Abstract
Singlequbit gates are essential components of a universal quantum computer. Without selective addressing of individual qubits, scalable implementation of quantum algorithms is extremely challenging. When the qubits are discrete points or regions on a lattice, selectively addressing magnetic spin qubits at the nanoscale remains a challenge due to the difficulty of localizing and confining a classical divergencefree field to a small volume of space. Herein we propose a technique for addressing spin qubits using voltagecontrol of nanoscale magnetism, exemplified by the use of voltage control of magnetic anisotropy. We show that by tuning the frequency of the nanomagnet’s electric field drive to the Larmor frequency of the spins confined to a nanoscale volume, and by modulating the phase of the drive, singlequbit quantum gates with fidelities approaching those for faulttolerant quantum computing can be implemented. Such singlequbit gate operations require only tens of femtoJoules per gate operation and have lossless, purely magnetic field control. Their physical realization is also straightforward using foundry manufacturing techniques.
Introduction
Current physical implementations of quantum processors utilize qubits based on trapped ions^{1}, neutral atoms^{2}, nuclear spins^{3,4,5}, topological qubits^{6}, superconducting circuits^{7}, quantum dots^{8,9}, semiconductor spin qubits^{10}, NV centers in diamond^{11} as well as solidstate qubits made from other color centers^{12}. Spin qubits were among the first experimental realizations towards proposed quantum processors due to their long coherence times and available control methods in magnetic resonance experiments^{4,5}. In order to build quantum devices with spin qubits, a scalable design that provides individual control and detection is needed^{9,13,14,15}.
Universal quantum computing can be achieved with a minimum set of quantum gates that allow for the implementation of arbitrary quantum algorithms^{16}. A robust implementation of quantum gates combined with error correction codes is the current prescription for faulttolerant quantum computing^{17}. The creation of highfidelity single and twoqubit gates remains a challenge in every implementation, especially those involving spin qubits that are spatially localized at the atomic to nanoscales. At those length scales, the selective control of spin qubits is demanding because of the difficulty in creating strong, localized control fields that affect only the qubits in the volume of interest, while minimizing crosstalk with neighboring regions.
In this work, we show that for an isolated electron system, individual control of spin qubits can be realized using nanomagnets. Nanoscale magnets present two key advantages in controlling spin qubits: (1) Unlike collective application of microwaves in magnetic resonance experiments, they allow for the application of highly localized magnetic fields that minimize the effect on neighboring qubits. (2) They offer an extremely energy efficient pathway for the control of qubits. This leverages spintronic methods for energyefficient manipulation of magnetization through the use of spinorbittorque (SOT)^{18,19,20}, voltage control of magnetic anisotropy (VCMA)^{21,22,23,24,25,26}, strain mediated voltage control or “straintronic” based methods^{27,28,29,30,31,32} and other paradigms for voltage control of magnetism^{33}. Energy efficiency is achieved through voltage control, rather than current control, thereby avoiding current dissipation losses^{34}(I^{2}R) associated with the generation of magnetic fields. For example, the energy dissipation per bit for VCMA^{35} and voltage induced strain from a piezoelectric layer is less than 1 fJ and 100 aJ, respectively, making them 100 and 1,000 times more efficient than stateoftheart spintransfer torque (STT) methods^{36}, which consume ~ 100 fJ per bit^{34}. Thus, the use of VCMA in controlling the magnetization of nanomagnets^{37,38,39,40,41} results in an energy efficient method for controlling qubits. Another interesting candidate is strainmediated voltage control. Prior work has shown that one can use surface acoustic waves to drive a magnetic film at resonance, which emits magnons in a wide frequency band, some of which produce microwaves that drive transitions in NV centers^{42}. However, this does not result in coherent rotations of the qubits. More recently, coherent rotation of single spin qubits in a NV center^{43} by spinwaves propagating adjacent to it has been demonstrated. Nanoscale manipulation of silicon qubits^{9,14,15} including of flying qubits^{13} have also been demonstrated.
Herein we demonstrate the feasibility of scalable, small footprint, highfidelity, energyefficient quantum gates based on VCMA. Here, we use electron spins with gfactor of 2.0 as a model system to simulate qubit dynamics in the presence of a static external field whose magnitude is comparable to the stray field of the nanomagnets. This intermediatefield regime is considered more challenging due to the more pronounced effects of spatial inhomogeneities (i.e. spatially varying Larmor frequency and axis of quantization) and the lack of rotatingwave approximation. We also consider control of qubit ensembles located in a finitesize nanoscale volume, where field inhomogeneities degrade gate fidelity when averaged over the volume.
The choice of implementing nanoscale control of spin ensembles in this work is motivated by recent proposals^{44,45,46,47,48,49,50,51,52} for quantum entanglers, bona fide qubits, quantum sensing and quantum memory. In all cases, high fidelity gate operations are needed. However, this comes at a cost, as gates implemented by an ensemble of spins distributed over a volume would suffer lower gate fidelity due to field inhomogeneities. This is studied here to ensure that we derive the benefits of spin ensembles while still achieving high gate fidelities. We shall use the term "qubit volume” to refer to the mesoscopic region enclosing the spin ensemble of interest. In the rest of the paper, we perform simulations of magnetization dynamics in voltage controlled nanomagnets and spin dynamics in spins proximal to such nanomagnets. We show that the spins can be individually addressed and driven at their Larmor frequency by the magnetization dynamics of nanomagnets, to implement singlequbit gates with fidelities approaching those for faulttolerant quantum computing.
Results and discussion
Voltage control of nanomagnets to apply control pulses to the qubit
The magnetization dynamics of the nanomagnets — as simulated by solving the LandauLifshitzGilbert (LLG) equation (see Methods section) — leads to a time varying induced magnetic field in the qubit volume (also assumed to be nanoscale). A schematic diagram of the simulation setup of the qubit volume (5 nm × 5 nm × 1 nm) with a nanomagnet in each side is shown in Fig. 1a–c. The qubit volume consists of a planar array of 25 spins defined as s_{ij} (i, row number; j, column number) in each cell so that each spin is separated from its neighbor by 1 nm.
The nanomagnets that drive the Rabi oscillations in these spins by inducing a resonant AC magnetic field due to their magnetization dynamics are elliptical in shape and have length, a = 60 nm, width, b = 20 nm, and thickness, t = 1 nm. Since the qubit volume is nearfield as it is very close to the nanomagnets (distance is ~10 nm, which is a fraction of the wavelength ~15 cm), we calculate AC magnetic field at the qubit volume from the magnetostatic field induced by the nanomagnet, which changes as a function of time due to the magnetization dynamics.
The nanomagnets and the qubit volume are assumed to be placed in a uniform external magnetic field pointing along the direction of the zaxis. Due to perpendicular magnetic anisotropy (PMA) as well as the global bias magnetic field (along + z) the magnetization of the nanomagnets are outofplane (and points along + z). To alter the magnetizations of the two identical nanomagnets, PMA is varied through the application of VCMA.
Note that in our case, the VCMA makes the inplane direction easy, the shape anisotropy due to elliptical nanomagnet shape drives the magnetization to the easy (either ± x) axis of the nanomagnet with equal probability^{53}. To preferentially orient the magnetization along + x, an exchange bias from an underlying antiferromagnet (AFM) can be applied, resulting in a highly localized exchange bias field, \({B}_{{{{{{{{\rm{bias}}}}}}}},x}^{{{{{{{{\rm{ext}}}}}}}}}\) along (+ x) in each nanomagnet. This exchange bias field can be realized at a ferromagnet/antiferromagnet (e.g., CoFeB/IrMn) interface as shown in Fig. 1b. The rotation of the magnetization to the + xdirection due to VCMA induces a magnetic field along + x in the qubit volume, which is located between the two nanomagnets with a distance of 10 nm from each of them indicated as d in Fig. 1a. The magnetization is restored to the zdirection when the PMA is increased.
By applying a sinusoidal voltage to the nanomagnets to induce VCMA, a periodic (sinusoidal with higher harmonics due to nonlinear response) magnetic field is induced along xaxis which is applied to the spins in the qubit volume and causes Larmor precession of these spins when frequency of this induced field drives the spins at resonant condition for a particular value of the effective magnetic bias field in the zdirection (due to the effective global bias magnetic field).
A ferromagnet/oxide interface below the qubit volume plays two roles. It creates a PMA in the film, which is magnetized to point along − z axis that cancels part of the external magnetic bias field along + z to produce an effective field which corresponds to the Larmor precession of the spins in the qubit volume at 0.5 GHz (or 2 GHz). The qubit volume can also be initialized by applying a spintransfertorque (STT) current where the MgO acts as a tunnel barrier layer.
The parameters used in the simulations are listed in Table 1. The effective bias magnetic field applied in the nanomagnet accounts for an external bias field of 0.3 T along the zdirection and the field along the zdirection due to PMA.
Induced field profiles
We simulated and obtained the magnetic field in the qubit volume for two cases: for a single nanomagnet and for two nanomagnets. The histogram plots in Fig. 2 show magnetic field gradients in both the x and z directions with a single nanomagnet and with two nanomagnets. The row and column numbers in the x and y axis correspond to the position of the spin in the qubit volume. In Fig. 2a and b, the maximum induced magnetic field along the xaxis \({B}_{\max ,x}\) and the zaxis \({B}_{\max ,z}\) are given for each of the 25 cells considered in the qubit volume for the case of a single nanomagnet. The maximum amplitude achieved is 0.007 T and the field gradient is 0.003 T or 42.86% in the xdirection. This field gradient creates inhomogeneity and leads to low fidelity of quantum gate operations. The simulation result shows a reduced magnetic field gradient and improved amplitude in both x and z directions with two nanomagnets. The maximum amplitude \({B}_{\max ,x}\) achieved is 0.011 T which is comparatively higher, and the field gradient is 0.001 T or 9.09%, which is comparatively lower than for the case with a single nanomagnet.
The VCMAinduced sinusoidal variation of PMA, magnetization dynamics due to this PMA variation, and time varying magnetic field in the qubit volume due to this magnetization dynamics are shown in Fig. 3 along with corresponding frequency domain plots. A purely sinusoidal PMA variation of 500 MHz is shown in time domain and frequency domain in Fig. 3a,b. The magnetization in the nanomagnet (Fig. 3c) and the induced magnetic field (Fig. 3e) contains higher harmonics (1 GHz, 2 GHz etc.) due to the nonlinear response of the nanomagnet to VCMA as shown in Fig. 3d,f. The magnetization of the nanomagnet pointing in the + z axis induces a magnetic field in negative z direction due to the dipole effect.
The induced magnetic field (B_{x}, B_{y}, B_{z}) in the qubit volume in response to a 2 GHz sinusoidal VCMA applied in the nanomagnets and its frequency domain plot are shown in Fig. 4. The xcomponent (B_{x}) contains 2 GHz as well as higher order harmonics such as 4 GHz, 6 GHz etc.
For a single frequency control pulse, perfect gate implementation is possible in theory. It can be shown that as other harmonics add to the control field, reaching the same clean rotations become more challenging and we expect a drop in the gate fidelity as the number of Fourier components increases. Fourier decomposition is performed for the induced magnetic field at 500 MHz and 2 GHz (see Methods section IV.B). Since the 2 GHz field has a smaller number of components we expect gate operations with larger fidelities in comparison with the 500 MHz drive.
Spin dynamics
We show how to use the induced field of a nanomagnet for implementation of singlequbit gates on electronspin qubits. By describing the evolution of spins, we show that despite the complex nature of the induced field profile, robust implementation of quantum gates is achievable.
The Hamiltonian, \({{{{{{{\mathcal{H}}}}}}}}(t)={\gamma }_{e}{{{{{{{\bf{B}}}}}}}}(t)\cdot {{{{{{{\bf{S}}}}}}}}\), for spin interaction with a magnetic field B(t) = B_{0} + B_{1}(t) is
where \({\omega }_{0}={\gamma }_{e}{B}_{0}\hat{z}\) is the angular velocity of electron spins (the Larmor frequency) when subjected to the external static field B_{0}, γ_{e} is the gyromagnetic ratio of the electron and \({\omega }_{1\alpha }(t)={\gamma }_{e}{{{{{{{{\bf{B}}}}}}}}}_{1}(t)\cdot \hat{\alpha }\) for \(\hat{\alpha }\in \{\hat{x},\hat{y},\hat{z}\}\), is proportional to the strength of the control field in each direction. The timeindependent portion of the Hamiltonian \({{{{{{{{\mathcal{H}}}}}}}}}_{0}={\omega }_{1x}^{{{{{{{{\rm{st}}}}}}}}}{\hat{S}}_{x}{\omega }_{1y}^{{{{{{{{\rm{st}}}}}}}}}{\hat{S}}_{y}({\omega }_{0}+{\omega }_{1z}^{{{{{{{{\rm{st}}}}}}}}}){\hat{S}}_{z}\) includes the external field B_{0} and the timeindependent part of the nanomagnet induced field \({{{{{{{{\bf{B}}}}}}}}}_{1}^{{{{{{{{\rm{st}}}}}}}}}\). This is a result of the bias field applied to fix the rotation direction of the magnetization vector in the nanomagnet. \({{{{{{{{\mathcal{H}}}}}}}}}_{1}(t)={\omega }_{1x}^{{{{{{{{\rm{var}}}}}}}}}(t){\hat{S}}_{x}{\omega }_{1y}^{{{{{{{{\rm{var}}}}}}}}}(t){\hat{S}}_{y}{\omega }_{1z}^{{{{{{{{\rm{var}}}}}}}}}(t){\hat{S}}_{z}\) represents the timedependent part of the induced magnetic field, which is used to control qubits, in place of radio frequency (RF) or microwave pulses.
Spin dynamics in the lab frame, is described with the Liouville vonNeumann equation, Eq. (4), with a unitary propagator defined as
where τ is the Dyson time ordering operator.
The induced magnetic field of nanomagnet has a pronounced static field along x and z directions. These timeindependent field components are part of the \({{{{{{{{\mathcal{H}}}}}}}}}_{0}\) Hamiltonian and as a result, the spins precess around an effective field defined by these fields, which is in the x − z plane, slightly deviating from the zaxis. The angular velocity for this precession is
Timeindependent components of the induced field, \({\omega }_{1x}^{{{{{{{{\rm{st}}}}}}}}}\) and \({\omega }_{1z}^{{{{{{{{\rm{st}}}}}}}}}\) are evaluated using the time average of field components. Considering that ω_{r} should be in resonance with the drive frequency of the nanomagnet, the amplitude of the static external field is chosen such that ω_{0} satisfies this equation.
The unitary propagator is evaluated for the continuous application of drive voltage using Eq. (1). Spin evolution shows the stepwise rotation of spins modulated with the rotation along the effective field with angular velocity ω_{r}. Depiction of spin dynamics is done by initializing one electron spin along the x, y, and z axes and projecting it on the x, z, and y axes after its rotation. Figure 5 shows the average observed signal for both drive frequencies at 500 MHz and 2 GHz. Since the largest timedependent field component is along x, as we apply these pulse segments in resonance with the Larmor frequency of electron spin, we observe x rotations. As expected, the density matrix initialized along the xaxis only precess around the effective field with no change in time. Density matrices initialized in the y − z plane, on the other hand, are affected by the x rotations.
These results are similar to the spin rotations in the traditional magnetic resonance experiments where spin control is implemented using RF pulses in resonance with the Larmor frequency of the spins in the external magnetic field. In the 2 GHz drive example, since the spin rotation happens in smaller steps, there is a smoother transition and we have more control for singlequbit gate implementation. The X/2 gate which is a π/2 rotation along the xaxis can be achieved by stopping the drive when ρ_{y} rotates to z, or equivalently when ρ_{z} rotates to y. This rotation happens at 3.872 ± 0.001 ns for 500 MHz case and at 4.498 ± 0.001 ns in the case of 2 GHz drive. A sudden change of drive voltage, especially mid pulse, will cause oscillatory residual magnetic fields a.k.a. ringing effect. Ideally, we would like to implement gates that last an integer number of pulse segments to minimize the ringing effect. Rotations along the yaxis, or any other orientation in the x − y plane, are implemented by shifting the phase of these X pulses, which is done by applying delays before the start of the pulse train. The X gate is achieved at 7.882 ± 0.001 ns for 500 MHz and at 8.998 ± 0.001 ns for 2 GHz. Figure 6 shows the field profile of two rotations necessary for the implementation of Clifford gates, for the 2 GHz drive case.
In quantum information processing, error correction codes are used to prevent the loss of quantum information due to imperfections of quantum control. The idea is that if the gates are implemented with enough fidelity or with acceptable error probability per gate (or simply error per gate, EPG), they can be effectively used for quantum information processing. This threshold for EPG is determined by further assumptions of the error model and device parameters and is often between 10^{−6} and 3 × 10^{−3} (see refs. ^{9,54,55,56}). The typical value used as a threshold for the experimental implementation of quantum computers is the EPG of less than 10^{−4} ^{57}. In the case of twolevel systems, average gate fidelity can be computed by comparing the ideal and noisy implementation of unitary maps^{58}. For a general, linear, and tracepreserving map \({{{{{{{\mathcal{M}}}}}}}}\), and its ideal counterpart unitary U, the gate fidelity averaged over initial states is defined in terms of the HilbertSchmidt inner product
As indicated in Fig. 6, the fidelity of the X/2 gate implemented by the nanomagnet, is close to the required EPG threshold at 99.97 ± 0.01%. This fidelity value applies to a single spin located at the center of the qubit volume. When the qubit has a finite volume, the fidelity will degrade due to field inhomogeneities over the volume. To assess the impact of inhomogeneities, we view the fidelity as a function of position (i.e. \(\bar{F}\equiv \bar{F}({{{{{{{\bf{r}}}}}}}})\)) and average over the lattice sites:
where Ω is the set of lattice points, \(\bar{F}({{{{{{{{\bf{r}}}}}}}}}_{i})\) is the fidelity of the gate at lattice site r_{i} ∈ Ω and N = ∣Ω∣ is the number of lattice sites. For the X/2 gate the volume averaged fidelity drops to 99.90 ± 0.01%. The fidelity for the longer X gate (99.87 ± 0.01% at the central spin), on the other hand, falls short of the required threshold. Longer pulses may be improved by using composite pulses that compensate locally for errors in the rotation angles. This is a wellestablished method in nuclear magnetic resonance for the design of robust pulses^{59,60}. The socalled Knill pulse is a composite π pulse designed specifically to be robust against frequency offset and pulse amplitude fluctuation errors. It is a symmetric combination of the following five pulses
An X gate implemented with Knill composite pulse shows significant improvement and reaches a gate fidelity of 99.93 ± 0.01% at the central spin and 99.79 ± 0.01% when averaged over the designated qubit volume.
Volume averaging was done on a 2D planar lattice Ω consisting of N = 25 spins separated 1 nm apart. As expected, performance degrades compared to the case of 1 spin. This merely reflects the wellknown fact that it is impossible to be perfectly on resonance with all the spins simultaneously. The B_{1} inhomogeneity thus degrades the average gate fidelity. Although the resulting fidelities dip below the typical value for acceptable error rates, they remain reasonably close to it. A map of average gate fidelity for qubit volume, with frequency detuning of ±0.3% and pulse amplitude errors of ±1% is plotted in Fig. 6. The results show high fidelity regions that depict the robustness of our gates.
Conclusion
In conclusion, the use of nanoscale magnets allows the production of highly localized AC magnetic fields to implement single qubit gates. While not all nanomagnets end up identical during fabrication, variations in anisotropy can be shown to have a negligible effect on the results presented here when employing stateoftheart lithographic processes. Despite the highly nonlinear response of the magnetization of the nanoscale magnets to an electric field, we are able to achieve high single qubit gate fidelity through the appropriate use of robust composite pulses. The significance of using nanomagnets for quantum control is that we can achieve local control over spin qubits. The power required to oscillate nanomagnets at 500 MHz and 2 GHz by voltage control is lower than that required to power a coil by current control.
We note that the two frequencies studied here were chosen to show flexibility of operation. The operating frequency of these nanomagnets could be increased by a few hundreds megahertz, but not much higher due to the dissipative properties of the magnets. Apart from qubit lifetime, there are no theoretical lower bounds on the minimum operating frequency. To control the spin ensemble, the limiting factor for driving the nanomagnet to higher frequencies is not the capacitance and resistance but the ferromagnetic resonance (FMR) frequency which can be increased up to 31 GHz^{61} with associated increase in nonlinearity and higher harmonics. The FMR frequency of the nanomagnet ultimately limits the frequency of the AC control field. For example, a VCMA controlled nanomagnet with oxide barrier 1 nm thick and radius 50 nm would have a capacitance of 1.7 × 10^{−15} F and resistance of 2000 Ω^{62,63}. This corresponds to an RC time constant of ~3.5 ps that enables operation at ~100 GHz while FMR limits operation to 31 GHz^{61}. Also, no attempt was made here to optimize the results, as optimization efforts would depend on the details of qubits and architecture used. With some effort, the geometric arrangement could be improved, for example, to increase the homogeneity and/or minimize the stray field affecting neighboring qubits. The use of composite pulses allows for qubit control that is robust with respect to field inhomogeneity (Fig. 6). Combining the burgeoning spintronic field of energy efficient voltage control of magnetism with quantum computing with robust spin qubits, will stimulate further experiments in energy efficient, robust quantum computing devices at temperatures of a few K.
Methods
Micromagnetics
The simulations of the magnetization dynamics in the nanomagnets are performed by solving the LLG equation
at 0 K using a micromagnetic framework (MuMax3^{64}). We note that singlequbit^{65} and twoqubit^{66} control is demonstrated at temperature above 1 K^{65,66,67,68,69} in silicon quantum dots and operation around 1 K does not introduce significant noise to alter the dynamics of the nanomagnet magnetization results presented herein (see Fig. 7). The root mean square (RMS) error of magnetization along the xdirection is only 0.0039 at 500 MHz and 0.0044 at 2 GHz. Here, α is the Gilbert damping coefficient, γ is the gyromagnetic ratio, \({{{{{{{\bf{m}}}}}}}}=\frac{{{{{{{{\bf{M}}}}}}}}}{{M}_{s}}\) is the normalized magnetization, where M is the magnetization and M_{s} is the saturation magnetization. The effective magnetic field, H_{eff} in this case consists of the fields due to the exchange interaction, uniaxial anisotropy of the nanomagnets, and the demagnetizing field.
where H_{an} is the effective field due to the uniaxial perpendicular magnetic anisotropy (PMA) which can be modulated using voltage control of magnetic anisotropy (VCMA), H_{ex} is the effective field due to Heisenberg exchange coupling and H_{d} is the field due to the demagnetization energy (shape anisotropy).
The effective field due to the perpendicular magnetic anisotropy, H_{an} is given as:
Here, the first order uniaxial anisotropy constant is K_{u1}, the magnetic permeability of free space is μ_{0}, and \(\hat{z}\) is the unit vector corresponding to the anisotropy direction.
While PMA is created from the interaction between the ferromagnet’s hybridized d_{xz} and oxygen’s p_{z} orbital at a ferromagnet/oxide interface^{70}, by the application of voltage pulse, the interface electron density as well as perpendicular anisotropy can be changed^{71}. This phenomenon is called VCMA^{72,73,74}.
The cell sizes are chosen to be 1 nm^{3}, so that all dimensions are well within the limit of ferromagnetic exchange length calculated by \(\sqrt{2{A}_{{{{{{{{\rm{ex}}}}}}}}}/{\mu }_{0}{M}_{s}^{2}}\approx 4.99\) nm.
Quantum control with periodic, polychromatic, inhomogeneous field
Spin dynamics in magnetic resonance experiments is governed by the timedependant part of the Hamiltonian resulting from the application of RF pulses. Consider the Zeeman interaction between spin S and external static field (\({{{{{{{{\bf{B}}}}}}}}}_{0}={B}_{0}\hat{z}=({\omega }_{0}/\gamma )\hat{z}\), where γ: gyromagnetic ratio, gμ_{B}/ℏ for electrons or g_{n}μ_{N}/ℏ for nuclei) and also timedependent RF fields
is:
where
Denoting operators transformed to the rotating frame by a tilde, we write:
By differentiating the latter expression with respect to time, we find the evolution of density matrix in the interaction representation, a.k.a. Liouville vonNeumann equation:
where
The solution to Eq. (4) is given in terms of timeordered exponentials:
Consider a single frequency RF pulse
and let ω_{z}(t) = 0 for simplicity. At the resonance condition ω_{0} = ω_{r}, the spin evolution in the interaction representation is described with
This is a rotation with the Rabi frequency w_{1}. We can extend the analogy for a periodic control field with period T, containing multiple frequencies, expressed as a sum over Fourier components
Here f_{0} = 1/T is the fundamental frequency and the coefficient in the Fourier space are defined as
Similar expressions also exist for y and z. Substitution into Eq. (6) gives:
By setting the fundamental frequency 2πf_{0} = ω_{r} = ω_{0} the control Hamiltonian becomes:
Using the ladder operators, \({\hat{S}}_{+}={\hat{S}}_{x}+i{\hat{S}}_{y}\) and \({\hat{S}}_{}={\hat{S}}_{x}i{\hat{S}}_{y}\), and \([{\hat{S}}_{z},{\hat{S}}_{\pm }]=\pm {\hat{S}}_{\pm }\) we have
and the Hamiltonian becomes:
where we used the shorthand notation:
The real parameters c_{α}[n] are the components of the control field oscillating at the Larmor frequency ω_{0}. The component c_{z}[0] is the timeaverage of the field (z component). If the zcomponent is sinusoidal, it has no d.c. component and c_{z}[0] = 0. If there is a d.c. offset (nonzero background field), this will cause a shift in the resonance frequency away from ω_{0} by the amount c_{z}[0].
The x, y and z components of one control pulse segment, induced by the nanomagnet at the 500 MHz and 2 GHz drive frequencies are shown in Fig. 8. For each direction α, Fourier components \({\omega }_{\alpha }[n]={c}_{\alpha }{{{{{{{{\rm{e}}}}}}}}}^{i2\pi {n}_{\alpha }t/T}\), are added to reconstruct the original timedomain field profile. Notice that by adding higher number of Fourier components, a better approximation of the induced field is achieved. These components may be used in Eq. (8) to evaluate the unitary propagator for the periodic control field. From this we conclude that although the presence of harmonics (Fig. 8) in the control field introduces significant deviations from a sinusoid shape, the presence of an external field comparable to the peak B_{1} field is sufficient to average away these components and yield a high gate fidelity. Normally, the rotating wave approximation is only applied in the limit of high fields.
Data availability
Data points used to construct the graphs can be obtained by contacting the lead authors (chowdhurymf@vcu.edu and mniknam@gmail.com).
Code availability
There is no custom code required to produce the results.
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Acknowledgements
J.A., M.F.C., and M.M.R. were supported in part by National Science Foundation (NSF) grants 1815033 and 1909030. The research at UCLA was partially supported by NSF awards 2137984 and 1936375. J.A., M.F.C., L.S.B., M.N. also acknowledge support from NSF expandQISE grant 2231356.
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J.A., K.L.W., R.N.S., and L.S.B. conceived the idea. All authors discussed the results and commented on the paper. J.A. defined the nanomagnet magnetization control and field inhomogeneity problem and M.F.C. performed the micromagnetic simulations with help from M.M.R. and W.A.M. L.S.B. defined the spin evolution problem and M.N. performed the spin dynamics simulations and quantum gate calculations. J.A., M.F.C., L.S.B. and M.N. wrote the paper.
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Niknam, M., Chowdhury, M.F.F., Rajib, M.M. et al. Quantum control of spin qubits using nanomagnets. Commun Phys 5, 284 (2022). https://doi.org/10.1038/s42005022010418
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DOI: https://doi.org/10.1038/s42005022010418
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