Spin-qubits device with nanomagnets for shuttling-based quantum gates and methods of operating the spin-qubits device

Abstract

The disclosure relates to a spin-qubits device (1) for hosting a two-dimensional array of quantum-dot regions (300) for hosting a plurality of spin-qubits. The spin-qubits device (1) comprises: a semiconductor multilayer (30) disposed on a substrate (10); a gate layer (50) disposed on the semiconductor multilayer (30), the gate layer (50) comprising a plurality of control gates (500) configured to control the two-dimensional array of quantum-dot regions (300) in the semiconductor multilayer (30); the device further comprising: a periodic two-dimensional array of nanomagnets (700) configured to provide a spatially-periodic magnetic-stray-field configuration in the two-dimensional array of quantum-dot regions (300), the spatially-periodic magnetic-stray-field configuration for performing shuttling-based quantum gates that involve shuttling one or more spin-qubits through quantum-dot region. The invention has applications in quantum computing and quantum simulations, for example by a quantum computer that includes one or more instances of the spin-qubits devices.

Description

[0001] Spin-qubits device with nanomagnets for shuttling-based quantum gates and methods of operating the spin-qubits device

[0002] Field of the invention

[0003] The invention relates to a spin-qubits device with nanomagnets for shuttling-based quantum gates and methods of operating the spin-qubits device.

[0004] Background art

[0005] Chien-An Wang et al demonstrated shuttling-based quantum gates [Wang et al], in which spin-qubits hosted in a spin-qubits device are controllably manipulated by shutting the spin-qubits through quantum-dot regions formed in a semiconductor multilayer of the spinqubits device.

[0006] For example, the shuttling-based quantum gates can involve shuttling spin-qubits back and forth one or more times between different quantum-dot regions, wherein an external magnetic field is applied to have Zeeman splitting and control a Larmor frequency. A variation of the spin-quantization axes occurs between the quantum-dot regions, resulting from microscopic variations in the material due to for example strain, and said variation enables implementation of shuttling-based quantum gates.

[0007] Such controlled shuttling / hopping of spin-qubits between quantum-dot regions with site-dependent spin-quantization axis is a promising candidate of engineering a semiconductorbased quantum processor.

[0008] Therefore, there is a need to devise spin-qubits devices that enable efficient control of spin-qubits by shuttling-based quantum gates.

[0009] Summary of the invention

[0010] A task set forth by the inventors is to provide an improved spin-qubits device and enable efficient control of spin-qubits by shuttling-based quantum gates.

[0011] The inventors solved the task by providing a spin-qubits device according to the appended independent claims, with advantageous aspects as set out in the dependent claims. Furthermore, methods of operating the spin-qubits device are provided. The invention has applications in quantum computing and quantum simulations, for example by a quantum computer that includes one or more instances of the spin-qubits device according to the present invention. Technical advantages of the provided spin-qubits device include, among other things, at least one of facilitated controllability, fabrication flexibility, and scalability. Technical advantages are further set out in the detailed description below. Brief description of the drawings

[0012] The present invention is discussed in more detail below, with reference to the attached drawings, in which:

[0013] Fig. la shows a side view of a prior-art spin-qubits device.

[0014] Fig. lb shows a top view of a prior-art spin-qubits device.

[0015] Fig. 2a shows a side view of a spin-qubits device of the present disclosure.

[0016] Fig. 2b shows a top view of a spin-qubits device of the present disclosure.

[0017] Fig. 2c illustrates a distance between nanomagnets and a quantum-dot layer of a semiconductor multilayer of the spin-qubits device.

[0018] Fig. 2d illustrates elongated shapes of the nanomagnets.

[0019] Fig. 2e shows an arrangement of nanomagnets and quantum-dot regions based on a square-tiling type.

[0020] Fig. 2f shows an arrangement of nanomagnets and quantum-dot regions based on a triangular-tiling type.

[0021] Fig. 2g shows a variation of quantization axis tip in between neighbouring quantum dots at the example of a spin-qubits device with a triangular-tiling type.

[0022] Fig. 2h shows a quantization axis tip and decoherence gradient each varying along nearest-neighbouring quantum-dot regions at the example of a spin-qubits device with a squaretiling type.

[0023] Figs. 2i-j illustrate aspects of rectangular shaped nanomagnets.

[0024] Fig. 3a is a schematic flow diagram for a method of operating the spin-qubits device.

[0025] Fig. 3b is a diagram of a quantum processor comprising one or more instances of the spin-qubits device.

[0026] Detailed description

[0027] 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.

[0028] 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.

[0029] 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.

[0030] 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.

[0031] 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.

[0032] 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.

[0033] The terms used in describing the various embodiments of the present disclosure are for the purpose of describing particular embodiments and are not intended to limit the present disclosure. As used herein, the singular forms are intended to include the plural forms as well, unless the context clearly indicates otherwise. All of the terms used herein including technical or scientific terms have the same meanings as those generally understood by an ordinary skilled person in the related art unless they are defined otherwise. The terms defined in a generally used dictionary should be interpreted as having the same or similar meanings as the contextual meanings of the relevant technology and should not be interpreted as having ideal or exaggerated meanings unless they are clearly defined herein. According to circumstances, even the terms defined in this disclosure should not be interpreted as excluding the embodiments of the present disclosure. The person skilled in the art will understand that the features described above and / or below may be combined in any way deemed useful. The drawings of the present disclosure show examples / embodiments of the invention, which will be described in detail hereinafter. It is to be understood that one or more of elements / components shown and / or described in one or more of these examples / embodiments and not in others may be used in those others too unless mechanical or other limitations prevent such an implementation. Moreover, describing features of different examples / embodiments in a single passage does not automatically mean that those features are inextricably linked. They may be applied separately from one another.

[0034] Fig. la illustrates a prior-art spin-qubits device (la) for hosting a two-dimensional array of quantum-dot regions (300) for hosting a plurality of spin-qubits (900).

[0035] The spin-qubits device (la) comprises a semiconductor multilayer (30) that is disposed on a substrate (10). A gate layer (50) is disposed on the semiconductor multilayer (30) and comprises a plurality of control gates (500) that are configured to control the two-dimensional array of quantum-dot regions (300) in the semiconductor multilayer (30). The semiconductor multilayer (30) may comprise one or more layers, which may also be referred to as sub-layers.

[0036] Fig. lb shows a top view of the prior-art spin-qubits device (la) to illustrate the plurality of quantum-dot regions (300) and a shuttling-based quantum gate (Ga) configured to controllably manipulate a spin-qubit (e.g. 913).

[0037] Fig. lb shows that the plurality of quantum-dot regions (300) forms a two-dimensional array of individual quantum-dot regions, of which some are labelled, for illustration purposes, as 311, 313, 315, ...; 322, 324, 326, ...; 331, 333, 335, ...; 342, 344, 346, ...; 351, 353, 355, ...; 371, 373, 3735, ...; 391, 393, 395.

[0038] A quantum-dot region can be understood as an electrostatically-confined region created and controlled by control gates of the plurality of control gates (500). For example, the plurality of control gates (500) may comprise plunger gates configured to control potential wells and barrier gates configured to control barriers between the potential wells. By tuning voltages on control gates, the potential wells may be defined and controlled so as to provide a confining region in which spin-qubits can be hosted. The gate layer (50) may comprise one or more sublayers of gates.

[0039] Spin-qubits may be carried by a suitable spin-carrier, such as an electron or a hole. The way of implementing spin-qubits may depend on for example the semiconductor material. Each quantum-dot region can be populated with a respective spin qubit. For example, as illustrated in Fig. lb, a spin-qubit (913) may occupy quantum-dot region (331), another spin-qubit (935) may occupy another quantum-dot region (335), yet another spin-qubit (971) may occupy yet another quantum-dot region (371); etc.

[0040] In Fig. lb, an external magnetic field (Bext) is applied to energetically split spin states of the spin-qubits. Microscopic variations in the material, due to for example strain, can create a variation of the spin-quantization axes between two nearest-neighbouring quantum-dot regions so that there is a sufficiently large and suitable angle between a first spin-quantization axis of a first quantum-dot region (e.g., 331) and a second spin-quantization axis of a second quantum-dot region (e.g., 322) that is a nearest-neighbour of the first quantum-dot region.

[0041] The spin-quantization axis corresponds to a direction in which a spin’s magnetic moment is pointing to. The spin-quantization axis may be related to a magnetic field direction, with a relation depending on involved materials. At the example of a silicon-based semiconductor for hosting the quantum-dot regions, the spin-quantization axis at a quantumdot region is in good approximation colinear with a magnetic field orientation at that quantumdot region. At the example of a germanium-based semiconductor for hosting the quantum-dot regions, microscopic details such as electrostatics and material strain conditions of the respective quantum-dot region as well as an orientation of the magnetic field can influence the direction of the spin-quantization axis.

[0042] Fig. lb further illustrates a shuttling-based quantum gate (Ga) by showing a related pathway. At the example of spin-qubit (913), a shuttling-based quantum gate (Ga) can be implemented by shuttling the spin-qubit (913) from the quantum-dot region (331) to a nearest neighbouring quantum-dot region (322) and back, with the effect that a spin-state of the spinqubit (913) is rotated around an axis n with a certain angle 9. Specific quantum gates can be implemented for example by tuning waiting times at the involved quantum-dot regions, a number of times the shuttling steps are performed, combinations thereof, etc. Such implementation details may depend on the spin-quantization axis variation between the two involved quantum-dot regions and on which quantum gate one wants to perform. Accordingly, depending on the spin-quantization axis variation, the shuttling-based quantum dots can be implemented by tuning parameters of shuttling operations accordingly.

[0043] Loss and DiVincenzo envisioned implementing quantum gates based on shuttling electrons between quantum dots [Loss & DiVincenzo]. Shuttling-based quantum gates contrast methods that are based on electric-dipole spin resonance (EDSR), in which a spin-qubit is merely moved locally at a quantum-dot region, e.g. [Aldeghi et al, 1; Aldeghi et al, 2; Philips et al; Tadokoro et al]. More details and background on such shuttling-based quantum gates for controllably manipulating one or more spin-qubits are outlined in [Wang et al]. The inventors devised a spin-qubits device (1) that simplifies control by shuttling-based quantum gates. Figs. 2a-2j illustrate the spin-qubits device (1) and aspects thereof by example. Reference signs are added in the description below for illustration purposes for intelligibility reasons, but the disclosure is not limited to the illustrations.

[0044] The spin-qubits device (1) is for hosting a two-dimensional array of quantum-dot regions (300) for hosting a plurality of spin-qubits (900). The spin-qubits device (1) comprises: a semiconductor multilayer (30) disposed on a substrate (10); a gate layer (50) disposed on the semiconductor multilayer (30), the gate layer (50) comprising a plurality of control gates (500) configured to control the two-dimensional array of quantum-dot regions (300) in the semiconductor multilayer (30); characterized by further comprising: a periodic two-dimensional array of nanomagnets (700) configured to provide a spatially-periodic magnetic-stray -field configuration in the two-dimensional array of quantumdot regions (300), the spatially-periodic magnetic-stray -field configuration for performing shuttling-based quantum gates (Gi; G2) that involve shuttling one or more spin-qubits (911; 942) through quantum-dot regions (311-322; 342-333).

[0045] Such a spin-qubits device (1) provides a technical advantage of enabling simplified shuttling-based quantum operations on spin-qubits, because a periodic two-dimensional array of nanomagnets (700) is devised that exhibits the spatially-periodic magnetic-stray-field configuration suitable for performing the shuttling-based quantum operations.

[0046] The spin-qubits device (1) has the dedicated periodic two-dimensional array of nanomagnets (700) that provides a suitable spatially-periodic magnetic-stray-field configuration. Thereby, for example, shuttling-based quantum operations can be implemented without the need of using a solenoid for creating a controlled external magnetic field, which reduces the amount of required control. Furthermore, shuttling-based quantum operations can be implemented without relying on microscopic variations in the quantization axis which may be uncontrollable. Instead, the suitable variations in the quantization axis are controllably created by the periodic two-dimensional array of nanomagnets (700).

[0047] Moreover, the spatial periodicity of the magnetic-stray-field configuration allows to employ similar control parameters for controlling different regions of the spin-qubits device (1). For example, the periodic configuration enables to establish similar magnetic conditions throughout the array of quantum-dot regions, thereby further simplifying the controlled manipulation of spin-qubits in the spin-qubits device (1). For example, a variation of the quantization axis between two nearest-neighbour quantum-dot regions may be similar in different parts of the array of quantum-dot regions.

[0048] The two-dimensional array of quantum-dot regions (300) can be flexibly populated by a plurality of spin-qubits (900). For example, leaving more quantum-dot regions vacant facilitates shuttling-based quantum gates and moreover allows shuttling to peripheral sensing regions. The spin-qubits device (1) may further include various other components in or close to the semiconductor multilayer (30) for different purposes such as readout, initialization, etc.

[0049] Fig. 2c illustrates that the semiconductor multilayer (30) may comprise a plurality of layers (30-1, 30-2, 30-3). How many and which type of layers may depend for example on materials involved. A quantum-dot layer (30-2) is a semiconductor layer of the semiconductor multilayer (30) in which the quantum-dot regions (300) are formed and thus in which the spinqubits can be hosted. Accordingly, the quantum-dot layer (30-2) is also referred to as “the quantum well” or “the quantum well region”. The quantum-dot layer (30-2) is optionally surrounded by one or more other layers (30-1, 30-3) that each may comprise one or more sublayers. Non-limiting examples of semiconductor multilayers (30) can be found in [Scappucci et al], which shows different arrangements of semiconductor multilayers (30) that can also be used for the spin-qubits device (1) of the present disclosure. Figure 1 in [Scappucci et al], shows some examples based on a Silicon-substrate as a lowest substrate layer (30-1); “Si-MOS” has28Si as a quantum-dot layer (30-2) and a28SiO2 layer as a further layer (30-3) on top of the quantum-dot layer (30-2); ”Si / SiGe”, in which a quantum-dot layer (30-2) is implemented as a28Si layer surrounded by a lower layer (30-1) comprising Sii-xGexand Sio.7Geo.3 sub-layers, and by an upper layer (30-3) comprising Sio.7Geo.3 and AI2O3 sub-layers; and “Ge / SiGe” with a Ge-based quantum-dot layer (30-2) and also surrounded by other layers. On top of the semiconductor multilayer (30) is the gate layer (50).

[0050] Preferably, the semiconductor multilayer (30) comprises a quantum-dot layer (30-2), which is preferably surrounded by one or more layers (30-1, 30-3), each preferably comprising one or more sub-layers. Typically, the quantum-dot regions (300) are then formed just below a surface of the quantum-dot layer (30-2), which can contact an upper layer (30-3) disposed on top of and in contact with the quantum-dot layer (30-2), or, as the upper layer (30-3) is optional, directly contact the gate layer (50). Preferably, the semiconductor multilayer (30) is configured to implement the quantum-dot regions (300) based on a two-dimensional electron gas (2DEG) or a two-dimensional hole gas (2DHG). Preferably, a distance (d) between a bottom (700b-l) of each respective nanomagnet (700b) and a surface of the quantum-dot layer (30-2) of the semiconductor multilayer (30) is between 65nm and 200nm, more preferably between 70nm and 180nm, more preferably between 80nm and 180nm, even more preferably between 90nm and 170nm, most preferably between lOOnm and HOnm, wherein the quantum-dot layer (30-2) is configured to host the two-dimensional array of quantum-dot regions (300).

[0051] Specifically devising the distance (d) in such dimensions has the technical advantage that the total magnetic field strength is suitably below a threshold so that spin-qubits (900) hosted in the quantum-dot layer (30-2) exhibit a Larmor frequency of below 1GHz. Thereby, shuttling-based quantum gates can be efficiently performed. At the same time, the magnetic- stray-field configuration is suitable for a providing a sufficiently strong magnetic field for Zeeman splitting so as to split spin-states of the spin-qubits hosted in the quantum-dot regions, so that an external magnetic field strength can be reduced or even put to zero if desired. The specific distances (d) thus enable to create a particularly suitable magnetic-stray-field configuration that efficiently enables the shuttling-based quantum gates, even at zero external magnetic field. The specific distance (d) has the further technical advantage of increasing fabrication flexibility for patterning electrostatic gates in the gate layer (50), as the distance (d) can create space in the gate layer (50) for flexibly positioning gates therein.

[0052] Fig. 2c illustrates a number of nanomagnets (700a, 700b, 700c, 700d) to illustrate that the distance (d) is between a bottom part (700b-l) of a nanomagnet (700b) and a surface of the quantum-dot layer (30-2). In case that the semiconductor multilayer (30) contains another layer (30-3) on top of the quantum-dot layer (30-2), the distance (d) is thus not to a surface of the semiconductor multilayer (30) but additionally includes a width (w-30-3) of the another layer (30-3).

[0053] Preferably, relative from a top view of the spin-qubits device (1), each nanomagnet has an elongated shape (7001a-b, 7002).

[0054] A nanomagnet having an elongated shape, from the top view, means that the nanomagnet is longer in one dimension than in another. In other words, its length (f ) in a first direction (yi, y2) is larger than its width (w) in a second direction (xi, X2'). Various elongated shapes may be considered. For example, the nanomagnet may have a substantially rectangular or substantially oval shape, but the elongated shape is not limited thereto. For example, a width of the nanomagnet may vary along its length dimension in a non-monotonic manner. Width and length dimensions of the nanomagnets may vary and may range for example in between 30nm and 200nm, as illustrated further below in the context of example layouts and tables. The nanomagnets having an elongated shape, rather than a square or circle, provides the technical advantage of a shape-induced anisotropy that causes magnetization to persist at ambient conditions. Devising the nanomagnets to have elongated shapes is, in the context of the present spin-qubits device (1), a particularly effective way of omitting the need of an externally applied magnetic field when operating the spin-qubits device (1). Moreover, the elongated shape supports a persisting magnetization and thus reduces a chance of flipping of the magnetization and thus the magnetic-stray -field even at room temperatures. Moreover, the shape-induced anisotropy supports magnetization to persist when operating the spin-qubits device (1) at low temperatures.

[0055] Thus, the nanomagnets having an elongated shape further enhances stable operation capabilities and utility of the spin-qubits device (1). Hence, creating the magnetic-stray-field by nanomagnets of elongated shape is, in the context of the spin-qubits device (1), particularly advantageous and effective.

[0056] Preferably, the elongated shape is a rectangular shape (7001a-b) or oval shape (7002). Such shapes are effective in creating the spatially-periodic magnetic-stray -field configuration. However, as outlined above, other elongated shapes may be considered as well.

[0057] Figs. 2b, 2e, 2f illustrate the nanomagnets having a bar-shape, but various other elongated shapes may be implemented as well. Moreover, corners may be smoother and may also depend on manufacturing aspects. Preferred dimensions of the length and width may generally depend on material properties and material choices of the nanomagnets. Preferably, the length of a nanomagnet is at least twice as large as the width, which is an effective way of keeping a magnetization of the nanomagnets. The important aspect of the preferred elongated shape is the nanomagnet substantially having a length (longer dimension) and a width (shorter dimension), in contrast to a symmetric shape such as a circle or square. Fig. 2d illustrates examples of elongated shapes (7001a, 7001b, 7002) with indicated length (f) and width (w) dimensions. A chosen scale of the nanomagnets may depend on the materials and other choices, and examples are provided for illustration further below.

[0058] Preferably, the nanomagnets are configured to be magnetized along a length dimension of their elongated shape. Such magnetization is illustrated further below in Figs. 2i-2j, in particular Fig. 2i(b).

[0059] Preferably, the nanomagnets (700) are placed at vertices of a first periodic tiling, and the quantum-dot regions (300) are located at vertices of a second periodic tiling.

[0060] Tilings can also be referred to as tessellations. A periodic tiling involves a basis shape (or “basic shape”) formed by one or more convex regular polygons that form the basis of the respective tiling / tessellation. Fig. 2e illustrates a basis shape SI and a basis shape S2, Fig. 2f shows basis shapes T1 and T2, respectively.

[0061] Basing the positioning of both the nanomagnets (700) and the quantum-dot regions (300) on vertices of respective periodic tilings is an efficient way of devising the nanomagnets in relation to the quantum-dot regions so as to provide the magnetic-stray-field configuration that is suitable for the shuttling-based quantum gates. That is, specifically devising both the positions of nanomagnets and also the positions of the quantum-dot regions on vertices of periodic tilings is an efficient way of building suitable arrangements in which shuttlings between nearest-neighbouring quantum dots are enabled starting from any quantum dot.

[0062] For example, the magnetic-stray -field configuration may allow for performing a shuttling-based quantum gate (Gi; G2) starting from any one of the quantum-dot regions. For example, a spin-qubit (911; 942) initialized at a first quantum-dot region (311; 342) can be controlled by a shuttling-based quantum gate (Gi; G2) by shuttling the spin-qubit to a nearest- neighbour quantum-dot region (322; 333) directly adjacent to the first quantum-dot region.

[0063] Preferably, the first and second tilings are regular tilings. In other words, the tilings are based on regular polygons.

[0064] When the quantum-dot regions are located / positioned according to vertices of a regular tiling, each quantum-dot region may be controlled on an equal footing, i.e., in a similar manner. For example, quantum-dot regions then have a same amount of suitable nearest-neighbour quantum dots, so that utility of the spin-qubits device (1) is further increased. So, placing the nanomagnets and quantum-dot regions on such periodic spots of regular tilings is an efficient way of devising suitable periodic patterns for the creating of the magnetic-stray-field configuration that is suitable for the shuttling-based quantum gates.

[0065] Preferably, the first and second tilings have a same tiling type. Thereby, a periodicity of the nanomagnet arrangement / positioning can be adapted to a periodicity of the quantum-dot region arrangement / positioning. Examples are illustrated in Figs. 2e and 2f.

[0066] Preferably, the tiling type is a square tiling, a triangular tiling, or a hexagonal tiling. Such tiling types are regular tilings that provide an advantageous way of devising the arrangements of nanomagnets and quantum-dot regions that are suitable for creating the magnetic-stray-field suitable for the performing of the shuttling-based quantum gates, as a periodicity can be flexibly devised.

[0067] Preferably, the first and second tilings are a triangular tiling or a hexagonal tiling. The triangular tiling has the advantage that a quantum-dot region has more nearest-neighbours, which increases control options for shuttling-based quantum gates. For illustration purposes, Figs. 2b and 2e show an arrangement based on a square tiling type, and Fig. 2f shows an arrangement based on a triangular tiling type. In general, various other tilings may be considered. For example, a hexagonal tiling type can also be devised.

[0068] Preferably, the tiling type is the triangular tiling, and the first regular tiling is based on a first basis triangle (Tl) that is larger than and / or tilted relative to a second basis triangle (T2) of the second regular tiling. An example is illustrated in Fig. 2f.

[0069] Preferably, each nanomagnet (701c) is placed centrally with respect to a respective triangle (T2) of the second regular tiling, the respective triangle (T2) with vertices at which respective three quantum-dot regions (501a, 501b, 501c) are placed. Such relative placing is particularly advantageous for creating the magnetic-stray -field configuration in which each pair of nearest-neighbour quantum-dot regions can be used to perform the shuttling-based quantum gates, by the spin quantization axis suitably varying in between the quantum-dot regions. Fig. 2g illustrates such suitable variation at an example of such a triangular tiling type. In Fig. 2g, the quantization axis tip is shown with respect to a quantum-dot region 0 (Fig. 2g(A)), a quantum-dot region 1 (Fig. 2g(B)), and a quantum-dot region 2 (Fig. 2g(C)), the quantum-dot regions 0, 1, 2 forming a basis triangle.

[0070] Preferably, the nanomagnets have an elongated shape and elongate in a direction (y2) that is different from directions (yz , X2') of the second basis triangle (T2).

[0071] Preferably, the tiling type is the square tiling, the first regular tiling is based on a first basis square (SI) extending in two perpendicular directions (xf, yi’), and each nanomagnet’s length elongates in a direction (yi) different from the two perpendicular directions (xf, yf), preferably at an angle of substantially 45 degrees. Such a preferred configuration is illustrated in Figs. 2b and 2e. By orienting the nanomagnets in such a way, the magnetic-stray-field configuration enables performing the shuttling-based quantum gates between any pair of quantum-dot regions, so that the spin-qubits device (1) exhibits increased utility and flexibility. Preferably, the second regular tiling is based on a second basis square (S2) extending in the same two perpendicular directions (xf, yf), as illustrated in Figs. 2b and 2e.

[0072] Preferably, an inter-nanomagnet distance between each pair of nearest-neighbour nanomagnets is larger than an inter-quantum-dot-region distance between each pair of nearest- neighbour quantum-dot regions. Thereby, the magnetic-stray -field configuration has a variation in between neighbouring quantum-dot regions that is greater than 0 degrees and lower than 180 degrees, which is suitable for shuttling-based quantum gates. In general, inter-nanomagnet distances may be chosen depending on material properties. In a preferred variation, the inter- nanomagnet distance is smaller than twice the inter-quantum-dot-region distance. Combinations of the preferred aspects are even more preferred. For example, exploiting the specific distance (d) range between the bottom of the nanomagnets and the surface of the quantum-dot layer (30-2) in combination with elongated-shaped nanomagnets is particularly effective in creating the magnetic-stray -field for performing the shuttling-based quantum gates. Further combining with the preferred tiling aspects is yet more advantageous. As another example, employing regular tilings for the positioning of the nanomagnets and quantum-dot regions, and using elongated shapes of the nanomagnets, is in combination particularly advantageous, and even more advantageous when further orienting the elongated shape along a direction different from the pattern underlying the quantum-dot region tiling, and yet even more advantageous when yet further combining with the specific distance (d).

[0073] Preferably, the two-dimensional array of nanomagnets (700) is disposed in a nanomagnets-hosting region (70) of the spin-qubits device (1), the nanomagnets-hosting region (70) being located within the gate layer (50) or being located on top of the gate layer (50). Thereby, the spin-qubits device (1) can be compact while still saving the need of an external magnetic field when operating that spin-qubits device (1). Whether the nanomagnets (700) are placed at a height relative to the quantum-dot layer (30-2) similar to the gate layer (50) or not may be chosen considering a width (w-30-3) of an eventual layer (30-3) and a desired specific distance (d) as outlined further above. For example, when the width (w-30-3) is relatively larger, the nanomagnets (700) may be placed relatively closer to the gate layer (50) or may be arranged within the gate layer (50) so as to achieve the specific distance (d). For example, when the semiconductor multilayer (30) has a relatively thin layer (30-3) on top of the quantum-dot layer (30-2), the nanomagnets-hosting region (70) may be located on top of the gate layer (50).

[0074] Preferably, the spatially-periodic magnetic-stray-field configuration exhibits a quantization-axis angle variation in between two nearest-neighbouring quantum-dot regions (311-322) of between 10-170 degrees, preferably between 45-135 degrees, more preferably between 80 and 100 degrees. Such angles are particularly suitable for performing the shuttlingbased quantum gates, optionally also without the need of applying an external magnetic field. Preferably, the variation is in between 10-80 degrees, yet more preferably in between 20-60 degrees.

[0075] Preferably, the plurality of control gates (500) comprises: a plurality of electrostatic plunger gates configured to control potential-wells of the quantum-dot regions (300), and a plurality of electrostatic barrier gates configured to control potential-barriers between adjacent quantum-dot regions (300). Preferably, the plurality of quantum-dot regions (300) is configured to host a plurality of spin-qubits (900) carried by a respective plurality of spin-qubit carriers, such as electrons or holes, that can be controllably shuttled in between quantum-dot regions by controlling the plurality of control gates (500). Which type of spin-qubit carriers are employed may depend on material choices. As non-limiting and merely illustrative examples: electrons may be controlled in silicon-based semiconductors, and holes may be controlled in germanium-based semiconductors.

[0076] Below, the illustrations shown in the figures are briefly described for showcasing the invention. However, the disclosure is not limited to the illustrations and below descriptions.

[0077] Fig. 2a shows a side view of a spin-qubits device (1) of the present disclosure. The spinqubits device (1) is shown with the substrate (10), the semiconductor multilayer (30) for hosting the quantum-dot regions (300), the gate layer (50) for hosting the gates (500), and a nanomagnets-hosting-region (70) for hosting the nanomagnets (700).

[0078] Fig. 2b shows a top view of a spin-qubits device (1) of the present disclosure. The top view illustrates various advantageous aspects such as the elongated shape of the nanomagnets and their relative orientation (along yi direction) different from the two perpendicular directions (xf, yi') that form the basis for the second regular tiling of the quantum-dot regions. A number of nanomagnets and quantum-dot regions are labelled for illustration purposes. Fig. 2d illustrates examples of elongated shapes of the nanomagnets.

[0079] Fig. 2e shows an arrangement of nanomagnets and quantum-dot regions based on a square-tiling type, in which the quantum-dot regions are positioned based on a basic square (S2) and the nanomagnets (700) are positioned based on a basic square (SI), each basic square suitable for generating the respective tiling underlying the positions of nanomagnets and quantum-dot regions, respectively. As can be seen in this example, quantum-dot regions are positioned on each vertex of the regular tiling arising from the basic square (S2), so that each quantum-dot region can be controlled in a similar manner. Example spin-qubits (911, 935, 942, 971) are illustrated. For instance, spin-qubit 911 can be controlled by a shuttling-based quantum gate Gi that involves shuttling the spin-qubit 911 from its starting position 311 to a neighbouring quantum-dot region 322 and back, one or more times and including respective waiting times. Similarly, spin-qubit 942 can be controlled by shuttling-based quantum gate G2.

[0080] Fig. 2c shows a distance between nanomagnets and a quantum-dot layer of a semiconductor layer of the spin-qubits device. In particular, the distance between a surface of the quantum-dot layer (30-2) and a bottom of a nanomagnet (700b) is illustrated. Fig. 2f shows an arrangement of nanomagnets and quantum-dot regions based on a triangular-tiling type. The tiling underlying the nanomagnets can be seen as tilted relative to the tiling underlying the quantum-dot regions. Moreover, a basic triangle (Tl) underlying the nanomagnet-tiling is larger than a basic triangle (T2) underlying the quantum-dot-region-tiling. Moreover, nanomagnets are positioned centrally on triangles of the quantum-dot regions, providing a magnetic-stray -field configuration that varies periodically in a such a manner that each quantum-dot region can be controlled in a similar manner.

[0081] Fig. 2g shows a variation of quantization axis tip in between neighbouring quantum dots at the example of a spin-qubits device with a triangular-tiling type. The panels (A), (B), (C) show quantization axis tips relative to quantum-dot regions “0”, “1”, “2” of a basis triangle of the second regular tiling based on the triangle tiling type. Panel (A) shows the quantization axis at a position in the xy -plane relative to quantum-dot region 0, panel (B) relative to quantum-dot region 1, and panel (C) relative to quantum-dot region 2. Fig. 2g showcases the advantage that the relative change in the quantization axis between any two of the three quantum-dot regions is sufficient for implementing shuttling-based quantum gates.

[0082] Fig. 2h shows a quantization axis tip and decoherence gradient each varying along a line of nearest-neighbouring quantum-dot regions at the example of a spin-qubits device with a square-tiling type. The top part of the figure shows the quantization axis tip varying periodically when starting from a first quantum-dot region (e.g. 342) and moving along a direction yf along nearest-neighbouring quantum-dot regions (e.g. 342->333 etc.). The lower part of the figure shows a variation of a decoherence gradient along the line.

[0083] Figs. 2i-j show aspects of rectangular shaped nanomagnets. Panel (a) shows a Larmor frequency in a zoomed-in region with quantum-dot regions indicated by small circles. As can be seen, the Larmor frequency is below 1GHz and thus particularly suitable for shuttling-based quantum gates. Panel (b) shows the magnetization of one particular nanomagnet (placed in the center of the figure). The arrows showcase that the magnetization is substantially aligned with an elongation direction of the nanomagnet. Panel (c) shows the magnetic field lines around one particular nanomagnet in the plane of the quantum-dot layer (30-2) and panel (d) shows the magnetic field lines in a cross-section of the quantum-dot layer (30-2). Panels (c) and (d) show the stray magnetic field changing direction and, therefore, giving rise to the pattern of quantization axis tip variation for shuttling control.

[0084] Provided is furthermore a quantum processor (1000) that comprises one or more instances of the spin-qubits device (1) according to the present invention, each spin-qubits device (1) preferably comprising one or more or all of the above-outlined preferred aspects. Provided is furthermore a method of operating the spin-qubits device (1) according to the present invention, preferably comprising one or more or all of the above-outlined preferred aspects. The method comprises: magnetizing (Al) the periodic two-dimensional array of nanomagnets (700) to create the spatially-periodic magnetic-stray-field configuration in the two-dimensional array of quantum-dot regions (300); and controlling (A2) the plurality of control gates (500) to perform a shuttling-based quantum gate acting on one or more spin-qubits.

[0085] The method exploits the nanomagnets (700) to create the magnetic-stray-field configuration, thereby enabling performing the shuttling-based quantum gates, if desired even without the need of applying an external magnetic field during the performing, which reduces the amount of required control.

[0086] Preferably, the magnetizing comprises applying an external magnetic field acting on the periodic two-dimensional array of nanomagnets (700), wherein the controlling of the plurality of control gates (500) is performed based on the spatially-periodic magnetic-stray- field configuration without external magnetic fields. In other words, shuttling-based quantum gates can be performed without external magnetic fields, a possibility enabled by the spin-qubits device (1).

[0087] Preferably, the magnetizing comprises applying an external magnetic field acting on the periodic two-dimensional array of nanomagnets (700), preferably in a direction substantial parallel with a length dimension of an elongated shape of the nanomagnets.

[0088] Preferably, the method further comprises populating the quantum-dot regions by a plurality of spin-qubits (900) but leaving at least one quantum-dot region empty, so as to enable the performing of shuttling-based quantum gates. In other words, an occupancy density of the quantum-dot regions (300) can be flexibly adapted, and spin-qubits and quantum-dot regions may be configurable. For example, one may leave at least one quantum-dot region empty in between two nearest-neighbouring spin-qubits.

[0089] Preferably, a total magnetic-field strength at the quantum-dot regions is kept below a threshold so that spin-qubits hosted in the quantum-dot regions each have a respective Larmor frequency of below 1GHz. Then shuttling-based quantum gates can be efficiently implemented.

[0090] Preferably, the performing of the shuttling-based quantum gate is performed without applying an external magnetic field, or is performed with applying an external magnetic field but while keeping a total magnetic-field strength at the quantum-dot regions below a threshold so that spin-qubits hosted in the quantum-dot regions each have a respective Larmor frequency of below 1GHz.

[0091] Provided is furthermore a computer program product comprising instructions which, when the program is executed by a computer, cause the computer to carry out the method, preferably comprising one or more or all of the above-outlined preferred aspects.

[0092] Provided is furthermore a computer-readable storage medium having stored thereon the computer program product.

[0093] Provided is furthermore an apparatus comprising a processor and a memory storing instructions which, when executing by the processor, cause the processor to perform the method, preferably comprising one or more or all of the above-outlined preferred aspects.

[0094] Next, some example-layouts are described for illustration purposes. The examplelayouts were numerically simulated in an area of 3990nm x 3815nm with a mesh size of 5nm x 5nm x 5nm with variations in terms of nanomagnet parameters (width, length, thickness tmagnet) and material parameters (exchange stiffness Aex, saturation magnetization Ms, magnetocrystalline anisotropy Kmc):

[0095] The exchange stiffness, saturation magnetization, and magneto-crystalline anisotropy are known technical parameters for characterizing magnets. Cobalt (Co) and Iron (Fe) are known materials for magnets with, for example, the following parameter values:

[0096] • Co: o Saturation magnetization 1.4 MA / m, [Tadokoro et al]. o Saturation magnetization 1.450 MA / m, [Legrand et al]. o Saturation magnetization 1.467 MA / m, exchange stiffness of 29.06 pj / m, [Neumann, Schreiber].

[0097] • Fe: o Saturation magnetization 1.7 MA / m, exchange stiffness of 21 pj / m [Aldeghi et al, 2]

[0098] In the examples, the parameter values are varied starting from such known parameter values to simulate the examples of Co and Fe but also deviations. Furthermore, the distance between the nanomagnets and the quantum wells was taken as lOOnm. The example-layouts showcase the versatility of the spin-qubits device (1) of the present invention, which is however not limited to the illustrated examples. The tables give indications on which parameter combinations indicate more satisfactory results in terms of a stable periodic magnetic-stray- field configuration. A “+-“ means a working example, a “+” means a working and more satisfactory example, and a “++” means a particularly well-working and therefore more preferred example. The following tables are based on width = 70nm, length = 175nm, Kmc=0.

[0099] When working with Ms= 1700, higher values of As are preferred. The following table is based on width = 70nm, length = 175n

[0100] The following table is based on width = 60nm, length = 120n

[0101] The following tables are based on width = 60nm, length = 12

[0102] The following table is based on width = 70nm, length = 165n

[0103] The following table is based on width = 70nm, length = 140n

[0104] The following table is based on width = 40nm, length = 120n

[0105] Moreover, by decreasing the magnet thickness, one can observe a tendency of a lower qubit frequency and lower decoherence gradient. Moreover, a variation in distance from the quantum well in the range of 90-140nm showed similar good results as well. Next, two examples of a nanomagnet triangular-layout are shown for illustration purposes as well. Similarly as before, a distance between the nanomagnets and the semiconductor of lOOnm was chosen for illustration.

[0106] The following table is based on width = 30nm, length = 60n The following table is based on width = 30nm, length = 60n

[0107] The following list of references is referred to in the present document and is incorporated herein by way of reference.

[0108] List of references [Aldeghi et al, 1] Michele Aldeghi, Rolf Allenspach, Andriani Vervelaki, Daniel letter, Kousik Bagani, Floris Braakman, Martino Poggio, and Gian Salis. Simulation and measurement of stray fields for the manipulation of spin-qubits in one and two-dimensional arrays. arXiv:2410.08865vl. 11 October 2024.

[0109] [Aldeghi et al, 2] Michele Aldeghi, Rolf Allenspach, Gian Salis. Modular nanomagnet design for spin qubits confined in a linear chain. Appl. Phys. Lett. 122, 134003 (2023).

[0110] [Legrand et al] W. Legrand, S. Lopes, Q. Schaeverbeke, F. Montaigne, and M.M. Desjardins. Optimal design of nanomagnets for on-chip field gradients. Phys. Rev. Applied 20, 044062 (2023).

[0111] [Loss & DiVincenzo] Daniel Loss and David P. DiVincenzo. Quantum computation with quantum dots. Phys. Rev. A 57, 120. 1 January 1998.

[0112] [Neumann, Schreiber] R. Neumann; L. R. Schreiber. Simulation of micro-magnet stray-field dynamics for spin qubit manipulation. J. Appl. Phys. 117, 193903 (2015).

[0113] [Philips et al] Philips, S.G.J., M^dzik, M.T., Amitonov, S.V. et al. Universal control of a six- qubit quantum processor in silicon. Nature 609, 919-924 (2022).

[0114] [Tadokoro et al] Masahiro Tadokoro, Takashi Nakajima, Takashi Kobayashi, Kenta Takeda, Akito Noiri, Kaito Tomari, Jun Yoneda, Seigo Tarucha, and Tetsuo Kodera. Designs for a two-dimensional Si quantum dot array with spin qubit addressability. Sci Rep 11, 19406 (2021). https: / / doi.org / 10.1038 / s41598-021-98212-4. 2021.

[0115] [Scappucci et al] Scappucci, G., Taylor, P.J., Williams, J.R. et al. rystalline materials for quantum computing: Semiconductor heterostructures and topological insulators exemplars. MRS Bulletin 46, 596-606 (2021). https: / / doi.org / 10.1557 / s43577-021-00147-8.

[0116] [Wang et al] Chien-An Wang, Valentin John, Hanifa Tidjani, Cecile X. Yu, Alexander Ivlev, Corentin Deprez, Floor van Riggel en-Doelman, Benjamin D. Woods, Nico W. Hendrickx, Will I. L. Lawrie, Lucas E. A. Stehouwer, Stefan Oosterhout, Amir Sammak, Mark Friesen, Giordano Scappucci, Sander L. de Snoo, Maximilian Rimbach-Russ, Francesco Borsoi, Menno Veldhorst. Operating semiconductor quantum processors with hopping spins. ArXiv, arXiv:2402.18382vl. February 2024.

Claims

What is claimed is:

1. Spin-qubits device (1) for hosting a two-dimensional array of quantum-dot regions (300) for hosting a plurality of spin-qubits (900), the spin-qubits device (1) comprising: a semiconductor multilayer (30) disposed on a substrate (10); a gate layer (50) disposed on the semiconductor multilayer (30), the gate layer (50) comprising a plurality of control gates (500) configured to control the two-dimensional array of quantum-dot regions (300) in the semiconductor multilayer (30); characterized by further comprising: a periodic two-dimensional array of nanomagnets (700) configured to provide a spatially-periodic magnetic-stray -field configuration in the two-dimensional array of quantumdot regions (300), the spatially-periodic magnetic-stray -field configuration for performing shuttling-based quantum gates (Gi; G2) that involve shuttling one or more spin-qubits (911; 942) through quantum-dot regions (311-322; 342-333).

2. The spin-qubits device (1) of claim 1, wherein a distance (d) between a bottom (700b-l) of each respective nanomagnet (700b) and a surface of a quantum-dot layer (30-2) of the semiconductor multilayer (30) is between 65nm and 200nm, preferably between 70nm and 180nm, more preferably between 80nm and 180nm, even more preferably between 90nm and 170nm, most preferably between lOOnm and HOnm, wherein the quantum-dot layer (30-2) is configured to host the two-dimensional array of quantum-dot regions (300).

3. The spin-qubits device (1) of any one of the preceding claims, wherein, relative from a top view of the spin-qubits device (1), each nanomagnet has an elongated shape (7001a-b, 7002).

4. The spin-qubits device (1) of claim 3, wherein the nanomagnets are configured to be magnetized along a length dimension of their elongated shape.

5. The spin-qubits device (1) of any one of the preceding claims, wherein: the nanomagnets (700) are placed at vertices of a first periodic tiling, and the quantum-dot regions (300) are located at vertices of a second periodic tiling.

6. The spin-qubits device (1) of claim 5, wherein the first and second tilings are regular tilings.

7. The spin-qubits device (1) of claim 5 or 6, wherein the first and second tilings have a same tiling type, preferably a triangular tiling or a hexagonal tiling.

8. The spin-qubits device (1) of claim 7, wherein the tiling type is the triangular tiling, and the first regular tiling is based on a first basis triangle (Tl) that is larger than and / or tilted relative to a second basis triangle (T2) of the second regular tiling.

9. The spin-qubits device (1) of claim 8, wherein each nanomagnet (701c) is placed centrally with respect to a respective triangle (T2) of the second regular tiling, the respective triangle (T2) with vertices at which respective three quantum-dot regions (501a, 501b, 501c) are placed.

10. The spin-qubits device (1) of claim 7, wherein the tiling type is the square tiling, the first regular tiling is based on a first basic square (SI) extending in two perpendicular directions (xf, yf), and each nanomagnet’s length elongates in a direction (yi) different from the two perpendicular directions (xf, yf ), preferably at an angle of substantially 45 degrees.

11. The spin-qubits device (1) of claim 10, wherein the second regular tiling is based on a second basic square (S2) extending in the same two perpendicular directions (xf , y ).

12. The spin-qubits device (1) of any one of the preceding claims, wherein an inter-nanomagnet distance between each pair of nearest-neighbour nanomagnets is larger than an inter-quantum- dot-region distance between each pair of nearest-neighbour quantum-dot regions.

13. The spin-qubits device (1) of any one of the preceding claims, wherein the two-dimensional array of nanomagnets (700) is disposed in a nanomagnets-hosting region (70) of the spin-qubits device (1), the nanomagnets-hosting region (70) being located within the gate layer (50) or being located on top of the gate layer (50).

14. The spin-qubits device (1) of any one of the preceding claims, wherein the spatially-periodic magnetic-stray -field configuration exhibits a quantization-axis angle variation in between two nearest-neighbouring quantum-dot regions (311-322) of between 10-170 degrees, preferably between 45-135 degrees, more preferably between 80 and 100 degrees.

15. The spin-qubits device (1) of any one of the preceding claims, wherein the plurality of control gates (500) comprises: a plurality of electrostatic plunger gates configured to control potential-wells of the quantum-dot regions (300), and a plurality of electrostatic barrier gates configured to control potential-barriers between adjacent quantum-dot regions (300).

16. The spin-qubits device (1) of any one of the preceding claims, wherein the plurality of quantum-dot regions (300) is configured to host a plurality of spin-qubits (900) carried by a respective plurality of spin-qubit carriers, such as electrons or holes, that can be controllably shuttled in between quantum-dot regions by controlling the plurality of control gates (500).

17. Quantum processor (1000) comprising one or more instances of the spin-qubits device (1) of any one of the preceding claims.

18. Method of operating the spin-qubits device (1) of any one of the preceding device claims, the method comprising: magnetizing (Al) the periodic two-dimensional array of nanomagnets (700) to create the spatially-periodic magnetic-stray-field configuration in the two-dimensional array of quantum-dot regions (300); and controlling (A2) the plurality of control gates (500) to perform a shuttling-based quantum gate acting on one or more spin-qubits.

19. The method of the preceding method claim, wherein the magnetizing comprises applying an external magnetic field acting on the periodic two-dimensional array of nanomagnets (700), preferably in a direction substantial parallel with a length dimension of an elongated shape of the nanomagnets.

20. The method of any one of the preceding method claims, further comprising populating the quantum-dot regions (300) by a plurality of spin-qubits (900) but leaving at least one quantumdot region empty so as to enable the performing of shuttling-based quantum gates.

21. The method of any one of the preceding method claims, wherein a total magnetic-field strength at the quantum-dot regions is kept below a threshold so that spin-qubits hosted in the quantum-dot regions each have a respective Larmor frequency of below 1GHz.

22. The method of any one of claims 18-20, wherein the performing of the shuttling-based quantum gate is performed without applying an external magnetic field, or is performed with applying an external magnetic field but while keeping a total magnetic-field strength at the quantum-dot regions below a threshold so that spin-qubits hosted in the quantum-dot regions each have a respective Larmor frequency of below 1GHz.

23. 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.

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

25. 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.

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