System and method for qubit initialization and control

Abstract

A method for initializing a predetermined target spin state of multi-donor quantum dots in a semiconductor substrate, the method comprising: i) loading electrons into the multi-donor quantum dots in a spin-down state; ii) performing a 0 or 1 electron reset pulse; iii) applying an RF signal to drive at least one EDSR transition; and iv) repeating steps ii) to iii) N times to achieve a predetermined target spin state.

Description

Technical Field

[0001] Aspects of the present disclosure relate to advanced processing systems, and more particularly to quantum processing systems and methods and systems for initializing and / or controlling processing elements.

Background Art

[0002] The developments described in this section are known to the inventors. However, unless otherwise indicated, any development described in this section should not be regarded as prior art simply because they are included in this section, or regarded as known to those skilled in the art.

[0003] Large-scale quantum processing systems are a promising technological revolution and have the potential to solve problems beyond the reach of classical machines. To date, several different structures, materials, and architectures for implementing qubits (or quantum bits) as well as corresponding quantum control and processing systems have been proposed.

[0004] Before such large-scale quantum computers can be commercially manufactured, several hurdles need to be overcome. One such essential requirement is to accurately measure qubit states at any given time in a quantum processing device. In the art, different types of sensors as well as qubit measurement and initialization techniques have been proposed. However, some of these techniques can be error-prone. Therefore, improved techniques for measuring qubit states and / or initializing qubits in a quantum processing system are desirable.

Summary of the Invention

Means for Solving the Problems

[0005] According to a first aspect of the present invention, there is provided a method for initializing a predetermined target spin state of multi-donor quantum dots in a semiconductor substrate, the method comprising: i) loading electrons into the multi-donor quantum dots in a spin-down state; ii) executing a 0 or 1 electron reset pulse; iii) applying an RF signal for driving at least one EDSR transition; and iv) repeating steps ii) to iii) N times to achieve a predetermined target spin state.

[0006] According to a second aspect of the present invention, there is provided a method for initializing a predetermined target spin state of multi-donor quantum dots in a semiconductor substrate, the method comprising: i) determining an initialization pulse sequence that depends on a predetermined target spin state, the initialization pulse sequence including at least one electron reset pulse and at least one EDSR transition; ii) loading electrons into the multi-donor quantum dots in a spin-down state; iii) applying the initialization pulse sequence; and iv) repeating step iii) N times to achieve a predetermined target spin state.

[0007] According to a third aspect of the present invention, a method for initializing a predetermined target spin state of multi-donor quantum dots in a semiconductor substrate is provided, the method comprising: i) loading electrons into the multi-donor quantum dots in a spin-down state; ii) performing an ESR measurement to determine the total nuclear spin state; according to the determination that the total nuclear spin state of the multi-donor quantum dots is not the target spin state; iii) determining and executing an initialization pulse sequence, the initialization pulse sequence including at least one ESR spin reset pulse and at least one EDSR transition, determining and executing; performing an ESR measurement to determine the total nuclear spin state; initializing the nuclear spin according to the determination that the total nuclear spin state of the multi-donor quantum dots is the target spin state; and returning to step iii) according to the determination that the total nuclear spin state of the multi-donor quantum dots is not the target spin state.

[0008] According to a fourth aspect of the present invention, a quantum processing element configured to initialize a predetermined target spin state in multi-donor quantum dots is provided, the quantum processing element comprising a semiconductor substrate and a dielectric material forming an interface with the semiconductor substrate; multi-donor quantum dots embedded in the semiconductor substrate, the multi-donor quantum dots including at least two donor atoms, the at least two donor atoms sharing at least one electron; and a control element for controlling the multi-donor quantum dots; electrons are loaded into the multi-donor quantum dots in a spin-down state; the control element is configured to apply an RF signal for driving at least one EDSR transition; and to apply at least one electron spin reset pulse; whereby the target spin state is achieved and the quantum dots are initialized.

[0009] The features and advantages of the present invention will become apparent from the following description of embodiments of the present invention by way of example only, with reference to the accompanying drawings.

Brief Description of the Drawings

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[0011] Overview The spin states of electrons or nuclei in semiconductor materials are good candidates for transmitting quantum information and serving as qubits (or quantum bits) for a quantum computer system. To perform quantum computing, the following three important steps are required: initialization of the qubits, control of the qubits, and readout of individual qubits. Some aspects of the present disclosure provide new and improved techniques for qubit initialization and control.

[0012] One type of quantum computer system is based on the spin states of individual qubits, where the qubits are electrons and / or nuclear spins localized inside a semiconductor quantum chip. These electrons and / or nuclear spins are confined either within quantum dots defined by gates or on donor atoms positioned within a semiconductor substrate. As a result of the weak magnetic coupling to the environment inherent in donor qubits in silicon, the coherence time is long. Single-qubit and two-qubit operations higher than the error correction threshold have been demonstrated using an AC magnetic field.

[0013] FIG. 1 shows an example of a spin qubit device 100 formed within a silicon substrate using donor atoms. FIG. 1A is a top view of the qubit device 100. FIG. 1B is a cross-sectional side view. The qubit device 100 may be used for a quantum computer that includes a plurality of these qubits. As shown in the drawings, the qubit device 100 is formed as a structure that includes a semiconductor substrate 102 and a dielectric 104. In this example, the substrate is isotopically purified silicon (silicon 28), and the dielectric is silicon dioxide. In other examples, the substrate may be silicon (Si). An interface 107 is formed where the substrate 102 and the dielectric 104 contact. In this example, the interface 107 is a Si / SiO2 interface. To form a qubit, a donor atom 108 is positioned within the substrate 102 inside the region 109 under the gate 106. A donor atom, such as a phosphorus atom 108, can be introduced into the substrate using nanofabrication techniques, such as hydrogen lithography provided by, for example, a scanning tunneling microscope, or industry-standard ion implantation techniques. In this example, the qubit device 100 includes a single atom 108 embedded in a silicon 28 crystal. However, the methods described herein may be applied to a qubit device 100 that includes a cluster of two or more embedded atoms 108.

[0014] Electrons 120 are loaded into the device 100 by the gate electrode 106. The physical state of the electrons 120 is described by a wave function 121, which is defined as the probability amplitude of finding an electron at a particular location. A donor qubit in silicon relies on using a potential well that is naturally formed by a donor atomic nucleus to confine an electron spin.

[0015] The gate electrode 106 is positioned over the region 109 and is operable to interact with the donor atom 108. For example, the gate electrode 106 may be used to induce an AC electric field in the region between the interface 107 and the donor atom 108 in order to modulate the hyperfine interaction between the electron 120 and the nucleus of the donor atom 108, where the hyperfine interaction is an interaction between an electron spin and a nuclear spin of the donor atom.

[0016] Figure 2A shows a multi-donor quantum dot device 200. To form the qubit device 200, two donor atoms 202A and 202B are located within the quantum dot 201 in the semiconductor substrate 204. In some examples, the donor atoms 202A, 202B are phosphorus atoms. In this example, the qubit 200 includes two donor atoms 202, which are generally referred to herein as 2P quantum dots. The silicon substrate 204 is covered by a barrier material / dielectric 206 such as silicon dioxide. Further, a gate 208 and an antenna 210 may be located on the dielectric 206 in the region above the multi-donor quantum dot 201. A voltage may be applied to the gate 208 to confine the electron 220 within the quantum dot 201. The electron 220 may be shared by the two donor atoms 202A, 202B.

[0017] Figure 2B shows another example of a multi-donor quantum dot device 250. This is similar to the device shown in Figure 2A. The only difference is the gate arrangement. In Figure 2A, the gate was shown as being disposed on top of the dielectric 206. In Figure 2B, the gate is located within the semiconductor substrate 204. In some embodiments, one or more gates 212 are disposed in the same plane as the donor dots. This in-plane gate may be connected to the surface of the substrate via a metal via (not shown). In some examples, the multi-donor quantum dot device may include gates both on the surface (as in Figure 2A) and within the semiconductor substrate (as in Figure 2B).

[0018] To confine one or more electrons 220 within the quantum dot 201, a voltage may be applied to the gate electrode 212. A 2P quantum dot having one electron has three spins (two nuclear spins from the donors and the electron spin). Thus, each of the three spins may be regarded as a qubit.

[0019] FIG. 2C shows another example of a multi-donor quantum dot device 270. To form the qubit device 270, three donor atoms 202A, 202B, and 202C are located within the quantum dot 201 in the semiconductor substrate 204. In some examples, the donor atoms 202A, 202B, and 202C are phosphorus atoms. In this example, the qubit 270 includes three donor atoms 202, which are generally referred to herein as 3P quantum dots. The electrons 220 may be shared by the three donor atoms 202A, 202B, and 202C.

[0020] In other examples, there may be m donor atoms, where m is an integer. In some examples, there may be up to 10 donor atoms within the quantum dot device. If there are m donor atoms that are phosphorus atoms, the multi-donor quantum dot may be referred to as an mP quantum dot. A general quantum dot system having m donors and one electron has a total of m + 1 spins, and thus m + 1 possible qubits.

[0021] The spin of the electron 220 in the 2P quantum dot 201 is strongly coupled to the nuclear spins of the two donor atoms via the hyperfine interaction. Here, the hyperfine interaction is the interaction between the electron spin and the nuclear spins of the donor atoms. This hyperfine interaction ranges from tens to hundreds of MHz depending on the donor configuration in the quantum dot. Thus, the frequency at which the electron qubit operates varies for different nuclear spin configurations. For example, a 2P quantum dot has four nuclear spin configurations, namely

[0022]

Number

[0023] exists. Here, the first arrow indicates the nuclear spin state of the first donor 202A, and the second arrow indicates the nuclear spin state of the second donor 202B, or vice versa. This variability in the qubit operating frequency in the multi-donor quantum dot poses an important challenge for single qubit operation of electron spins.

[0024] An important component of a scalable quantum computer is the ability to individually address qubits and apply quantum gates. However, since quantum gates typically require a mechanism for discrimination in the frequency domain, the qubit addressing ability has been a challenge for spin qubit devices.

[0025] One particular aspect of qubit addressing ability is the initialization protocol. Preparing the qubit in a well-defined initial state is one of the important requirements for performing any quantum computation. Existing multi-qubit initialization methods for donor quantum dot devices are inefficient and may be difficult to scale up.

[0026] One option for controlling the nuclear spin of the donor atom is to use nuclear magnetic resonance (NMR). In particular, if the nuclear spins are individually addressable, NMR enables control over the nuclear spins. Thus, NMR can be used to initialize the nuclear spins into any desired configuration. However, NMR brings limitations and requires more hardware.

[0027] NMR operates in a very different frequency band from ESR (10 - 100 MHz as opposed to 20 - 40 GHz for ESR). For this reason, a transmission line (not shown) for sending signals is required for broadband operation. Thus, this requirement for broadband operation results in performance limitations because different devices for the radio frequency range are needed. A further disadvantage of using NMR includes limitations on the devices for sending radio frequency (RF) signals, and this disadvantage is usually resolved by using two separate devices, thus also resulting in hardware overhead here.

[0028] Another prior art is the use of post - selection. Here, a single frequency corresponding to one nuclear spin configuration is used to address the electron qubits. The measurements are repeated multiple times and the measurement results are averaged. Then the data is filtered, and for one nuclear spin configuration corresponding to the selected single qubit frequency, all measurements made when the nuclear spin is not in the correct configuration are discarded. This avoids the requirement of initializing the quantum system to a desired initial state. However, this post - selection method is very inefficient. The post - selection method results in at least 2 m times the time overhead, where m is the number of donors in the system.

[0029] Another method used is frequency - division multiplexing. This method addresses the qubits using the frequencies corresponding to each nuclear spin configuration instead of addressing the qubits at one frequency. Thus, the electron qubits can be addressed regardless of the initial nuclear spin configuration. Effective qubit operation gains benefits by maximizing the available RF drive power, so this method is also very inefficient. However, the total RF power results in a heating effect and is thus limited by the cooling power of the refrigerator used to hold the device. For this reason, with division multiplexing, 2 -mOnly the

[0030] Aspects of the present disclosure propose and demonstrate an initialization protocol for multi-donor spin qubits in semiconductors that can be used for quantum computing in current and future implementations of quantum processing / computing systems. The initialization protocol according to aspects of the present disclosure can initialize the m+1 spins of the multi-donor dot system to a specific state as a requirement for any computational step. This initialization protocol offers several advantages over known techniques. By being able to initialize the nuclear configuration to a specific configuration with high fidelity by a series of pulses within the ESR frequency band, it is possible to perform efficient operation of the electron qubit at a single frequency using maximum drive power without hardware overhead and with minimal time overhead.

[0031] According to some other aspects of the present disclosure, a mechanism for controlling the Rabi frequency is provided so that qubit initialization and control can be performed faster and more efficiently than in techniques known in the past. In particular, aspects of the present disclosure control the Rabi frequency by changing the angle of the electric field applied to the quantum device.

[0032] Electron Spin Resonance (ESR) and EDSR in Multi-Donor Quantum Dots FIG. 3A shows an energy level diagram for an example of a 2P quantum dot.

[0033] For a 2P quantum dot, there are 2 2 =4 possible nuclear spin configurations, namely

[0034]

Number

[0035] There exists. For a 2P quantum dot having one electron, there are eight total spin states. They are

[0036]

Number

[0037] where the first single - arrow indicates the electron spin. For a general multi - donor quantum dot system having m donors and one shared electron, there are a total of 2×2 m total spin states. In FIG. 3A, eight total spin states arranged by relative energy are shown. The four total spin states corresponding to down - spin electrons are lower in energy than the four total spin states corresponding to up - spin electrons.

[0038] FIG. 3A also shows eight possible transitions between spin states driven by electron spin resonance (ESR) or electron - dipole spin resonance (EDSR). In particular, there are four possible EDSR transitions labeled EDSR1 - EDSR4 and four possible ESR transitions labeled ESR1 - ESR4.

[0039] ESR is a direct means for driving an electron between its two spin states. In the presence of an external magnetic field B0, the spin - energy levels of the electron are no longer degenerate. The two spin states |↑〉, |↓〉 are separated by an energy difference ΔE. Thus, by applying an AC magnetic field, the electron spin can be changed from the spin - down state |↓〉 to the spin - up state |↑〉, or vice versa. ESR is due to the coupling of the electron's intrinsic magnetic moment to the external magnetic field B0.

[0040] In particular, ESR is a transition between opposite electron spin states but with the same nuclear spin configuration. For example, the four ESR transitions in the 2P system are as follows.

[0041]

Number

[0042] On the other hand, EDSR in the donor system is due to the modulation of the hyperfine coupling between the electron spin and the nuclear spin of the donor atom in the system. EDSR is mediated by an electric field that simultaneously flips the electron spin and one of the nuclear spins in a multi-donor system. For the 2P system, there are four possible EDSR transitions as shown in FIG. 3A.

[0043] FIG. 3B shows the frequencies of the ESR and EDSR spin-up transitions of FIG. 3A for a 2P quantum dot. For example, to excite a 2P quantum dot system from spin state

[0044]

Number

[0045] to spin state

[0046]

Number

[0047] a single frequency corresponding to, for example, an ESR frequency (ESR4) of about 40.7 GHz may be applied to gate 208. This ESR frequency (ESR4) is shown as the vertical peak 302D in FIG. 3B. The ESR frequency and the EDSR frequency are proportional to the applied magnetic field and can be varied over a wide range.

[0048] The EDSR pulse may be generated by a single tone similar to ESR using an on-chip microwave antenna (gate 208) or by using a gate electrode (210 or 212). The EDSR transition is an electron-nuclear flip-flop transition that can occur via modulation of the hyperfine interaction between the nuclear spin and the electron spin. This modulation of the hyperfine interaction may be achieved by applying an electric field that shifts the electron wave function away from the donor nucleus.

[0049] To drive one of the EDSR transitions, it is necessary to modulate the hyperfine interaction at a frequency corresponding to the energy between two allowed states. For example, Figure 3A shows four EDSR transitions, each having a different corresponding drive frequency indicated by peaks 1, 2, 3, and 4 in Figure 3B. Figure 3B also shows four ESR peaks, and each ESR transition corresponds to a different drive frequency indicated by peaks 302A, 302B, 302C, and 302D.

[0050] Figure 4A shows the EDSR energy diagram for a 3P quantum dot. For the 3P quantum dot, there are a total of 2 3 = 8 nuclear spin configurations and a total of 16 total spin states. For the example of the 3P quantum dot, the hyperfine coupling for the first nuclear spin is 201 MHz, the hyperfine coupling for the second nuclear spin is 77 MHz, and the hyperfine coupling for the third nuclear spin is 42 MHz. The eight nuclear spin states are indicated by relative energy levels. The bottom row of the states corresponds to the eight nuclear spin states with spin-down electrons, and the top row corresponds to the eight nuclear spin states with spin-up electrons. There are 12 EDSR transitions (1 - 12), which are shown as lines connecting the upper and lower states (EDSR transitions 5, 6, 7, and 8 flip the spin state of the first nuclear spin flip, EDSR transitions 2, 3, 10, and 11 flip the spin state of the second nuclear spin, and EDSR transitions 1, 4, 9, and 12 flip the spin state of the third nuclear spin flip). There are also eight ESR transitions (not shown) in Figure 4A. The ESR transitions flip only the electron spin and leave the nuclear spin unaffected. Thus, similar to Figure 3A, the ESR transitions connect the lower state to the upper state directly above it.

[0051] To individually address the three nuclear spins, different NMR frequencies are required. The NMR frequencies are related to the hyperfine interaction by the following equation.

[0052] [Number]

[0053] Here, the nuclear Zeeman depends on the static magnetic field, and ± depends on whether the electron spin is up or down. The addressing frequency of each individual nuclear spin depends on its environment. The frequency for addressing nuclear spins is in the MHz range, while the EDSR and ESR frequencies are in the GHz range. Therefore, the requirements for broadband operation by a transmission line are not needed here. Figure 4B shows a plot of the EDSR spectrum for a 3P quantum dot (the ESR spectrum is not shown). The x-axis is the frequency in units of GHz, and the y-axis shows the probability of spin-up of the nuclear spin. The frequencies of all 12 EDSR transitions for the electron spin-up state are shown. All EDSR transitions (1 - 12) are on the order of dozens of GHz, more specifically in the range of 40 - 41 GHz. The ESR transitions are also in the range of 40 - 41 GHz but are not shown. Therefore, a narrow frequency range is required to operate both ESR and EDSR. Thus, there are no specific performance requirements for the transmission line, and no additional frequency generation equipment is needed.

[0054] This EDSR drive protocol may be applied to an mP quantum dot or any multi-donor quantum dot having up to 10 donor atoms.

[0055] Initialization Protocol In accordance with aspects of the present disclosure, an ultra-fine based EDSR may be used to drive electron-nuclear transitions in multi-donor qubits in silicon. The initialization protocol disclosed herein is a deterministic high-fidelity initialization protocol for nuclear spin configurations. Thus, this initialization protocol obviates the need for NMR, frequency division multiplexing, or post-selection for electron qubit operations in order to enable quantum gate operations on donor electron qubits at a single frequency. In summary, EDSR-based polarization constitutes a powerful tool for the operation of multi-qubit systems within a single quantum dot required for qubit count scaling.

[0056] FIG. 5A is a flowchart illustrating an example of an initialization protocol 500A according to aspects of the present disclosure. The initialization protocol 500A consists of a series of EDSR pulses interleaved with at least one ESR pulse to bring about a target state. In one example, the target state may be a total spin-down state. This total spin-down state may then be used as an initial state for quantum measurement or qubit operation. For example, the initial target state in a 2P quantum dot system may be

[0057]

Number

[0058] a state. In other examples, the target state may be any spin state of a multi-donor quantum dot system.

[0059] Method 500A begins at step 502, where an initialization pulse sequence is determined depending on the target state. This initialization pulse sequence is independent of the initial state.

[0060] For the example of a 2P quantum dot system, the following are examples of pulse sequences for achieving different predetermined target states. Refer to FIG. 3A.

[0061]

Table 1

[0062] Here, the ESR for 2P quantum dots ALL includes the following ESR transitions: ESR1, ESR2, ESR3, and ESR4. Refer to Figure 3A. The ESR ALL pulse may be executed in two ways. The ESR ALL One way to execute the pulse is to continuously apply all of the ESR1, ESR2, ESR3, and ESR4 pulses. The ESR ALL Another way to execute it is to simultaneously apply the ESR pulses using frequency division multiplexing (a standard RF technique). This method may save time. The ESR for a multi-donor quantum dot system ALL will be recognized to include all ESR transitions.

[0063] Next, in step 504, electrons are deterministically loaded into the multi-donor quantum dots in the spin-down state. In some examples, electrons are loaded into the mP quantum dots by applying a voltage to the gate electrodes. The voltage may be applied to at least one of the surface gate electrodes 208 and 210 of Figure 2A, or may be applied via the in-plane gate electrode 212 of Figure 2B or Figure 2C.

[0064] In an example of the 2P system, the electron 220 is deterministically loaded in the spin-down state. Therefore, this system

[0065]

Number

[0066] may be one of the four combined spin states of. Refer to Figure 3A. For the initialization protocol 500A, it is not necessary to determine the initial state.

[0067] In some examples, multiple electrons may be loaded into a multi-donor quantum dot system, and the spins of the unpaired electrons can be used as electron spin qubits in the device.

[0068] Next, in step 506, a 0 or 1 electron spin reset pulse is executed. When a 0 electron reset pulse is required, nothing needs to be done, and this method may proceed to step 508. The electron spin reset pulse may be executed using an ESR pulse, or by unloading and loading electrons in the quantum dot, or alternatively the electron spin reset pulse may be executed by applying a voltage to the gate electrode. To drive an ESR transition using the electron spin reset pulse, an RF signal is applied to at least one of the gate electrode or the transmission line. The ESR transition flips the electron spin while leaving the nuclear spin unchanged. For example,

[0069]

Number

[0070] from the spin state of

[0071]

Number

[0072] for the transition to the spin state of, ESR2 in FIG. 3A can be used.

[0073] Next, in step 508, an RF signal is applied to the gate electrode or the transmission line to drive at least one EDSR transition. The applied signal corresponds to an EDSR transition that brings the total spin state closer to the target state. The EDSR transition is driven to perform a controlled SWAP gate from spin-up to spin-down or vice versa for one of the nuclear spins, while simultaneously slipping the electron spin.

[0074] For example, if the initial state is the lowest energy spin state

[0075]

Number

[0076] then the EDSR signal may correspond to an EDSR4 transition to bring about the

[0077]

Number

[0078] state. Here, the EDSR4 transition drives a controlled SWAP gate for the second nuclear spin and electron spin. In another example, if the initial state is the lowest energy spin state

[0079]

Number

[0080] then the RF signal applied to the gate electrode or transmission line may

[0081]

Number

[0082] correspond to an EDSR3 transition to bring about the state. Here, the EDSR3 transition drives a controlled SWAP gate for the first nuclear spin.

[0083] Next, in step 510, steps 506 and 508 are repeated N times in order to achieve the target state with high probability. For this reason, due to the high fidelity of the process and the very high probability that the EDSR signal(s) achieve the target state, the target state is achieved. Here, N is any integer greater than or equal to 1. In principle, steps 506 and 508 may be repeated for the required number of times. Since the device can be designed to have a large EDSR efficiency and thus high fidelity, it should also be possible to achieve initialization with N = 1.

[0084] Method 500A operates without requiring any feedback during the process, i.e., it is not necessary to measure the spin state during the initialization protocol, so this is an unconditional initialization protocol. For this reason, for any mP quantum dot system, a robust process with high-probability target state achievement may be brought about by only a subset of the total number of transitions. For example, in the 2P quantum dot system of step 510, the total spin-down target state is achieved and this system is initialized. Now the qubit device can perform quantum operations. The probability of achieving the target state may be increased by the device design and it may be very efficient (probability of about 1).

[0085] For example, if the target state is

[0086]

Number

[0087] then, in step 502, an initialization pulse sequence for achieving this target state is determined. From Table A, the initialization sequence is as follows. i. Electron reset pulse ii. EDSR4 iii. EDSR1 iv. Electron reset pulse v. EDSR2 vi. Electronic reset pulse

[0088] It will be recognized that the pulse sequence corresponding to each target state is designed to reach that target state regardless of the initial state. For example, if the initial state is

[0089]

Number

[0090] then, using method 500A, in step 502, the initialization pulse sequence is determined depending on the target state being

[0091]

Number

[0092] For example, refer to the above pulse sequences i - vi.

[0093] Next, in step 504, an electron is loaded into the 2P quantum dot in the spin - down state. Next, in step 506, a 0 - electron reset is performed because the electron is in the spin - down state. Next, in step 508, an RF signal is applied to drive the EDSR4 and EDSR1 transitions. In this case, the EDSR4 transition is appropriate / valid. If the state is

[0094]

Number

[0095] then the EDSR1 transition would be appropriate. The spin state at the end of step 508 is

[0096]

Number

[0097] It is. Next, in step 510, steps 506 and 508 are repeated N times to achieve the target state with high probability. Thus, returning to step 504, an electronic reset pulse is executed. The electron spin reset pulse is executed by using an ESR transition, unloading and loading electrons, or applying a voltage to the gate electrode. The spin state here is

[0098] [Number]

[0099] It is. Next, in step 508, an RF signal is applied to drive the EDSR2 transition, and the spin state

[0100] [Number]

[0101] is brought about. Returning to step 506 again, an electron spin reset pulse is executed, and the final target state

[0102] [Number]

[0103] is brought about.

[0104] Similarly, using this example of the initialization pulse sequence, any initial spin state can be robustly converted to the target state

[0105] [Number]

[0106] with robustness.

[0107] FIG. 5B is a flowchart showing an alternative deterministic series of method steps 500B for an example of an initialization protocol according to an aspect of the present disclosure.

[0108] Method 500B begins at step 520, where electrons are deterministically loaded into the multi-donor quantum dot in a spin-down state. In some examples, electrons are loaded into the mP quantum dot by applying a voltage to a gate electrode. The voltage may be applied to at least one of the surface gate electrodes 208, 210 of FIG. 2A or via the in-plane gate electrode 212 of FIG. 2B.

[0109] Next, at step 522, an ESR measurement may be performed to determine whether the system is in the target state. When the spin state is determined to be the target state, then at step 524, the nuclear spin of the multi-donor quantum dot system is initialized.

[0110] For example, at step 522, an ESR measurement is performed on the 2P quantum dot, and the result of the ESR measurement is that the total spin configuration is

[0111]

Number

[0112] in the state of. If this is the target state, then the nuclear spin is initialized and method 500B ends at step 524.

[0113] However, if the overall spin state determined in step 522 is not the target state, then method 500B proceeds to step 526. In step 526, an initialization pulse sequence that will convert the initial state to the target state is calculated. The initialization pulse sequence includes ESR transition(s) / electron spin reset pulse(s) and EDSR transition(s). The initialization pulse sequence corresponds to a sequence of ESR and EDSR transitions that convert a known initial state to a known target state, such as the transitions shown in FIGS. 3A and 4A.

[0114] Returning to the example of the 2P quantum dot system, if the overall spin state is

[0115]

Number

[0116] determined to be and this is not the target state, then in step 526 an initialization pulse sequence including ESR, EDSR, and electron spin reset pulses is calculated and then executed.

[0117] For the example of the 2P quantum dot system, the target state may be the

[0118]

Number

[0119] overall spin-down state. Thus, a pulse sequence is calculated to effect the

[0120]

Number

[0121] conversion of the initial state to.

[0122]

Number

[0123] One possible initialization pulse sequence for performing the conversion is as follows. Refer to Figure 3A.

[0124]

Number

[0125] Once this initialization pulse sequence is calculated, it is executed. To drive the EDSR transition, an RF signal is applied to a gate electrode (e.g., gates 208, 210, and / or 212) or a transmission line. In this example, the applied signal corresponds to an EDSR transition that brings the overall spin state closer to the overall spin-down state. The EDSR transition is driven to perform a controlled SWAP gate on one of the nuclear spins.

[0126] By performing an ESR transition, an electron spin reset pulse may be executed. Alternatively, an electron spin reset pulse may be executed by unloading and loading electrons in a quantum dot. Alternatively, an electron reset pulse may be executed by applying a voltage to a gate electrode. In one example, step (i) may be performed using an ESR transition, step (iii) may be performed by unloading and loading electrons in a multi-donor quantum dot, and step (v) may be performed by applying a voltage pulse to a gate electrode. In other examples of the initialization pulse sequence, electron reset may be performed using different methods and combinations thereof.

[0127] After the initialization pulse sequence is executed, method 500B proceeds back to step 522, where an ESR measurement is performed to determine the overall spin state and whether the target state has thereby been achieved.

[0128] Returning to the example of the 2P quantum dot, an ESR measurement is performed at step 522, and the total spin state is

[0129]

Number

[0130] determined to be. In this example, since this is the target spin state, the method ends at step 524 and the nuclear spin is initialized.

[0131] After completion of the initialization protocol, the multi-donor quantum dot system is initialized and becomes available for qubit operations. Methods 500A and 500B are described as starting from an initial state

[0132]

Number

[0133] but this protocol succeeds regardless of the initial spin state.

[0134] In some examples, the initialization protocol 500B may be executed multiple times to ensure that the state is initialized with high fidelity. Thus, this initialization protocol is robust against errors.

[0135] In another embodiment, this initialization protocol may be used for a quantum processing system including a plurality of multi-donor quantum dots.

[0136] Experimental Results Figures 6A and 6B show the experimental results for a first example of 2P and 3P quantum dot systems, respectively. The ESR peak amplitude measured over time is a direct measure of the efficiency of the initialization protocol. When the initialization protocol is not applied, the system is in any one of the nuclear spin configurations with a similar probability.

[0137] Figure 6A shows the ratio of electron spin-up as a function of the nuclear spin configuration for a first example of a 2P quantum dot system. The first example of the 2P quantum dot system has the following hyperfine interactions. A1 = 65 MHz and A2 = 103 MHz. There are four peaks at specific frequencies corresponding to the four possible nuclear spin configurations for this 2P quantum dot system. The height of each peak corresponds to the probability that the system is in that nuclear spin configuration.

[0138] Figure 6B shows the ratio of electron spin-up as a function of the nuclear spin configuration for a first example of a 3P quantum dot system. This 3P quantum dot system has the following hyperfine interactions. A1 = 42 MHz, A2 = 77 MHz, and A3 = 201 MHz. There are eight peaks at specific frequencies corresponding to the eight possible nuclear spin configurations for the 3P quantum dot system. The height of each peak corresponds to the probability that the system is in that nuclear spin configuration.

[0139] It is expected that the successful initialization protocol will cause a significant suppression of all but one of the ESR peaks. The target state for the example of the 2P quantum dot

[0140]

Number

[0141] This result when using an initialization protocol such as those disclosed herein to initialize to the target state for the example of the 2P quantum dot is shown in FIGS. 7A and 7B.

[0142] Figure 7A shows the ESR measurement after executing the initialization protocol 500A for a first example of a 2P quantum dot system, showing the frequency in GHz versus the measured ratio of electron spin-up. Thus, this 2P quantum dot system is in a nuclear spin state

[0143]

Number

[0144] It can be seen that

[0145] Figure 7B shows the ESR measurement after executing the initialization protocol 500A for the first example of the 3P quantum dot system, and shows the frequency in GHz with respect to the measured ratio of electron spin-up. This 3P quantum dot system has a nuclear spin state

[0146]

Number

[0147] It can be seen that

[0148] Figure 7C is a plot showing the peak amplitude by EDSR for the 3P quantum dot. In particular, this plot shows the ratio of spin-up on the y-axis and the frequency of the EDSR pulse on the x-axis. BG in the figure shows the background measurement, which is intentionally applied at a frequency away from any resonance to see how the measurement changes over time and as a reference for the height of different peaks.

[0149] In the case of 3P, since the number of EDSR transitions (1 - 8) is large, there may be several different efficient initialization protocols. Three different initialization pulse sequences are shown here, which are 702 (for large hyperfine spins), 704 (for intermediate hyperfine spins), and 706 (for small hyperfine spins), and all of these result in similar results of efficient initialization of the first

[0150]

Number

[0151] peak. These three pulse sequences bring about

[0152]

Number

[0153] It corresponds to the same pulse sequence as that used to initialize to the state. This pulse sequence involves the initialization of different nuclear spins. For example, the pulse sequence 702 is for qubits that are initially in a state with a large hyperfine value, the pulse sequence 704 is for qubits that are initially in a state with an intermediate hyperfine value, and the pulse sequence 706 is for qubits that are initially in a state with a small hyperfine value.

[0154] After the nuclear spin configuration is initialized using the initialization protocol disclosed herein, the electron qubits become addressable at a single frequency. The initialization protocol results in stable electron qubits and enables coherent control of the qubits.

[0155] By applying an AC electric field to the gate electrode, the electron qubits

[0156]

Number

[0157] can be driven between. This results in Rabi oscillations that constitute the oscillatory behavior of a two-level system.

[0158] Figures 8A and 8B show the results of Rabi - Shevron experiments for the first examples of 2P and 3P quantum dot systems, respectively. The x - axis is the duration and the y - axis is the frequency. For the 2P quantum dot system, coherent Rabi oscillations are observed at approximately 39.0826 GHz (indicated by the brighter - colored part of the plot). For the 3P quantum dot system, coherent Rabi oscillations are observed at approximately 40.40225 GHz.

[0159] Figure 9A shows the EDSR energy diagram for a first example of a 3P quantum dot system. As discussed above with reference to Figure 4A, there are 12 EDSR transitions (1 - 12) for the 3P quantum dot, which are shown as lines connecting the upper and lower states. These EDSR transitions correspond to the angular momentum that conserves the electron - nuclear flip - flop transition. There are also 8 vertical ESR transitions (not shown).

[0160] To individually address each of the three nuclear spins I1, I2, and I3, different NMR frequencies are required. These frequencies were directly determined from the measured ESR spectrum for the 3P quantum dot. As mentioned above, the addressing ability frequencies of the individual nuclear spins depend on their environment. Thus, these NMR frequencies are different from those required for the 3P device in Figure 4A. Four of these transitions (EDSR transitions 4, 5, 8, and 9) correspond to the flip of the first nuclear spin I1, where the second and third nuclear spins remain unchanged, for example

[0161]

Number

[0162] and so on. Four of these transitions (EDSR transitions 2, 3, 10, and 11) correspond to the flip of the second nuclear spin I2, where the first and third nuclear spins remain unchanged. The remaining four transitions (EDSR transitions 1, 6, 7, and 12) correspond to the flip of the third nuclear spin I3, where the first and second nuclear spins remain unchanged.

[0163] Figure 9B shows a plot of the measured EDSR spectra corresponding to the 12 different transition frequencies for the first example of the 3P quantum dot. The x - axis shows the frequency in GHz, and the y - axis shows the proportion of electron spin - up. The central frequency values for each of the 12 EDSR transitions (1 - 12) are shown. Thus, Figure 12B shows the frequencies in GHz that drive each of the 12 EDSR transitions. Each of the 12 EDSR transitions is labeled at the top of the plot.

[0164] To characterize the 3P quantum dots, an EDSR spectrum is measured. To ensure that all EDSR transitions are captured in this measurement, a pulse sequence that randomizes the nuclear configuration is introduced. First, electrons with random spin states are loaded into the 3P quantum dots by rapidly pulsing the output across the 0→1 charge transition line of the quantum dots. Next, an electron-nuclear SWAP gate is executed by applying four EDSR transitions corresponding to the same nuclear spin. This drives the nuclear spin into a random initial state. For example, applying EDSR transitions 4, 5, 8, and 9 affects the first nuclear spin state. Then this process is repeated for the remaining nuclear spins. For example, an electron-nuclear SWAP gate is executed by applying four EDSR transitions that affect the second nuclear spin, namely EDSR transitions 2, 3, 10, and 11. Similarly, EDSR transitions 1, 6, 7, 12 are applied to execute another electron-nuclear SWAP gate for the third nuclear spin. After executing this pulse sequence, the nuclear spin configuration is randomized and initialized.

[0165] Next, using the voltage applied to the gate, the electrons are re-initialized to the spin-down state |↓〉. Then an adiabatic inversion EDSR pulse is applied, where the frequency of the pulse is adiabatically swept around the expected EDSR frequency. This flips the electrons into the conditional |↑〉 manifold with respect to the random initial nuclear spin configuration. The measured EDSR spectrum shown in Figure 12B is consistent with the calculated EDSR values determined using the hyperfine couplings obtained from the ESR measurement.

[0166] The EDSR initialization protocol relies on a controlled SWAP gate between the electron and nuclear spins and is realized using an adiabatic inversion pulse. As a result, the achieved initialization fidelity strongly depends on the efficiency of the adiabatic inversion pulse, i.e., the efficiency of flipping the conditional electron spin with respect to the initial nuclear spin configuration. To characterize the inversion efficiency, a Landau-Zener interferometry is performed for three EDSR transitions (one transition for each nuclear spin). In this experiment, EDSR transitions 9, 11, and 12 are used. First, the system is

[0167] [Number]

[0168] initialized to the state of. Next, an adiabatic inversion pulse in the target EDSR transition is applied. In particular, the frequency is linearly swept (chirped) around the transition frequency. For example, refer to the frequencies shown for transitions 9, 11, and 12 in Figure 12B. The inversion rate is given by the following Landau-Zener formula.

[0169] [Number]

[0170] Furthermore, the inversion rate strongly depends on the speed of the frequency sweep ω / τ (τ is the inversion pulse duration and ω is the frequency width of the sweep) compared to the drive strength f r (expressed here in terms of the corresponding Rabi frequency).

[0171] Figure 9C shows the initial state

[0172] [Number]

[0173] Shows the probability of inversion for the EDSR transitions 9, 11, and 12 corresponding to the flips of the first, second, and third nuclear spins, respectively, when starting from r,i . The x-axis represents the pulse duration (τ) in milliseconds (ms), and the y-axis represents the probability of electron spin-up. The probability of electron spin-up was measured at a microwave output power of 0 dBm. The dashed lines 902A, 902B, and 902C are the fits to Equation 1 for each of the three EDSR transitions 9, 11, and 12, respectively. From the fitted data, the corresponding Rabi frequency f r,i can be obtained. The Rabi frequency for the EDSR transition 9 is 6.47 kHz, for the EDSR transition 11 is 23.78 kHz, and for the EDSR transition 12 is 11.12 kHz.

[0174] Figure 9D is a plot showing the dependence of the Rabi frequency on the square root of the microwave power for each of the three EDSR transitions 9, 11, and 12. The x-axis is the square root of the power in units of m√W. The y-axis is the Rabi frequency in kHz. For each EDSR transition, the Rabi frequency depends linearly on the square root of the power of the microwave output. The slopes of the lines 908A (corresponding to transition 9), 908B (corresponding to transition 11), and 908C (corresponding to transition 12) are different for each nuclear spin. This linear dependence is expected.

[0175] Based on the practical assumption that the microwave-induced electric field (E) is equal for all three transition frequencies and donor positions, the different slopes represent that the dipole moments of the individual donors are different, coupling them to different strengths of the electrical drive. The coupling strength is given primarily (To first order) by the amplitude of the Stark shift caused by the electric field, i.e., δA i =η i E. The Stark shift is the splitting of the energy levels by an external electric field. Here, η i is the Stark coefficient, which is a measure of how the electric field E splits for each hyperfine coupling.

[0176] Using atomic tight-binding modeling, the Stark coefficients for each donor nuclear spin may be determined. The Stark coefficients for each donor nuclear spin are shown in the inset 906. These coefficients are combined with the measured Rabi frequency f r,i to constrain the direction and amplitude of the AC electric field at the qubit location. Thus, both the direction and amplitude of the AC electric field may be obtained. In this example, the amplitude of the AC electric field is |E| = 33.6 ± 0.5 kV / m (at 0 dBm). This is consistent with the electric field simulated using finite element modeling of an antenna, such as gate 208 in FIG. 2A. In some examples, the AC electric field may be applied through an in-plane gate, such as gate 212 in FIG. 2B or FIG. 2C.

[0177] Future redesign of the device architecture may apply microwave pulses to in-plane gate electrodes to induce stronger and more local fields, enabling faster and still more coherent operation.

[0178] Finally, the optimal pulse duration may be determined while addressing the trade-off between device heating and the pulse duration contributing to the overhead of the initialization protocol, using the Landau-Zener calibration curve (FIG. 9C).

[0179] In one example, it was observed that more frequent device instabilities were also observed when the microwave power exceeded 0 dBm. Additionally, the pulses and pulse durations contribute to the overhead of the initialization protocol. At 0 dBm, the inversion rates of I2 and I3 approach 1 (unity) for τ > 4 ms, but only about 70% for I1. Thus, in this example, τ = 5 was selected for the actual initialization sequence to avoid an extremely long sequence duration.

[0180] FIGS. 10A and 10B are respectively

[0181]

Number

[0182] shows the initialization sequence for the nuclear spin state. FIGS. 10A and 10B also show the measured ESR spectra after applying the initialization sequence.

[0183] Starting from a random nuclear configuration, the nuclear spin configuration

[0184]

Number

[0185] is initialized. To do this, the electron spin is initialized to the spin-down state |↓〉, and four EDSR pulses are applied. In this example, the EDSR pulses corresponding to EDSR transitions 4, 5, 8, and 9 are applied. This sequence transfers the spin-down polarization from the electron to the nuclear spin I1 without conditioning on the states of the other two nuclear spins. The electron spin initialization is repeated, i.e., the electron is flipped from |↑〉 to |↓〉. Then, the EDSR pulses corresponding to EDSR transitions 2 and 3 are applied to initialize spin I2 (conditioned on spin I1 already

[0186]

Number

[0187] being in that state). To end the nuclear initialization sequence, the electron is initialized again, and the EDSR pulse corresponding to EDSR transition 1 is applied to initialize spin I3 (conditioned on spins I1 and I2 already

[0188]

Number

[0189] (conditioned to be). This sequence of pulses is shown in the inset 1002 of FIG. 13A. After completion of this initialization process using EDSR, the ESR spectrum may be measured. FIG. 13A shows the measured ESR spectrum. The x-axis indicates the frequency in GHz and the y-axis indicates the ratio of electron spin-up. The measured ESR spectrum has a main peak at the lower end of the shown frequency region. This peak corresponds to the ESR transition when the nuclear spin is

[0190] [Number]

[0191] Thus, this ESR spectrum confirms that the initialization to the predetermined

[0192] [Number]

[0193] of the nuclear spin has been successful.

[0194] The initialization process for initializing the nuclear spin to the spin-up state

[0195] [Number]

[0196] is similar, but here the electrons are initialized to the spin-up state |↑〉 before each EDSR step. The pulse sequence is shown in the inset 1004 of FIG. 10B. In some examples, the electrons may be initialized to the spin-up state |↑〉 by applying ESR inversion pulses simultaneously to all eight ESR frequencies.

[0197] Figure 10B shows the measured ESR spectrum after the initialization process has been executed. The x-axis indicates the frequency in GHz, and the y-axis indicates the ratio of electron spin-up. The measured ESR spectrum has a main peak at the upper end of the frequency region shown. This peak corresponds to the ESR transition when the nuclear spin is in the

[0198]

Number

[0199] state. Therefore, this ESR spectrum confirms that the initialization of the nuclear spin to the predetermined

[0200]

Number

[0201] has been successful.

[0202] To further investigate and quantify the initialization fidelity, a series of repeated randomized initialization sequences interleaved with nuclear state readout to determine the nuclear configuration before and after initialization were performed. By interleaved here, it means that the readout is performed after the initialization sequence has been executed, and this may be repeated multiple times. First, the nuclear spin configuration of the 3P quantum dot is randomized. Next, the inversion and readout of the electron spin at each of the 8 ESR frequencies are performed, and the first nuclear readout sequence is executed by averaging 60 times. This is followed by l initialization attempts, and then the second nuclear readout sequence follows. By knowing the nuclear configuration before and after initialization, it becomes possible to use a Markov chain model to construct the normalized transition probability matrix M. Then, the initialization fidelity from the initial state i to the final state j is given by the matrix element M j,i as given.

[0203] Figure 10C starts from the initial state (i) to the target state

[0204]

Number

[0205] Indicates the initialization fidelity that transitions to. The x-axis is the number l of initialization trials in the range of 0 to 10. The y-axis shows the initialization fidelity. In particular, for each of the seven possible initial states (excluding the case where i = j, i.e., the initial state is already the target state), starting from each one, the target state

[0206]

Number

[0207] The initialization fidelity for the target state is shown as a function of the number l of initialization trials. Similarly, Figure 10D shows the target state starting from each of the seven possible initial states (excluding the case where i = j, i.e., the initial state is already the target state)

[0208]

Number

[0209] Shows the initialization fidelity for the target state as a function of the number l of initialization trials.

[0210] These curves are normalized by the probability of starting from the target state and remaining there (when i = j), which removes approximately 5% of the state readout and electron initialization errors. The seven different initial configurations can be divided into two categories defined by the state of the nuclear spin i1. From Figure 12C, it is observed that I1 has the lowest inversion rate because η1 is smaller.

[0211] As a result, from both Figure 10C and Figure 10D, the fidelity for the flip of I1 is lower for l < 5. The overall effective initialization fidelity may be determined by averaging different initial nuclear spin configurations (black dashed line).

[0212]

Number

[0213] The overall effective initialization fidelity for both is, for repetitions with l > 5

[0214]

Number

[0215] and is per initialization shot

[0216]

Number

[0217] It has been determined that this is the case. By optimizing the inversion efficiency of EDSR to perform higher-fidelity electron spin initialization, the single-shot fidelity can be further improved to > 99%.

[0218] Figure 11A shows the probability of electron spin-up as a function of the nuclear spin configuration for a second example of a 2P quantum dot system, which has the following hyperfine interactions: A1 = 33 MHz and A2 = 77 MHz. The dots represent experimental data, while the solid lines represent the fit. The x-axis shows δf in MHz, and the y-axis represents the probability of spin-up. At specific frequencies corresponding to four nuclear spin configurations for the second example of the 2P quantum dot system, four spin-up probability peaks 902A - 902D exist. In particular, peaks 1102A, 1102B, 1102C, and 1102D occur when the nuclear spins are respectively

[0219]

Number

[0220] indicates the probability of being in the state. The height of each of the peaks 1102A - D corresponds to the probability that the system is in its nuclear spin configuration, and as can be seen from FIG. 11A, the heights of all four probabilities are approximately the same.

[0221] Before the initialization protocol is applied, the probabilities that the nuclear spin exists in any one of the four nuclear spin states are substantially equal.

[0222] FIG. 11B shows the ESR measurement after the initialization protocol according to method 500A has been executed for a second example of a 2P quantum dot system. In particular, FIG. 11B shows the probability of spin - up at different frequencies. The probability that the nuclear spin is

[0223]

Number

[0224] in the state has peak 1104, while the probability that the nuclear spin is in any other state is found to be approximately 0. Thus, the second 2P quantum dot system is found to be initialized to the nuclear spin state

[0225]

Number

[0226] is clear.

[0227] FIG. 12A shows the ratio of electron spin - up as a function of the nuclear spin configuration for a second example of a 3P quantum dot system, which has the following hyperfine interactions: A1 = 12 MHz, A2 = 15 MHz, and A3 = 119 MHz. At specific frequencies corresponding to the eight nuclear spin configurations for the second example of the 3P quantum dot system, eight peaks exist. The height of each peak corresponds to the probability that the system is in its nuclear spin configuration. Before the initialization protocol is applied, the probabilities that the nuclear spin exists in any one of the eight nuclear spin states are substantially equal. Peak 1202 corresponds to the case where the nuclear spin is

[0228]

Number

[0229] indicates the probability of being in the state of.

[0230] Figure 12B shows the ESR measurement after the initialization protocol according to method 500A is executed for the second 3P quantum dot system, and the frequency in GHz is shown for the measured proportion of electron spin up. Therefore, the second 3P quantum dot system is in the nuclear spin state

[0231]

Number

[0232] is found to be initialized to. Refer to peak 1204.

[0233] Figure 13A shows the proportion of electron spin up as a function of the nuclear spin configuration for a 4P quantum dot system. This 4P quantum dot system has the following hyperfine interactions. A1 = 30 MHz, A2 = 68 MHz, A3 = 97 MHz, and A4 = 180 MHz. At specific frequencies corresponding to 16 nuclear spin configurations for the 4P quantum dot system, 16 peaks exist. The height of each peak corresponds to the probability that the system is in its nuclear spin configuration. Before the initialization protocol is applied, the probability that the nuclear spin exists in any one of the 16 nuclear spin states is substantially equal. Peak 1302 indicates that the nuclear spin is

[0234]

Number

[0235] in the state of.

[0236] FIG. 13B shows the ESR measurement after the initialization protocol according to method 500A has been executed for the 4P quantum dot system, with the frequency in GHz shown against the measured proportion of electron spin-up. Thus, it can be seen that the 4P quantum dot system has been initialized to the nuclear spin state

[0237]

Number

[0238] See peak 1304.

[0239] Electrical Control of Electron and Nuclear Spins Electrical control of the spin is desirable for qubit operation. An oscillating electric field may be applied locally or globally to the multi-donor quantum dot system. For example, an AC electric field may be applied via one or more surface gate electrodes such as gate 210 and / or gate 208 in FIG. 2A. In other embodiments, an AC electric field may be applied via an in-plane gate such as gate 212 in FIG. 2B. Electrical control of the nuclear spin can deterministically polarize the nuclear spin. This electrical control of the nuclear spin may then be used for many purposes. For example, performing two-qubit gate operations, generating a local magnetic field, etc.

[0240] The present inventors have discovered that the driving intensity of EDSR can be varied by changing the direction of the oscillating electric field. This is because the change in the direction of the electric field affects the coupling between the electron spin and the electric field, causing a change in the intensity of the EDSR signal required to initialize or control the corresponding mP qubit. Furthermore, the inventors have found that the driving intensity of EDSR for the mP qubit can be optimized at a specific direction of the electric field.

[0241] By deriving the Rabi frequency driven by EDSR, an estimation of the AC electric field radiated from the antenna at the dot position can be found. For example, an ultrafine-coupled donor-electron system such as a 3P system with electrons has a static spin Hamiltonian in the following form. H0 = ω n I z + ω e S z + AI·S (2) Here, ω n (ω e ) is the donor nucleus (electron) Larmor frequency, I(S) is the nuclear (electron) spin operator, and A is the hyperfine constant. EDSR causes a nuclear-electron spin flip-flop transition, and the relevant I z + S z = 0 subspace has the following Hamiltonian.

[0242]

Number

[0243] When an AC electric field is applied, it couples to the spin levels through the Stark effect. For a system with two or more donor atoms, due to the lack of spherical symmetry, a finite dipole moment is generated, and these systems have a linear Stark shift. Since the linear Stark shift for a small AC electric field is the main term, (I z + S z = 0 subspace, the AC electric drive can be expressed as follows.

[0244]

Number

[0245] Here, η is the linear Stark coefficient, and ε is the amplitude of the AC electric field. Shifting to the eigenbasis of H0 with the rotating wave approximation and selecting an appropriate energy origin, the full Hamiltonian of this system may be expressed as follows.

[0246]

Number

[0247] Here, ω0 is the intrinsic splitting of H0 and the approximation

[0248]

Number

[0249] is the intrinsic splitting. Therefore, the Rabi frequency of the drive is ηε / 2 at the resonance frequency.

[0250] Since the donors are close, the AC electric field is equal at all donor positions. This means that the Stark coefficient is proportional to the Rabi frequency, and the proportionality constant is the amplitude of the AC electric field.

[0251] For a small electric field ε, the hyperfine constant A i (ε) of donor i can be linearly described as follows. A i (ε)=A i (0)+ε▽ ε A i | ε=0 ⇒ΔA i (ε)=ε·▽ ε A i | ε=0 (6) Here, ΔA i (ε) is the hyperfine Stark shift of donor i, and ▽ ε A i | ε=0 =η i is defined as the Stark coefficient vector for donor i. Therefore, the electric field response to the electron density at the donor site depends on the direction of the field. When the field is

[0252]

Number

[0253] When along, the Stark coefficient is

[0254]

Number

[0255] as follows.

[0256] Therefore, by changing the angle of the applied AC electric field, the Rabi frequency for each nuclear spin configuration may be optimized. By doing so, the speed and efficiency of the initialization protocol may be improved.

[0257] Using an atomistic tight-binding modeling tool, the best-fit donor configuration was determined by changing its orientation under an electric field. Figure 14A shows the amplitude and direction of the AC electric field applied to an example of a 3P quantum dot.

[0258] Since the dot has a planar configuration, only dipoles along the xy plane are important, and thus the x and y components of the electric field are considered. The assumption of the linearity of the Stark shift is valid for an electric field with a height of 100 kV / m. These simulations show that for the electric field applied at an angle of vertical to 24.20° in Figure 14A, a Stark coefficient proportional to the Rabi frequency observed experimentally is given. From this, it was determined that the AC electric field amplitude is 33.6 kV / m at 0 dBm.

[0259] Figure 14B shows the Rabi frequency for an electric field of 33.6 kV / m. As shown in Table B, at the aforementioned angle of 24.2, the calculated Rabi frequency is in good agreement with the experiment.

[0260]

Table 2

[0261] Furthermore, while the quantum processing systems described herein are shown with gate electrodes and transmission lines for controlling corresponding qubits, these may not always be necessary. In other embodiments and examples, other control means may be used without departing from the scope of the present disclosure.

[0262] Accordingly, this embodiment should be considered illustrative and non-limiting in all respects.

[0263] Unless the context requires otherwise, the term "comprise" as used herein, and variations of that term such as "comprising," "comprises," and "comprised," are not intended to exclude further additions, components, integers, or steps.

[0264] Any reference to prior art in the specification does not admit or suggest that this prior art forms part of the common general knowledge in any jurisdiction, or that this prior art would be understood and regarded as relevant by a person skilled in the art, and / or could reasonably be expected to be combined with other prior art.

Claims

**Claim 1** A method for initializing a predetermined target spin state of multi-donor quantum dots in a semiconductor substrate, the method comprising: i) loading electrons in a spin-down state into the multi-donor quantum dots; ii) executing a 0 or 1 electron reset pulse; iii) applying an RF signal for driving at least one EDSR transition; and iv) repeating steps ii) to iii) N times to achieve the predetermined target spin state. **Claim 2** The method according to claim 1, further comprising determining an initialization pulse sequence depending on the predetermined target spin state before performing step (i), and applying the pulse sequence. **Claim 3** A method for initializing a predetermined target spin state of multi-donor quantum dots in a semiconductor substrate, the method comprising: i) determining an initialization pulse sequence depending on the predetermined target spin state, the initialization pulse sequence including at least one electron reset pulse and at least one EDSR transition; ii) loading electrons in a spin-down state into the multi-donor quantum dots; iii) applying the initialization pulse sequence; and iv) repeating step iii) N times to achieve the predetermined target spin state. **Claim 4** The method according to claim 2 or claim 3, wherein the initialization pulse sequence is configured to achieve the target spin state without depending on the initial spin state. **Claim 5** The method according to claim 4, wherein each target spin state has an associated initialization pulse sequence. **Claim 6** The method according to any one of claims 2 to 5, wherein the initialization pulse sequence is determined based on the initial spin state and the target spin state. **Claim 7** performing an ESR measurement to determine the overall spin state of the multi-donor quantum dots; and repeating steps (ii) to (iii) until the target spin state is achieved according to the determination that the overall spin state of the multi-donor quantum dots is not the target spin state. The method according to any one of claims 1 to 6, further comprising **Claim 8** A method for initializing a predetermined target spin state of multi-donor quantum dots in a semiconductor substrate, the method comprising: i) loading electrons in a spin-down state into the multi-donor quantum dots; ii) performing an ESR measurement and determining the total nuclear spin state according to a determination that the total nuclear spin state of the multi-donor quantum dots is not the target spin state; iii) determining and executing an initialization pulse sequence, the initialization pulse sequence including at least one ESR spin reset pulse and at least one EDSR transition; iv) performing an ESR measurement, determining the total nuclear spin state, ending the method according to a determination that the total nuclear spin state of the multi-donor quantum dots is the target spin state, and returning to step iii) according to a determination that the total nuclear spin state of the multi-donor quantum dots is not the target spin state. **Claim 9** The method according to any one of claims 1 to 8, wherein the target spin state is a total nuclear spin-down state. **Claim 10** The method according to any one of claims 1 to 9, wherein the RF signal for driving the at least one EDSR transition is applied to a transmission line. **Claim 11** The method according to any one of claims 1 to 9, wherein the RF signal for driving the at least one EDSR transition is applied to at least one gate electrode among a plurality of gate electrodes. **Claim 12** The method according to any one of claims 1 to 11, wherein the multi-donor quantum dots include at least two phosphorus atoms. **Claim 13** The method according to any one of claims 1 to 12, wherein the donor-based quantum dots are 2P quantum dots. **Claim 14** The method according to any one of claims 1 to 12, wherein the multi-donor quantum dots are 3P quantum dots. **Claim 15** The method according to any one of claims 1 to 12, wherein the multi-donor quantum dots have 4 to 10 phosphorus donor atoms. **Claim 16** The multi-donor quantum dots are 2P quantum dots, and the predetermined target spin state is the spin state 【Number 1】 wherein the initialization pulse sequence is i) An electronic reset pulse ii) EDSR4 iii) EDSR1 iv) An electronic reset pulse v) EDSR2 vi) An electronic reset pulse The method according to claim 5, comprising:

17. The multi-donor quantum dot is a 2P quantum dot, and the predetermined target spin state is the spin state 【Number 2】 wherein the initialization pulse sequence is i) An electronic reset pulse ii) ESR1, ESR2, ESR3, ESR4 iii) EDSR4 iv) EDSR1 v) An electronic reset pulse vi) EDSR3 vii) An electronic reset pulse The method according to claim 5, comprising:

18. The multi-donor quantum dot is a 2P quantum dot, and the predetermined target spin state is the spin state 【Number 3】 wherein the initialization pulse sequence is i) An electronic reset pulse ii) EDSR4 iii) EDSR1 iv) An electronic reset pulse v) ESR1, ESR2, ESR3, ESR4 vi) EDSR2 viii) An electronic reset pulse The method according to claim 5, comprising:

19. The multi-donor quantum dot is a 2P quantum dot, and the predetermined target spin state is the spin state 【Number 4】 wherein the initialization pulse sequence is i) An electronic reset pulse ii) ESR1, ESR2, ESR3, ESR4 iii) EDSR4 iv) EDSR1 v) An electronic reset pulse vi) ESR1, ESR2, ESR3, ESR4 vii) EDSR3 viii) An electronic reset pulse The method according to claim 3, comprising:

20. A quantum processing element configured to initialize a predetermined target spin state in a multi-donor quantum dot, the quantum processing element comprising: A semiconductor substrate and a dielectric material forming an interface with the semiconductor substrate, A multi-donor quantum dot embedded in the semiconductor substrate, the multi-donor quantum dot including at least two donor atoms, the at least two donor atoms sharing at least one electron, A control element for controlling the multi-donor quantum dot, The electron is loaded into the multi-donor quantum dot in a spin-down state, The control element is configured to Apply an RF signal for driving at least one EDSR transition and Apply at least one electronic spin reset pulse A quantum processing element by which the target spin state is achieved and the quantum dot is initialized.

21. The quantum processing element according to claim 20, wherein the control element includes at least one gate electrode.

22. The quantum processing element according to claim 20, wherein the control element includes a transmission line for applying the RF signal for driving at least one EDSR transition.

23. The quantum processing element according to any one of claims 20 to 22, wherein the donor atoms of the multi-donor quantum dot are phosphorus atoms and the semiconductor substrate is a silicon substrate.

24. The quantum processing element according to any one of claims 20 to 23, wherein the at least one gate electrode is fabricated in the semiconductor substrate.

25. The quantum processing element according to any one of claims 20 to 23, wherein the at least one gate electrode is patterned on the surface of the semiconductor.

26. A quantum processing element configured to initialize a predetermined target spin state in a multi-donor quantum dot, the quantum processing element comprising: a semiconductor substrate and a dielectric material forming an interface with the semiconductor substrate; a multi-donor quantum dot embedded in the semiconductor substrate, the multi-donor quantum dot including at least two donor atoms, the at least two donor atoms sharing at least one electron; a control element for controlling the multi-donor quantum dot; the electron is loaded into the multi-donor quantum dot in a spin-down state; the control element is configured to execute the method according to any one of claims 1 to 19.

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