of the respective first 100(1) and second 100(3) quantum-dot chains. In other words, the involved quantum-dot chains 100(1) and 100(3) in Fig. 4b share a common superconductor 2 comprising the superconducting link 2(2) and the superconducting couplers 2(j) that may be included in the respective linking elements 6(1) and 6(3) of the quantum-dot chains 100(1) and 100(3). For example, each of the quantum-dot chains 100(1) and 100(3) may correspond to a quantum-dot chain 100B in which all involved superconducting parts are connected to form a single superconductor. In that case, the superconducting link 2(2) in Fig. 4b indicates that all superconducting couplers of both the quantum-dot chains 100(1) and 100(3) are connecting to a form the single superconductor 2. As further examples, one or more of superconducting couplers of the quantum-dot chain 100(1) may be connected with one or more of superconducting couplers of the quantum-dot chain 100(3) to form the superconductor 2.

[0081] An example of such a single connected superconductor 2 in relation to superconducting couplers involved in linking elements is also illustrated in the context of Fig. 7b

[0082] Employing a single connected superconductor 2 so that all involved superconducting couplers are connected to from a single superconductor is preferred, because thereby tuning the quantum-dot chains 100(1), 100(3) to their respective Majorana sweet spot is facilitated (as described above with reference to 100B). Moreover, a global charging energy may be efficiently added to the superconductor 2 and the respective quantum-dot chain. Further preferred examples of how to efficiently add and control a global charging energy are discussed further below in the context of Figs. 1 la-1 Id.

[0083] Preferably, a total of one or two auxiliary quantum dots 9(1), 9(3) per readout system 7(1) are employed.

[0084] In summary, a quantum-dot chain 100 comprises a pair of two outer quantum dots (1(1), 1(2)) and is configured to host a pair of Majorana zero modes (yx, y2) in the pair of two outer quantum dots (1(1), 1(2)).

[0085] Preferably, the quantum-dot chain 100 is configured to implement a Kiteav chain. Implementing a Kitaev chain can be a feasible way of realizing the pair of Majorana zero modes.

[0086] In general, the qubit 300 is based on quantum-dot chains configured to host Majorana zero modes (y15y2, y3, y4) and comprises: a first quantum-dot chain (e.g., 100(1), 100A, 100B) comprising a first pair of two outer quantum dots and configured to host a first pair of Majorana zero modes in the first pair of two outer quantum dots; a second quantum-dot chain (e.g., 100(3), 100A, 100B) comprising a second pair of two outer quantum dots and configured to host a second pair of Majorana zero modes in the second pair of two outer quantum dots; and a readout system (e.g., 7(1)) comprising one or more auxiliary quantum dots and configured to provide a coupling between the Majorana zero modes.

[0087] Preferably, the provided coupling comprises coupling two Majorana zero modes to an auxiliary quantum dot of the readout system 7(1). Thereby, the coupled Majorana zero modes are not coupled directly but via a quantum-dot energy level of the involved auxiliary quantum dot, which increases fidelity and avoids an overlap of the Majorana wave functions.

[0088] Preferably, each of the first (100(1)) and second (100(3)) quantum-dot chains is configured to implement a respective Kitaev chain, which is an effective way of providing the Majorana zero modes.

[0089] Preferably, the quantum-dot chain 100 comprises superconducting couplers (2(1), 2(3)). For example, the quantum-dot chain 100 may comprise a sequence of alternating quantum dots 5(1), 5(3) and semiconductor-superconductor hybrid elements involving the superconducting couplers 2(1), 2(3).

[0090] Preferably, the quantum-dot chain 100 comprises a single connected superconductor 2 that comprises the superconducting couplers (2(1), 2(3)). Examples are illustrated in Figs. 2a and 2b. By designing the quantum-dot chain 100 to have such a single connected superconductor 2, the tuning to its Majorana sweet spot in which the two Majorana zero modes are realized in the two outer quantum dots 1(1), 1(2) can be facilitated.

[0091] Preferably, the qubit 300 is based on quantum-dot chains involving superconducting couplers and includes a single connected superconductor 2 comprising all of the involved superconducting couplers. For instance, the illustrated example of Fig. 4b, in which a superconductor 2 forms a single connected superconductor 2 and comprises a superconducting link 2(2) and superconducting couplers involved in the linking parts 6(1) and 6(3), is preferred over the illustrated example of Fig. 4a in which superconducting couplers between the different linking elements 6(1) and 6(3) are not connected via an overarching superconducting link 2(2).

[0092] In the context of the qubit 300, employing the single connected superconductor 2 has the additional technical advantage that parity flips during qubit operations can be more easily suppressed. Thereby, it is less likely to leave a qubit subspace when operating the qubit, and thus performing quantum computing is facilitated.

[0093] The inventors have identified a further particularly advantageous configuration of quantum-dot chains for efficiently providing the qubit 300 based on Majorana zero modes and for efficiently controlling the qubit 300.

[0094] The particularly advantageous configuration is based on devising a different geometric configuration that involves curved quantum-dot chains as opposed to the straight / linear quantum-dot chains 100, 100A, 100B, 100(1), 100(3) shown in Figs. 1-4.

[0095] Figs. 5a-5c show examples of curved quantum-dot chains 200A, 200B, 200, each with a pair of two outer quantum dots 1(1), 1(2) for hosting a pair of Majorana zero modes. Fig. 6 shows schematically a curved quantum-dot chain 200 with a pair of two outer quantum dots for hosting a pair of Majorana zero modes.

[0096] Fig. 5a shows an example of a curved quantum-dot chain 200A that has a similar configuration as the straight quantum-dot chain 100A shown in Fig. lb and as schematically shown in Fig. 3, but in which the elements involved in the linking element (6(1) in Fig. 3, 11(1) in Fig. 6) are not arranged linearly, i.e. are not arranged in a straight line. Instead, the superconducting couplers 2(1) and 2(3) and the quantum-dot 5(1) are arranged in a curved geometric configuration. Such a curved geometric configuration may be achieved by accordingly manufacturing the superconducting couplers 2(1) and 2(3) and the quantum-dot 5(1). For example, the involved control gates 3(k) (k=l, 3, 5, ..., 19) may be arranged accordingly so as to establish elements of the linking element to follow a curved line different from a straight line. The curved line may have a non-zero curvature. The curved line may have a constant or a non-constant / varying curvature. Different curved geometric configurations may be considered when realizing the curved quantum-dot chain 200A.

[0097] Fig. 5b shows an example of a curved quantum-dot chain 200B that has a similar configuration as the straight quantum-dot chain 100B shown in Fig. 2a and as schematically shown in Fig. 3, but in which as well the elements involved in the linking element are not arranged linearly / not arranged in a straight line. Similarly as in Fig. 5a, the respective elements are instead arranged in a curved geometric configuration. In this example, the superconducting couplers 2(1) and 2(3) are moreover connected via a superconducting link 2(2), so that a single connected superconductor 2 comprising the superconducting link 2(2) and both superconducting couplers 2(1) and 2(3) is formed. The curved quantum-dot chain 200 may also comprise control gates 3(k) (k=l, 3, 5, ..., 21) for defining and controlling quantum dots 1(1), 1(2), 5(1) and for controlling couplings between neighbouring quantum dots and respective superconducting couplers. The control gates 3(k) are however not illustrated in Fig. 5b as they may be similarly arranged as already shown in Fig. 5a.

[0098] Fig. 5c shows an example of a curved quantum-dot chain 200B that has a similar configuration as the straight quantum-dot chain 100B shown in Fig. 2b and as schematically shown in Fig. 3, but in which as well the elements involved in the linking element are not arranged linearly / not arranged in a straight line and are instead also arranged in a curved manner. Reference signs correspond and the illustration of control gates is omitted for brevity.

[0099] In Figs. 5a-5c, the quantum dots 1(1), 1(2), 5(1), 5(3) of the curved quantum-dot chain 200 can be defined and controlled in the semiconductor 4(1) for example via control gates (not illustrated in Figs. 5b-5c).

[0100] The examples of a curved quantum-dot chain shown in Figs. 5a-5c show that when viewed for example from a top view, a straight line connecting the first outer quantum dot 1(1) and quantum dot 5(1) does not cross the second outer quantum dot 1(2). Similarly, a straight line connecting a third outer quantum dot 1(3) and quantum dot 5(3) does not cross the fourth outer quantum dot 1(4). In other words, a geometric configuration of the curved quantum-dot chainsis different from a linear / straight quantum-dot configuration. For example, a line starting from the first outer quantum dot 1(1) and passing consecutively through the elements of the linking element (e.g., through the chain of alternating superconducting couplers 2(k) and quantum dots 5(j)) and ending at the second outer quantum dot 1(2) may have a non-zero curvature.

[0101] For example, the curved quantum-dot chain 200 A, 200B, 200 may be formed to resemble a half-ring structure in which its two outer quantum dots 1(1) and 1(2) are positioned closer to each other as compared to in the straight quantum-dot chain 100A, 100B, 100.

[0102] In other words, by exploiting the design flexibility of quantum-dot chains, the two outer quantum dots 1(1), 1(2) configured to host the pair of Majorana zero modes may be brought closer together.

[0103] Preferably, the semiconductor 4(1) is based on a two-dimensional electron gas or a two-dimensional hole gas or nanowire networks. Thereby, a higher design flexibility is achieved and quantum dots can be efficiently defined via control gates (e.g., via electrostatic gates). For example, curved quantum-dot chains 200 such as illustrated in Figs. 5a-5c may be fabricated based on a two-dimensional electron gas / two-dimensional hole gas / nanowire networks semiconductor 4(1). This allows for improved interconnectivity of the various Majorana zero modes and for more design flexibility of qubit geometries, as discussed further below with reference to Fig. 7.

[0104] Fig- 6 shows schematically a curved quantum-dot chain 200 in which, relative to the illustration of a straight quantum-dot chain 100, the linking element 11(1) is shown curved and not straight. The linking element 11(1) shown in Fig. 6 may otherwise however contain the same elements as the linking element 6(1) shown in Fig. 3, i.e. one or more quantum dots 5(j) interleaved with one or more superconducting couplers 2(k), and control gates 3(j). Examples of the quantum-dot chain 200 of Fig. 6 are shown in Figs. 5a-5c (200 A, 200B).

[0105] Preferably, the qubit 300 is implemented based on one or more curved quantum-dot chains 200, which is particularly advantageous as it allows for design flexibility and may enable and facilitate coupling the Majorana zero modes.

[0106] Preferably, the qubit (300) is based on quantum-dot chains configured to host Majorana zero modes and comprises: a first curved quantum-dot chain comprising a first pair of two outer quantum dots and configured to host a first pair of Majorana zero modes in the first pair of two outer quantum dots; a second curved quantum-dot chain comprising a second pair of two outer quantum dots and configured to host a second pair of Majorana zero modes in the second pair of two outer quantum dots; and a readout system comprising one or more auxiliary quantum dots and configured to provide a coupling between the Majorana zero modes.

[0107] Preferably, each of the first and second curved quantum-dot chains is configured to implement a respective Kitaev chain. Examples of the qubit 300 based on curved quantum-dot chains are discussed next.

[0108] Fig. 7a shows a qubit 300 based on curved quantum-dot chains 200A(l) and 200A(3) each configured to host a respective pair of Majorana zero modes (y15y2) and (y3, y4) in their respective two outer quantum dots 1(1), 1(2) and 1(3), 1(4), respectively. The two curved quantum-dot chains 200A(l) and 200A(3) are coupled via a readout system 7(1) that is configured to provide a coupling between the Majorana zero modes (y15y2, y3, y4). The first and second curved quantum-dot chains 200A(l) and 200A(3) are each an example of a curved quantum-dot chain 200 as shown in Figs. 5a and schematically shown in Fig. 6.

[0109] The first curved quantum-dot chain 200A(l) comprises a first pair of two outer quantum dots 1(1), 1(2) and is configured to host a first pair of Majorana zero modes (y15y2) in the first pair of two outer quantum dots 1(1), 1(2). The second curved quantum-dot chain 200A(3) comprises a second pair of two outer quantum dots 1(3), 1(4) and is configured to host a second pair of Majorana zero modes (y3, y4) in the second pair of two outer quantum dots 1(3), 1(4). The readout system 7(1) of the qubit 300 comprises one or more auxiliary quantum dots 9(1), 9(3) and is configured to provide a coupling between the Majorana zero modes (y15y2, y3, y4). The first curved quantum-dot chain 200A(l) may further comprise one or more superconducting couplers 2(1), 2(3) and quantum dots such as the illustrated quantum dot 5(1) and control gates 3(k) as illustrated in Fig. 5a but that are not again illustrated in Fig. 7a. Similarly, the second curved quantum-dot chain 200A(3) may also further comprise one or more superconducting couplers 2(5), 2(7) and quantum dots such as the illustrated quantum dot 5(3) and as well control gates 3 (k) as illustrated in Fig. 5a but that are not again illustrated in Fig. 7a.

[0110] For providing the coupling between the Majorana zero modes (y15y2, y3, y4), the readout system 7(1) comprises one or more auxiliary quantum dots 9(1), 9(3) and may comprise a number of tunnelling-control gates 8(1), 8(3), 8(5), 8(7) and on-site control gates 10(1), 10(3), 10(5) for controlling the respective one or more auxiliary quantum dots 9(1), 9(3). Each of the tunnelling-control gates 8(1), 8(3), 8(5), 8(7) may be configured to control a respective tunnelling amplitude between a corresponding auxiliary quantum dot 9(1), 9(3) and a corresponding outer quantum dot 1(1), 1(2), 1(3), and / or 1(4). For example, the tunnelling- control gate 8(5) may be configured to control a tunnelling amplitude between one or more auxiliary quantum dots and the outer quantum dot 1(3) that is configured to host a Majorana zero mode y3. Similarly, the tunnelling-control gate 8(3) may be configured to control a tunnelling amplitude between one or more auxiliary quantum dots and the outer quantum dot 1(2) that is configured to host a Majorana zero mode y2. Thereby, a coupling between Majorana zero modes y2and Ks may be provided, similarly as for the examples of Figs. 4a and 4b based on straight quantum-dot chains 100(1), 100(3).

[0111] However, due to the involvement of curved quantum-dot chains 200A(l) and 200 A(3) in the qubit 300, couplings between more pairs of the Majorana zero modes (y15y2, y3, y4) are enabled.

[0112] For example, the coupling between the Majorana zero modes (yx, y2, y3, y4) may additionally comprise a first coupling between the first yxand the second y2Majorana zero modes of the first curved quantum-dot chain 200(1), because the curved quantum-dot chain 200(1) allows said first coupling via the readout system 7(1) and the involved two outer quantum dots 1(1) and 1(2), which are relatively closer to each other and / or are simultaneously closer to the readout system 7(1). Furthermore, the coupling may additionally comprise a second coupling between Majorana zero modes yxand y3, via controlling the tunnelling-control gates 8(1) and 8(5) and the one or more auxiliary quantum dots. Furthermore, the coupling may additionally comprise a third coupling between Majorana zero modes y2and y3as described further above.

[0113] The first, second and third couplings may thus all be realized thanks to the readout system 7(1) being positioned in proximity to all four outer quantum dots 1(1), 1(2), 1(3), 1(4). The proximity is enabled by implementing the curved quantum-dot chains 200A(l) and 200A(3) instead of straight quantum-dot chains 100.

[0114] Together, the first, second and third couplings enable the implementation of a universal set of single-qubit quantum gates acting on the qubit 300. The first coupling between yxand y2allows to implement the Pauli operator <JX= iyxy2. The second coupling between yxand y3allows to implement the Pauli operator oy= iy^. The third coupling between y2and y3allows to implement the Pauli operator <JZ= iy2y3. The geometric arrangement further allows to implement other single-qubit gates such as phase gates and Hadamard gates to achieve universality. The qubit 300 may thus be regarded as an example implementation of the Majorana box qubit and the above-described couplings enable not only to realize but also to controllably manipulate the Majorana box qubit. Some more details regarding the Majorana box qubit may be found in [Plugge et al] and references therein.

[0115] Fig. 7b shows a preferred example of the qubit 300 based on curved quantum-dot chains 200B(l) and 200B(3) configured to host the Majorana zero modes (y15y2, y3, y4). While the elements and capabilities are as illustrated in and described with reference to Fig. 7a and are therefore not repeated, the example shown Fig. 7b is preferred because in addition the involved curved quantum-dot chains 200B(l) and 200B(3) share a common superconductor 2. The common superconductor 2 comprises the superconducting couplers 2(1) and 2(3) involved in the first curved quantum-dot chain 200(1) and further comprises the superconducting couplers 2(5) and 2(7) involved in the second curved quantum-dot chain 200(3). The common superconductor 2 further comprises a superconducting link 2(2) that connects all the superconducting couplers.

[0116] The qubit 300 comprising a single connected superconductor 2 comprising all superconducting couplers 2(l)-2(7) and the superconducting link 2(2) has advantages as already described further above in the context of the quantum-dot chains 100B, 200B. Among other things, also a global charging energy may be obtained that can conserve a global parity of the qubit 300. Therefore, stability of the qubit 300 can be enhanced.

[0117] Figs. 8a and 8b show schematically a respective generalization of the examples of Figs. 7a and 7b, respectively. In these generalizations, the curved quantum-dot chains 200(1) and 200(3) are each of arbitrary length. For example, each of the linking elements 11(1) and 11(3) in Figs. 8a and 8b may comprise as many superconducting couplers 2(i), quantum dots 5(j) and control gates 3(k) as illustrated in the foregoing figures, but may also comprise more of such elements for example so as to implement longer Kitaev chains. One or more of the curved quantum-dot chains 200(1), 200(3) may each correspond to a curved quantum-dot chain 200A in which the superconducting couplers are not connected or to a curved quantumdot chain 200B in which the superconducting couplers are connected to form a single connected superconductor.

[0118] As shown in Fig. 8b, also when the length of each curved quantum-dot chain 200(1), 200(3) is longer, it is still preferred to fabricate a shared superconductor 2 comprising a superconducting link 2(2) and all superconducting couplers 2(k) involved in the curved quantum-dot chains 200(1) and 200(3). A maximum length of the involved curved quantumdot chains 200(1), 200(3) may be related to feasibility of controlling a corresponding number of control gates and / or a preferred maximum size of the respective curved quantum-dot chain. In general, utilizing longer chains can improve resilience of the Majorana zero modes against noise and may thereby improve qubit coherence.

[0119] The qubit 300 may also be extended to 6 Majoranas by joining three quantum-dot chains. For example, three curved quantum-dot chains may be implemented to a single parityconserving geometry.

[0120] Furthermore, combinations of different quantum-dot chains may be considered for the implementing of the qubit 300. For example, a first quantum-dot chain for hosting the first pair of Majorana zero modes may be any one 100 A, 100B, 100, 200 A, 200B, 200 and a second quantum-dot chain for hosting the second pair of Majorana zero modes may as well be any one of 100A, 100B, 100, 200A, 200B, 200.

[0121] Figs. 9a-9c show examples of two-qubit systems 500 based on combining instances of qubits 300, referred to by 300(1) and 300(3) and a readout system 7(3). The qubits 300(1) and 300(3) are each an instance of the qubit 300 illustrated with reference to Figs. 4-8, preferably also involving curved quantum-dot chains.

[0122] The two-qubit system 500 illustrated in Fig. 9a comprises one or more readout systems 7(1), 7(3) (which may also be regarded as comprised in a readout system 7) with one or more auxiliary quantum dots 9(1), 9(3), 9(5), 9(7) and tunnelling-control gates 8(k), k = 1, 3, ..., 17. Furthermore, the two-qubit system 500 may comprise control gates involved in the (curved) quantum-dot chains and on-site control gates for the one or more auxiliary quantum dots 9(l)-9(7). However, such control gates and on-site control gates are not again illustrated in Figs.9a-9c as they may be implemented similarly as already described in the aforementioned and described figures.

[0123] The first qubit 300(1) in the illustration of Fig. 9a comprises a first curved quantumdot chain 200(1) and a second curved quantum-dot chain 200(3). Each of the curved quantum-dot chains 200(1), 200(3) is configured to host a respective pair of Majorana zero modes (y15y2; Ts- IT) intheir respective pair of outer quantum dots (1(1), 1(2); 1(3), 1(4)). Similarly, the second qubit 300(3) comprises a third curved quantum-dot chain 200(5) and a fourth curved quantum-dot chain 200(7). Each of the curved quantum-dot chains 200(5), 200(7) is configured to host a respective pair of Majorana zero modes (y5, y6; y7, y8) in their respective pair of outer quantum dots (1(5), 1(6); 1(7), 1(8)).

[0124] Since in each qubit 300(1), 300(3), the respective curved quantum-dot chains (200(1), 200(3); 200(5), 200(7)) are each comprising a single common superconductor (2-1; 2-3), each qubit 300(1), 300(3) possesses a charging energy and can conserve total parity. For example, the single common superconductor 2-1 of the qubit 300(1) comprises the superconducting link 2(2) and the superconducting couplers involved in the links 11(1) and 11(3). Similarly, the single common superconductor 2-3 of the qubit 300(3) comprises the superconducting link 2(4) and the superconducting couplers involved in the links 11(5) and 11(7). Each of the superconductors 2-1, 2-3 corresponds to a superconductor 2 as described above, and a detailed description is therefore not repeated.

[0125] The readout system 7 (or the one or more readout systems 7(1), 7(3)) comprising the one or more auxiliary quantum dots 9(l)-9(7) may provide a coupling between the Majorana zero modes (y1, y2, y3,y4, y3, y6, y?,ys). For example, by enabling coupling of auxiliary quantum dots 9(1), 9(3) and auxiliary quantum dots 9(5), 9(7) via a tunnelling control gate 8(17), one can measure a two-qubit parity entangling the two qubits 300(1), 300(3), such as °zi°z2=~YeY?Y2Y3- Other two-qubit operations based on coupling different Majorana zero modes via auxiliary quantum dots of the readout system 7 can be considered and are efficiently enabled by the implementation based on quantum-dot chains.

[0126] Fig. 9b illustrates schematically that a lower part of Fig. 9a involving Majorana zero modes y4— y4and an upper part of Fig. 9a involving Majorana zero modes y5— y8each implement a respective qubit 300(1) [qubit 1], 300(3) [qubit 2],

[0127] That is, by employing curved quantum-dot chains 200(1), 200(3), 200(5), 200(7) that are each configured to host a respective pair of Majorana zero modes Y2k-i> Y2k), k=l, . .., 4, and a readout system 7 coupling the Majorana zero modes, a two-qubit system 500 hosting two qubits 300(1), 300(3) can be realized and moreover can be controlled to measure two- qubit parity.

[0128] Thereby, qubits 300, 500 can be efficiently provided together with a universal set of quantum gates.

[0129] Fig. 9c illustrates yet another example of an arrangement involving a first qubit 300(1) and a second qubit 300(3) coupled via a readout system 7 (7(1), 7(3)). In this example, the readout system is arranged differently so as to facilitate the coupling of different Majorana zero modes as compared to the arrangement shown in Figs. 9a-9b. Furthermore, the number of auxiliary quantum dots is not limited to the illustrated number but can be one or more. Fig. 9d illustrates for example a case of one auxiliary quantum dot (9(11) and 9(13), respectively) per involved readout system. Furthermore, a two-qubit system is not limited to employing curved quantum-dot chains 200, but may also include straight quantum-dot chains 100. Further geometric arrangements different than those illustrated may be considered as well.

[0130] Fig. 9d illustrates that the readout-system 7 may comprise one auxiliary quantum dot for enabling coupling and reading out of a plurality of outer quantum dots and / or a plurality of Majorana zero modes. Different parts of the readout-system 7, such as 7(1), 7(3), may each comprise one or more auxiliary quantum dots. Fig. 9d illustrates one quantum dot per part.

[0131] Furthermore, combinations of different quantum-dot chains may be considered for the implementing of the two-qubit system 500. For example, any involved quantum-dot chain for hosting a first pair of Majorana zero modes may be any one of 100 A, 100B, 100, 200 A, 200B, 200. Similarly, combinations of different instances of the qubit 300 may be involved in a two- qubit system. Figs. lOa-lOb show examples of multi-qubit systems 700 involving qubits 300(k) (k=l, 3, 5, . . 19), based on employing a mix of straight and curved quantum-dot chains each configured to host respective Majorana zero modes.

[0132] As can be seen from the illustrations in Figs. lOa-lOb, the qubits proposed in the present document may be incorporated in a quantum computing chip comprising a plurality of qubits. Furthermore, one can implement combinations of different local arrangements and mixtures of curved quantum-dot chains — as in qubits 300(1), 300(3), 300(5) — and straight quantum-dot chains — as in qubit 300(7).

[0133] Furthermore, the inventors found that controllability of the qubits can be increased when employing a gate-tuneable junction so that the qubits can be operated with and / or without a charging energy. This is illustrated in Figs, lla-d, which illustrate employing a Josephson junction 13(1) for tuning an electron density. The invention is not limited to the illustrated Josephson junction 13(1) however, but also a Normal-metal-superconductor junction may be employed.

[0134] Fig. Ila shows the quantum-dot chain 100B of Fig. 2a, but in which additionally the superconductor 2 is connected to a Josephson junction 13(1). The Josephson junction 13(1) in this example is configured to couple the superconductor 2 and a grounded superconductor 15(1). The grounded superconductor 15(1) may also be referred to as a grounded superconducting lead 15(1). One or more control gates 14(1), 14(3) may be employed to enable tuning the connection between the superconductor 2 of the quantum-dot chain 100 and the superconducting lead 15(1).

[0135] By coupling the superconductor 2 to the superconducting lead 15(1) via the gate- tuneable Josephson junction 13(1), the quantum-dot chain 100 may be efficiently provided with a gate-tuneable charging energy. Thereby, qubits based on a quantum-dot chain 100 may be better protected from quasi-particle poisoning and a charging energy can be efficiently employed.

[0136] The Josephson junction 13(1) may involve part of the semiconductor 4(1), as illustrated in Fig. 11b in the context of the quantum-dot chain 100 of Fig. la. In Fig. 11b, additionally normal leads (’’Normal leads”) are shown, and quantum dots (“Quantum dots”) corresponding to outer quantum dots 1(1) and 1(2) of Fig. la and with control gates (“Kitaev chain gates”) corresponding to control gates 3(k) of Fig. la, k=l, 3, 5, ..., 13. The superconductor 2(1) in Fig. la is illustrated to be grounded, but in Fig. 11b the corresponding superconductor is extended to form a superconducting bridge (“Superconducting bridge”) that forms part of a gate-tuneable Josephson junction (“Gate-tunable Josephson junction”), which is an example of the Josephson junction 13(1). Control gates (“Josephson junction gates”) can be used to tune the connection between the superconducting bridge and a superconducting lead (“Superconducting lead”), resulting in a gate-tuneable charging energy of the quantumdot chain.

[0137] The addition of a gate-tuneable junction to couple the superconductor 2 to a superconducting lead 15(1) is not limited to the prior-art quantum-dot chains illustrated in Figs, la-lc, but is preferably also employed in the context of a curved quantum-dot chain 200.

[0138] Fig. 11c illustrates for example the quantum-dot chain 200 of Fig. 5b in which additionally the superconductor 2, in this illustration more specifically the superconducting link 2(2), is connected to a superconducting lead 15(1) via a Josephson junction 13(1). The superconducting lead 15(1) may be grounded. The connection between the superconductor 2 and the superconducting lead 15(1) may be controlled via one or more control gates 14(1), 14(3) (also: Josephson junction gates 14(1), 14(3)).

[0139] Similarly, Fig. lid illustrates that the qubit 300 comprising curved quantum-dot chains 200(1) and 200(3) preferably further includes a Josephson junction 13(1). The illustration of Fig. lid shows the qubit 300 as shown in Fig. 8b, but additionally comprising a gate-tuneable Josephson junction 15(1). As in the other illustrations of Figs, lla-llc, the superconductor 2 is connected to a superconducting lead 15(1) via the gate-tuneable Josephson junction 13(1). The connection between the superconductor 2 and the superconducting lead 15(1) may be tuned via one or more control gates 14(1), 14(3). The superconducting lead 15(1) may be grounded.

[0140] Similarly, a coupling to a superconducting lead 15(1) via a Josephson junction 13(1) may be employed in any other configuration of the qubit 300, 300(j), curved quantum-dot chains 200, 200(k), two-qubits 500, multi-qubit systems, etc.. For example, one or more of involved superconductors in the respective curved quantum-dot chains 200 may each be separately coupled to a respective superconducting lead 15(1) via a respective Josephson junction 13(1) that can be tuned via respective one or more control gates 14(1), 14(3).

[0141] In general, the superconductor 2 in any one of the above illustrations in Figs 1-11 is not directly grounded but is provided with a charging energy.

[0142] The addition of a Josephson junction 13(1) coupling to a grounded superconducting lead 15(1) such as illustrated in Figs, lla-lld is an example of weakly and controllably coupling the superconductor 2 to a grounded superconducting lead. The addition may facilitate a tune-up procedure of the respectively involved quantum-dot chains 100 and curved quantum-dot chains 200. Furthermore, the addition may provide an efficient way of protecting the respective qubits 300, 300(j), 500, from interactions with the environment, which may efficiently reduce for example quasi-particle poisoning.

[0143] For example, with the gate-tuneable Josephson junction 13(1) one can control a resistance of the involved device / quantum-dot chain 100 / curved quantum-dot chain 200 to the ground. On the one hand, the Josephson junction 13(1) can be set to a high transmission which corresponds to low resistance to the ground. On the other hand, the Josephson junction 13(1) can be depleted leaving the involved device / quantum-dot chain 100 / curved quantumdot chain 200 floating. This ground tunability may form a crucial part for an initial qubit tune- up procedure and for protection against accidental parity switches.

[0144] Next, examples of tuning up (A) and floating (B) are briefly described.

[0145] A) A qubit tune-up may achieve non-Abelian excitations in the involved quantum dots.

[0146] 1) First, the gate-tunable Josephson junction 13(1) is may be set to a high transparency such that the resistance to the ground is minimal.

[0147] 2) Second, hybrid segments of the involved quantum-dot chain 100 / curved quantum-dot chain 200 (i.e., the parts involving the superconducting couplers 2(1), 2(3), 2(5), etc.) may be tuned to host Andreev Bound States. The quantum dots 1(1), 1(2), 1(3), etc, 5(1), 5(3), etc. may be formed in the semiconductor 4(1) using the respective control gates 3(k), k = 1, 3, etc., in a few electron regime with large (> 2.5 meV) level spacing. Tune up can be realized via tunneling spectroscopy using normal leads or via dispersive gate sensing.

[0148] 3) Third, one may tune / increase the interaction of the quantum dots 1(1), 1(2), 1(3), etc, 5(1), 5(3), etc. of the involved quantum-dot chain 100 / curved quantum-dot chain 200 via the Andreev bound states. This may be specifically done by tuning (e.g., increasing) for example tunneling rates from the respective quantum dots to the respective Andreev bound states through increasing the transparency of the barrier next to the superconducting couplers, for example by lowering a voltage applied to tunnelling-control gates.

[0149] 4) By tuning up the chemical potential in the hybrid segments / superconducting couplers one can reach the Majorana sweet-spot of the Kitaev chains wherein Majorana zero modes are formed in the outer quantum dots. Details of reaching the Majorana sweet spot are not described here because they are known [Dvir et al],

[0150] B) Floating by lifting the ground connection: While being a Majorana sweet-spot, transparency of the connection between the superconductor 2 and the superconducting lead 15(1) via the gate-tuneable Josephson junction 13(1) may be reduced through electrostatic gating. Simultaneously, one may pinch-off the normal leads with electrostatic gates. This step may have a two-fold result. By floating, the involved quantum-dot chain 100 / curved quantum-dot chain 200 may gain a finite charging energy. If a charging energy is added to the system of four Majorana zero modes, a global parity of the system can be conserved and the four Majorana operators can be mapped to a degenerate system with spin-1 / 2 degrees of freedom. In such a system, the measurement of the parity of different pairs of Majorana zero modes can correspond to the three Pauli operators that can implement single-qubit gates in a measurement-based protocol. Examples thereof are also described with reference to Figs. 7a-7b. Additionally, the charging energy makes it energetically unfavorable to add stray quasiparticles to the system, which may otherwise poison the qubit.

[0151] The above-described floatable ground is not exclusive to the qubits 300, 500 based on curved quantum-dot chains 200 described in here, but can also be used for other proposals regarding topological qubits.

[0152] As mentioned further above, the invention is not limited to the illustrated Josephson junction 13(1) however, but also a Normal-metal-superconductor junction may be employed. In this case, the grounded superconducting lead 15(1) may be replaced by a grounded metal and the Josephson junction 13(1) may be replaced by a Normal-metal-superconductor junction.

[0153] The present document thus discloses at least the following. Technical advantages of the various parts may be understood by the skilled person from the above-described illustrations.

[0154] Qubit based on quantum-dot chains configured to host Majorana zero modes, the qubit comprising: a first quantum-dot chain comprising a first pair of two outer quantum dots and configured to host a first pair of Majorana zero modes in the first pair of two outer quantum dots; a second quantum-dot chain comprising a second pair of two outer quantum dots and configured to host a second pair of Majorana zero modes in the second pair of two outer quantum dots; and a readout system comprising one or more auxiliary quantum dots and configured to provide a coupling between the Majorana zero modes.

[0155] Preferably, each of the first and second quantum-dot chains is configured to implement a respective Kitaev chain. Preferably, wherein at least one of the first and second quantum-dot chains is a curved quantum-dot chain.

[0156] Preferably, the readout system is configured to provide the coupling between the Majorana zero modes by: coupling, via its one or more auxiliary quantum dots, at least two quantum dots of the four outer quantum dots, and / or coupling, via its one or more auxiliary quantum dots, at least one quantum-dot of the first pair of two outer quantum dots with at least one quantum-dot of the second pair of two outer quantum dots.

[0157] Preferably, the readout system further comprises one or more tunnelling-control gates each configured to tune a respective tunnelling amplitude between a respective auxiliary quantum-dot of the readout system and a respective outer quantum-dot.

[0158] Preferably, at least one of the first pair of two outer quantum dots and the second pair of two outer quantum dots is located in proximity to the readout system.

[0159] Preferably, the provided coupling between the Majorana zero modes comprises: a first coupling between a first and second Majorana zero mode of the first pair of Majorana zero modes, a second coupling between the first Majorana zero mode of the first pair of Majorana zero modes and a third Majorana zero mode of the second pair of Majorana zero modes, and a third coupling between the second (y2) Majorana zero mode of the first pair of Majorana zero modes and the third Majorana zero mode of the second pair of Majorana zero modes.

[0160] Preferably, each of the first, second and third couplings is provided via coupling the respective outer quantum dots to one or more of the one or more auxiliary quantum dots.

[0161] Preferably, for the implementing of the respective Kitaev chain, each or both of the first and second quantum-dot chains comprises a respective linking part that involves a chain of alternating respective superconducting couplers and respective quantum dots.

[0162] Preferably, the qubit further comprises a superconductor comprising the respective superconducting couplers and a respective superconducting link so as to form a single connected superconductor so that all involved superconducting couplers in the linking parts of the first and second quantum-dot chains are part of the single connected superconductor.

[0163] Preferably, the superconductor is connected to a grounded superconducting lead via a gate-tuneable Josephson junction or is connected to a grounded metal lead via a Normal- metal-superconductor junction. Preferably, each of the first and second quantum-dot chains is implemented based on one of a two-dimensional electron-gas, 2DEG, a two-dimensional hole gas, 2DHG, or a nanowire network.

[0164] Two-qubit system comprising a first and a second instance of the qubit and a readout system comprising the readout systems of the first and second qubits, respectively, the first qubit involving a first group of Majorana zero modes and the second qubit involving a second group of Majorana zero modes, wherein the readout system is configured to provide a coupling between the first and second groups of Majorana zero modes.

[0165] Preferably, the provided coupling between the first and second groups of Majorana zero modes comprises a fourth coupling between a first sub-group of two Majorana zero modes of the first group and a second sub-group of further two Majorana zero modes of the second group.

[0166] Preferably, the provided coupling between the first and second groups of Majorana zero modes involves a fifth coupling of respective involved outer quantum dots, configured to host respective Majorana zero modes, to one or more auxiliary quantum dots of the readout system.

[0167] Quantum-computing chip for implementing topological quantum computing, comprising one or more instances of the qubit and / or comprising one or more instances of the two-qubit system.

[0168] Quantum-dot chain comprising: a pair of two outer quantum dots configured to host a pair of Majorana zero modes in the first pair of two outer quantum dots; a chain of alternating quantum-dots and superconducting couplers; and a superconductor comprising a superconducting link and the superconducting couplers so as to form a single connected superconductor.

[0169] Curved quantum-dot chain comprising: two outer quantum dots configured to host a pair of Majorana zero modes in the first pair of two outer quantum dots; and a chain of alternating quantum-dots and superconducting couplers, wherein a line starting at one outer quantum dot and going through the chain and ending at the other outer quantum dot is a curved line.

[0170] In general, for the present disclosure, any semiconductor material for engineering quantum dots may be employed and any superconductor material may be employed.

[0171] Preferably, the semiconductor is based on a two-dimensional electron gas (2DEG), a two-dimensional hole gas (2DHG), and / or nanowire networks. Preferably, the semiconductor is based on one or more of Indium Antimonide Arsenide (InSbAs), Indium Arsenide (InAs), Indium Antimonide (InSb), Germanium, and / or van der Waals materials such as Graphene, Tungsten diselenide (WSe2), Niobium diselenide (NbSe2), Boron nitride (hBN), and the semiconductor 4(1) may comprise stacks and / or mixtures and / or combinations thereof. As an example, the semiconductor may be a Germanium 2DEG or a Germanium 2DHG. Preferably, the semiconductor allows for spinorbit interaction. The spin-orbit interaction may be internal and / or induced.

[0172] Preferably, the superconductor and / or any of the superconducting elements 2(k), k=l, 2, . . ., are based on Aluminium, Vanadium, Lead, Tin, Indium, Niobium and / or Tantalum.

[0173] Preferably, the qubits and quantum-dot chains are operated at relatively low temperatures, e.g. in a milli-Kelvin range such as below lOOOmK, so as to be operated in a low energy regime.

[0174] The following list of references is referred to in the present document and is incorporated herein by way of reference.

[0175] List of references

[0176] [Aseev et al] Pavel Aseev, Guanzhong Wang, Luca Binci, Amrita Singh, Sara Marti -Sanchez, Marc Botifoll, Lieuwe J. Stek, Alberto Bordin, John D. Watson, Frenk Boekhout, Daniel Abel, John Gamble, Kevin Van Hoogdalem, Jordi Arbiol, Leo P. Kouwenhoven, Gijs de Lange, and Philippe Caroff, Ballistic InSb Nanowires and Networks via Metal-Sown Selective Area Growth, Nano Lett. 2019, 19, 12, 9102-9111, November 15, 2019 http s : / / doi . org / 10.1021 / acs . nanol ett .9b 04265.

[0177] [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).

[0178] [Bordin et al, 2] Alberto Bordin, Xiang Li, David van Driel, Jan Cornells 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).

[0179] [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).

[0180] [Jason Jung et al] Jason Jung, Sander G. Schellingerhout, Markus F. Ritter, Sofieke C. ten Kate, Orson A.H. van der Molen, Sem de Loijer, Marcel A. Verheijen, Heike Riel, Fabrizio Nichele, Erik P.A.M. Bakkers, Selective Area Growth of PbTe Nanowire Networks on InP, Advanced Functional Materials Volume 32, Issue 51 2208974.

[0181] [Karzig et al] Torsten Karzig, Christina Knapp, Roman M. Lutchyn, Parsa Bonderson, Matthew B. Hastings, Chetan Nayak, Jason Alicea, Karsten Flensberg, Stephan Plugge, Yuval Oreg, Charles M. Marcus, and Michael H. Freedman, Scalable designs for quasiparticle- poisoning-protected topological quantum computation with Majorana zero modes, Physical Review B 95.23 (2017): 235305.

[0182] [Kitaev] Alexei Kitaev, Unpaired Majorana fermions in quantum wires, arXiv, Physics- Uspekhi 44, 131 (2001).

[0183] [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).

[0184] [Op het Veld et al] Roy L. M. Op het Veld, Di Xu, Vanessa Schaller, Marcel A. Verheijen, Stan M. E. Peters, Jason Jung, Chuyao Tong, Qingzhen Wang, Michiel W. A. de Moor, Bart Hesselmann, Kiefer Vermeulen, Jouri D. S. Bommer, Joon Sue Lee, Andrey Sarikov, Mihir Pendharkar, Anna Marzegalli, Sebastian Koelling, Leo P. Kouwenhoven, Leo Miglio, Chris J. Palmstrom, Hao Zhang & Erik P. A. M. Bakkers, In-plane selective area InSb Al nanowire quantum networks, Commun Phys 3, 59 (2020). https: / / doi.org / 10.1038 / s42005-020-0324-4.

[0185] [Plugge et al] Stephan Plugge, Asbjorn Rasmussen, Reinhold Egger and Karsten Flensberg, Majorana box qubits, New Journal of Physics 19.1 (2017): 012001.

[0186] [Sarnia et al] Sankar Das Sarma, Michael Freedman, and Chetan Nayak, Majorana zero modes and topological quantum computation, Npj Quantum Information 1, 15001 EP - (2015).

[0187] [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).

[0188] [Q. Wang et al] Qingzhen Wang, Sebastiaan LD ten Haaf, Ivan Kulesh, Di Xiao, Candice Thomas, Michael J Manfira, and Srijit Goswami, Triplet cooper pair splitting in a two- dimensional electron gas, arXiv:2211.05763 (2022). Aspects are disclosed in the following clauses 1-17.

[0189] 1. Qubit (300) based on quantum-dot chains configured to host Majorana zero modes (y15y2, y3, Y4 comprising: a first quantum-dot chain (100(1), 200(1)) comprising a first pair of two outer quantum dots (1(1), 1(2)) and configured to host a first pair of Majorana zero modes (y15y2) in the first pair of two outer quantum dots (1(1), 1(2)); a second quantum-dot chain (100(3), 200(3)) comprising a second pair of two outer quantum dots (1(3), 1(4)) and configured to host a second pair of Majorana zero modes (y3, y4) in the second pair of two outer quantum dots (1(3), 1(4)); and a readout system (7(1)) comprising one or more auxiliary quantum dots (9(1), 9(3)) and configured to provide a coupling between the Majorana zero modes (y15y2, y3, y4).

[0190] 2. The qubit (300) of clause 1, wherein each of the first and second quantum-dot chains is configured to implement a respective Kitaev chain.

[0191] 3. The qubit (300) of any one of the preceding clauses, wherein at least one of the first and second quantum-dot chains is a curved quantum-dot chain (200).

[0192] 4. The qubit (300) of any one of the preceding clauses, wherein the readout system (7(1)) is configured to provide the coupling between the Majorana zero modes (y15y2, y3, y4) by: coupling, via its one or more auxiliary quantum dots (9(1), 9(3)), at least two quantum dots of the four outer quantum dots (1(1), 1(2), 1(3), 1(4)), and / or coupling, via its one or more auxiliary quantum dots (9(1), 9(3)), at least one quantum-dot of the first pair of two outer quantum dots (1(1), 1(2)) with at least one quantumdot of the second pair of two outer quantum dots (1(3), 1(4)).

[0193] 5. The qubit (300) of clause 4, wherein the readout system (7(1)) further comprises one or more tunnelling-control gates (8(1), 8(3), 8(5), 8(7)) each configured to tune a respective tunnelling amplitude between a respective auxiliary quantum-dot of the readout system (7(1)) and a respective outer quantum-dot (1(1), 1(2), 1(3), 1(4)).

[0194] 6. The qubit (300) of any one of the preceding clauses, wherein at least one of the first pair of two outer quantum dots (1(1), 1(2)) and the second pair of two outer quantum dots (1(3), 1(4)) is located in proximity to the readout system 7(1).

[0195] 7. The qubit (300) of any one of the preceding clauses, wherein the provided coupling between the Majorana zero modes (y15y2, y3, y4) comprises: a first coupling (<J%= iy^) between a first (yx) and second (y2) Majorana zero mode of the first pair of Majorana zero modes (y15y2), a second coupling (ay= 1 / 1 / 3) between the first Majorana zero mode ( / - of the first pair of Majorana zero modes (y15y2) and a third Majorana zero mode (y3) of the second pair of Majorana zero modes (y3, y4), and a third coupling (<JZ= i / 2 / 3) between the second (y2) Majorana zero mode of the first pair of Majorana zero modes (y15y2) and the third Majorana zero mode (y3) of the second pair of Majorana zero modes (y3, y4), wherein each of the first, second and third couplings is provided via coupling the respective outer quantum dots to one or more of the one or more auxiliary quantum dots.

[0196] 8. The qubit (300) of clause 2, wherein for the implementing of the respective Kitaev chain, each or both of the first and second quantum-dot chains comprises a respective linking part (11(1), 11(3)) that involves a chain of alternating respective superconducting couplers (2(1), 2(3), 2(5)) and respective quantum dots (5(1), 5(3)).

[0197] 9. The qubit (300) of clause 8, further comprising a superconductor (2) comprising the respective superconducting couplers (2(1), 2(3), 2(5)) and a respective superconducting link (2(2)) so as to form a single connected superconductor so that all involved superconducting couplers in the linking parts (6(1), 6(3), 11(1), 11(3)) of the first and second quantum-dot chains are part of the single connected superconductor.

[0198] 10. The qubit (300) of clause 9, wherein the superconductor (2) is connected to a grounded superconducting lead (15(1)) via a gate-tuneable Josephson junction 13(1) or is connected to a grounded metal lead via a Normal -metal-superconductor junction.

[0199] 11. The qubit (300) of any one of the preceding clauses, wherein each of the first and second quantum-dot chains is implemented based on one of a two-dimensional electron-gas, 2DEG, a two-dimensional hole gas, 2DHG, or a nanowire network.

[0200] 12. Two-qubit system (500) comprising a first qubit (300(1)) according to clause 1 and a second qubit (300(3)) according to clause 1 and a readout system (7(1), 7(3)) comprising the readout systems of the first and second qubits (300(1), 300(3)), respectively, the first qubit (300(1)) involving a first group of Majorana zero modes (y15y2, y3, y4) and the second qubit (300(3)) involving a second group of Majorana zero modes (y5, y6, y7, y8), wherein the readout system (7(3)) is configured to provide a coupling between the first (Yi> ¥2, ¥3, ¥4) and second (y5, y6, y7, y8) groups of Majorana zero modes.

[0201] 13. The two-qubit system (500) of clause 12, wherein the provided coupling between the first (¥i, ¥2, ¥3, ¥4) and second (y5, y6, y7, y8) groups of Majorana zero modes comprises a fourth coupling (crzicrz2= — Y6Y7Y2Y3) between a first sub-group (¥2 / 3) of two Majorana zero modes of the first group (y15y2, y3, y4) and a second sub-group (y6y7) of further two Majorana zero modes of the second group (y5, y6, y7, y8)-

[0202] 14. The two-qubit system (500) of clause 12, wherein the provided coupling between the first (Yi> Y2> YS> Y4) and second (y5, y6, y7, y8) groups of Majorana zero modes involves a fifth coupling of respective involved outer quantum dots, configured to host respective Majorana zero modes, to one or more auxiliary quantum dots of the readout system (7(1), 7(3)).

[0203] 15. Quantum-computing chip for implementing topological quantum computing, comprising one or more instances of the qubit (300) of any one of clauses 1-10 and / or comprising one or more instances of the two-qubit system (500) of any one of clauses 11-14.

[0204] 16. Quantum-dot chain (100B, 200B) comprising: a pair of two outer quantum dots (1(1), 1(2)) configured to host a pair of Majorana zero modes (y15y2) in the first pair of two outer quantum dots (1(1), 1(2)); a chain of alternating quantum-dots (1(1), 1(2), 5(1)) and superconducting couplers (2(1), 2(3)); and a superconductor (2) comprising a superconducting link (2(2)) and the superconducting couplers (2(1), 2(3)) so as to form a single connected superconductor.

[0205] 17. Curved quantum-dot chain (200, 200 A, 200B) comprising: two outer quantum dots (1(1), 1(2)) configured to host a pair of Majorana zero modes (Yi, y2) in the first pair of two outer quantum dots (1(1), 1(2)); and a chain of alternating quantum-dots (1(1), 1(2), 5(1)) and superconducting couplers (2(1), 2(3)), wherein a line starting at one outer quantum dot (1(1)) and going through the chain and ending at the other outer quantum dot (1(2)) is a curved line.

Claims

What is claimed is:

1. Qubit (300) based on quantum-dot chains configured to host Majorana zero modes (y15y2, y3, Y4 comprising: a first quantum-dot chain (100(1), 200(1)) comprising a first pair of two outer quantum dots (1(1), 1(2)) and configured to host a first pair of Majorana zero modes (y15y2) in the first pair of two outer quantum dots (1(1), 1(2)); a second quantum-dot chain (100(3), 200(3)) comprising a second pair of two outer quantum dots (1(3), 1(4)) and configured to host a second pair of Majorana zero modes (y3, y4) in the second pair of two outer quantum dots (1(3), 1(4)); and a readout system (7(1)) comprising one or more auxiliary quantum dots (9(1), 9(3)) and configured to provide a coupling between the Majorana zero modes (y15y2, y3, y4), wherein at least one of the first and second quantum-dot chains is a curved quantumdot chain (200), wherein at least one of the first pair of two outer quantum dots (1(1), 1(2)) and the second pair of two outer quantum dots (1(3), 1(4)) is located in proximity to the readout system 7(1).

2. The qubit (300) of claim 1, wherein each of the first and second quantum-dot chains is configured to implement a respective Kitaev chain.

3. The qubit (300) of any one of the preceding claims, wherein the readout system (7(1)) is configured to provide the coupling between the Majorana zero modes (y15y2, y3, y4) by: coupling, via its one or more auxiliary quantum dots (9(1), 9(3)), at least two quantum dots of the four outer quantum dots (1(1), 1(2), 1(3), 1(4)), and / or coupling, via its one or more auxiliary quantum dots (9(1), 9(3)), at least one quantum-dot of the first pair of two outer quantum dots (1(1), 1(2)) with at least one quantumdot of the second pair of two outer quantum dots (1(3), 1(4)).

4. The qubit (300) of claim 3, wherein the readout system (7(1)) further comprises one or more tunnelling-control gates (8(1), 8(3), 8(5), 8(7)) each configured to tune a respective tunnelling amplitude between a respective auxiliary quantum-dot of the readout system (7(1)) and a respective outer quantum-dot (1(1), 1(2), 1(3), 1(4)).

5. The qubit (300) of any one of the preceding claims, wherein the provided coupling between the Majorana zero modes (y15y2, y3, y4) comprises: a first coupling (<J%= iy1y2) between a first (y- and second (y2) Majorana zero mode of the first pair of Majorana zero modes (y15y2), a second coupling= iyxy3) between the first Majorana zero mode (y- of the first pair of Majorana zero modes (y15y2) and a third Majorana zero mode (y3) of the second pair of Majorana zero modes (y3, y4), and a third coupling (<JZ= iy2y3) between the second (y2) Majorana zero mode of the first pair of Majorana zero modes (y15y2) and the third Majorana zero mode (y3) of the second pair of Majorana zero modes (y3, y4), wherein each of the first, second and third couplings is provided via coupling the respective outer quantum dots to one or more of the one or more auxiliary quantum dots.

6. The qubit (300) of claim 2, wherein for the implementing of the respective Kitaev chain, each or both of the first and second quantum-dot chains comprises a respective linking part (11(1), 11(3)) that involves a chain of alternating respective superconducting couplers (2(1), 2(3), 2(5)) and respective quantum dots (5(1), 5(3)).

7. The qubit (300) of claim 6, further comprising a superconductor (2) comprising the respective superconducting couplers (2(1), 2(3), 2(5)) and a respective superconducting link (2(2)) so as to form a single connected superconductor so that all involved superconducting couplers in the linking parts (6(1), 6(3), 11(1), 11(3)) of the first and second quantum-dot chains are part of the single connected superconductor.

8. The qubit (300) of claim 7, wherein the superconductor (2) is connected to a grounded superconducting lead (15(1)) via a gate-tuneable Josephson junction 13(1) or is connected to a grounded metal lead via a Normal -metal-superconductor junction.

9. The qubit (300) of any one of the preceding claims, wherein each of the first and second quantum-dot chains is implemented based on one of a two-dimensional electron-gas, 2DEG, a two-dimensional hole gas, 2DHG, or a nanowire network.

10. Two-qubit system (500) comprising a first qubit (300(1)) according to claim 1 and a second qubit (300(3)) according to claim 1 and a readout system (7(1), 7(3)) comprising the readout systems of the first and second qubits (300(1), 300(3)), respectively, the first qubit (300(1)) involving a first group of Majorana zero modes (yx, y2, y3, y4) and the second qubit (300(3)) involving a second group of Majorana zero modes (y5, y6, y7, y8), wherein the readout system (7(3)) is configured to provide a coupling between the first (Yi> Y2> YS> Y4)and second (y5, y6, y7, y8) groups of Majorana zero modes.

11. The two-qubit system (500) of claim 10, wherein the provided coupling between the first (Yi> Y2> YS> Y4)and second (y5, y6, y7, y8) groups of Majorana zero modes comprises a fourth coupling (crzicrZ2 = — Y6Y7Y2Y3) between a first sub-group (Y2Y3) of two Majorana zero modes of the first group (y15y2, y3, y4) and a second sub-group (Y6Y7) of further two Majorana zero modes of the second group (y5, y6, y7, y8).

12. The two-qubit system (500) of claim 10, wherein the provided coupling between the first (Yi> Y2> YS> Y4)and second (y5, y6, y7, y8) groups of Majorana zero modes involves a fifth coupling of respective involved outer quantum dots, configured to host respective Majorana zero modes, to one or more auxiliary quantum dots of the readout system (7(1), 7(3)).

13. Quantum-computing chip for implementing topological quantum computing, comprising one or more instances of the qubit (300) of any one of claims 1-9 and / or comprising one or more instances of the two-qubit system (500) of any one of claims 10-12.

14. Quantum-dot chain (100B, 200B) comprising a pair of two outer quantum dots (1(1), 1(2)) and configured to host a pair of Majorana zero modes (y15y2) in the pair of two outer quantum dots (1(1), 1(2)); wherein the quantum-dot chain (100B, 200B) comprises: a chain of alternating quantum-dots (1(1), 1(2), 5(1)) and superconducting couplers (2(1), 2(3)); and a superconductor (2) comprising a superconducting link (2(2)) and the superconducting couplers (2(1), 2(3)) so as to form a single connected superconductor.

15. Curved quantum-dot chain (200, 200 A, 200B) comprisingtwo outer quantum dots (1(1), 1(2)) and configured to host a pair of Majorana zero modes (y15y2) in the first pair of two outer quantum dots (1(1), 1(2)); and wherein the curved quantum-dot chain (200, 200A, 200B) comprises a chain of alternating quantum-dots (1(1), 1(2), 5(1)) and superconducting couplers (2(1), 2(3)), wherein a line starting at one outer quantum dot (1(1)) and going through the chain and ending at the other outer quantum dot (1(2)) is a curved line.

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