Controlled-rotation quantum gates for controlling a target nuclear spin by an electron spin hosted in an electron-spin host

Abstract

Provided are methods of performing a controlled-rotation quantum-gate configured to rotate a target nuclear spin by a target rotation controlled by an electron spin, wherein the electron spin is hosted in an electron-spin host and is surrounded by a plurality of nuclear spins comprising the target nuclear spin The methods comprise a sequence of controlled-sub-rotation quantum gates interleaved with temporary-storage quantum gates, each controlled-sub-rotation quantum gate configured to provide a respective sub-rotation of the target nuclear spin and each respective temporary-storage quantum gate configured to, temporarily for a respective time period, store a respective spin state of the electron spin in a quantum memory while putting the electron spin in a respective electron-spin eigenstate, wherein the respective sub-rotations sum up to the target rotation.

Description

[0001] Controlled-rotation quantum gates for controlling a target nuclear spin by an electron spin hosted in an electron-spin host

[0002] Field of the invention

[0003] The invention relates to controlled-rotation quantum gates for controlling a target nuclear spin by an electron spin hosted in an electron-spin host.

[0004] Background art

[0005] A promising building block for quantum technology is an electron spin hosted in a colour centre of a solid-state such as a diamond, for example a nitrogen-vacancy centre (NV), a tin-vacancy centre (SnV), or a silicon-vacancy centre (SiV) in diamond. In these examples, the electron spin is related to one or more electrons that are locally confined at the colour centre and that give rise to an effective spin system. Enabling precise control of the electron spin has, among other things, applications in quantum-computing, -sensing and -networks.

[0006] For example, the electron spin can be used to control and / or sense a plurality of nuclear spins surrounding the electron spin. The surrounding nuclear spins can be the spins of carbon isotopes13C in diamond that interact with the electron spin via spin-spin interactions involving hyperfine couplings. In the context of quantum sensing, the plurality of nuclear spins may relate to a target sample such as a molecule that is brought in close proximity to the electron spin which thereby can function as a quantum sensor.

[0007] The interactions between the electron spin and the nuclear spins can also be exploited so as to use the nuclear spins for example as qubits. For example, an individual target nuclear spin can be controlled by a controlled-rotation quantum gate involving the electron spin, in which the target nuclear spin rotates in dependence on a spin state the electron spin, which functions as a control qubit. A goal of such controlled-rotation quantum gates is to implement a controlled rotation on the target nuclear spin conditional on the electron-spin state while also protecting coherence of the electron spin.

[0008] Known techniques for performing controlled-rotation quantum gates on a target nuclear spin include Dynamic Decoupling control [e.g.: Taminiau et al], Dynamically Decoupling Radio Frequency control [e.g.: Bradley et al], Uhrig Dynamical Decoupling control [E.g.: [Takou et al, Uhrig et al], Robust Dynamic Decoupling control [e.g.: Casanova et al], which leverage a difference in hyperfine couplings between the electron spin and other nuclear spins to selectively control the target nuclear spin. These techniques have been demonstrated already on various platforms such as color centers in a diamond [Cramer et al, Smarak et al, Nguyen et al, Bradley et al], color centers in silicon carbide [Babin et al], rare-earth ions [Uysal et al]. A general problem with the known techniques is that while they attempt to target a particular nuclear spin with the quantum gate, other nuclear spins may also be affected by the quantum gate, which leads to undesired cross-talk and can negatively impact the selectivity.

[0009] Summary of the invention

[0010] A task set fourth by the inventors is to improve controlled-rotation quantum gates.

[0011] The inventors solved the task by providing methods of performing controlled-rotation quantum gates according to the appended independent claims, which exhibit increased selectivity for a target nuclear spin. A controlled-rotation quantum-gate is configured to rotate a target nuclear spin by a target rotation controlled by an electron spin, wherein the electron spin is hosted in an electron-spin host and is surrounded by a plurality of nuclear spins comprising the target nuclear spin. The methods comprise a sequence of controlled-sub-rotation quantum gates interleaved with temporary-storage quantum gates, each controlled-sub-rotation quantum gate configured to provide a respective sub-rotation of the target nuclear spin and each respective temporary-storage quantum gate configured to, temporarily for a respective time period, store a respective spin state of the electron spin in a quantum memory while putting the electron spin in a respective electron-spin eigenstate, wherein the respective sub-rotations sum up to the target rotation. The invention can have applications in quantum-computing, - simulations, -sensing, -networks. Further technical advantages are discussed further below in the detailed description.

[0012] / / 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.

[0013] 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.

[0014] 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.

[0015] 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.

[0016] 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.

[0017] 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.

[0018] 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.

[0019] 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. / / Brief description of the drawings

[0020] The present invention is discussed in more detail below, with reference to the attached drawings, in which:

[0021] Fig. la illustrates an electron spin (10) hosted in an electron-spin host (100) and being surrounded by a plurality of nuclear spins (200). Fig. la further illustrates, as an example of the electron-spin host (100), a tin-vacancy centre (SnV).

[0022] Fig. lb illustrates a target sample (2000) brought in proximity to an electron spin (10) of an electron-spin host (100) located at a surface of a solid-state material (1000), for sensing individual nuclear spins of the target sample (2000) by the electron spin (10).

[0023] Fig. 1c shows schematically two prior-art controlled-rotation quantum gates.

[0024] Fig- 2 showcases a disadvantage of a prior-art controlled-rotation quantum gate by an exemplary numerical simulation.

[0025] Fig- 3 shows a controlled-rotation quantum gate (C-Ri) according to the present disclosure, involving a temporary-storage quantum gate (Gi).

[0026] Fig. 4a show a preferred example of the temporary-storage quantum gate (Gi) used in the controlled-rotation quantum gate according to the present disclosure.

[0027] Fig. 4b shows an example of a transfer quantum gate (TG) according to the present disclosure.

[0028] Figs. 5a-5d show examples of quantum memories used in the controlled-rotation quantum gates according to the present disclosure.

[0029] Figs. 6a-6b show another example of the controlled-rotation quantum gate (C-Ri).

[0030] Figs. 7a-7d shows a further controlled-rotation quantum gate (C-Rk) according to the present disclosure, involving k temporary-storage quantum gates (Gi, ..., Gk).

[0031] Fig- 8 showcases, by an illustrative example, reduced cross-talk and enhanced selectivity of a controlled-rotation quantum gate (C-Rk=s) of the present disclosure.

[0032] Figs. 9a-9b are flowcharts showing the methods according to the present disclosure.

[0033] Fig. 9b shows a quantum processor (1) for performed methods according to the present disclosure.

[0034] Detailed description

[0035] Fig. la illustrates an electron spin (10) hosted in an electron-spin host (100) and being surrounded by a plurality of nuclear spins (200) of a solid-state material (1000). The electron-spin host (100) hosting the electron spin (10) may be based on a colour centre in diamond [Abobeih et al, Pompili et al] or rare-earth ions [Ruskuc et al]. The electron-spin host (100) may be located in the solid-state material (1000) such as a diamond or crystal, or in a molecule [Bayliss et al, Fursina et al, Morton et al].

[0036] The electron spin (10), in the context of the solid-state material (1000), is typically related to one or more electrons that are disposed in the electron-spin host (100) of the solid- state material (1000). For example, the electron spin (10) may relate to an electron (e‘) that is locally-confined at a tin-vacancy (SnV) centre in diamond or at a silicon-vacancy (SiV) centre in diamond. In these examples, the electron spin (10) relates to a spin-'A system. Other examples an electron spin (10) relating to a spin-’A system are a T-centre in silicon [Photonic Inc] and a vanadium centre in silicon carbide [Astner et al]. As another example, the electron spin (10) may relate to two electrons that are locally-confined at a nitrogen-vacancy (NV) centre and that give rise to a spin-1 system.

[0037] In general, the electron spin (10) may relate to any spin-S system with spin quantum number S=l / 2, 1, 3 / 2, 2, 5 / 2, etc., that may depend on the involved electron-spin host (100) and / or the solid-state material (1000).

[0038] The plurality of nuclear spins (200) is shown to surround the electron spin (10). In other words, the plurality of nuclear spins (200) is in proximity to the electron spin (10). In the illustration of Fig. la, the plurality of nuclear spins (200) comprises individual nuclear spins labelled as (201, 202, 203, 204, 205, 206, 207). There may be any number of nuclear spins: 1, 2, 3, ..., 50, 51, ..., 100, 101, .., 500, 501, ..., 1000, 1001, etc. The number of nuclear spins is thus not limited to the illustration of Fig. la, in which a limited number of nuclear spins is depicted for illustration purposes and intelligibility only.

[0039] The plurality of nuclear spins (200) is also depicted as included in the solid-state material (1000). In the example of a solid-state material (1000) relating to a diamond, the plurality of nuclear spins (200) may be related to nuclei of carbon isotopes13C that are distributed in the diamond, each nucleus exhibiting a spin-’A quantum state constituting a respective nuclear spin.

[0040] The plurality of nuclear spins (200) may however also be carried by other types of nuclei and / or may also be located outside of the solid-state (1000), for example in spectroscopy applications in which the electron spin (10) is envisioned as a quantum sensor. Fig. lb illustrates a target sample (2000) brought in proximity to an electron spin (10) of an electronspin host (100) located at a surface of a solid-state material (1000), for sensing individual nuclear spins of the target sample (2000) by the electron spin (10). So, the plurality of nuclear spins (200) can be part of a solid-state material (1000), for example related to nuclei of carbon isotopes13C, but may also relate to a target sample (2000) such as a target molecule or target virus that one wishes to sense by controlling the electron spin (10).

[0041] In general, each nuclear spin (201-207) of the plurality of nuclear spins (200) may relate to any spin-I system with spin quantum number 1=1 / 2, 1, 3 / 2, 2, 5 / 2, etc., that may depend on the involved solid-state material (1000) or the target sample (2000).

[0042] The electron spin (10) is surrounded by the plurality of nuclear spins (200) so that spinspin interactions between the electron spin (10) and nuclear spins (200) can be controlled to an advantage. This is discussed next at the example of a tin-vacancy (SnV) centre.

[0043] Fig. la shows a SnV centre in diamond as an example of the electron-spin host (100). The SnV centre refers to two vacancies in the diamond lattice structure in between which a tin atom (Sn) is located. The SnV centre can bind an electron (e‘) and exhibits a spin-1 / 2 system. The SnV centre is therefore also referred to as a negatively charged SnV centre.

[0044] For controllably realizing the electron spin (10) in the SnV centre, an external magnetic field B is typically applied so as to cause a Zeeman splitting of lowest energy levels of the electron (e‘) to split in two states - labelled in Fig. la as |0) and | 1) - that are separated by an energy gap related to a certain frequency a>a. The two states |0) and | 1) relate to spin eigenstates and can also be referred to as electron-spin eigenstates.

[0045] The notation of |0) and 11) may be used to emphasize that the electron spin (10) can be controlled in a qubit-subspace spanned by two spin eigenstates. The electron spin (10) can be initialized into one of its electron-spin eigenstates and can be controllably manipulated by microwave driving, preferably at temperatures below 1.5K. For example, exposing the electron spin (10) to electromagnetic radiation with frequency o>acauses Rabi oscillations, by which the electron spin (10) can be controlled into a superposition of its electron-spin eigenstates and / or can be controlled to transition into another electron-spin eigenstate, et cetera. A pulse configured to flip an electron-spin state of the electron spin (10) is referred to as a TI -pulse. That is, 7 -pulse acts on eigenstates as |0) <— > 11) and | 1) ■-> |0), and on a superposition state accordingly as + b |0)). A spin state of the electron spin (10) may be denoted as |me) = a|0) + b| l). The electron spin (10) can be read out for example by exploiting that energy gaps ~a>band ~a>cstarting from |0) or | 1), respectively, are different, i.e. exploiting spin-dependent fluorescence. Such control operations on the electron spin (10) are well-known and are therefore not recalled in detail here. In other examples of electron-spin hosts (100), the electron spin (10) may be controlled within a spin subspace spanned by two particular spin eigenstates, i.e., using two spin-projections of a higher-spin system. For example, an NV-centre relates to a spin-1 system with overall three spin states, in which the electron spin (10) may be controlled in a sub-space of two of these three spin states.

[0046] Fig. la further illustrates a spin-spin interaction (10-201) between the electron spin (10) and an individual nuclear spin (201), by example. Without being bound by theory, but for illustration, the spin-spin interaction (10-201) may be described by the Hamiltonian in which a>Ldenotes the Larmor frequency, Izand Ixspin operators of the nuclear spin (201), Szthe spin operator of the electron spin (10), and A and A± the parallel and perpendicular components of the hyperfine coupling / interaction between the electron spin (10) and the nuclear spin (201), respectively. The components A|| and A± may also be referred to as hyperfine coupling parameters. The z-direction may refer to the direction of the magnetic field B. The magnetic field may be applied in any direction, for example aligned to a quantization axis of the electron spin (10), and may have a magnitude suitable for causing Zeeman splitting, for example lOOmT. Under the external magnetic field B, the Larmor frequency fL= ycBzrefers to a precession frequency of the nuclear spin around the external magnetic field B.

[0047] The Hamiltonian dynamics related to H results in a precession axis of the nuclear spin (201) depending on the spin state of the electron spin (10), as c , = (si A±, 0, fL+ Si A ), wherein Si is the spin projection of the electron spin (10) for i=0, 1 relating to |0) and | 1), respectively. More background may also be found in [Zahedian et al] and the references therein.

[0048] Spin-spin interactions such as (10-201) may also be referred to as a spin-spin coupling or electron-spin nuclear-spin coupling, which can be exploited to implement quantum gates that manipulate a particular nuclear spin controlled by the electron spin (10), i.e., depending on the electron spin (10). Examples thereof are discussed next in terms of prior-art controlled-rotation quantum gates (C-R: C-R-DD, C-R-DDRF, C-R-UDD).

[0049] Fig. 1c shows schematically two prior-art controlled-rotation quantum gates for rotating a target nuclear spin (e.g., 201, 204) by a target rotation (R) controlled by the electron spin (10).

[0050] A goal of a controlled-rotation quantum gate (C-R) is to implement a rotation R=R(ai) on a particular nuclear spin conditional on the electron spin state (10) while also protecting coherence of the electron spin (10). The rotation refers to an evolution of a state of the nuclear spin in the Bloch-sphere representation relating to a subspace of two nuclear-spin eigenstates in which the nuclear spin is controlled. At the example of the plurality of nuclear spins (200) relating to nuclei of carbon isotopes13C, a nuclear spin is a spin-U state and its two nuclear- spin eigenstates may for example be denoted as | f) and | T). The terminology “Dynamical Decoupling” refers to the aspect of protecting the coherence of the electron spin (10). The target nuclear spin may for example be nuclear spin (201) in Fig. la, but may also be another individual nuclear spin of the plurality of nuclear spins (200) surrounding the electron spin (10), such as any one of nuclear spins (202-207). Which nuclear spins can be targeted may depend on factors such as the distance between the targeted nuclear spin and the electron spin (10), so that a respective coupling is strong enough to implement a quantum gate.

[0051] A first prior-art example recalled in Fig. 1c is known as Dynamical Decoupling (DD) control, that was introduced in [Taminiau et al]. The DD control involves applying 7t-pulses to the electron spin (10) that are interleaved with inter-pulse delay periods of a certain length, shown in Fig. 1c as 2T, in which the system is freely evolving. The DD control sequence performs a conditional rotation on a target nuclear spin if r fulfils a certain resonance condition of the target nuclear spin. The parameter N is an integer that relates to the total number of 7t- pulses that are applied in a DD sequence. The target rotation (R) achieved by the DD sequence thus depends on the parameters T and N, R= R(T, N), and hence can be controlled by, for example, first finding a suitable T and subsequently tuning the number of pulses N so as to coherently rotate the target nuclear spin over a desired angle. As an example, a controlled NOT gate, CNOT, may be implemented, up to single-qubit rotations, by a maximal entangling operation that rotates the nuclear spin over an angle ± 7t / 2, wherein the sign is controlled by the electron spin state. A CNOT implements a rotation on the target nuclear spin controlled by the electron spin (10) that can be considered as having the functional role of a control qubit; for example, no rotation on the target nuclear spin if the electron spin (10) was in |0) and full rotation if the electron spin (10) was in 11). More such known details are not discussed here but can be found in [Taminiau et al],

[0052] A second prior-art example is known as Dynamically Decoupled Radio Frequency control, DDRF control, that was introduced in [Bradley et al]. The DDRF control involves, in addition to DD control, well-chosen radio-frequency, RF, pulses applied to a targeted nuclear spin, which is schematically illustrated in Fig. 1c by additional sinusoidal waves. DDRF sequences also allow to control a target rotation (R) with a tuneable rotation angle and rotation axis. More details are not discussed here but are known and can be found in [Bradley et al].

[0053] In addition to the sketched example of DD control and DDRF control, there are further controlled-rotation quantum gates such as Uhrig Dynamical Decoupling [E.g.: Uhrig et al] in which the time between rr-pulses is varied according to a formula and Robust Dynamic Decoupling [e.g.: Casanova et al], which are referenced but not detailed here. In summary, the prior-art examples illustrate that conditional -rotation quantum gates can be implemented that rotate a certain nuclear spin by a certain rotation controlled by an electron spin. The rotation depends on parameters such as T, N, phases, etc, by which the rotation can be tuned. For example, an angle of the rotation can be tuned in DD control by suitable choice of T, N. For example, an angle and axis of the rotation can be tuned in DDRF control by T, N, phases <ptand amplitudes Ai of RF pulses as outlined in [Bradley et al].

[0054] When targeting a particular nuclear spin with a prior-art controlled-rotation quantum gate, there may be couplings with other nuclear spins that are not intended to be targeted, for example because the other nuclear spins have similar hyperfine couplings. The inventors have found a disadvantage that, when targeting via prior-art gates a particular nuclear spin having a particular hyperfine coupling, there may be considerably more couplings with other nuclear spins that are not intended to be targeted than expected; in particular, for an electron spin (10) related to an electron-spin / i system and using DDRF control. Such additional coupling to other nuclear spins is unwanted and is referred to as cross-talk.

[0055] Fig- 2 showcases the disadvantage of a prior-art controlled-rotation quantum gate by an exemplary numerical simulation.

[0056] In the exemplary numerical simulation, a nuclear spin-'A with hyperfine coupling Ataris targeted by a conditional -rotation quantum gate performed by a DDRF sequence and an electron spin (10) with spin-’A. The hyperfine coupling Atarof the targeted nuclear spin is indicated by Atarin a frequency diagram. In this example, the targeted nuclear spin has a perpendicular hyperfine coupling of A = 50 kHz and a parallel hyperfine coupling of A =100 kHz. The Larmor frequency used in the simulation is 1071 kHz and the strength of the RF drive involving in the DDRF sequence ([Bradley et al]) is 1.2kHz applied on resonance with f i. The number of decoupling pulses was N=24 and an inter-pulse delay parameter of r=26.6ps was chosen, thereby realizing a fully entangling gate. The gate is performed with the electron spin (10) being in a superposition state

[0057] The crosstalk can be quantified by calculating the electron-spin coherence <J%) in the presence of a potential bystander nuclear spin that is in a fully mixed state, wherein a value of 1 indicates no crosstalk and a value of 0 indicates maximal cross talk. Fig. 2 plots values of the electron-spin coherence (<J%) for a potential bystander nuclear spin with other hyperfine couplings, so that the plot of Fig. 2 illuminates which other hyperfine-parameter regime of nuclear spins would also be coupled by the conditional -rotation quantum gate performed by the DDRF sequence. Fig- 2 shows that even though a target nuclear spin with a hyperfine coupling parameter of Atar is targeted, a much wider parameter regime of potential hyperfine couplings is affected, as indicated by regions in which (<J%) is closer to 0. For example, when targeting Atarby the conditional -rotation quantum gate performed by DDRF, other nuclear spins with hyperfine couplings Aunwanted, i or Aunwanted, 2 are also coupled by this conditional -rotation quantum gate, i.e., are also interacting with the electron spin (10). As a consequence, a nuclear-spin system in which one intends to target a nuclear spin with Atarbut in which other nuclear spins are present with Aunwanted, i or Aunwanted, 2 that are coupling to / interacting with the electron spin (10) as well, controlling the target nuclear spin purely via prior-art conditional -rotation quantum gates is less effective, especially if the system is a electron spin-1 / 2 as is the case in this numerical example.

[0058] The inventors have devised improved controlled-rotation quantum gates, “C-Ri”, in which unwanted cross-talk is further reduced, so that nuclear spins can be targeted more selectively. Below, references to the figures are included for illustration and for intelligibility reasons.

[0059] Fig- 3 shows a controlled-rotation quantum gate (C-Ri) according to the present disclosure, involving a temporary-storage quantum gate (Gi).

[0060] The present disclosure provides a method for performing a controlled-rotation quantumgate (C-Ri) configured to rotate a target nuclear spin (201) by a target rotation (Ri) controlled by an electron spin (10), wherein the electron spin (10) is hosted in an electron-spin host (100) and is surrounded by a plurality of nuclear spins (200) comprising the target nuclear spin (201). The method comprises: applying a first controlled-rotation quantum gate (C-Rn) configured to rotate the target nuclear spin (201) by a first sub-rotation (Rn) controlled by the electron spin (10); applying a temporary-storage quantum gate (Gi) configured to, temporarily for a time period (ti), store a spin state (|me)) of the electron spin (10) in a quantum memory (300) while putting the electron spin (10) in an electron-spin eigenstate (|0); | 1)); and applying a second controlled-rotation quantum gate (C-R12) configured to rotate the target nuclear spin (201) by a second sub-rotation (R12) controlled by the electron spin (10), the second sub-rotation (R12) adding with the first sub-rotation (R11) to the target rotation (Ri).

[0061] The controlled-rotation quantum gate (C-Ri) can be understood as effectively splitting a target rotation (Ri) into two sub-rotations (Rn, R12) that are separated by a time period (ti) during which the electron spin (10) is deliberately put into an eigenstate while storing the electron spin’s quantum state in a quantum memory (300). It has been found that including the temporary-storage gate (Gi), to cast the electron spin (10) in an eigenstate for a certain time period (ti) that effectively splits a target rotation (Ri) into two sub-rotations (Rn, R12), reduces unwanted cross-talk and thus increases selectivity.

[0062] The electron spin (10) in the electron-spin host (100) may be related to various types of colour centres (100) in a solid-state material (1000), as already outlined further above. The electron spin (10) may be initially in any superposition state or in an entangled state with other nuclear spins.

[0063] The plurality of nuclear spins (200) may be disposed in the solid-state material (1000) but may also relate to a target sample (2000) disposed on the solid-state material (1000) in the context of using the electron spin (10) as a quantum sensor, as also outlined further above.

[0064] Each of the sub-rotations (Rn, R12) may be implemented for example by one of the above-recited prior-art quantum gates or by quantum gates for rotation. That is, the first and second controlled-rotation quantum gates (C-Ri, C-R2) may be implemented based on for example DD control, DDRF control, UDD control, RDD control, and the like. The first and second controlled-rotation quantum gates may also be referred to as controlled-sub-rotation quantum gates.

[0065] The target rotation (Ri) can thus be performed by performing two sub-rotations (Rn and R12) that are interleaved by the temporary-storage quantum gate (Gi), the two sub-rotations adding up to the target rotation (Ri).

[0066] The electron-spin eigenstate into which the electron spin (10) is put for the duration of the time period (ti) may be any one of the electron-spin’s eigenstates. For example, one may control the electron spin (10) in a qubit subspace of two electron-spin eigenstates |0) and | 1) and the electron spin (10) may be cast into one of them during the time period (ti). As another example, in case the electron spin (10) relates to a higher-spin system, one may also cast the electron spin (10) into another spin eigenstate outside the qubit subspace. In any such case, the time period (ti) in which the electron spin (10) is deliberately put into an electron-spin eigenstate causes different nuclear spins to pick up different phases, thereby contributing to an increased selectivity of the whole controlled-rotation quantum gate (C-Ri). The electron-spin eigenstate into which the electron spin (10) is put is preferably different from a spin projection with ms=0, in which case there would be no hyperfine interactions with other nuclear spins.

[0067] The time period (ti) may be chosen depending on the specific system involved, such as the electron spin (10), the electron-spin host (100), a solid-state material (1000), and / or the quantum memory (300). Already including a relatively short time period in the temporary - storage quantum gate (Gi) increases selectivity. Preferably, the time period (ti) is at least 10ns, more preferably at least 100ns, even more preferably at least Ips.

[0068] Preferably, the time period (ti) is smaller than a decoherence time of the quantum memory (300). The decoherence relates to a time of preserving an arbitrary superposition state. For example, the quantum memory (300) may be suitable for storing a quantum state for a certain amount of time, i.e., the quantum memory (300) may have a certain decoherence time. The time period (ti) is preferably chosen smaller than such decoherence time so as to effectively protect the coherence of the spin state of the electron spin (10) when performing the controlled- rotation quantum gate (C-Ri). A decoherence time of the quantum memory (300) may also be extended by pulses or control sequences acting on the quantum memory (300) and configured to extend the decoherence time of the quantum memory (300). In such a case, the time period (ti) may be chosen accordingly in a wider range, but preferably smaller than an extended decoherence time of the quantum memory (300). Preferably, the time period (ti) is chosen shorter than the depolarizing time Ti of the electron spin (10) and the decoherence time or an extended decoherence time of the quantum memory (300). The depolarizing time Ti refers to a time duration during which the electron spin (10) can preserve an eigenstate. In addition, the time period (ti) may also be optimized by choosing different values and checking their associated selectivity.

[0069] Without being bound by theory, one can understand that including the temporarystorage quantum gate (Gi) with the time period (ti) allows different nuclear spins to pick up a respective different phase proportional to the time period (ti), the picked-up phase being first order in the parallel hyperfine parameter. As a consequence, selectivity of the whole controlled- rotation quantum gate (C-Ri) can be increased due to exploiting differences in the parallel hyperfine parameters. The temporary-storage quantum gate (Gi) may be understood to introduce a certain amount of asymmetry in a specific manner, the asymmetry leading to a higher selectivity of the whole controlled-rotation quantum gate (C-Ri). The technical effect of increased selectivity is particularly pronounced for the electron spin (10) relating to a spin-'A system.

[0070] The two sub-rotations are preferably having a substantially same rotation axis. That may be achieved for example by fine-tuning the time period (ti) so that the second sub-rotation (R2) continues on a same rotation axis as the first sub-rotation (Ri). Such choosing may be made dependent on for example a precession frequency of the target nuclear spin. For DDRF control the phase of the RF pulses may be adapted similarly as known from DDRF control itself in which the phase of the RF pulses are dependent on a phase of a nuclear spin. Preferably, one or more of the two controlled sub-rotation quantum gates (C-Rn, C-R12) are performed based on DD control, DDRF control, UDD control, or RDD control. For example, each sub-rotation may be tuned by choosing parameters involved in DDRF control such as T, N, etc. For example, when a target rotation (Ri) can be implemented by DDRF by particular parameters r and a total number of rr-pulses Ni, the first controlled sub-rotation quantum gate may be implemented with the same parameter r and half the number of TI -pulses, N11 = Ni / 2, and the second controlled sub-rotation quantum gate may be implemented with the same parameter T and also half the number of TI -pulses, N12 = Ni / 2. Such two controlled subrotation quantum gates may then be interleaved by a temporary-storage quantum gate (Gi) with a particular time period (ti). For example, the time period (ti) may be chosen substantially equal to or similar to the parameter T. The time period (ti) may also be chosen substantially equal to a time duration of the first or second controlled sub-rotation quantum gate, e.g. a particular multiple of the parameter T. The time period (ti) may also be chosen longer than a duration of a sub-rotation quantum gate.

[0071] So, by including the temporary-storage quantum gate (Gi), the target rotation (Ri) of the target nuclear spin can be performed with a higher selectivity as opposed to an attempt of performing the target rotation directly with for example only one DD sequence or one DDRF sequence.

[0072] The temporary-storage quantum gate (Gi) exploits the quantum memory (300) into which a current spin state of the electron spin (10) is temporarily stored for the duration of the time period (ti). The electron spin (10) being temporarily put into one of its electron-spin eigenstates means that the spin state stored in the quantum memory (300) is put back on the electron spin (10) after expiry of the time period (ti) and thus just before the applying of the second controlled-rotation quantum gate (C-R12).

[0073] Preferably, the method according to the first aspect further comprises initializing a qubit (301) of the quantum memory (300) in a qubit-eigenstate, and the temporary-storage quantum gate (Gi) comprises transferring the spin state back and fourth between the qubit (301) and the electron spin (10), the back and fourth transferring being separated by the time period (ti).

[0074] The initializing of the qubit (301) facilitates the temporary storage of the spin state of the electron spin (10). The initializing of the qubit (301) may be performed before, simultaneously, or after the first controlled-rotation quantum gate (C-Rn), depending on for example a decoherence time of the qubit (301) and / or the quantum memory (300), and / or a time scale and control resources needed for the initializing and the controlled rotation, respectively. Preferably, the qubit (301) is initialized before transferring the spin state onto it. Preferably the applying of the temporary-storage quantum gate (Gi) further comprises: applying a first transfer quantum gate (TGn) configured to transfer the spin state of the electron spin (10) to the quantum memory (300); and applying, after the time period (ti) has lapsed, a second transfer quantum gate (TG12) configured to transfer back the spin state from the quantum memory (300) to the electron spin (10).

[0075] Fig. 4a shows such a preferred example of the temporary-storage quantum gate (Gi) used in the controlled-rotation quantum gate according to the present disclosure.

[0076] In Fig. 4a, the first controlled-rotation (or: “sub-rotation”) quantum gate (C-Rn) (brief: “controlled rotation”) and the second controlled-rotation (or: “sub-rotation”) quantum gate (C- R12) are interleaved by the temporary-storage quantum gate (Gi) with the time period (ti). After the applying of the first controlled rotation (C-Rn), the first transfer quantum gate (TGn) is applied, which has the effect of storing the spin state of the electron spin (10) in the quantum memory (300). After the applying of the first transfer quantum gate (TGn), the quantum memory (300) carries the spin state of the electron spin (10), which may be denoted as |me). For the duration of the time period (ti), the electron spin (10) is in an electron-spin eigenstate, for example |0). That may be an automatic consequence of the transfer quantum gate (TGn) or may be achieved by re-initializing the electron spin (10) after the applying of the first transfer quantum gate (TGn). The electron-spin eigenstate may be of the qubit subspace in which the electron spin (10) is controlled, but may, for an electron spin (10) with higher spin number, also relate to another eigenstate outside such a qubit subspace. After the time period (ti) has lapsed, the second transfer quantum gate (TG12) is applied which transfers back the spin state (|me)) carried by the quantum memory (300) to the electron spin (10). For such second transfer quantum gate (TG12), the electron spin (10) may be initialized in an electron-spin eigenstate for receiving the spin state (|me)), for example in |0), if not already being in such eigenstate.

[0077] Fig. 4b shows, for illustration, a preferred example of a transfer quantum gate (TG) according to the present disclosure in a quantum circuit notation.

[0078] In Fig. 4b, a first qubit is initially in a superposition state |+) = + | 1)) and a second qubit is initially in a qubit-eigenstate |0). In this case, the controlled-NOT acts on a two- qubit basis state \ab) as \a, a © b), wherein a, b are 0 or 1, © denotes addition modulo 2, and \ab) is short-hand notation for the tensor product of a first-qubit state |a) and a second-qubit state \b). For example, the state 111) is mapped by the C-NOT to 110). The Hadamard gate H maps a basis state |0) to |+) and the other basis state | 1) to |— ) = — (|0) — | 1)). The measurement-symbol indicates measuring the respective qubit. The Z-gate conditional on a measurement result (“Z-gate conditional on measurement result”) adds a phase, that is 10) <— > |0) and 11) i-> — 11). The quantum circuit may be understood to transfer the first-qubit state onto the second qubit as follows, wherein normalization factors -1 / 2 are omitted for brevity.

[0079] The initial two-qubit state is | + 0). That is, the first qubit is initially in |+) and the second qubit is initially in |0).

[0080] The C-NOT maps the initial two-qubit state to 100) + 111).

[0081] The Hadamard gate further maps to 100) + 110) + 101) — 111).

[0082] The measurement maps either to |00) + 101) or to 110) — 111).

[0083] The Z gate conditional on measurement result maps further to either 100) + |01) or to |10> + | 11).

[0084] The final two-qubit state is either 10)(| 0) + | 1)) = |0+) or 11)(| 0) + | 1)) = |0+). Independent of the outcome of the measurement, the second qubit is finally in |+). So, the initial state of the first qubit is transferred to the second qubit.

[0085] Both the superposition state |+)~|0) + | 1) and the second-qubit initial state |0) are merely chosen for illustration purposes. The initial state of the first qubit may also be in a more general state c|0) + d| l), | c |2+ | d |2= 1. The initial state of the second qubit may also be in the other initial state 11), in which case the C-NOT is a C-NOT controlled on 0 instead of on 1; i.e., the C-NOT would then be configured to act as (|0b) ■-> |0, b © 1)) and (lb) ■-> | lb)).

[0086] At the example of the first transfer quantum gate (TGn), the electron spin (10) may take the role of the first qubit in Fig. 4b and a qubit (301) of the quantum memory (300) may take the role of the second qubit in Fig. 4b.

[0087] Preferably, the applying of the temporary-storage quantum gate (Gi) comprises: applying a first transfer quantum gate (TGn) configured to transfer the spin state of the electron spin (10) to the quantum memory (300); and applying, after the time period (ti) has lapsed, a second transfer quantum gate (TGn) configured to transfer back the spin state from the quantum memory (300) to the electron spin (10). In other words, the second transfer quantum gate (TGn) is applied at expiry of a time window starting from the applying of the first transfer quantum gate (TGn) and ending after the first time period (ti) has lapsed.

[0088] As outlined above in Figs. 4a-4b, transferring a spin state back and fourth between the electron spin (10) and quantum memory (300) by such transfer quantum gates (TG) is a particular effective way of temporarily storing the spin state of the electron spin (10) while also putting the electron spin (10) in an electron-spin eigenstate during the time period (ti). Preferably, the second transfer quantum gate (TG12) comprises initializing the electron spin (10) in a particular electron-spin eigenstate (|0), | 1)) before the transferring of the spin state from the quantum memory (300) back to the electron spin (10). Such preparing of the electron-spin (10) may facilitate control as same transfer instructions may be used for transferring back a state of the quantum memory (300) onto the electron spin (10).

[0089] Preferably, the temporary-storage quantum gate (Gi) comprises measuring the electron spin (10), which, after the measuring, is in an electron-spin eigenstate. A sequence of gates that involves measuring the electron spin (10) is an effective way of putting the electron spin (10) in an electron-spin eigenstate. An illustration thereof is shown in Fig. 4b.

[0090] Preferably, the first transfer quantum gate (TGn) comprises: a controlled-NOT gate (denoted as C-®) applied to the pair of electron-spin (10) and a qubit (301) of the quantum memory (300), configured to entangle the electron spin (10) and the qubit (301); a Hadamard gate (H) applied to the electron-spin (10); a measurement gate configured to measure out / read out the electron spin (10); and a Z gate conditional on measurement result (denoted as C-Z) configured to apply a Z-gate on the qubit (301) controlled by / depending on the electron spin (10). Such a preferred transfer quantum gate (TGi) is illustrated in Fig. 4b. The qubit of the quantum memory (300) may be initialized to an eigenstate before performing the controlled- rotation quantum gate (C-Ri) or before performing the respective transfer quantum gate (TGn).

[0091] Preferably, the second transfer quantum gate (TG12) is similarly based on Fig. 4b, in which the electron spin (10) takes the role of the second qubit and the qubit memory (300) takes the role of the first qubit whose state is to be transferred to the electron spin (10).

[0092] A transfer of a spin state of the electron spin (10) to the quantum memory (300) may also involve more qubits of the quantum memory (300). The quantum memory (300) may in general comprise one or more qubits for temporarily storing a respective spin state of the electron spin (10). For example, a teleportation quantum gate involving a total of three qubits may be performed for transferring a spin state of the electron spin (10) to a particular qubit of the quantum memory (300).

[0093] Examples of the quantum memory (300) are shown next with reference to Figs. 5a-5d.

[0094] Figs. 5a-5d show examples of quantum memories (300) used in the controlled-rotation quantum gates according to the present disclosure.

[0095] Fig. 5a shows that the quantum memory (300) may be related to a nuclear spin of the electron-spin host (100). For example, a qubit (301) of the quantum memory (300) may be realized as a nuclear spin (101) carried by a nucleus of a colour centre. Fig. 5a shows two examples thereof: a tin-vacancy (SnV) centre and a silicon-vacancy (SiV) centre. An interaction between the electron spin (10) and the nuclear spin (101) of the Sn-atom or of the Si-atom can be exploited so as to use the Sn-atom or the Si-atom as a quantum memory (300). For example, quantum gates between an electron spin (10) and a29Si-atom of a SiV centre with a quantum memory time exceeding two seconds have been demonstrated in [Stas et al]. In general, a nuclear spin of a colour centre that couples suitably strongly to the electron spin (10) may be used as a quantum memory (300).

[0096] Figs. 5b-5d show further examples of components that may be used as quantum memories (300). Fig. 5b shows that for example another nuclear spin (204) of the plurality of nuclear spins (200) may be used as the quantum memory (300). Fig. 5c shows that the quantum memory (300) may be located outside of a solid-state material (1000) but that has a coupling (10-300) to the electron spin (10), e.g. a photonic, that can be used to transfer quantum states. Fig. 5d shows as yet another examples that the solid-state material (1000) may comprise another electron-spin host (102) coupled to the electron spin (10) by a coupling (10-300) for controllably exchanging states so that the other electron-spin host (102) can be used as the quantum memory (300). For example, the other electron-spin host (102) may also relate to a colour centre with a Sn-atom or a Si-atom whose nuclear spin couples suitably to the electron spin (10), so that a similar situation as in Fig. 5a arises that can be exploited. Preferably, an electron-electron coupling between electron spins of the respective electron-spin hosts is used.

[0097] Figs. 6a-6b illustrate further examples of the controlled-rotation quantum gate (C-Ri).

[0098] Preferably, the temporary-storage quantum gate (Gi) comprises: applying a first swap quantum gate (Gn) configured to swap states of a qubit (301) of the quantum memory (300) and of the electron spin (10), so that the electron spin (10) is in one of its electron-spin eigenstates and the qubit (301) carries the spin state of the electron-spin, and applying, after the time period (ti), a second swap quantum gate (G12) configured to swap back the swapped states of the qubit (301) and of the electron spin (10), so that the electron spin (10) is back in the spin state.

[0099] The electron spin (10) can for example be initialized after the first swap so as to realize that the electron spin (10) is in an eigenstate. Using swap gates is a particular effective way of transferring the spin state back and fourth between the qubit (301) and the electron spin (10), the back and fourth transferring being separated by the time period (ti).

[0100] Preferably, the method further comprises initializing, before or as part of the temporarystorage quantum gate (Gi), the qubit (301) in an initial qubit-eigenstate. An advantage thereof is that then, as a consequence of the first swap, the electron spin (10) is automatically in an eigenstate already, so that an initialization-sequence on the electron spin (10) can be saved.

[0101] Fig. 6a shows an example in which the quantum memory (300) is initialized in an eigenstate. For example, as exemplified in the context of Figs. 5a-5d, the quantum memory (300) may comprise a qubit (301) that can be initialized in an eigenstate. An initial state of the electron spin (10) in this example is denoted as |me). After the first controlled-sub-rotation quantum gate (C-Rn), the first swap gate (Gn) is performed. As a consequence of the first swap gate, the states of the quantum memory (300) and the electron spin (10) are swapped / exchanged, so that the electron spin (10) is automatically in an eigenstate and the former spin state of the electron spin (10), |me), is stored in the quantum memory (300). After the first time period (ti), the second swap gate (Gn) is performed, which swaps back the states between the quantum memory (300) and the electron spin (10).

[0102] Preferably, one or more or all of the swap quantum gates (e.g. Gn, Gn) may be implemented by involving a plurality of controlled-NOT gates. The swap quantum gates (Gn, Gn) (also referred to as swap gates) may be implemented for example by involving three controlled-NOT gates, as also demonstrated in [Stas et al] at the example of a SiV centre. Such a SiV centre can be used as a quantum memory (300) for implementing the temporary-storage quantum gate (Gi) by means of swap gates (Gn, Gn). When an initial state of a qubit (301) of the quantum memory (300) is an eigenstate, for example by initializing the qubit (301), one may also implement a swap quantum gate by two controlled-NOT gates.

[0103] Figs. 7a-7d shows a further controlled-rotation quantum gate (C-Rk) according to the present disclosure, involving k temporary-storage quantum gates (Gi, ..., Gk).

[0104] A further method is provided for performing a controlled-rotation quantum-gate (C-Ri) configured to rotate a target nuclear spin (201 ) by a target rotation (Ri) controlled by an electron spin (10), wherein the electron spin (10) is hosted in an electron-spin host (100) and is surrounded by a plurality of nuclear spins (200) comprising the target nuclear spin (201). The method comprises: applying a chain of 2k alternating controlled-rotation quantum gates (C-Ri) and temporary-storage quantum gates (Gi), wherein k > 2, followed by applying a final controlled- rotation quantum gate (C-Ri,k+i), wherein: each of the controlled-rotation quantum gates is configured to rotate the target nuclear spin (201) by a respective sub-rotation (Rii, Ri,k+i) controlled by the electron spin (10); each of the temporary-storage quantum gates is configured to, temporarily for a respective time period (ti), store a respective spin state ((|me)) of the electron spin (10) in a quantum memory (300) while putting the electron spin (10) in a respective electron-spin eigenstate (|0), | 1)); and the respective sub-rotations (Rn, R12,..., Ri,k+i) sum up to the target rotation (Ri).

[0105] As illustrated by the quantum circuit of Fig. 7a, the further method means that a target rotation (Ri) can be split up in even more sub-rotations (Rn, R12, ..., Rik, Ri,k+i) interleaved with temporary-storage quantum gates (Gi, G2, ..., Gk). The integer k may be any integer, i.e.,

[0106] 1, 2, 3, 4, 5, ..., 20, ... 30, etc., and may be chosen depending on preferences. For k=l, the previously discussed case is obtained.

[0107] A target rotation (Ri) can thus be split up in k+1 sub-rotations (Rn, R12, ..., Ri,k+i) and involving k temporary-storage quantum gates (Gi, G2, ..., Gk). Employing the chain of 2k alternating sub-rotations and temporary-storage gates, followed by a final sub-rotation, allows for further differentiating between different nuclear spins, so that the selectivity of the controlled-rotation quantum gate (C-Ri) is further enhanced.

[0108] All of the quantum gates shown in Fig. 7a may be implemented and combined, including one or more or all of the preferred aspects, as already outlined above in the description relating to Figs. 1-6.

[0109] For example, each of the controlled sub-rotation quantum gates (Rn, R12, ..., Ri,k+i) may be implemented based on for example DD control, DDRF control, UDD control, RDD control, and the like. Also each of the temporary-storage quantum gates (Gi, ..., Gk) may be implemented as outlined above in the description relating to Figs. 1-6. Moreover, each temporary-storage quantum gate may have a different choice of respective time period (ti), i=l,

[0110] 2, ... , k. Preferably, each respective time period (ti) is the same to simplify control. However, the general method is not limited to such choice. Moreover, transfer quantum gates and swap gates may also be combined. For example, a first temporary-storage quantum gate may be performed based on transfer quantum gates and a second temporary-storage quantum gate may be performed based on swap gates. Any combination of transfer and swap gates can be used. In general, swap gates are preferred, as their implementation does not involve measurements and real-time feedback.

[0111] Preferably, the respective electron-spin eigenstates, into which the electron spin (10) is put by the respective temporary-storage quantum gate, have a same spin-projection sign (positive or negative). More preferably, they are equal to one another. Choosing the same signs further increases selectivity. Choosing the same eigenstates can even further increase selectivity, in particular when for example choosing a spin-eigenstate with a highest available spin quantum number. As an example, one may consider a target rotation (Ri) that may be implemented by a DDRF control involving a certain inter-pulse delay parameter r and a total number of decoupling pulses N. A controlled-rotation quantum gate (C-Ri) may be obtained in various ways by splitting up the DDRF-control.

[0112] For example, one preferably evenly divides the total number of decoupling pulses among a plurality of sub-rotations. Each sub-rotation may be implemented based on DDRF with the same inter-pulse delay parameter r but with a smaller number of decoupling pulses N / (k+l), with N / (k+l) being an integer. In this case, all sub-rotations are identical and the sum of decoupling pulses of all sub-rotations is equal to the total number of decoupling pulses N. As another example, one may choose to divide the number of decoupling pulses not evenly among the sub-rotations but may consider an uneven distribution; for example, a first subrotation with N / ji TT -pulses, a second sub-rotation with N / j? TT -pulses, ..., a k+l-th sub-rotation with N / jk+i Tt -pulses, such that (N / ji) + (N / j2)+ ... + (N / jk+i) = N, wherein each N / jmis an integer relating to the number of TT -pulses of the m’th sub-rotation.

[0113] For example, considering N=24, one may choose three sub-rotations each with N=8. Or, considering N=36, one may choose 9 sub-rotations each with N=4. As another example, considering N=24, one may choose a total of four sub-rotations, the first sub-rotation with N=4, the second and third sub-rotations with N=8, and the fourth sub-rotation with N=4.

[0114] Preferably, bulk sub-rotations are identical, so that the quantum-circuit / the controlled- rotation quantum gate (C-Ri) is symmetric. The bulk sub-rotations refer to the second to penultimate sub-rotation quantum gate. In the example of considering N=24 and a total of four sub-rotations, the first sub-rotation with N=4, the second and third sub-rotations with N=8, and the fourth sub-rotation with N=4, the bult sub-rotations refer to the second and third subrotations.

[0115] More preferably, all involved sub-rotations are equal to one another, in other words, evenly splitting a target rotation into a plurality of equal sub-rotations.

[0116] Fig- 8 showcases, by an illustrative example, reduced cross-talk and enhanced selectivity of a controlled-rotation quantum gate (C-Rk=s) of the present disclosure. In this example gate, a DDRF control sequence is split into four DDRF control sequences separated by respective temporary-storage quantum gates. The example gate may be denoted in the following way: (DDRF(N=4) - 1 -DDRF(N=4)) * 3. The example gate was performed with t = 60 / J.S and r=26.6ps as in Fig. 2. Comparing with the example of Fig. 2, which was based on DDRF(N=24), one can recognize a reduced cross-talk. For illustration, when targeting Atar, other nuclear spins Aunwanted,i and Aunwanted,2 are also interacting in Fig. 2 but are not in Fig. 8. Therefore, the examples illustrate an enhanced selectivity.

[0117] Figs. 9a-9b are flowcharts showing the methods according to the present disclosure with a temporary-storage quantum gate (Gi). The flowcharts may be further considered by including one or more of all of the preferred aspects disclosed in the present document.

[0118] Fig. 9c shows a quantum processor (1) for performing methods according to the present disclosure. The quantum processor (1) may comprise one or more instances of an electron spin (10) hosted in an electron-spin host (100), e.g. (1-1), (1-2). Moreover, the quantum processor (1) may comprise components for performing the method, such as a magnetic-fields generator (1-3), electromagnetic-waves generator (1-4), a processor (1-5), a computer-readable storage medium (1-6), measurement devices, etc., depending on the type of electron spin (10), electronspin host (100) and eventual solid-state material (1000) involved. Such architectural aspects suitable for performing quantum gates such as the controlled-rotation quantum gates, controlled-sub-rotation quantum gates, temporary-storage quantum gates, etc., are known and therefore not detailed here.

[0119] The quantum processor (1) according to the present invention comprises one or more instances of an electron spin (10) hosted in an electron-spin host (100) and is configured to perform the method according to the present invention, the method preferably including one or more or all of the preferred aspects discussed above.

[0120] The quantum processor (1) may also be referred to as a quantum-processor system (1) or a quantum computer (1).

[0121] The computer-readable storage medium (1-6) may comprises instructions, which, when executed by the quantum processor (1), cause the quantum processor (1) to perform one or more of the methods preferably including one or more of the preferred aspects discussed above.

[0122] Aspects of the present disclosure are also put forward in the following clauses.

[0123] Clause 1. Method of performing a controlled-rotation quantum-gate (C-Ri) configured to rotate a target nuclear spin (201) by a target rotation (Ri) controlled by an electron spin (10), wherein the electron spin (10) is hosted in an electron-spin host (100) and is surrounded by a plurality of nuclear spins (200) comprising the target nuclear spin (201), the method comprising: applying a first controlled-rotation quantum gate (C-Rn) configured to rotate the target nuclear spin (201) by a first sub-rotation (Rn) controlled by the electron spin (10); applying a temporary-storage quantum gate (Gi) configured to, temporarily for a time period (ti), store a spin state (|me)) of the electron spin (10) in a quantum memory (300) while putting the electron spin (10) in an electron-spin eigenstate (|0); | 1)); and applying a second controlled-rotation quantum gate (C-R12) configured to rotate the target nuclear spin (201) by a second sub-rotation (R12) controlled by the electron spin (10), the second sub-rotation (R12) adding with the first sub-rotation (R11) to the target rotation (Ri).

[0124] Clause 2. The method according to clause 1, further comprising: initializing a qubit (301) of the quantum memory (300) in a qubit-eigenstate; and wherein the temporary-storage quantum gate (Gi) comprises transferring the spin state back and fourth between the qubit (301) and the electron spin (10), the back and fourth transferring being separated by the time period (ti).

[0125] Clause 3. The method according to any one of the preceding clauses, wherein the applying of the temporary-storage quantum gate (Gi) comprises: applying a first transfer quantum gate (TGn) configured to transfer the spin state of the electron spin (10) to the quantum memory (300); and applying, after the time period (ti) has lapsed, a second transfer quantum gate (TG12) configured to transfer back the spin state from the quantum memory (300) to the electron spin (10).

[0126] Clause 4. The method of the preceding clause, wherein the second transfer quantum gate (GT12) comprises initializing the electron spin (10) in a particular electron-spin eigenstate (|0)) before the transferring of the spin state from the quantum memory (300) back to the electron spin (10).

[0127] Clause 5. The method according to any one of the preceding clauses, wherein the temporary-storage quantum gate (Gi) comprises measuring the electron spin (10), which, after the measuring, is in an electron-spin eigenstate (|0), | 1)).

[0128] Clause 6. The method according to any one of the preceding clauses, wherein the temporary-storage quantum gate (Gi) comprises: applying a first swap quantum gate (Gn) configured to swap states of a qubit (301) of the quantum memory (300) and of the electron spin (10), so that the electron spin (10) is in one of its electron-spin eigenstates and the qubit (301) carries the spin state of the electron-spin, and applying, after the time period (ti), a second swap quantum gate (G12) configured to swap back the swapped states of the qubit (301) and of the electron spin (10), so that the electron spin (10) is back in the spin state.

[0129] Clause 7. The method of the preceding clause, further comprising: initializing, before or as part of the temporary-storage quantum gate (Gi), the qubit (301) in an initial qubit-eigenstate.

[0130] Clause 8. Method of performing a controlled-rotation quantum-gate (C-Ri) configured to rotate a target nuclear spin (201) by a target rotation (Ri) controlled by an electron spin (10), wherein the electron spin (10) is hosted in an electron-spin host (100) and is surrounded by a plurality of nuclear spins (200) comprising the target nuclear spin (201), the method comprising: applying a chain of 2k alternating controlled-rotation quantum gates (C-Ri) and temporary-storage quantum gates (Gi), wherein k > 2, followed by applying a final controlled- rotation quantum gate (C-Ri,k+i), wherein: each of the controlled-rotation quantum gates is configured to rotate the target nuclear spin (201) by a respective sub-rotation (Rn, Ri,k+i) controlled by the electron spin (10); each of the temporary-storage quantum gates is configured to, temporarily for a respective time period (ti), store a respective spin state ((|me)) of the electron spin (10) in a quantum memory (300) while putting the electron spin (10) in a respective electron-spin eigenstate (|0), | 1)); and the respective sub-rotations (Rn, R12,..., Ri,k+i) sum up to the target rotation (Ri).

[0131] Clause 9. The method according to any one of the preceding clauses, wherein the quantum memory (300) comprises one or more qubits for temporarily storing a respective spin state of the electron spin (10).

[0132] Clause 10. The method according to any one of the preceding clauses, wherein the electron-spin host (100) corresponds to a colour centre, a rare-earth ion, or a molecule.

[0133] Clause 11. The method according to any one of the preceding clauses, wherein the electron spin (10) corresponds to a spin / i system, preferably to a tin-vacancy in diamond (1000) or a silicon-vacancy in diamond (1000).

[0134] Clause 12. The method according to any one of the preceding clauses, wherein the plurality of nuclear spins (200) relates to spin / i nuclei of carbon isotopes13C in diamond (1000).

[0135] Clause 13. The method according to any one of the preceding clauses, wherein at least one of the controlled-rotation quantum gates (C-Rn, C-R12, C-Rii) and / or of the swap quantum gates (Gn, G12, GH) is implemented based on a dynamic decoupling, DD, sequence, a DD radio frequency, DDRF, sequence, a Uhrig Dynamical Decoupling, UDD, sequence, and / or a Robust Dynamic Decoupling, RDD, sequence. Clause 14. Quantum processor (1) comprising one or more instances of an electron spin (10) hosted in an electron-spin host (100), the quantum processor (1) configured to perform the method according to any one of the preceding clauses.

[0136] Clause 15. Computer-readable storage medium (1-6) comprising instructions, which, when executed by the quantum processor (1), cause the quantum processor (1) to perform the method according to any one of the preceding method clauses.

[0137] The following list of references is referred to in the present document and is incorporated herein by way of reference.

[0138] List of references

[0139] [A] Authors. Title. Publisher. Date of publication.

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[0142] [Babin et al] Babin, C., Stbhr, R., Morioka, N. et al. Fabrication and nanophotonic waveguide integration of silicon carbide colour centres with preserved spin-optical coherence. Nat. Mater. 21, 67-73 (2022). 2022.

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[0144] [Bradley et al] Bradley CE, Randall J, Abobeih MH, Berrevoets RC, Degen MJ, Bakker MA, et al. A Ten-Qubit Solid-State Spin Register with Quantum Memory up to One Minute. Phys Rev X. 2019 Sep 11 ;9(3):031045. 2019.

[0145] [Casanova et al] J. Casanova, Z.-Y. Wang, J. F. Haase, and M. B. Plenio. Robust dynamical decoupling sequences for individual-nuclear-spin addressing. Phys. Rev. A 92, 042304. 5 October 2015.

[0146] [Fursina et al] Alexandra A. Fursina, Alexander Sinitskii. Toward Molecular Spin Qubit Devices: Integration of Magnetic Molecules into Solid-State Devices. ACS Applied Electronic MaterialsVol 5 / Issue 7. 13 July 2023. [Morton et al] Morton, J., Tyryshkin, A., Ardavan, A. et al. Bang bang control of fullerene qubits using ultrafast phase gates. Nature Phys 2, 40-43 (2006). https: / / doi.org / 10.1038 / nphysl92. 2006.

[0147] [Nguyen et al] C. T. Nguyen, D. D. Sukachev, M. K. Bhaskar, B. Machielse, D. S. Levonian, E. N. Knall, P. Stroganov, C. Chia, M. J. Burek, R. Riedinger, H. Park, M. Loncar, and M. D. Lukin. An integrated nanophotonic quantum register based on silicon-vacancy spins in diamond. Phys. Rev. B 100, 165428. 2019.

[0148] [Photonic Inc] Francis Afzal et al. Distributed Quantum Computing in Silicon. arXiv:2406.01704vl [quant-ph], 3 June 2024.

[0149] [Pompili et al] M. Pompili et al. Realization of a multinode quantum network of remote solid- state qubits. Science 372,259-264(2021).DOI: 10.1126 / science.abgl919. 2021.

[0150] [Ruskuc et al] Andrei Ruskuc, Chun- Ju Wu, Emanuel Green, Sophie L. N. Hermans, Joonhee Choi, Andrei Faraon. Scalable Multipartite Entanglement of Remote Rare-earth Ion Qubits. arXiv:2402.16224vl. 2024.

[0151] [Smarak et al] Smarak Maity, Linbo Shao, Young-Ik Sohn, Srujan Meesala, Bartholomeus Machielse, Edward Bielejec, Matthew Markham, and Marko Loncar. Spectral Alignment of Single-Photon Emitters in Diamond using Strain Gradient. Phys. Rev. Applied 10, 024050. 2018.

[0152] [Stas et al] P.-J. Stas et al. Robust multi-qubit quantum network node with integrated error detection. Science 378, 557-560(2022). DOI: 10.1126 / science.add9771. 2022.

[0153] [Taminiau et al] Taminiau TH, Wagenaar JJT, van der Sar T, Jelezko F, Dobrovitski VV, Hanson R. Detection and Control of Individual Nuclear Spins Using a Weakly Coupled Electron Spin. Phys Rev Lett. 2012 Sep 25; 109(13): 137602. 2012.

[0154] [Takou et al] Evangelia Takou, Edwin Barnes, and Sophia E. Economou. Precise Control of Entanglement in Multinuclear Spin Registers Coupled to Defects. Phys. Rev. X 13, 011004. 18 January 2023.

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Claims

What is claimed is:

1. Method of performing a controlled-rotation quantum-gate (C-Ri) configured to rotate a target nuclear spin (201) by a target rotation (Ri) controlled by an electron spin (10), wherein the electron spin (10) is hosted in an electron-spin host (100) and is surrounded by a plurality of nuclear spins (200) comprising the target nuclear spin (201), the method comprising: applying a first controlled-rotation quantum gate (C-Rn) configured to rotate the target nuclear spin (201) by a first sub-rotation (Rn) controlled by the electron spin (10); applying a temporary-storage quantum gate (Gi) configured to, temporarily for a time period (ti), store a spin state (|me)) of the electron spin (10) in a quantum memory (300) while putting the electron spin (10) in an electron-spin eigenstate (|0); | 1)); and applying a second controlled-rotation quantum gate (C-Rn) configured to rotate the target nuclear spin (201) by a second sub-rotation (R12) controlled by the electron spin (10), the second sub-rotation (R12) adding with the first sub-rotation (R11) to the target rotation (Ri).

2. The method according to claim 1, further comprising: initializing a qubit (301) of the quantum memory (300) in a qubit-eigenstate; and wherein the temporary-storage quantum gate (Gi) comprises transferring the spin state back and fourth between the qubit (301) and the electron spin (10), the back and fourth transferring being separated by the time period (ti).

3. The method according to any one of the preceding claims, wherein the applying of the temporary-storage quantum gate (Gi) comprises: applying a first transfer quantum gate (TGn) configured to transfer the spin state of the electron spin (10) to the quantum memory (300); and applying, after the time period (ti) has lapsed, a second transfer quantum gate (TG12) configured to transfer back the spin state from the quantum memory (300) to the electron spin (10).

4. The method of the preceding claim, wherein the second transfer quantum gate (GT12) comprises initializing the electron spin (10) in a particular electron-spin eigenstate (|0)) before the transferring of the spin state from the quantum memory (300) back to the electron spin (10).

265. The method according to any one of the preceding claims, wherein the temporary-storage quantum gate (Gi) comprises measuring the electron spin (10), which, after the measuring, is in an electron-spin eigenstate (|0), | 1)).

6. The method according to any one of the preceding claims, wherein the temporary-storage quantum gate (Gi) comprises: applying a first swap quantum gate (Gn) configured to swap states of a qubit (301) of the quantum memory (300) and of the electron spin (10), so that the electron spin (10) is in one of its electron-spin eigenstates and the qubit (301) carries the spin state of the electron-spin, and applying, after the time period (ti), a second swap quantum gate (G12) configured to swap back the swapped states of the qubit (301) and of the electron spin (10), so that the electron spin (10) is back in the spin state.

7. The method of the preceding claim, further comprising: initializing, before or as part of the temporary-storage quantum gate (Gi), the qubit (301) in an initial qubit-eigenstate.

8. Method of performing a controlled-rotation quantum-gate (C-Ri) configured to rotate a target nuclear spin (201) by a target rotation (Ri) controlled by an electron spin (10), wherein the electron spin (10) is hosted in an electron-spin host (100) and is surrounded by a plurality of nuclear spins (200) comprising the target nuclear spin (201), the method comprising: applying a chain of 2k alternating controlled-rotation quantum gates (C-Ri) and temporary-storage quantum gates (Gi), wherein k > 2, followed by applying a final controlled- rotation quantum gate (C-Ri,k+i), wherein: each of the controlled-rotation quantum gates is configured to rotate the target nuclear spin (201) by a respective sub-rotation (Rii, Ri,k+i) controlled by the electron spin (10); each of the temporary-storage quantum gates is configured to, temporarily for a respective time period (ti), store a respective spin state ((|me)) of the electron spin (10) in a quantum memory (300) while putting the electron spin (10) in a respective electron-spin eigenstate (|0), | 1)); and the respective sub-rotations (Rn, R12,..., Ri,k+i) sum up to the target rotation (Ri).

9. The method according to any one of the preceding claims, wherein the quantum memory (300) comprises one or more qubits for temporarily storing a respective spin state of the electron spin (10).

10. The method according to any one of the preceding claims, wherein the electron-spin host (100) corresponds to a colour centre, a rare-earth ion, or a molecule.

11. The method according to any one of the preceding claims, wherein the electron spin (10) corresponds to a spin / i system, preferably to a tin-vacancy in diamond (1000) or a silicon- vacancy in diamond (1000).

12. The method according to any one of the preceding claims, wherein the plurality of nuclear spins (200) relates to spin / i nuclei of carbon isotopes13C in diamond (1000).

13. The method according to any one of the preceding claims, wherein at least one of the controlled-rotation quantum gates (C-Rn, C-R12, C-Rii) and / or of the swap quantum gates (Gn, G12, GH) is implemented based on a dynamic decoupling, DD, sequence, a DD radio frequency, DDRF, sequence, a Uhrig Dynamical Decoupling, UDD, sequence, and / or a Robust Dynamic Decoupling, RDD, sequence.

14. Quantum processor (1) comprising one or more instances of an electron spin (10) hosted in an electron-spin host (100), the quantum processor (1) configured to perform the method according to any one of the preceding claims.

15. Computer-readable storage medium (1-6) comprising instructions, which, when executed by the quantum processor (1), cause the quantum processor (1) to perform the method according to any one of the preceding method claims.

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