Controlling a semiconductor single-spin qubit

Abstract

A method for operating a single-spin qubit in a quantum processor is described wherein the method comprises: preparing a single-spin qubit in a first quantum dot of the quantum processor, wherein information in the single-spin qubit is encoded in the single spin states of a charge carrier, preferably the single spin states of a hole, in the first quantum dot, preferably a type IV semiconductor quantum dot such as a silicon quantum dot or a silicon-germanium quantum dot; and, executing a single qubit gate operation on the single-spin qubit by modifying the quantization axis and / or energy splitting of the single-spin qubit for a predetermined time, the modification of the quantization axis and / or energy splitting being based on controlling an anisotropic g-tensor associated with the single-spin qubit and / or a magnetic field locally applied to the single-spin qubit.

Description

[0001] Controlling a semiconductor single-spin qubit

[0002] Field of technology

[0003] The invention relates to controlling one or more semiconductor single-spin qubits, and, in particular, though not exclusively, to methods and a controller for controlling of one or more single-spin qubits in a semiconductor quantum dot structure or in an array of semiconductor quantum dots and a quantum processor comprising such controller.

[0004] Background

[0005] Methods for coherent single-spin control in qubits include electron spin resonance and electron dipole spin resonance using either micromagnets or spin-orbit interaction. These methods however rely on resonant Rabi driving and require high-power, high-frequency, and analog control signals that already limit qubit performance in small quantum processors. Hence, to scale quantum circuits for a quantum processor based on single-spin semiconductor qubits, the development of local, efficient, and low-power control mechanisms for these single-spin qubits is a key objective.

[0006] Burkard et al. describe in their article Semiconductor Spin Qubits arXiv:2112.08863 different semiconductor spin qubit implementations including Loss- Divincenzo single-spin qubits based single electron or hole spins, donor spin qubits based on nuclear spins of31P donor atoms in silicon, multiple-spin (single-triple) qubits encoded in a coupled double quantum dot system and exchange-only spin qubits based on coupled triple quantum dots. As described in Burkard et al., control of a single-spin qubit may be based on the magnetic field or the g-factor at the site of the single-spin qubit. Currently, such control is based on a combination of static and oscillatory electro-magnetic fields, also referred to as resonant Rabi driving. For example, in Hendrickx et al, A four-qubit germanium quantum processor, Nature 591, 580 (2021) a microwave tone that is resonant with the Zeeman- splitting of the qubit is used to generate coherent Rabi oscillations which can be used for single-qubit rotation operations. This requires high-power control signals and is typically inefficient at low magnetic field, where qubit coherence is optimal.

[0007] Multi-spin encoded qubits resolve this issue but have the disadvantage that susceptibility to leakage outside of the computational subspace is increased, sensitivity to the signal level of the applied pulses is strong, and control sequences are typically more complicated. As an example, operations based on an exchange-only qubit requires four exchange pulses to achieve an arbitrary single qubit gate and over 12 exchange pulses for a single two-qubit gate, effectively limiting their performance. Hence, from the above it follows that there is a need in the art for efficient and low-power control of a single-spin qubit which elevate the problems related to known singlespin qubit control schemes. In particular, there is a need in the art for efficient and low-power methods to control single-spin qubits using digital control signals, compatible with a scalable qubit architecture.

[0008] As will be appreciated by one skilled in the art, aspects of the present invention may be embodied as a system, method or computer program product. Accordingly, aspects of the present invention may take the form of an entirely hardware embodiment, an entirely software embodiment (including firmware, resident software, micro-code, etc.) or an embodiment combining software and hardware aspects that may all generally be referred to herein as a "circuit," "module" or "system." Functions described in this disclosure may be implemented as an algorithm executed by a microprocessor of a computer. Furthermore, aspects of the present invention may take the form of a computer program product embodied in one or more computer readable medium(s) having computer readable program code embodied, e.g., stored, thereon.

[0009] The embodiments in this disclosure relate to control schemes for a single-spin qubit allowing control of a single-spin qubit based on discrete digital control signals. Here, the term single-spin qubit refers to a qubit wherein the information is encoded in the single spin states |T) and |l) of a charge carrier (electron or hole) in a single quantum dot (one site). As explained in the above-referred article by Burkard et al. Semiconductor Spin Qubits such single-spin qubit encoding should be distinguished from other spin qubit encodings requiring multiple spin states and / or multiple quantum dots (multiple sites).

[0010] In an aspect, the embodiments in this disclosure may relate to a method for operating a single-spin qubit in a quantum processor, wherein the method may comprise preparing a single-spin qubit in a first quantum dot of the quantum processor, preferably the single-spin qubit being configured to encode information in the single spin states of a single charge carrier, preferably a single hole, in the first quantum dot, preferably a semiconductor quantum dot such as a silicon quantum dot, a germanium quantum dot or a silicongermanium quantum dot; and, executing an operation on the single-spin qubit by modifying the quantization axis and / or energy splitting of the single-spin qubit for a predetermined time.

[0011] In an embodiment, the modification of the quantization axis and / or energy splitting may include controlling an anisotropic g-tensor associated with the single-spin qubit. In another embodiment, the modification of the quantization axis and / or energy splitting may include controlling the magnetic field locally applied to the single-spin qubit. A key advantage in the operation of the single-spin qubit according to the embodiments, is that the spin rotation frequency is determined by the Larmor precession rate, which remains substantial even at small magnetic fields at which quantum coherence is substantially improved. The embodiments enable universal quantum control with error rates that are well below the thresholds for quantum error correction, using a minimal number of low-power, low-frequency, discrete pulses applied to the electrodes of the quantum dot.

[0012] The single-spin qubit system may have an anisotropic g-tensor. For example, in an embodiment, the single-spin qubit system may be implemented as a hole spin in silicon-germanium quantum dot. Such single-spin qubit system has a strong spin-orbit interaction, giving rise to anisotropic g-tensor that can be electrically controlled. The large anisotropy of the g-tensor may lead to a spin quantization axis that varies by tens of degrees.

[0013] In an embodiment, the information may be encoded in a multitude of single spins qubits. For example, the quantum information may be encoded in a set of single spin qubits.

[0014] In an embodiment, controlling the anisotropic g-tensor further may include: shuttling the single-spin qubit from the first quantum dot to a second quantum dot wherein the single-spin qubit in the first quantum dot is associated with a first g-tensor and the singlespin qubit in the second quantum dot is associated with a second g-tensor, which is different from the first g-tensor.

[0015] Hence, the variation in the spin quantization axis can be implemented between neighboring quantum dots that can host a single-spin qubit. Such spatial dot-dot variation in the g-tensor of a single-spin qubit may be used to achieve high-fidelity single-spin qubit control using discrete pulses by shuttling the spin between quantum dot sites.

[0016] Shuttling allows variation in the g-tensor by designing quantum dots differently, e.g. by using different (semiconductor) materials or layers. Further, controlling single-spin qubits (single qubit operations, readout operations and initialization operations) based on shuttling are less sensitive to variations (noise) on the control signal. Finally, shuttling also provides a higher qubit connectivity which can be combined with executing single-qubit shuttling operations such as rotation operations.

[0017] In an embodiment, the second quantum dot spatially neigbours the first quantum dot. In an embodiment, the second quantum dot and the first quantum dot are separated by a barrier region which can be controlled by one or more barrier electrodes.

[0018] In an embodiment, the single-spin qubit may be a single hole spin qubit.

[0019] In an embodiment, the controlling of the anisotropic g-tensor may further include: modifying the g-tensor of the single-spin qubit for the predetermined time.

[0020] In an embodiment, the controlling of the anisotropic g-tensor may further include: applying one or more control pulses to control electrodes, preferably one or more gate electrodes and / or barrier electrodes, of the first and second quantum dot, the one or more control pulses being configured to shuttle the single-spin qubit from the first quantum dot to the second quantum dot, preferably the one or more control pulses being configured as one or more digital signals comprising two or more discrete levels.

[0021] In an embodiment, the controlling of the anisotropic g-tensor may further include: manipulating the charge carrier occupation of the quantum dot hosting the spin qubit.

[0022] In an embodiment, the controlling of the anisotropic g-tensor may further include: manipulating the shape of the quantum dot hosting the spin qubit.

[0023] In an embodiment, the controlling of the anisotropic g-tensor may further include: manipulating the size of the host quantum dot hosting the spin qubit.

[0024] In an embodiment, the controlling of the anisotropic g-tensor may further include: manipulating the electric field in the host quantum dot hosting the spin qubit.

[0025] In an embodiment, the difference between first g-tensor and the second g- tensor may be based on implementing one or more first semiconductor layers in the first single spin quantum dot and one or more second semiconductor layers in the second single spin quantum dot, which are different from the one or more first semiconductor layers.

[0026] In an embodiment, the difference between the first and second g-tensor may be controlled by manipulating the strain in the quantum dot hosting the single-spin qubit. In an embodiment, the semiconductor material, the dielectric material, and / or the gate metal material may be locally varied and / or structured to impose a strain on the quantum dot region.

[0027] In an embodiment, the qubit quantization axis may be modified by applying a local magnetic field variation to the single-spin qubit. In an embodiment, the local magnetic field variation may be amplified through an anisotropic g-tensor of the single-spin qubit.

[0028] In an embodiment, the local field variation may be applied to the single-spin qubit by running a current through a local stripline conductor in the vicinity of the single-spin qubit or by local magnetic or superconducting structures in the vicinity of the single-spin qubit.

[0029] In an embodiment, the operation of the single-spin qubit may define a single qubit rotation operation, the single qubit rotation operation may be achieved by controlling the precession of the single-spin qubit around the modified quantization axis for the predetermined time.

[0030] In an embodiment, the effective g-factor of the single-spin qubit is may be reduced to zero to enhance quantum coherence, wherein the effective g-factor is defined as the size of the g-tensor in the direction of the applied magnetic field. In an embodiment, the operation of the single-spin qubit may include a readout of the single-spin qubit, wherein the readout is optimized by controlling the modification in spin quantization axis or in qubit energy splitting.

[0031] In an embodiment, the operation of the single-spin qubit may include a qubit initialization, wherein the qubit initialization is optimized by controlling the modification in spin quantization axis or in qubit energy splitting.

[0032] In a further aspect, the embodiments may relate to a quantum processor comprising an array of semiconductor quantum dots and a plurality of control electrodes connected to a controller for controlling one or more single-spin qubits hosted in at least part of the array of quantum dots, wherein the controller may be configured to perform the method steps according any of the above-mentioned embodiments.

[0033] In yet a further aspect, the embodiments may relate to a quantum processor comprising an array of semiconductor quantum dots and a plurality of control electrodes connected to a controller for controlling one or more single-spin qubits hosted in at least part of the array of quantum dots, wherein the controller may be configured to perform the steps of: preparing single-spin qubits in at least part of the array of quantum dot, wherein information in the single-spin qubits is encoded in the spin states of a charge carrier in the quantum dots; and, executing a single qubit gate operation on at least part of the single-spin qubits by modifying the quantization axis and / or energy splitting of the single-spin qubits for a predetermined time, the modification of the quantization axis and / or energy splitting being based on controlling an anisotropic g-tensor associated with the single-spin qubit and / or a magnetic field locally applied to the single-spin qubits.

[0034] In an embodiment, controlling the anisotropic g-tensor further may include: shuttling each of at least part of the single-spin qubits from a first quantum dot to a second quantum dot, preferably the second quantum dot neighboring the first quantum dot, wherein the single-spin qubit in the first quantum dot is associated with a first g-tensor and the singlespin qubit in the second quantum dot is associated with a second g-tensor, which is different from the first g-tensor.

[0035] In an embodiment, the shuttling of each of at least part of the single-spin qubits from the first quantum dot to the second quantum dot may include applying a control pulse to control the barrier between the first and second quantum dot.

[0036] In an embodiment, the control pulse may be a digital block pulse comprising two discrete levels.

[0037] In an embodiment, the pulse duration of the control pulse may determine the time that the quantization axis and / or energy splitting of the single-spin qubit is modified.

[0038] In an embodiment, the array of semiconductor quantum dots may form a dense two-dimensional grid and the single-spin qubits may be sparsely arranged in the array of quantum dots and the controller being configured to shuttle at least part of the single-spin qubits from quantum dot to quantum dot to form high qubit-to-qubit connectivity.

[0039] The embodiments may also relate to a program product comprising software code portions configured for, when run in the memory of a computer, executing the method steps according to any of the embodiments described in this application. The invention will be further illustrated with reference to the attached drawings, which schematically will show embodiments according to the invention. It will be understood that the invention is not in any way restricted to these specific embodiments.

[0040] Brief description of the drawings

[0041] Fig. 1 depicts schematics of part of a quantum dot structure comprising an electrode structure for controlling single-spin qubits in the quantum dot structure;

[0042] Fig. 2 illustrates shuttling and squeezing a single-spin qubit in a quantum dot structure according to various embodiments;

[0043] Fig. 3 illustrates shuttling a single-spin qubit in a quantum dot structure based on a digital signal to realize a rotation operation around a predetermined axis according to an embodiment;

[0044] Fig. 4 depicts a flow diagram for a method of controlling single-spin qubits in the quantum dot structure according to an embodiment.

[0045] Fig. 5 illustrates methods of manipulating the qubit quantization axis according various embodiments;

[0046] Fig. 6 illustrates methods of manipulating the qubit quantization axis according further embodiments;

[0047] Fig. 7 illustrates methods of manipulating the qubit quantization axis according further embodiments;

[0048] Fig. 8 depicts an architecture of a sparsely occupied dense quantum dot array that allow gate operations based on shuttling single-spin qubits

[0049] Fig. 9 depicts an architecture of a quantum dot array in which local magnetic fields can be generated;

[0050] Fig. 10 illustrates the sensitivity of the quantization axis of a heavy hole singlespin qubit as a function of the rotation of its g-tensor or an external magnetic field;

[0051] Fig. 11 depicts experimental data demonstrating high quality single qubit gates operations.

[0052] Fig. 12A and 12B illustrate methods for readout and initialisation of a singlespin qubit according various embodiment. Detailed description

[0053] The embodiments in this disclosure describe structures and methods for controlling a single-spin qubit wherein quantum information may be encoded in the spin state of the qubit. Fig. 1A and 1 B depict top view and cross-sectional schematics respectively of part of a quantum dot structure comprising an electrode structure including gate electrodes 102I,2 and barrier electrodes 104I-7. As shown by the cross-sectional figure, the quantum dot structure includes a stack of semiconducting layers including one or more semiconductor layers 120,118 formed on a substrate 122 in which quantum dot regions 106i,2 may be formed. The electrodes may be formed over the stack of semiconducting layers wherein the electrodes may be used to form and control the quantum dot regions in the stack of semiconductor layers.

[0054] The stack of semiconductor layers is designed to create quantum dots which can host single-spin qubits, including single hole spin qubits. Suitable systems for forming such quantum dots include amongst others silicon-germanium heterostructures and silicon metal-oxide-semiconductor (SiMOS) structures. Examples of such structures are described in the article by Lawrie et al, Quantum Dot Arrays in Silicon and Germanium, Appl. Phys. Lett. 116, 080501 (2020), which is hereby incorporated by reference into this application. For example, in an embodiment, the semiconductor layer stack may include a Silicon substrate, an intrinsic Silicon layer, an isotopically purified Silicon (28Si) epitaxial layer and a SiO2 layer.

[0055] In some embodiments, the stack of semiconductor layers may be designed for realizing quantum dots that can host qubits with an anisotropic g-tensor. Suitable candidates include quantum dots that are based on SiGe. For example, in an embodiment, the semiconductor layer stack may include a Si / SiGe heterostructure formed on a Silicon substrate, wherein the Si / SiGe heterostructure may include a graded Sii.xGexlayer and an isotopically purified Silicon (28Si) epitaxial layer between two SiGe layers. In another embodiment, the semiconductor layer stack may include a Ge / SiGe heterostructure formed on a Silicon substrate, wherein the Ge / SiGe heterostructure includes a Germanium layer formed on the Silicon or Germanium substrate followed by a reverse or forward graded Sii-xGexand a Ge epitaxial layer between two SiGe layers. In an embodiment, the Ge and / or SiGe layers may include isotopically purified Ge and / or isotopically purified SiGe respectively. In an embodiment, the concentration of isotopes with a finite nuclear spin may be reduced below 100 ppm.

[0056] The electrodes may be electrically isolated from the semiconductor layers by one or more insulating layers (not shown) between the semiconductor layers and the electrode structures. The gate electrodes 102I,2 may be used to form and control the quantum dot areas 106I,2 in the stack of semiconductor layers and the barrier electrodes 104I-7 may be used to control barrier regions located between the quantum dot areas. By applying voltages 112i,2to the gate electrodes quantum dot regions (quantum wells) may be formed under the gates in the one or more semiconductor layers in which charge carriers are laterally confined.

[0057] The gate voltage may be tuned such that exactly one or more charge carriers 110 (e.g. one or more electrons or holes) are confined in a quantum well. This voltage will typically determine the working voltage of the quantum dots. The controllable quantum dot region comprising a single spin-carrying charge carrier may be configured as a single-spin qubit. Fig. 1C schematically illustrates the potential landscape which may be formed in the semiconductor layers by controlling the electrodes. In particular, the figure illustrates a charge carrier 110 which is trapped in one of the potential wells formed under the gate electrodes wherein the potential wells are separated by barrier potentials located between the quantum dot regions. The height of a potential barrier may be controlled (e.g. lowered or increased) by a barrier electrode so that either interaction (low potential barrier) or isolation (high potential barrier) between neighbouring quantum dots may be achieved.

[0058] The quantum dots can be designed differently to accommodate a different qubit quantisation axis. One method to achieve this is by accommodating a different g-tensor in the quantum dots. Here, the g-tensor defines the interaction of the electronic spin of the charge carrier in the quantum dot with an external magnetic field. Certain spin-qubit systems, such as a hole in a SiGe quantum dot, can give rise to large anisotropic g-tensor that can be controlled electronically (e.g. using gates and / or current biases) or structurally (e.g. using different materials and or layouts for the quantum dots. Another method to achieve this is by accommodating a different local magnetic field direction in the quantum dots.

[0059] Fig. 2A-2C illustrate spin qubit operations according to various embodiments. Fig. 2A depicts a cross-sectional view of a quantum dot structure comprising a first quantum dot 205i and a second quantum dot 2052 separated by a barrier as discussed with reference to Fig. 1A. The quantum dots may be designed to host spin qubits with a qubit quantisation axis, which can be controlled by a controllable anisotropic g-tensor and / or a (local) magnetic field in the vicinity of the single-spin qubit. The anisotropic g-tensor can be tuned so that quantum dots host single-spin qubits with different qubit quantisation axis. Further, the electrodes may be controlled such that a spin-carrying charge 203i (a single-spin qubit) is formed in the first quantum dot. Next, the gate and barrier electrodes may be controlled to move the single-spin qubit from the first quantum dot site to the second quantum dot site.

[0060] The transfer of the single-spin qubit from the first to the second quantum dot is further illustrated in Fig. 2B, which provides view of a potential energy landscape of the quantum dot system. The black line 206 indicates the initial situation wherein a single-spin qubit is located in the first quantum dot. Then, the electrodes may be controlled to increase the height of the potential well of the first quantum dot and lower the potential barrier and potential well of the second quantum dot (as shown by the arrows 2IO1-3). The dotted line 208 indicates the temporal modification of the potential energy landscape. The potential change may cause the single-spin qubit to move from the first quantum dot associated with a first qubit quantisation axis to the second quantum dot associated with a second qubit quantisation axis, which is different from the first qubit quantisation axis. The process of moving the single-spin qubit is referred to as shuttling. As a result of the difference in qubit quantisation axis between the spin qubit in the two quantum dot sites, the charge spin, e.g. hole spin, will precess around a quantisation axis which is different from the quantisation axis of the charge spin in the first quantum dot. This operation may be used to define a singlespin qubit rotation operation. The speed at which the spin is shuttled defines the adiabaticity of the transfer. The axis around which the spin will rotate, will depend on this parameter. A fully adiabatic transfer will lead to precession around the initial quantization axis, with no effective spin rotation. A fully diabatic transfer will lead to precession around the quantisation axis of the qubit in the second quantum dot site. As a result, control over the transfer rate may be employed as a means to tune the rotation axis of the spin over the range defined by the two quantization axes of the qubit on either quantum dot site.

[0061] Fig. 2C illustrates a further method wherein the qubit quantisation axis of the spin qubit system may be modified in-place. By reshaping the quantum dot confinement potential from a first shape to a second shape, for example from circular to elongated, the qubit quantisation axis of the single qubit can be modified, leading to an effective change of quantisation axis.

[0062] Fig. 3A and 3B schematically illustrate a single-spin qubit operation according to an embodiment. In particular, Fig. 3B depicts the application of a rectangular pulse to a (combination of) electrostatic gate(s) of a quantum dot system comprising a first and a second quantum dot 302I,2 so that a spin qubit may be shuttled at the rising edge 3101 of the pulse (at time instance T1) from the first quantum dot 302i associated with a first g-tensor 304i to a (neighbouring) second quantum dot 304i associated with a second g-tensor 3042. As shown in Fig. 3A, quantum dot 302i and second quantum dot 3022 are designed to have a different g-tensors 304I,2. As a result, the quantisation axis 3082Of the qubit in second quantum dot is tilted with respect to quantisation axis 308i of the qubit in the first quantum dot, as indicated by the associated Bloch spheres 306I,2.

[0063] After the shuttling, the spin will therefore precess around the quantisation axis of the single-spin qubit in the second quantum dot for a predetermined time. At the falling edge 2102 of the pulse (at a time instance T2), the hole may be shuttled back to the first quantum dot. Alternatively, the single-spin qubit may be shuttled to a further quantum dot neighbouring the second quantum dot. As shown by the Bloch spheres 302I-3, application of a pulse for a predetermined time effectively results in a single qubit rotation operation, wherein the angle of rotation may be controlled by the width of the pulse. Due to the discrete nature of the charging of a quantum dot, the exact level (amplitude) of the control signal is less important, as long as it is large enough to enable a shuttling event from the first to the second dot.

[0064] Fig. 4 depicts a flow diagram of a method of controlling a spin-qubit according to an embodiment. As shown in the figure, the method may include a step 402 of preparing a single-spin qubit in a first quantum dot of the quantum processor, wherein information in the single-spin qubit is encoded in the single spin state of one or more charge carriers in the quantum dot in the first quantum dot. The single-spin qubit may be a single hole spin qubit. Further, the quantum does may be implemented as a type IV semiconductor quantum dot such as a silicon quantum dot or a silicon-germanium quantum dot. Then, a single qubit gate operation may be executed on the single-spin qubit by modifying the quantization axis and / or energy splitting of the single-spin qubit for a predetermined time. Here, the modification of the quantization axis and / or energy splitting may be based on controlling an anisotropic g- tensor associated with the single-spin qubit and / or a magnetic field locally applied to the single-spin qubit as will be described hereunder in more detail.

[0065] A key advantage in the operation of the single-spin qubit according to the embodiments, is that the spin rotation frequency is determined by the Larmor precession, which remains substantial even at small magnetic fields at which quantum coherence is substantially improved. As will be shown hereunder in more detail, the embodiments enable universal quantum control with error rates that are well below the thresholds for quantum error correction, using a minimal number of low-power, low-frequency, discrete pulses.

[0066] The variation in the spin quantization axis may be implemented between neighboring quantum dots that can host a single-spin qubit. Such spatial dot-dot variation in the spin quantization axis of a single-spin qubit may be used to achieve high-fidelity singlespin qubit control using discrete pulses by shuttling the spin between quantum dot sites. Shuttling allows variation in the spin quantization axis designing quantum dots having structural and / or material differences, e.g. by using different (semiconductor) materials or layers. Further, controlling single-spin qubits based on shuttling are less sensitive to variations (noise) on the control signal.

[0067] Fig. 5 depicts various methods based on modulating the electrostatic confinement of the quantum dot. As shown in this figure a hole g-tensor 502i,2can be controlled by engineering the size of the quantum dot 504I,2, the shape or aspect ratio of the quantum dot 506I,2, or the hole occupation of the quantum dot 508I,2. The shape of the quantum dot confinement can be extended beyond the ellipsoidal shapes depicted in this schematic. The size, shape and occupation of the quantum dot may be controlled by the gate voltages applied to the electrodes, e.g. gate and barrier electrodes, of a quantum dot as described with reference to Fig. 1 and 2. In other embodiment, the size and shape may be controlled by design and / or material parameters.

[0068] Fig. 6 depicts various methods based on varying material parameters at different quantum dot sites. In particular, schematics 604i,2and 608i,2Show top and cross- sectional views respectively of semiconductor structures for quantum dots wherein the hole g-tensor 602i,2Of a qubit hosted in such quantum dot can be controlled by engineering the properties of the semiconductor structure. For example, the material- or stoichiometric properties of the semiconductor structures may be varied from dot to dot 604I,2 . The schematics 608I,2 illustrate an example wherein a strain in (part of) the semiconductor structure, e.g. strain in layer 610I,2 as indicated by the arrays, may be varied from dot to dot 608I,2. For example, local strain variations may be introduced through the inclusion of materials 612I,2 with varying coefficients of thermal expansion.

[0069] Fig. 7 depicts various methods for varying magnetic field at different quantum dot sites 702I,2 . Schematics 704I,2 illustrate control of the spin quantization axis by modulating the effective magnetic field Boby applying a current (denoted by the arrow 705) nearby the quantum dot site which will generate an additional magnetic field component Bi. Schematics 706I,2 illustrate control of the spin quantization axis by including locally structured magnetic materials 707 which will generate an additional magnetic field component Bi. Schematics 708I,2 illustrate control of the spin quantization axis by including locally structured superconducting materials 709 that distort an externally applied magnetic field through the Meissner effect.

[0070] Fig. 8A-8C depicts an architecture for executing qubit operations using a semiconductor single-spin qubit quantum processor according to an embodiment. Fig. 8A illustrates a quantum processor, which may comprise of a grid of quantum dots 802 (denoted by circles 803), preferably a dense grid of quantum dots, wherein each quantum dot defines a quantum well which can host a single-spin qubit. Each quantum well may be bounded by a potential barrier, wherein the height of the potential barrier can be controlled thereby controlling the interaction between single-spin qubits in neighbouring quantum dots. As described with reference to Fig. 1 and 2, the potential landscape sensed by a single-spin qubit can be controlled by electrodes, allowing amongst others:

[0071] - control of interaction between neighbouring single-spin qubits to perform 2- qubit operations;

[0072] - shuttling of a single-spin qubit from on site to a neighbouring site to perform single qubit operations;

[0073] - initialization of a single-spin qubit, readout of a single-spin qubit; and / or, controlling the qubit quantisation axis and / or energy splitting of a single-spin qubit.

[0074] Fig. 8B schematically shows a quantum processor comprising a grid of quantum dots. Electrodes controlling the quantum dots of the quantum processer may be connected to one or more controllers 812I-4, which may be configured to generate signals, e.g. voltage- and current signals 814 to control the quantum dots. The controllers may be connected to a classical computer which may include an interface for receiving a sequence of qubit operations e.g. in the form of a quantum circuit and to instruct the one or more controllers to generate signals for the electrodes of the quantum processor in accordance with one or more qubit operation. Fig. 8C illustrates an example of such quantum circuit defining three qubits which are used to perform single and two qubit operations.

[0075] The quantum dot grid may be sparsely populated (i.e. ratio between populated and empty sites < 1 , preferably smaller than 0.8, more preferably smaller than 0.6) with single-spin qubits (each single-spin qubit is indicated in the figure by the arrow / ball 804). The sparse population enables a shuttling of the qubits throughout the array by passing through the empty sites. This leads to a larger connectivity between the qubits compared to a densely filled array, where only nearest neighbour connectivity is achieved. For example, it allows direct interactions between qubits that are initially far apart. The architecture enables the execution of single qubit operations by shuttling the single-spin qubit throughout the array, as described with reference to the embodiments in this disclosure.

[0076] An example of execution of the quantum operations as illustrated in Fig. 8C is indicated by the dashed line in the quantum dot array of Fig. 8A, where a single-spin qubit 8O6o may be shuttled from a starting position along quantum dots in the array, hopping between different quantum dot sites. The quantum dots may be configured (controlled and / or designed) such that the quantisation axis of a single-spin qubit changes during shuttling from one site to another. The quantisation axis may be changed by controlling the g-tensor of the single-spin qubit or the (local) magnetic field in the vicinity of the single-spin qubit as described with reference to the embodiments in this disclosure. This way, single-qubit rotations, e.g. a first single qubit rotation 8O81 and a subsequent second single qubit rotation 8O82, may be executed along the way, as schematically depicted by the slowly rotating spin of the single-spin qubit.

[0077] At a first site in the array, a first two-qubit gate 8101 may be executed between qubit 8O6o and a first target qubit 8O61, (as schematically depicted by the grey ellipse). Thereafter, single-spin qubit 8O6o may be shuttled further through the array, again executing (when desired) a single qubit rotation upon a shuttling event. Another, second two-qubit gate 810i may be performed between qubit 8O6o and a second target qubit 8O62. Finally, singlespin qubit 8O6o may be returned to its original position, where it can idle until next operations need to be performed. It is submitted, that implementation of the quantum dot array of Fig. 8A is just an example and can be implemented in many different ways including but not limited to: the grid symmetry (triangular, hexagonal, etc.), the density of quantum dot sites, or the exact filling of the array (less dense, more dense).

[0078] Fig. 9 depicts a schematic of a top view of an array of quantum dots wherein one or more conductive lines may be arranged over and / or around the quantum dots so that a local magnetic field can be generated which is different at different quantum dot sites. The figure depicts an example wherein conductive line 902 is arranged in meandering fashion around the qubits. The meandering line is used to induce a checkerboard shaped pattern of magnetic field orientations across a quantum dot array. In particular, running an electric current through the meandering strip line (as depicted by the arrows), will induce an effective magnetic field that points in alternating directions for neighbouring quantum dot sites (resembling a checkerboard pattern). This way, shuttling a qubit from one quantum dot site 904i associated with a first magnetic direction to another quantum dot site 9042 associated with a second magnetic direction which is different from the first magnetic direction a single qubit operation can be realized. It is noted that many different geometries of the conductive line may be used to achieve a magnetic field that is different at different quantum dot sites. Such magnetic field bias schemes may be used in combination with other embodiments in this disclosure to perform qubit operation.

[0079] Fig. 10 depicts the sensitivity of the quantisation axis of a single heavy hole spin qubit as a function of the rotation of its g-tensor or an external magnetic field. Due to the strongly anisotropic nature of the heavy hole spin, a small rotation in the g-tensor or the local magnetic field can lead to a large rotation of the qubit quantisation axis. The g-factors used in this simulation are gx=gy=0.05 and gz=10, corresponding to a typical single hole spin qubit system implemented in SiGe technology. The rotation is defined around the most sensitive magnetic field direction (the z-axis in this example) aligned with the small axes of the g- tensor (x and y axes in this example).

[0080] Fig. 11A-11D illustrate an experimental demonstration of the shuttle-based single-qubit operations as described with referend to the embodiments in this disclosure. Fig. 11D depicts a physical layout of a 2x2 quantum SiGe quantum processor. Details about the processor can be found in the article by Hendrickx et al, A four-qubit germanium quantum processor, Nature 591, 580 (2021). In this experiment quantum dots D1 and D4 are used for shuttling a single-spin qubit (D2 and D3 are not used in this example). In particular, quantum dot D1 is prepared to host a single hole spin qubit QA, while quantum dot D4 remains empty to allow for single qubit gate operations via shuttling. Quantum dot D4 is tuned such that when the single-spin qubit QA is shuttled to quantum dot D4, the orientation of the quantization axis of the qubit QA is different from when it resided in quantum dot D1. In this particular, experiment the quantization axis angle (i.e. the difference between the orientation of the quantization axis of the qubit in D1 and D4 is 014= 41.5°). The significant difference in g-tensor between the qubit in D1 and D4 allows for the construction of a single qubit rotation even at low magnetic fields.

[0081] Fig. 11A depicts the control signal used in the experiment, where single-spin qubit originally resides in quantum dot D1. The control signals to the quantum processor are pulsed such that the qubit is shuttled back and forth between D1 and D4. The full experiment encompasses a three-stage pulse. First, qubit QA is shuttled from QD1 to QD4 and left to evolve for time t4. Next, the qubit is shuttled back to QD1 and let to evolve for time ti , after which it is shuttled again to QD4 and let to evolve for time t4. Finally, qubit QA is returned to its original site QD1 and the state is measured. Measuring the state of QA may be performed using a Pauli spin blockade readout using quantum dots D3 and D4. The evolution of the qubit at the sites with different quantization axes leads to an effective single qubit rotation.

[0082] Fig. 11B depicts the spin-flip probability of the qubit, as a function of the different evolution times ti and t4. It can be observed that the final spin state depends on the time spent at the different quantum dot sites. By calibrating the length of pulses ti and t4, an effective X / 2 gate can be achieved, with an effective total gate time of 98 ns (35 ns) for QA corresponding to effective Rabi frequencies of 2.6 MHz, which is significant compared to the qubit resonance frequencies fA=42.6 MHz for an in-plane magnetic field of 25 mT.

[0083] The efficiency of the qubit control schemes according to the embodiments in this disclosure can be accessed by defining an efficiency = 2fRabi / fLarmorw ic largely determines the power dissipation. The efficiency for the experimental system is 77=23% for QA. This is a formidable improvement compared to previous demonstrations of high-fidelity qubit logic with spin qubits bits which have efficiencies between 0.04%-0.07%, see e.g. A. Noiri, et al., Nature 601 , 338 (2022), B. Undseth, et al., Phys. Rev. Appl. 19, 044078 (2023) and X. Xue, et al., Nature 601, 343 (2022). Consequently, while ESR and EDSR-based experiments are being constrained by power dissipation, the control of the single-spin qubits as described with reference to the embodiments in this disclosure have approximately 100 to more than 1000 times less power dissipation at a given drive amplitude.

[0084] Fig. 11C depicts the results of a benchmarking experiment of the shuttlebased quantum gate. The gate quality is assessed by performing randomized benchmarking. Here, N random Clifford gates are applied to the qubit wherein a Clifford gate is constructed from a combination of Xn / 2, Zn / and I gates, after which a recovery gate is applied that returns the qubit to its original state. By measuring the observed return probability as a function of N, the quality of the operation can be extracted. A single qubit gate fidelity of 99.967% is found. More generally, it has been established that moving spin qubits between quantum dots and (digital) baseband control allows for coherent shuttling with fidelities up to F=99.992%, single-qubit gate fidelities up to F=99.97%, and two-qubit gate fidelities up to F=99.3%. These operation fidelities of movable spin qubits, enables a qubit architecture consisting of sparse spin qubits and empty quantum dot sites, where shuttling operations may serve to move, rotate and entangle Loss-DiVincenzo qubits beyond the nearest neighbour as e.g. illustrated in Fig. 8 and 9.

[0085] Fig. 12A and 12B illustrate methods for readout and initialisation of a singlespin qubit according various embodiments. In particular, the boxes 1200i,2Of Fig. 12A illustrate a readout method for single qubits referred to as Pauli Spin Blockade readout. By applying voltage pulses to the electrostatic gates of the quantum device, a single-spin qubit 906i may be shuttled from a first quantum dot 904i to a second quantum dot 904i, where another single-spin qubit 906i resides. This will effectively perform a projective measurement, where the shuttling event only occurs when the two-spin state is antiparallel as shown in box 1200i. When both single-spin qubits are parallel no shuttling will take place due to the Pauli Spin Blockade as shown in box 12002. This shuttling event can be observed by a measurement apparatus, e.g. a charge sensor, resulting in a change of signal when the qubit is shuttled. As such, the spin state from the first qubit can be inferred. It is noted that due to spin-orbit interaction, the exact readout implementation may differ from the illustrated cases. As an example, a similar effect could be obtained but now with the parallel spin states.

[0086] Initialization of a single-spin qubit may be achieved through an inverted process, where two spin qubits are prepared in a single quantum dot and then split across two separate quantum dots, leaving two single-spin qubits with anti-parallel spin.

[0087] Fig. 12B illustrates the importance of the quantization axis in a projective measurement scheme. Depending on the exact experimental implementation, readout (or initialization) performance can be negatively affected by a misalignment between the quantization axes of the qubits the two quantum dots. In this example as shown in box 12101 projection of a first single-spin qubit 1212i onto the quantisation axis of a second single-spin qubit 12122 will lead to a false readout signal in part of the readout events if there is a misalignment between the quantization axis of the qubits, illustrated by the gray area above T. When the quantization axes are aligned as shown in box 12122, all readout events will return the correct signal, ‘O’. The ability to control the quantization axis in different quantum dots provides a means to improve initialization and / or readout performance.

[0088] Hence, in these embodiments, instead of controlling a variation of the quantization axis and / or energy splitting of a single-spin qubit when it shuttles from one quantum dot site to another neighboring quantum dot site, the quantization axis and / or energy splitting is controlled to have no variation during shuttling. For example, dot to dot variations that may exist due to local variations in the material and / or geometry of the dots may be compensated such that when shutting the quantization axis of the qubits remain parallel.

[0089] The terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of the invention. As used herein, the singular forms "a," "an," and "the" are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will be further understood that the terms "comprises" and / or "comprising," when used in this specification, specify the presence of stated features, integers, steps, operations, elements, and / or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and / or groups thereof.

[0090] The corresponding structures, materials, acts, and equivalents of all means or step plus function elements in the claims below are intended to include any structure, material, or act for performing the function in combination with other claimed elements as specifically claimed. The description of the present invention has been presented for purposes of illustration and description, but is not intended to be exhaustive or limited to the invention in the form disclosed. Many modifications and variations will be apparent to those of ordinary skill in the art without departing from the scope and spirit of the invention. The embodiment was chosen and described in order to best explain the principles of the invention and the practical application, and to enable others of ordinary skill in the art to understand the invention for various embodiments with various modifications as are suited to the particular use contemplated.

Claims

CLAIMS1. A method for operating a single-spin qubit in a quantum processor, the method comprising: preparing a single-spin qubit in a first quantum dot of the quantum processor, preferably the single-spin qubit being configured to encode information in the single spin states of a single charge carrier, preferably a single hole, in the first quantum dot, preferably a semiconductor quantum dot such as a silicon quantum dot, a germanium quantum dot or a silicon-germanium quantum dot; and, executing a single qubit gate operation on the single-spin qubit by modifying the quantization axis and / or energy splitting of the single-spin qubit for a predetermined time, the modification of the quantization axis and / or energy splitting including controlling an anisotropic g-tensor associated with the single-spin qubit and / or a magnetic field applied to the single-spin qubit.

2. Method according to claim 1 , wherein controlling the anisotropic g-tensor further includes: shuttling the single-spin qubit, from the first quantum dot to a second quantum dot, preferably the second quantum dot neighboring the first quantum dot, wherein the single-spin qubit in the first quantum dot is associated with a first g-tensor or a first magnetic field and the single-spin qubit in the second quantum dot is associated with a second g-tensor or a second magnetic field, the first g-tensor or first magnetic field being different from the second g-tensor or second magnetic field respectively.

3. Method according to claim 2 further comprising: applying a control pulse to a barrier electrode between the first and second quantum dot for shuttling the single-spin qubit from the first quantum dot to the second quantum dot, preferably the control pulse being a digital block pulse comprising two discrete levels, more preferably the pulse duration of the control pulse determining the time the quantization axis and / or energy splitting of the single-spin qubit is modified.

4. Method according to claims 2 or 3, wherein the difference between first g- tensor and the second g-tensor is controlled by implementing one or more first semiconductor layers in the first single spin quantum dot and one or more second semiconductor layers in the second single spin quantum dot, which are different from the one or more first semiconductor layers.

5. Method according to claim 2 or 3 wherein the difference between the first and second g-tensor is controlled by strain in a quantum dot hosting the single-spin qubit, thecontrol of the strain including locally varying and / or structuring semiconductor material, dielectric material, and / or metal material to impose a strain on the quantum dot hosting the single-spin qubit.

6. Method according to any of claims 1-5, wherein controlling the anisotropic g-tensor further includes: controlling the charge carrier occupation of a quantum dot hosting the spin qubit.

7. Method according to any of claims 1-6, wherein controlling the anisotropic g-tensor further includes: controlling the shape of a quantum dot hosting the spin qubit.

8. Method according to any of claims 1-7, wherein controlling the anisotropic g-tensor further includes: controlling the size or geometry of a host quantum dot hosting the spin qubit.

9. Method according to any of claims 1-8, wherein controlling the anisotropic g-tensor further includes: controlling the local electric field and / or magnetic field in a host quantum dot hosting the spin qubit.

10. Method according to any of claims 1-9 wherein the qubit quantization axis is modified by applying a local magnetic field variation to the single-spin qubit, the application of the local field variation including running a current through a local conductor in the vicinity of the single-spin qubit or by arranging local magnetic or superconducting structures in the vicinity of the single-spin qubit.

11. Method according to any of claims 1-10 wherein the single qubit operation defines a single qubit rotation operation, the single qubit rotation operation being achieved by controlling the precession of the single-spin qubit around a modified quantization axis for the predetermined time.

12. Method according to any of claims 1-11 where the effective g-factor of the qubit is reduced to zero to enhance quantum coherence; or to enhance the performance of one, two or multi-qubit gates.

13. Methods according to any of claims 1-12 wherein the method further includes:performing a readout of the single-spin qubit, wherein the readout is optimized by controlling the modification in the spin quantization axis or the modification in the qubit energy splitting.

14. Method according to any of claims 1-13 wherein the method further includes: performing a single-spin qubit initialization, wherein the qubit initialization is optimized by controlling the modification in the spin quantization axis or the modification in the qubit energy splitting.

15. A quantum processor comprising an array of semiconductor quantum dots and a plurality of control electrodes connected to a controller for controlling one or more single-spin qubits hosted in at least part of the array of quantum dots, wherein the controller is configured to perform the method steps according any of claims 1-14.

16. A quantum processor comprising an array of semiconductor quantum dots and a plurality of control electrodes connected to a controller for controlling one or more single-spin qubits hosted in at least part of the array of quantum dots, the controller being configured to perform the steps of: preparing single-spin qubits in at least part of the array of quantum dots, wherein information in the single-spin qubits is encoded in the spin states of a charge carrier in the quantum dots; and, executing a single qubit gate operation on at least part of the single-spin qubits by modifying the quantization axis and / or energy splitting of the single-spin qubits for a predetermined time, the modification of the quantization axis and / or energy splitting including controlling an anisotropic g-tensor associated with the single-spin qubit and / or a magnetic field locally applied to the single-spin qubits.

17. A quantum processor according to claim 16 wherein controlling the anisotropic g-tensor further includes: shuttling each of at least part of the single-spin qubits from a first quantum dot to a second quantum dot, preferably the second quantum dot neighboring the first quantum dot, wherein the single-spin qubit in the first quantum dot is associated with a first quantization axis and the single-spin qubit in the second quantum dot is associated with a second quantization axis, which is different from the first quantization axis.

18. A quantum processor according to claim 17 wherein the shuttling of each of at least part of the single-spin qubits from the first quantum dot to the second quantum dot includes applying a control pulse to control the barrier between the first and second quantum dot, preferably the control pulse being a digital block pulse comprising two discrete levels, more preferably the pulse duration of the control pulse determining the time that the quantization axis and / or energy splitting of the single-spin qubit is modified.

19. A quantum processor according to any of claims 16-18 wherein the array of semiconductor quantum dots forms a two-dimensional grid, preferably a dense two- dimensional grid, and wherein the single-spin qubits are sparsely arranged in the array of quantum dots and can be shuttled from quantum dot to quantum dot to form high qubit-to- qubit connectivity.

20. A computer program product comprising instructions which, when the program is executed by a computer, cause the computer to carry out the method steps according to any of claims 1-14.

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