Kitaev-chains device and methods of operating the device

Abstract

The disclosure relates to a Kitaev-chains device (1) and methods of operating the device The Kitaev-chains device (1) comprises: a plurality of gate-controlled semiconductor-superconductor-based Kitaev-chains modules (100), each Kitaev-chains module (101) comprising a plurality of respective Kitaev-chains (101-1; 101-2) and a respective superconductor element (301) forming part of the respective Kitaev-chains (101-1; 101-2); a superconductor (300) connecting the superconductor elements (301, 303) of the plurality of Kitaev-chains modules (100); and a signal module (500) configured to control radio-frequency, RF, signals on the superconductor (300) to control one or more of the plurality of Kitaev-chains modules (100). The invention has applications in quantum computing and quantum simulations, for example by a quantum computer that includes one or more instances of the Kitaev-chains device.

Description

[0001] Kitaev-chains device and methods of operating the device

[0002] Field of the invention

[0003] The invention relates to a Kitaev-chains device and methods of operating the device.

[0004] Background art

[0005] Alexei Kitaev proposed a certain one-dimensional chain of fermions as a building block suitable for topological quantum computing [Kitaev, 1-2], The one-dimensional chain is commonly referred to as the Kitaev chain and a Hamiltonian for describing the basic dynamics of the Kitaev chain with N sites may be denoted as

[0006]

[0007] wherein fk

[0008]

[0009] denote fermionic annihilation and creation operators, respectively, the real-valued parameter skrelates to an on-site energy of site k, the parameters tk relate to amplitudes for single-particle coupling, the complex-valued parameters Ak relate to Andreev-type couplings between neighbouring sites, and h.c. denotes “Hermitian conjugate”.

[0010] Kitaev found that when tuning a Kitaev chain to a parameter regime in which sk= 0 and | Ak| = tk> 0, the chain hosts two Majorana bound states (also referred to as “Majorana zero modes” or “Majorana modes”) at the two boundary sites k=l and k=N of the chain. These Majorana bound states are particular quasi -particle quantum states that can be used for implementing decoherence-free quantum computing that is inherently fault-tolerant by its physical nature [Nayak et al; Sarma et al], A parameter regime suitable for forming two Majorana bound states at the boundary of a Kitaev chain is also called a Majorana sweet spot or simply sweet spot. Kitaev chains exhibiting Majorana bound states at their boundary were successfully implemented by a gate-controlled quantum-dot chain that comprises quantum dots and superconducting couplers [Bordin et al, 1-3; van Loo], Kitaev chains with larger number of sites are desirable because they can provide better protected Majorana bound states.

[0011] In view of technical advantages including the prospect of decoherence-free quantum computing, there is a vast interest in engineering a Kitaev-chains device that can be controlled in an efficient manner.

[0012] Summary of the invention

[0013] A task set forth by the inventors is to provide a Kitaev-chains device and methods of operating the device that enable control in an efficient manner.

[0014] The inventors solved the task by providing a Kitaev-chains device according to appended independent claim 1, with advantageous aspects as set out in the dependent claims.Furthermore, methods of operating the Kitaev-chains device are provided according to further appended independent method claims. The methods are for operations of tuning-up, initializing, and reading-out.

[0015] The invention has applications in quantum computing and quantum simulations, for example by a quantum computer that includes one or more instances of the Kitaev-chains device according to the present invention. Technical advantages include, among other things, at least one of simplifying tune-up and initialization because a signal module of the Kitaev-chains device enables control by fewer control signals and control lines, thereby making control more efficient and scaling-up feasible. Technical advantages are further set out in the detailed description below.

[0016] Brief description of the drawings

[0017] The present invention is discussed in more detail below, with reference to the attached drawings showing examples, in which:

[0018] Fig. la is an illustration of a Kitaev-chains device with Kitaev-chains modules.

[0019] Fig. lb is an illustration of a Kitaev-chains module of the Kitaev-chains device.

[0020] Figs. Ic-lf are illustrations of Kitaev chains.

[0021] Fig. 1g is another illustration of the Kitaev-chains device.

[0022] Fig. Ih is an illustration of a Kitaev-chains device with signal-and-frequency-control modules based on resonators.

[0023] Fig- li is another illustration of a Kitaev-chains device signal-and-frequency-control modules based on tuneable Josephson junctions.

[0024] Figs. 2a-2b are illustrations of signal modules.

[0025] Fig- 3 is an illustration of tuning-up a Kitaev chain.

[0026] Fig- 4 shows an aspect in the context of initializing a Kitaev-chains.

[0027] Fig. 5a-c are schematic flow diagrams of methods for operating the Kitaev-chains device.

[0028] Fig. 6 is an illustration of a quantum processor comprising one or more instances of the Kitaev-chains device.

[0029] Figs. 7a-7d show measurement results of an example of a Kitaev chain.

[0030] Detailed description

[0031] 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.

[0032] 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.

[0033] 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.

[0034] 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.

[0035] 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.

[0036] 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.

[0037] 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 skilledperson 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.

[0038] 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.

[0039] The inventors devised a Kitaev-chains device that can be efficiently controlled. Below, reference signs are added in the description below for illustration purposes and for intelligibility reasons.

[0040] The Kitaev-chains device (1) provided by the present disclosure comprises:

[0041] a plurality of gate-controlled semiconductor-superconductor-based Kitaev-chains modules (100), each Kitaev-chains module (101) comprising a plurality of respective Kitaev-chains (101-1; 101-2) and a respective superconductor element (301) forming part of the respective Kitaev-chains (101-1; 101-2);

[0042] a superconductor (300) connecting the superconductor elements (301, 303) of the plurality of Kitaev-chains modules (100); and

[0043] a signal module (500) configured to control radio-frequency, RF, signals on the superconductor (300) to control one or more of the plurality of Kitaev-chains modules (100).

[0044] By the combination of the signal module (500) and the superconductor (300) connecting the plurality of Kitaev-chains modules (100) specifically via their respective superconductor elements (301, 303), the Kitaev-chains device (1) enables efficient control of the Kitaev-chains modules via its signal module (500). For example, the signal module (500) can serve as a central control instance via which the Kitaev-chains modules can be controlled. Thereby, certain operations can be simplified by controlling the signal module (500) that composes a single / centralized unit through which control on and information about the Kitaev-chains modules can be obtained more efficiently as compared to needing many probes such as normal metallic leads at different parts of the Kitaev-chains device, which among other things allows to efficiently set individual gates of the Kitaev-chains modules to suitable voltage levels. The simplified control is possible by the specific arrangement of the signal module (500) with the connecting superconductor (300), in other words, by establishing a RF-signal connection specifically via the superconductor (300) to the respective superconductor elements so that the Kitaev-chains of the Kitaev-chains modules (100) can be addressed and measured via the signal module (500) by RF signals. The Kitaev-chains can thus be addressed, controlled, measured, manipulated, etc., by means of the signal module (500) that can use specifically the superconductor (300) as a platform to couple via RF signals with the Kitaev-chains via the respective superconductor elements and thereby enable control of the Kitaev-chains modules. The signal module (500) can thus be used to efficiently obtain information about a state of the Kitaev-chains modules.

[0045] The Kitaev-chains device (1) involves semiconductor and superconductor technology, which may be implemented in various ways. For example, a substrate may be employed on which semiconductor and superconductor materials can be formed, for example by creating a number of layers involving a semiconductor-superconductor layer that may comprise one or more layers of semiconductor and superconductor materials, which may be build in various ways as outlined in [Bordin et al, 1-3; Dvir et al; van Loo] and references therein. Preferably, the semiconductor-superconductor-based Kitaev-chains modules involve a 2-dimensional electron gas, 2DEG, a 2-dimensional hole gas, 2DHG, a nanowire, and / or a nanowire network grown for example by selective area growth; see also [Liu et al; Sumanta et al; ten Haaf et al; G. Wang et al]. For example, the Kitaev-chains device (1) may, for comprising the plurality of gate-controlled semiconductor-superconductor-based Kitaev-chains modules (100), further comprise: a substrate; a semiconductor-superconductor layer disposed on the substrate; and a gate layer disposed on the semiconductor-superconductor layer. The gate layer may comprise a plurality of control gates configured to control electrostatic potentials in the semiconductorsuperconductor layer. For example, as also outlined further below, the gate layer may comprise various control gates including plunger gates and barrier gates for defining and controlling quantum dots (or “quantum-dot regions”) in a quantum-dot-hosting-layer of the semiconductorsuperconductor layer. The arrangement of the control gates may be adapted for example to desired locations at which Kitaev-chains are to be implemented. For example, the semiconductor-superconductor layer may comprise one or more semiconductor layers. The quantum-dot-hosting layer may correspond to one of the semiconductor layers. One or moresuperconductor parts may be formed, for example, on top of the one or more semiconductor layers and / or one or more gates, for example by shadow-lithography. The Kitaev-chains may be arranged in various ways as part of their respective Kitaev-chains module. Examples are outlined further below.

[0046] The Kitaev-chains device (1) comprises the signal module (500) that allows to control the Kitaev-chains modules by control signals. The control may involve addressing the Kitaev-chains modules by radio-frequency, RF, signals, preferably high-frequency RF signals, on the superconductor (300) to interact with the Kitaev-chains modules. The control preferably involves detection based on reflectometry and / or transmission signals, as outlined further below. Methods of controlling the Kitaev-chains device (1) that make use of the signal module (500) are also described further below.

[0047] The Kitaev-chains device (1) enables to efficiently scale up to a larger number of Kitaev-chains modules (100), because the amount of control parameters can be reduced. For example, the signal module (500) enables to tune-up and initialize the Kitaev-chains modules by means of the superconductor (300) that connects the superconductor elements of the Kitaev-chains modules. Thereby, as compared to using various control lines within each Kitaev-chains module, the number of control parameters for tune-up and initialization can be greatly reduced, making the control of a larger number of Kitaev qubits more efficient and feasible. For example, one single central signal module (500) can be used and controlled instead of a much larger number of individual control elements for controlling the Kitaev-chains modules individually. For example, information about one or more of the Kitaev-chains modules can be obtained by the signal module (500), instead of needing to build additional control lines or employing such additional control lines when wanting to control the Kitaev-chains modules, e.g. tuning-up, initializing, measuring.

[0048] Fig. la illustrates a number of Kitaev-chains modules (101, 103, 105, 100-N). In principle, any number N of Kitaev-chains modules may be considered, depending on how many Kitaev-chains modules one wishes to implement. For example, N=10, 100, 1000, 10.000, 100.000, 1.000.000, etc. Fig. la is only schematically in that the Kitaev-chains modules may be distributed in various ways and not limited to the illustration. The important aspect shown by Fig. la is that the signal module (500) can control one or more of the plurality of Kitaev-chains modules (100) via the superconductor (300) that connects the superconductor elements (301, 303) of the Kitaev-chains modules.

[0049] Fig. lb further illustrates that each Kitaev-chains module (101) has a plurality of respective Kitaev-chains (101-1, 101-2). In general, each Kitaev-chains module may also havemore than two Kitaev-chains, depending on what aspect one wishes to implement. For example, a Kitaev qubit may be built involving two Kitaev-chains, or three Kitaev-chains, etc., depending for example on the type of qubit that is envisioned. Irrespective of which type of qubit is envisioned, however, the point of the Kitaev-qubits device (1) is that each Kitaev-chains module has a plurality of Kitaev-chains and further has a superconductor element that forms part of the respective Kitaev-chains. In other words, the signal module specifically exploits coupling to a superconductor element that forms part of a Kitaev chain of a Kitaev-chains module, via the superconductor. How many Kitaev-chains are employed, and how they are arranged, can depend on which qubit type one wishes to realize.

[0050] Fig. lb further illustrates that Majorana bound states can be formed, ^, yrl, yg2, yr2, by the Kitaev-chains and that the superconductor element (301) is involved in the Kitaev chains. Examples and aspects of tuning a Kitaev-chain to a Majorana sweet spot for forming Majorana bound states are shown further below.

[0051] Next, preferred aspects and examples are discussed.

[0052] Preferably, one or more, preferably all, of the Kitaev chains (101-1) comprises a respective array of respective gate-controlled semiconductor-based quantum dots (701-f, 701-1, 701-3, 701-r) and one or more respective superconductor coupling-segments (301-1, 301-3, 301-5) of the respective superconductor element (301).

[0053] Such arrays involving quantum dots and superconductor coupling-segments (or: “couplers”) are a preferred way of implementing the Kitaev chains, as such arrays have been successfully demonstrated to exhibit the dynamics of Kitaev chains for forming a pair Majorana bound states at their boundary when tuned to a Majorana sweet spot. Moreover, the signal module (500) is particularly effective in enabling control of the Kitaev-chains modules via the superconductor coupling-segments coupling to the quantum dots. Therefore, implementing the Kitaev chains based on such arrays is very much preferred. However, there may be further ways of implementing Kitaev chains, for example based on topological superconductors [Kitaev 2; Lutchyn et al; Oreg et al], and the Kitaev-chains device is not limited to such arrays.

[0054] Preferably, the array relates to an alternating chain of the quantum dots and the one or more superconductor coupling-segments, each superconductor coupling-segment (301-1) configured to enable, by Andreev bound states, a respective superconductivity coupling (A) between its neighbouring quantum dots (701-f, 701-1).

[0055] In other words, the array corresponds to a chain of alternating quantum dots and superconductor coupling-segments (quantum dots intertwined with superconducting couplers). When tuning the whole chain to its Majorana sweet spot, a pair of Majorana bound states ishosted on outer quantum dots (701-^, 701-r) of the array. The alternating chain is particularly effective in implementing a Kitaev chain, as the parameters for tuning the chain to its Majorana sweet spot can be effectively controlled by gates. The number of quantum dots and superconductor coupling-segments may be chosen arbitrarily depending on how long the chain is desired. As a rule of thumb, longer chains can host more stable Majorana bound states, but also require more parameters to be controlled.

[0056] Fig. 1c illustrates such an alternating chain. The illustrated alternating chain comprises quantum dots (701-f, 701-1, 701-3, ..., 701-r) and superconductor coupling-segments (301-1, 301-3, ..., 301-5).

[0057] The quantum dots (701-f, 701-1, 701-3, ..., 701-r) can be controlled by corresponding control gates (not illustrated) involving tunnelling-control gates (or: “barrier gates”) and on-site control gates (or: “plunger gates”). The quantum dots, formed in a semiconductor material, can be understood as regions defined by confining electric potentials that are created / applied via the gates. The gates can be implemented as electrostatic control gates on which voltages can be applied and may accordingly be referred to as on-site gates, control gates, plunger gates, potential-control gates, electrostatic gates, or finger gates, depending on their role or way of implementation. For example, on-site control gates can provide control of chemical potentials / depths of potential wells in the respective regions. Tunnelling-control gates can control tunnelling-amplitudes related to single-electron tunnelling between regions and respective quantum dots.

[0058] The superconductor coupling-segments (301-1, 301-3, ..., 301-5) may be implemented by manufacturing a superconducting material, such as Aluminium, on a semiconductor, for example by shadow-wall lithography. A superconductor coupling-segment may be referred to as a semiconductor-superconductor hybrid segment, hybrid segment, hybrid region, or the like.

[0059] The alternating chain may comprise further non-illustrated elements, such as one or more of leads, normal-metal leads, depletion gates, magnetic-field generators, and the like. For example, tunnelling-control gates can control tunnelling to leads / lead elements for performing measurements. However, such details regarding manufacturing are not outlined here, but are known and can be found for example in [Dvir et al; Bordin et al, 2-3; van Loo],

[0060] For illustration and without being bound by theory, consider Fig. 1c, in which the subchain of quantum dots (701-f) and (701-1) with superconductor coupling-segment (301-1) can be controlled via gates to realize a 2-site Kitaev chain model HK2 of Eqn. 1 with N=2. The onsite energy terms with parameters £fc, k=l and k=2, can be realized with the quantum dots (701-f ) and (701-1) controlled by corresponding on-site gates. By applying a magnetic field to induceZeeman splittings, spin-polarized quantum-dot orbitals occupied by electrons can realize the dynamics of spinless fermions. The single-electron coupling terms with tk and the Andreev-type couplings terms with k can be realized via Andreev bound states, ABSs. An ABS is an electron-hole superposition state / a quasi-particle state that can be formed due to the superconductor coupling-segment. The ABS is formed in a region of the superconductor coupling-segment between the quantum dots and can be used as a mediator for realizing couplings needed to implement a 2-site Kitaev chain. 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 quantum dot (e.g., 701-f) via the superconductor coupling-segment to the other quantum dot (e.g., 701-1). An ECT process can thus realize the single-electron coupling term wherein an electron is transitioning from site 1 (quantum-dot 701-f) to site 2 (quantum-dot 701-1), and the conjugate coupling tff f2, wherein an electron is transitioning from site 2 (quantum-dot 701-1) back to site 1 (quantum-dot 701-f). The ABS may further mediate a Cooper-pair coupling / splitting process referred to as Crossed-Andreev reflection, CAR. In CAR, two electrons in the superconductor coupling-segment (301-1) may form a Cooper pair and separately tunnel into its neighbouring quantum dots. Similarly, two electrons may tunnel from the two neighbouring quantum dots into the superconductor couplingsegment. Such a CAR may thus realize the terms

[0061]

[0062] and ^iA+^+, respectively. When being tuned to a Majorana sweet-spot (t=A) via control gates, the (minimal / reduced) quantum-dot chain of this example exhibits a pair of Majorana bound states at the two quantum dots (701 -f) and (701-1) (also referred to as “unpaired MBS”).

[0063] The Majoranas created by such a minimal 2-site Kitaev chain are protected by an energy gap in the 2-site system but can be susceptible to fluctuations as they can start overlapping due to global fluctuations, which splits them in energy and can cause unwanted qubit precession and errors. That is why such Majoranas are also referred to as poor man’s Majoranas. It is therefore desirable to create Majoranas in outer quantum dots that are separated by bulk quantum dots (e.g., 701-1, 701-3, ...) to increase the protection of the Majoranas against noise. When tuning the whole Kitaev chain (101-1) to its Majorana sweet spot, a first Majorana bound state (y^i) may be formed in one boundary / outer quantum dot (701-f) and a second Majorana bound state (yrl) may be formed in the other boundary / outer quantum dot (701-1).

[0064] Further ancillary and known aspects involved in a Kitaev chain based on a gate-controlled quantum-dot chain, such as controlling a magnetic field to realize the spin-lessdynamics of the Kitaev chain, cooling to milli-Kelvin ranges, and the like, are discussed for example in [Dvir et al; Bordin et al, 1-3] and are therefore not repeated here.

[0065] Preferably, the array relates to a consecutive chain of the quantum dots, each quantum dot (701-1) being proximitized by a respective superconductor coupling-segment (301-3) so as to enable, by inducing superconductivity to the respective proximate quantum dot (701-1), gate-controlled spin-preserving (t) and spin-flipping (tso) couplings between neighbouring quantum dots (701- 701-1).

[0066] Such an array is illustrated in Fig. Id. In this example, each quantum-dot has its own respective superconductor coupling-segment that, due to its proximity, induces a superconductivity coupling to the respective quantum dot. Thereby, spin-preserving (t) and spin-flipping (tso) couplings can be controlled by gates to tune the Kitaev chain to one of its Majorana sweet spots. In other words, in this example the array is not an alternating chains of quantum dots and couplers (intertwined chain), but is a consecutive chain of consecutive quantum dots in which each quantum dot is proximitized by a respective different coupler. Background on this example can be found in [Svensson and Leijnse; William et al; Miles et al]

[0067] Preferably, one or more, preferably all, of the superconductor elements (301, 303) are formed as a respective integrally-formed superconductor element that is shared by all the Kitaev-chains (101-1, 101-2) of the respective Kitaev-chains modules (101, 303).

[0068] That means that a superconductor element of a Kitaev-chains module is formed into one single connected piece / is made of a single piece. Thereby, coupling phases can be efficiently controlled. For example, Kitaev chains typically comprise different tunnelling terms and by employing an integrally-formed superconductor element in the Kitaev-chains module, phase control over the different tunnelling terms is efficiently enabled.

[0069] Preferably, the superconductor (300) is formed as an integrally-formed superconductor comprising the superconductor elements. Thereby, control by the signal module (500) is further enhanced, as a single integrally-formed superconductor is employed to provide a superconducting connection between the different Kitaev-chains modules.

[0070] Next, examples of arrangements of the Kitaev chains are described. A Kitaev qubit refers to a qubit that involves Kitaev chains, in particular Majorana bound states created by tuning Kitaev chains to their respective Majorana sweet spots. The Kitaev qubit may also be referred to as a “Kitaev qubit based on Kitaev chains”, “Kitaev-chains qubit”, or “Kitaev-chain-based qubit”, and the like. In general, there may be various ways of using Kitaev chains to form a Kitaev qubit, e g. [Bordin et al, 1-3; Nayak et al; Sarma et al; van Loo] and referencestherein. As another example, a Kitaev-transmon hybrid may be considered, which exploits Kitaev chains in a transmon circuit (“Kitaev-transmon” or “Kitmon”), [Pino et al]. In that context, preferably, one or more of the plurality of Kitaev-chains modules comprise a respective Josephson junction element and a respective capacitor element, for forming respective Kitaev-transmon hybrids (not illustrated).

[0071] A further example of making a Kitaev qubit is shown in Figs, le-lf.

[0072] Fig. le shows a curved Kitaev-chain, that is a curved chain of alternating quantum dots and superconductor coupling-segments such that the outer quantum dots of the chain can couple to a shared entity such as another quantum dot for read-out (not-illustrated in Fig. le, but in If). Such curved chains are preferably implemented based on a 2DEG and / or 2DHG.

[0073] Fig. If shows a preferred Kitaev-chains module (101). In this preferred variation, the Kitaev-chains module (101) has two curved Kitaev chains (101-1, 101-2) that are both curved such that all the outer quantum dots of the chains can couple to a shared entity for read-out, such as one or more quantum-dots (801) for read-out. Moreover, in this preferred variation, the superconductor element (301) is shared by both the Kitaev chains (101-1, 101-2), which efficiently enables phase control. The configuration shown in Fig. If showcases a Kitaev-chains module for hosting a Kitaev qubit, by means of curved Kitaev chains with boundary quantumdots coupling to a readout-module (801). Gates for controlling and coupling the quantum dots, superconductor coupling-segments, and read-out dots, respectively, are not illustrated for intelligibility reasons. In general, the length of the curved Kitaev-chains involved in the Kitaev-chains module is also not limited to the illustrated length, but may be varied. In general, preferably, one or more of, most preferably all of, the Kitaev-chains modules comprise, respectively, two or more curved Kitaev chains whose boundary quantum dots couple to a shared readout-module. Preferably, the shared readout-module comprises one or more quantum dots (801) for readout.

[0074] Preferably, the Kitaev-chains device (1) further comprises one or more signal-and-frequency-control modules (901, 903), each signal-and-frequency-control module (901) placed between the signal module (500) and a respective Kitaev-chains module (101) and configured to control whether a signal generated by the signal module can pass through to the respective Kitaev-chains module (101) and to control a frequency of a passed-through signal. Thereby, one can use frequency multiplexing for readout of different Kitaev-chains modules. A method of reading out a Kitaev-chains module is described further below.

[0075] Preferably, each of the one or more frequency-control modules (901, 903) relates to: a resonator (coi, C02) configured to exhibit a particular resonance frequency; ora gate-tuneable Josephson junction (JJ1, JJ3) configured to exhibit gate-tuneable resistance states including a zero-resistance state and a high-resistance state.

[0076] Frequency multiplexing and / or gate-tuneable Josephson junctions / field effect transistors (JoFETs) enable to address separate qubits.

[0077] Fig. Ih shows a number of resonators (coi, C02, (03, CON) that can be used to address respective Kitaev-chains modules (101, 103, 105, 100-N) at particular frequencies.

[0078] Fig- li shows a number of Josephson junctions (JJ1, JJ3, JJ5, JJ7) which similarly can be used to address respective Kitaev-chains modules (101, 103, 105, 100-N), for example by switching between respective zero-resistance and high-resistance states, wherein a zeroresistance state may be suitable for addressing a corresponding Kitaev-chains module via the signal module, and a high-resistance state may be suitable for decoupling a corresponding Kitaev-chains module from the signal module.

[0079] Preferably, the signal module (500) comprises at least one of

[0080] a reflectometry module (500a) configured to provide the control by reflectometry involving the RF signals applied to the superconductor (300); and / or

[0081] a transmission-setup module (500b) configured to provide the control by a transmission setup involving the RF signals applied to the superconductor (300).

[0082] The reflectometry and transmission setup are preferred ways of providing control of the Kitaev-chains modules by means of the signal module (500), as they allow precise control, generation, detection and measurements of signals on the superconductor (300), which is thereby exploited as a central coupling element to manipulate and / or obtain information about the Kitaev-chains modules. The signal module (500) may also be referred to as RF signal module, as the signals generated on the superconductor (300) relate to RF signals. The signal module (500) may comprise various elements and parts such as outlined for example in [Vigneau et al].

[0083] Figs 2a-b are illustrations of preferred elements of the reflectometry module (500a) and the transmission-setup module (500b). In brief, the reflectometry module (500a) may comprise a directional coupler (503a) while the transmission-setup module (500b) may instead work based on a transmission feedline (503b). The point is that RF signals and temporal changes thereof on the superconductor (300) can be generated and detected by means of the signal module (500), in the examples generated by a signal -generator module (501) and detected by a detection module (505). Preferably, the signal module (500) is based on the reflectometry module (500a) to control the Kitaev-chains modules by reflectometry involving measurements of RF signals reflected from the superconductor (300).Preferably, the reflectometry module (500a) comprises at least one of:

[0084] a signal -generator (501; VRF) configured to generate RF signals to be applied to the superconductor (300);

[0085] a directional coupler (503a) configured to transmit the RF signals, received from the signal-generator, to a resonator (509) and further configured to transmit reflected signals, reflected back to the directional coupler (503a) from the superconductor (300) and via the resonator (509), to a detection module (505) for reflected signals,

[0086] wherein the reflectometry module (500a) comprises the resonator (509) and the detection module (505),

[0087] wherein the resonator (509) is configured to control a resonance frequency of the RF signals to be applied to the superconductor (300), and

[0088] wherein the detection module (505) is configured to measure signals received from the directional coupler (503a).

[0089] Preferably, the transmission-setup module (500b) comprises at least one of:

[0090] a signal -generator (501; VRF) configured to generate RF signals to be applied to the superconductor (300);

[0091] a transmission feedline (503b) configured to transmit signals, received from the signalgenerator and a resonator (509), to a detection module (505) for detecting transmitted signals, wherein the reflectometry module (500a) comprises the resonator (509) and the detection module (505),

[0092] wherein the resonator (509) is configured to control a resonance frequency of the RF signals to be applied to the superconductor (300), and

[0093] wherein the detection module (505) is configured to measure signals received from the transmission feedline (503b).

[0094] In general, the signal module (500) preferably comprises a signal -generator (501; VRF) configured to generate RF signals to be applied to the superconductor (300) and a detection module (505) configured to detect RF signals indicative of a state of the superconductor (300). Thereby, the signal module (500) can control signals on the superconductor (300) to control one or more of the plurality of Kitaev-chains modules (100). Frequencies of the RF signals are preferably in the range of 50 Mega Hz (MHz) to 80 Giga Hz (GHz), more preferably between 100 MHz and 10GHz, more preferably between 100 MHz and 1 GHz. For a reflectometry module (500a) and a transmission-setup module (500b), the frequency is preferably in between 100 MHz to 10 GHz, more preferably in between 100 MHz to 1 GHz .In each of the reflectometry module (500a) and the transmission-setup module (500b), the combination of the signal-generator (501) and detection module (505) allows to efficiently obtain information on the superconductor (300) and thereby ultimately on the Kitaev-chains modules via RF signals. At the example of the reflectometry module (500a), the directional coupler (503a) allows to focus the analysis directly on the reflected signals detected by the detection module (505), while at the example of the transmission-setup module (500b), the transmission feedline (503b) forwards a signal to the detection module (505) that involves source signals of the signal-generator and reflected signals coming in from the resonator. Both the reflectometry module (500a) and the transmission-setup module (500b) are examples of the signal module (500) that enables detection measurements based on detecting reflections of one or more control signals generated on the superconductor.

[0095] Preferably, the resonator (509) is coupled to the directional coupler (503a) or to the transmission feedline (503b) via a capacitor element (507) or is coupled inductively. The resonator setup preferably corresponds to a inductance-capacitance (LC) resonator.

[0096] Provided is furthermore a quantum computer (1000) that comprises one or more Kitaev-chains devices (1) according to the present invention, each Kitaev-chains device (1) preferably comprising one or more or all of the above-outlined preferred aspects. The quantumprocessor (1000) may comprise further classical computing elements for controlling the one or more instances of the Kitaev-chains device (1).

[0097] Provided are furthermore methods of operating the Kitaev-chains device (1) according to the present invention, the device preferably comprising one or more or all of the above-outlined preferred aspects. A first method relates to tuning-up. A second method relates to initializing. A third method relates to read-out.

[0098] The first method is for tuning-up a Kitaev-chains module (101) of the Kitaev-chains device (1). Tuning-up refers to finding a set of parameters for tuning Kitaev-chains of a Kitaev-chains module to their respective Majorana sweet spots, which may be a first step to configuring / controlling the Kitaev-chains module to host a Kitaev qubit.

[0099] The method of tuning-up comprises:

[0100] decoupling (Al) the Kitaev-chains module (101) from the remaining Kitaev-chains modules (103, 105, ..., 100-N), by decoupling the superconductor (300) from the remaining Kitaev-chains modules (103, 105, ..., 100-N);

[0101] performing (A2), by the signal module (500), detection measurements based on detecting reflections of one or more control signals generated on the superconductor (300); andadapting (A3), based on the detection measurements, gate voltages on one or more control gates of the Kitaev-chains module (101) to tune-up the Kitaev-chains module (101).

[0102] By decoupling a Kitaev-chains module of interest from remaining Kitaev-chains modules, the signal module (500) can be used to address specifically that particular Kitaev-chains module of interest. The decoupling is performed by decoupling of the superconductor (300), which is the platform used by the signal module (500) to control the Kitaev-chains modules. Hence, after the decoupling, the detection measurements involving signals of the superconductor (300) can be used to specifically target the Kitaev-chains module of interest.

[0103] The performing (A2) may comprise generating, by the signal module (500), one or more control signals on the superconductor (300) and detecting, by the signal module (500), reflections of the one or more control signals. However, in general, the signals may also be generated by a separate entity and the signal module may be employed primarily for measuring the reflections. Preferably, however, the signal module is the primary module for generating the signals and detecting reflected signals.

[0104] Preferably, the performing of the detection measurements comprises:

[0105] varying one or more gate voltages on the one or more control gates of the Kitaev-chains module (101), to obtain information about a state of the Kitaev-chains module (101) by the signal module.

[0106] By varying gate voltages, one can cause the Kitaev-chains module (101) to change its response to signals on the superconductor (300) that couples to the Kitaev-chains module (101) of interest via the superconductor element (301). The caused / induced change can provide information on resonating energy levels that can be used for tuning-up. Thereby, the combination of decoupling the Kitaev-chains module (101) specifically by decoupling the superconductor (300) from remaining Kitaev-chains modules and then using the signal module (500) to control the same superconductor (300) as a platform for detection measurements , is a particularly effective way of using the signal module (500) and the superconductor (300). In other words, the detection measurements may thus involve altering a response-state of the Kitaev-chains module (101) of interest, relative to the superconductor (300). The altered response-state can be recognized in the detection measurements. Further below, examples thereof are discussed.

[0107] Preferably, the decoupling of the superconductor (300) from the remaining Kitaev-chains modules (103, 105, ..., 100-N) comprises decoupling quantum dots of the Kitaev-chains of the respective Kitaev-chains modules from the respective superconductor elements (303, 305, ..., 300-N) of the remaining Kitaev-chains modules (103, 105, ..., 100-N).The respective superconductor elements (301) forming part of the respective Kitaev-chains (101-1; 101-2) can be decoupled, thereby effectively decoupling from the superconductor.

[0108] Preferably, the decoupling of the quantum dots comprises at least one of: establishing, by barrier gates, Coulomb blockades between the quantum dots and the respective superconductor elements; and / or

[0109] depleting, by plunger and barrier gates, the quantum dots.

[0110] Using control gates, such as barrier gates and / or plunger gates, of the quantum dots of the to-be-decoupled Kitaev-chains, is an effective way of ensuring that the superconductor elements, and thereby the superconductor (300), are decoupled and the detection measurements for the tuning-up can focus on the Kitaev-chains module of interest.

[0111] Fig. 3, a-f, exemplifies tuning-up at the example of a Kitaev chain implemented by an alternating chain of quantum dots (701-f, 701-1, 701-r) intertwined with superconductor coupling-segments (301-1, 301-3) of a superconductor element (301). In the following description of further preferred aspects, reference signs to Fig.3, a-f, are added for intelligibility reasons and illustrative purposes, but the preferred aspects are not limited to the illustration.

[0112] Preferably, the method further comprises:

[0113] detecting, by involving the signal module (500), one or more first resonance energylevels of a first quantum dot (701-f) of a first Kitaev chain (101-1) of the Kitaev-chains module (101),

[0114] wherein during the detecting of the one or more first resonance energy-levels, the remaining quantum dots (701-1, 701-r) of the first Kitaev chain are decoupled from the superconductor element (301) of the Kitaev-chains module (101).

[0115] A resonance energy -level of the quantum-dot refers to an energy level that aligns with or is below the gap edge of the superconductor, at which the quantum-dot resonates with the superconductor, which can be detected by the signal module (500).

[0116] Preferably, the method further comprises:

[0117] after the detecting of the one or more first resonance energy-levels of the first quantum dot (701-f), further detecting, by involving the signal module (500), one or more further resonance energy-levels of each of the remaining quantum dots (701-1, 701-r) of the first Kitaev chain,

[0118] wherein during each detecting of one or more respective resonance energy-levels of a respective quantum dot (701-1), other quantum dots (701-f, 701-r) different from the respectivequantum dot (701-1) are decoupled from the superconductor element (301) of the Kitaev-chains module (101).

[0119] Thereby, suitable quantum-dot levels can be efficiently detected by a single signal module, simplifying a tune-up procedure of quantum dots as compared to using different gates for tune-up.

[0120] Preferably, the detecting, by involving the signal module (500), of one or more resonance energy-levels of a particular quantum dot comprises:

[0121] sweeping a gate voltage of a plunger gate of the particular quantum dot while detecting reflections of control signals of the signal module (500) to detect a modulation of the reflections, the modulation indicative of the one or more resonance energy-levels in which the particular quantum dot resonates with the superconductor.

[0122] The modulation is thus indicative of an energy -level of the first quantum dot (701-f) aligning with an energy gap edge of the superconductor (300). Thereby, a suitable energy level can be detected by the signal module.

[0123] Preferably, the method further comprises:

[0124] tuning-up quantum dots of a first pair of neighbouring quantum dots (701-f, 701-1) to respective resonance energy-levels and tuning-up, by involving the signal module (500), the first pair (701-f, 701-1) to a first Majorana sweet spot (t = A),

[0125] wherein during tuning-up the first pair (701-f, 701-1), other quantum dots (701-r) of the first Kitaev chain are decoupled from the superconductor element (301) of the Kitaev-chains module (101).

[0126] The tuning-up of the quantum dots may be performed by the preferred ways as outlined above, but may also be performed using gate measurements. However, a preferred way is to use the signal module (500) for both tuning-up of individual quantum dots as well as pairs, so that the control is unified at the signal module (500).

[0127] Preferably, the method further comprises:

[0128] after tuning-up to the first Majorana sweet spot (t = A) of the first pair of neighbouring quantum dots (701-f, 701-1), further tuning-up all respective remaining pairs (701-1, 701-r) of neighbouring quantum dots of the first Kitaev chain to their respective Majorana sweet spots, by involving the signal module (500),

[0129] wherein during each tuning-up of a respective pair (701-1, 701-r) of quantum dots, the respective other quantum dots (701 -f) are decoupled from the superconductor element (301) of the Kitaev-chains module (101).Preferably, the tuning-up, by involving the signal module (500), of a particular pair of quantum dots to its respective Majorana sweet spot, comprises:

[0130] sweeping a gate voltage of a control gate of the particular superconductor couplingsegment (301-1) that is between the particular pair while detecting reflections of control signals of the signal module (500) to detect a particular modulation of the reflections, the particular modulation indicative of the respective Majorana sweet spot.

[0131] Preferably, the method further comprises:

[0132] after tuning-up all pairs of neighbouring quantum dots of the first Kitaev-chain to their respective Majorana sweet spots, putting back the quantum dots of the first Kitaev-chain back in resonance by tuning them to their respective resonance energy-levels, to create a first pair of Majorana bound states (y , yrl) in outer quantum dots (701-f, 701-r) of the first Kitaev chain.

[0133] Thereby, the signal module (500) can be used to entirely tune-up a Kitaev chain of a Kitaev-chains module, without the need of other further measurement devices.

[0134] Preferably, the method further comprises:

[0135] after creating the first pair of Majorana bound states (y?1, yrl) of the first Kitaev chain, creating a second pair of Majorana bound states (y^2, yr2) ofasecond Kitaev chain (101-2) of the Kitaev-chains module (101).

[0136] The creating of the second pair may involve one or more, preferably all, of the abovereferenced preferred aspects described in the context of tuning-up the first pair.

[0137] As another example (not illustrated), instead of tuning a Kitaev chain to its Majorana sweet spot, the signal module (500) can also be exploited to tune up a quantum dots coupled via superconducting couplers, such as in the context of operating quantum dots as spin qubits in which the superconducting coupler can act as a coupling element for 2-qubit operations. A signal module may be used to implement gates on qubits / between two qubits; see [Leijnse and Flensberg] for spin qubits and superconducting couplers.

[0138] Provided is furthermore a method of tuning-up the Kitaev-chains device (1), comprising: tuning-up, one by one, each of the Kitaev-chains modules (101, 103, 105, ..., 100-N) by the method as outlined above, preferably including one or more of, preferably all of, the above-described preferred aspects.

[0139] Thus, the method of tune-up based on the signal module (500) simplifies tune-up and scales well to Kitaev-chains devices having a larger number of Kitaev-chains modules, by the signal module (500) serving as a unified control centre for tuning-up the different Kitaev-chains modules one by one.Provided is furthermore a method for initializing a Kitaev-chains module (101) of the Kitaev-chains device (1). The method of initializing comprises:

[0140] tuning-up (Bl) the Kiteav-chains module (101); and

[0141] applying (B2), by the signal module (500), an oscillating signal on the superconductor (300),

[0142] wherein an amplitude of the oscillating signal exceeds a superconducting gap of the superconductor (300).

[0143] Fig- 4 sheds light on why the oscillating signal is suitable for initializing the Kitaev-chains module. After tuning-up (Bl) of the Kitaev-chains module, the Kitaev chains of the Kitaev-chains module host respective Majoranas in their respective boundary quantum dots. In the context of a Kitaev qubit formed by four Majorana bound states, the total fermionic parity is fixed and the Kitaev qubit is encoded in the fermionic parity of different pairs of Majoranas. For example, if the total parity is even the qubit states |0> and |1> can be the states where iyiy2is even or odd, respectively. One could initialize in the even state by putting the chemical potentials of all quantum dots of the Kitaev chains above the superconducting gap, thereby allowing them to expel all unpaired quasiparticles. However, putting the chemical potentials of all quantum dots requires detuning via voltages on each of the corresponding electrostatic onsite gates. By the above-recited method of initializing, this is instead done using the superconductor (300) and the oscillating signal. The oscillating signal can be seen as a periodic modulation of the chemical potential of the superconductor with respect to the quantum dots, and thus has the same effect as detuning all quantum-dot energy levels. In order for the method to work, one needs to make sure the amplitude of the oscillating signals exceeds the superconducting gap of the superconductor. In that way, the quantum dots can lose their excess quasiparticles to the superconductor either by going above it (during one part of the periodic cycle) and ejecting (Fig. 4b) or by going below the negative gap and recombining with quasiparticles there (Fig. 4c). The superconducting gap exceeds induced superconducting gap values each relating to a respective superconducting gap between the superconductor (300) and a respective quantum dot of the Kitaev-chains module (101). In Fig. 4, when the amplitude exceeds the superconducting gap (b), any quasiparticles on the quantum dots (QD1, QD2) can be expelled into the superconductor (SCI) and drain away. This always leaves the quantum dots in an even occupancy and thus the even state of the Kitaev qubit. When the quantum dots are below the negative superconducting gap (c), the quantum dots can obtain quasiparticles from the superconductor and thereby end up in the total even state. In the middle of the cycle, whenthe quantum dots align with the Fermi level of the superconductor, Cooper pairs can form or split, but this leaves the parity of the quantum dots unchanged.

[0144] The amplitude of the oscillating signal is preferably relatively larger, as the efficiency and fidelity increase with increasing signal amplitude, but the amplitude preferably is also below a heating-threshold, beyond which the applied oscillating signal would heat / warm up the Kitaev-chains device (1) more than desired. The heating-threshold may depend on a coolingsystem used for the Kitaev-chains device (1) and aspects such as lines and attenuators of the cooling-system.

[0145] Preferably, the oscillating signal is a high-frequency signal with a frequency 50 Mega Hz (MHz) to multiple tens of Giga Hz (GHz), more preferably between 100 MHz and 10 GHz, more preferably between 100 MHz and 1 GHz.

[0146] A further striking advantage of the above method of initializing by the signal module (500) and superconductor (300) is that one can actually initialize more Kitaev-chains modules (101, 103) of the Kitaev-chains device (1) at once / in one go. Thus, provided is furthermore a method of initializing one or more Kitaev-chains hosted by respective one or more Kitaev-chains modules (101, 103) of the Kitaev-chains device (1). The method comprises:

[0147] tuning-up (Bl) the one or more Kitaev-chains modules (101, 103); and

[0148] applying (B2), by the signal module (500), an oscillating signal on the superconductor (300),

[0149] wherein an amplitude of the oscillating signal exceeds a superconducting gap between the superconductor (300) and the respective one or more Kitaev-chains modules (101, 103) By using the superconductor (300) and the signal module (500), the oscillating signal gets applied to all Kitaev chains of the Kitaev-chains modules, and thereby to all respective Kitaev qubits, at the same time. Thereby, the single oscillating signal can be used to simultaneously initialize a plurality of Kitaev-chains modules with a single operation. So, even more strikingly, the oscillating signal on the superconductor (300) can be used to initialize a plurality of Kitaev qubits all in one go.

[0150] Preferably, a relatively higher-power of the oscillating signal is used. Higher power in this case means a more efficient initialization and thus a higher fidelity. Again, the power is preferably limited by a heating-threshold that may depend on aspects of the cooling-system.

[0151] Provided is furthermore a method of reading-out a Kitaev-chains module (101) of the Kitaev-chains device (1). The method comprises:

[0152] performing (Cl), by the signal module (500), detection measurements based on detecting reflections of one or more control signals generated on the superconductor (300); anddetermining (C2), based on the detection measurements, a quantum capacitance of the Kitaev-chains module (101), the quantum capacitance being indicative of whether a Kitaev qubit is read-out as a first qubit state (|0)) or a second qubit state (| 1)).

[0153] The quantum capacitance depends on the parity of the Majorana pair in which qubit states are encoded [Liu et al]. For a qubit of two chains, the quantum capacitance is different for the two qubit states in the even subspace and can distinguish in which subspace the qubit lives. By the method, the two states being distinguishable by the quantum capacitance is detected by means of the signal module (500) and the superconductor (300).

[0154] Preferably, the performing of the detection measurements comprises using frequency multiplexing by sending, via the superconductor, one or more radio-frequency, RF, signals at respective resonance frequencies of the one or more signal-and-frequency-control modules (901, 903).

[0155] Preferably, the performing of the detection measurements comprises:

[0156] detuning all but two quantum dots of a Kitaev-chain of the Kitaev-chains module (101), the two quantum dots preferably corresponding to outer quantum dots of the Kitaev-chain that are hosting Majorana bound states.

[0157] In particular for longer chains, readout may include baseband pulses for detuning, preferably for detuning bulk quantum dots that are not hosting the unpaired Majoranas. The detuning is advantageous because the readout involves measuring the quantum capacitance of the system, which is the second derivative of energy with respect to chemical potential. In the case of a 2-site chain, the quantum capacitance is finite and different for the two parity states, but for longer chains (3 or more), the quantum capacitance becomes zero when the chemical potential is zero, because for every added site, a highest-order perturbative term drops out ([Bordin et al, 3; supplement section, “Enhanced protection”]). In the preferred way, by detuning the quantum dots of the Kitaev-chains module (101) that are not hosting Majorana bound states, one can read-out the chain as it is then effectively reduced to a 2-site chain.

[0158] In the above-described methods and its preferred aspects, the signal module (500) preferably comprises a reflectometry module (500a) or a transmission-setup module (500b), preferably including one or more of, preferably all of, the above-described preferred aspects of such modules. Accordingly, steps of generating of RF signals are preferably performed by a signal-generator and of detection of signals are preferably performed by a detection module.

[0159] Provided is furthermore a computer program product comprising instructions which, when the program is executed by a computer, cause the computer to carry out any one of theabove-described methods, preferably comprising one or more or all of the above-outlined preferred aspects.

[0160] Provided is furthermore a computer-readable storage medium having stored thereon the computer program product.

[0161] Provided is furthermore an apparatus comprising a processor and a memory storing instructions which, when executing by the processor, cause the processor to perform any one of the above-described methods, preferably comprising one or more or all of the above-outlined preferred aspects.

[0162] Figs. 7a-7d show measurement results of an example of a Kitaev chain. The example is elementary for further illustrative purposes but not for limiting the present disclosure.

[0163] Fig. 7a show the elementary example ( “Elementary device”), which relates to a minimal Kitaev chains based on a single superconducting coupler and two quantum dots. An LC resonator is connected to the superconductor and a high-frequency signal of 341MHz is applied, while measuring the reflected signal in a reflectometry setup.

[0164] Figs. 7b-d show tune-up, readout and initialization.

[0165] Fig. 7b shows the tune-up. Plunger gates (Vpi, Vp?) are varied to control the chemical potentials of the two quantum dots. When one of the two quantum dots is on resonance, the impedance of the elementary device changes which changes the properties of the resonator (such as quality factor, resonance frequency). This is observed as a change in the reflected signal which is measured in the reflected signal by reflectometry. In the data panel, this can be seen as horizontal stripes when QD1 is on resonance and as vertical stripes when QD2 is on resonance. When both are on resonance, a strong anti-diagonal line (along the direction “topleft to bottom-right” is present which indicates the quantum capacitance of the double-dot system. It is in the centre of one of these lines where the QDs can be tuned for the Majorana sweet spot, and thus for qubit operations.

[0166] Fig. 7c shows the readout. To that end, the quantum dots were tuned to the Majorana sweet spot. Thereafter, the reflected signal on the superconductor was detected as a function of time. The readout data shows that detected signal exhibits a telegraph process, which indicates that the system was changing in between two possible states. These two possible states can be used for qubit operations. A time-scale of qubit operations is typically much smaller than the time-scale of oscillations that can be seen in the readout data. In this example, the switching occurred due to quasiparticle poisoning from the superconductor on a millisecond timescale with a rate T, but qubit operations are faster and unaffected thereby. A desired qubit operation can be for example flipping the state, wherein the reflected signal will change as a result, whichcan be detected by the signal module. In the other subpanels of Fig. 7c, one can see the two signals forming two distributions in the I-Q plane (bottom-left) and principle component analysis projecting this as two gaussian distributions (top-right). Therefrom, one can estimate the population of the two states: when two states are degenerate, the population distribution will be 50% in one and 50% in the other. Calculating then the polarization results in a number ranging from -1 to +1, depending on the average distribution of the populations. Further analysis on the time-traces reveals the power spectral density and can be used to calculate the dwell time and Poissonian distribution for the switching events, both of which are useful metrics to characterize the system.

[0167] Fig. 7d shows the initialization. One can increase the applied power (e.g. higher amplitude) to initialize the system in one of the two states as shown in the initialization panel. As the power increases, it becomes more likely to eject single particles into the superconducting excitation gap, which results in an increasing switching rate T and a polarization that tends to 1 when the amplitude of the signal arriving at the device approaches the energy of the superconducting gap. For the panels in Fig. 7d, measurements were performed while initializing, so that the SNR dropped below 1 for increased applied power.

[0168] The following list of references is referred to in the present document and is incorporated herein by way of reference.

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Claims

What is claimed is:

1. Kitaev-chains device (1) comprising:a plurality of gate-controlled semiconductor-superconductor-based Kitaev-chains modules (100), each Kitaev-chains module (101) comprising a plurality of respective Kitaev-chains (101-1; 101-2) and a respective superconductor element (301) forming part of the respective Kitaev-chains (101-1; 101-2);a superconductor (300) connecting the superconductor elements (301, 303) of the plurality of Kitaev-chains modules (100); anda signal module (500) configured to control radio-frequency, RF, signals on the superconductor (300) to control one or more of the plurality of Kitaev-chains modules (100).

2. The Kitaev-chains device (1) of claim 1, wherein one or more, preferably all, of the Kitaev chains (101-1) comprises a respective array of respective gate-controlled semiconductor-based quantum dots (701-f, 701-1, 701-3, 701-r) and one or more respective superconductor coupling-segments (301-1, 301-3, 301-5) of the respective superconductor element (301).

3. The Kitaev-chains device (1) of claim 2, wherein the array relates to an alternating chain of the quantum dots and the one or more superconductor coupling-segments, each superconductor coupling-segment (301-1) configured to enable, by Andreev bound states, a respective superconductivity coupling (A) between its neighbouring quantum dots (701-f, 701-1).

4. The Kitaev-chains device (1) of claim 2, wherein the array relates to a consecutive chain of the quantum dots, each quantum dot (701-1) being proximitized by a respective superconductor coupling-segment (301-3) so as to enable, by inducing superconductivity to the respective proximate quantum dot (701-1), gate-controlled spin-preserving (t) and spin-flipping (tso) couplings between neighbouring quantum dots (701-f, 701-1).

5. The Kitaev-chains device (1) of any one of the preceding claims, wherein one or more, preferably all, of the superconductor elements (301, 303) are formed as a respective integrally-formed superconductor element that is shared by all the Kitaev-chains (101-1, 101-2) of the respective Kitaev-chains modules (101, 303).

276. The Kitaev-chains device (1) of the preceding claim, wherein the superconductor (300) is formed as an integrally-formed superconductor comprising the superconductor elements.

7. The Kitaev-chains device (1) of any one of the preceding claims, wherein the signal module (500) comprises at least one of:a reflectometry module (500a) configured to provide the control by reflectometry involving the RF signals applied to the superconductor (300); and / ora transmission-setup module (500b) configured to provide the control by a transmission setup involving the RF signals applied to the superconductor (300).

8. The Kitaev-chains device (1) of the preceding claim, wherein the signal module (500) comprises a signal -generator (501; VRF) configured to generate the RF signals to be applied to the superconductor (300) and a detection module (505) configured to detect RF signals indicative of a state of the superconductor (300).

9. The Kitaev-chains device (1) of the preceding claim, further comprising one or more signal-and-frequency-control modules (901, 903), each signal-and-frequency-control module (901) placed between the signal module (500) and a respective Kitaev-chains module (101) and configured to control whether a signal generated by the signal module can pass through to the respective Kitaev-chains module (101) and to control a frequency of a passed-through signal, wherein, preferably, each of the one or more frequency-control modules (901, 903) relates to:a resonator (coi, C02) configured to exhibit a particular resonance frequency; or a gate-tuneable Josephson junction (JJ1, JJ3) configured to exhibit gate-tuneable resistance states including a zero-resistance state and a high-resistance state.

10. The Kitaev-chains device (1) of the preceding claim, one or more, preferably all, of the plurality of Kitaev-chain modules (100) comprise a respective Josephson junction element and a respective capacitor element, for forming Kitaev-transmon qubits.

11. Quantum computer (1000) comprising one or more Kitaev-chains devices (1) of any one of the preceding device claims.

12. Method of tuning-up a Kitaev-chains module (101) of the Kitaev-chains device (1) of any one of the preceding device claims, the method comprising:decoupling (Al) the Kitaev-chains module (101) from the remaining Kitaev-chains modules (103, 105, ..., 100-N), by decoupling the superconductor (300) from the remaining Kitaev-chains modules (103, 105, ..., 100-N);performing (A2), by the signal module (500), detection measurements based on detecting reflections of one or more control signals generated on the superconductor (300); and adapting (A3), based on the detection measurements, gate voltages on one or more control gates of the Kitaev-chains module (101) to tune-up the Kitaev-chains module (101).

13. The method of tuning-up according to the preceding method claim, wherein the performing of the detection measurements comprises:varying one or more gate voltages on the one or more control gates of the Kitaev-chains module (101), to obtain information about a state of the Kitaev-chains module (101) by reflectometry.

14. The method of tuning-up according to any one of the preceding method claims, wherein the decoupling of the superconductor (300) from the remaining Kitaev-chains modules (103, 105, ..., 100-N) comprises decoupling quantum dots of the Kitaev-chains of the respective Kitaev-chains modules from the respective superconductor elements (303, 305, ..., 300-N) of the remaining Kitaev-chains modules (103, 105, ..., 100-N).

15. The method of tuning-up according to the preceding method claim, wherein the decoupling of the quantum dots comprises at least one of:establishing, by barrier gates, Coulomb blockades between the quantum dots and the respective superconductor elements; and / ordepleting, by plunger and barrier gates, the quantum dots.

16. The method of tuning-up according to any one of the preceding method claims, the method further comprising:detecting, by involving the signal module (500), one or more first resonance energylevels of a first quantum dot (701-f) of a first Kitaev chain (101-1) of the Kitaev-chains module (101),wherein during the detecting of the one or more first resonance energy-levels, the remaining quantum dots (701-1, 701-r) of the first Kitaev chain are decoupled from the superconductor element (301) of the Kitaev-chains module (101).

17. The method of tuning-up according to the preceding method claim, the method further comprising:after the detecting of the one or more first resonance energy-levels of the first quantum dot (701-f), further detecting, by involving the signal module (500), one or more further resonance energy-levels of each of the remaining quantum dots (701-1, 701-r) of the first Kitaev chain,wherein during each detecting of one or more respective resonance energy-levels of a respective quantum dot (701-1), other quantum dots (701-f, 701-r) different from the respective quantum dot (701-1) are decoupled from the superconductor element (301) of the Kitaev-chains module (101).

18. The method of tuning-up according to any one of claims 16-17, wherein the detecting, by involving the signal module (500), of one or more resonance energy-levels of a particular quantum dot comprises:sweeping a gate voltage of a plunger gate of the particular quantum dot while detecting reflections of control signals of the signal module (500) to detect a modulation of the reflections, the modulation indicative of the one or more resonance energy-levels in which the particular quantum dot resonates with the superconductor.

19. The method of tuning-up according to any one of the preceding method claims, further comprising:tuning-up quantum dots of a first pair of neighbouring quantum dots (701-f, 701-1) to respective resonance energy-levels and tuning-up, by involving the signal module (500), the first pair (701-f, 701-1) to a first Majorana sweet spot (t = A),wherein during tuning-up the first pair (701-f, 701-1), other quantum dots (701-r) of the first Kitaev chain are decoupled from the superconductor element (301) of the Kitaev-chains module (101).

20. The method of tuning-up according to the preceding method claim, the method further comprising:after tuning-up to the first Majorana sweet spot (t = A) of the first pair of neighbouring quantum dots (701-f, 701-1), further tuning-up all respective remaining pairs (701-1, 701-r) of neighbouring quantum dots of the first Kitaev chain to their respective Majorana sweet spots, by involving the signal module (500),wherein during each tuning-up of a respective pair (701-1, 701-r) of quantum dots, the respective other quantum dots (701 -f) are decoupled from the superconductor element (301) of the Kitaev-chains module (101).

21. The method of tuning-up according to any one of the preceding method claims, wherein the tuning-up, by involving the signal module (500), of a particular pair of quantum dots to its respective Majorana sweet spot comprises:sweeping a gate voltage of a control gate of the particular superconductor couplingsegment (301-1) that is between the particular pair while detecting reflections of control signals of the signal module (500) to detect a particular modulation of the reflections, the particular modulation indicative of the respective Majorana sweet spot.

22. The method of tuning-up according to any one of the preceding method claims, further comprising:after tuning-up all pairs of neighbouring quantum dots of the first Kitaev-chain to their respective Majorana sweet spots, putting back the quantum dots of the first Kitaev-chain back in resonance by tuning them to their respective resonance energy-levels, to create a first pair of Majorana bound states (y?1, yrl) in outer quantum dots (701-f, 701-r) of the first Kitaev chain.

23. The method of tuning-up according to the preceding method claim, further comprising: after creating the first pair of Majorana bound states (y?1, yrl) of the first Kitaev chain, creating a second pair of Majorana bound states (y^2, yr2) ofasecond Kitaev chain (101-2) of the Kitaev-chains module (101).

24. Method of tuning-up the Kitaev-chains device (1) of any one of the preceding device claims, the method comprising:tuning-up, one by one, each of the Kitaev-chains modules (101, 103, 105, ..., 100-N) by the method according to any one of the preceding method claims.

25. Method of initializing a Kitaev-chains module (101) of the Kitaev-chains device (1) of any one of the preceding device claims, the method comprising:tuning-up (Bl) the Kiteav-chains module (101); andapplying (B2), by the signal module (500), an oscillating signal on the superconductor (300),wherein an amplitude of the oscillating signal exceeds a superconducting gap of the superconductor (300).

26. Method of initializing one or more Kitaev-chains hosted by respective one or more Kitaev-chains modules (101, 103) of the Kitaev-chains device (1) of any one of the preceding device claims, the method comprising:tuning-up (Bl) the one or more Kitaev-chains modules (101, 103); andapplying (B2), by the signal module (500), an oscillating signal on the superconductor (300),wherein an amplitude of the oscillating signal exceeds a superconducting gap between the superconductor (300) and the respective one or more Kitaev-chains modules (101, 103).

27. Method of reading-out a Kitaev-chains module (101) of the Kitaev-chains device (1) of any one of the preceding device claims, the method comprising:performing (Cl), by the signal module (500), detection measurements based on detecting reflections of one or more control signals generated on the superconductor (300); and determining (C2), based on the detection measurements, a quantum capacitance of the Kitaev-chains module (101), the quantum capacitance being indicative of whether a Kitaev qubit is read-out as a first qubit state (|0)) or a second qubit state (| 1)).

28. The method of reading-out of claim 27, wherein the performing of the detection measurements comprises using frequency multiplexing by sending, via the superconductor, one or more radio-frequency, RF, signals at respective resonance frequencies of the one or more signal-and-frequency-control modules (901, 903).

29. The method of reading-out of any one of claims 27-28, wherein the performing of the detection measurements comprises:detuning all but two quantum dots of a Kitaev-chain of the Kitaev-chains module (101), the two quantum dots preferably corresponding to outer quantum dots of the Kitaev-chain that are hosting Majorana bound states.

30. Computer program product comprising instructions which, when the program is executed by a computer, cause the computer to carry out the method of any one of the preceding method claims.

31. Computer-readable storage medium having stored thereon the computer program product of the preceding claim.

32. Apparatus comprising a processor and a memory storing instructions which, when executing by the processor, cause the processor to perform the method according to any one of the preceding method claims.33

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