Also at: ]Center for Quantum Science and Engineering, UCLA Also at:
]Department of Electrical and Computer Engineering Also at: ]Center for
Quantum Science and Engineering, UCLA

# Quantum Control of Spin Qubits Using Nanomagnets

Mohamad Niknam† Department of Chemistry and Biochemistry, University of
California Los Angeles, 607 Charles E. Young Drive East, Los Angeles, CA
90095-1059, USA  [  Md. Fahim F. Chowdhury† Md Mahadi Rajib  Walid Al Misba
Department of Mechanical and Nuclear Engineering, College of Engineering,
Virginia Commonwealth University, Richmond, VA 23284-3068, USA  Robert N.
Schwartz  Kang L. Wang  Device Research Laboratory, Department of Electrical
and Computer Engineering, UCLA, 420 Westwood Plaza, Engineering IV, Los
Angeles, CA 90095, USA  Jayasimha Atulasimha  Department of Mechanical and
Nuclear Engineering, College of Engineering Virginia Commonwealth University,
Richmond, VA 23284-3068, USA  [<EMAIL_ADDRESS>Louis-S. Bouchard
Department of Chemistry and Biochemistry, University of California Los
Angeles, 607 Charles E. Young Drive East, Los Angeles, CA 90095-1059, USA  [
<EMAIL_ADDRESS>
###### Abstract

Single-qubit gates are essential components of a universal quantum computer.
Without selective addressing of individual qubits, scalable implementation of
quantum algorithms is not possible. When the qubits are discrete points or
regions on a lattice, the selective addressing of magnetic spin qubits at the
nanoscale remains a challenge due to the difficulty of localizing and
confining a classical divergence-free field to a small volume of space. Herein
we propose a new technique for addressing spin qubits using voltage-control of
nanoscale magnetism, exemplified by the use of voltage control of magnetic
anisotropy (VCMA). 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, single-qubit
quantum gates with fidelities approaching those for fault-tolerant quantum
computing can be implemented. Such single-qubit gate operations have the
advantage of remarkable energy efficiency, requiring only tens of femto-Joules
per gate operation, and lossless, purely magnetic field control (no E-field
over the target volume). Their physical realization is also straightforward
using existing foundry manufacturing techniques.

Spin qubit, mesoscopic qubit, quantum control, nanomagnet

††preprint: AAPM/123-QED

##  I 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 solid-state 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 [13,
14, 15, 9].

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 fault-tolerant quantum computing [17]. The
creation of high-fidelity single and two-qubit 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 cross-talk 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 energy-
efficient manipulation of magnetization through the use of spin-orbit-torque
(SOT) [18, 19, 20], voltage control of magnetic anisotropy (VCMA) [21, 22, 23,
24], strain mediated voltage control or “straintronic” based methods [25, 26,
27, 28] and other paradigms for voltage control of magnetism [29]. Energy
efficiency is achieved through voltage control, rather than current control,
thereby avoiding current dissipation losses [30] $(I^{2}R)$ associated with
the generation of magnetic fields. For example, the energy dissipation per bit
for VCMA [31] and voltage induced strain from a PZT layer is less than 1 fJ
and 100 aJ, respectively, making them 100 and 1,000 times more efficient than
state-of-the-art spin-transfer torque (STT) methods [32], which consume
$\sim$100 fJ/bit [30]. Thus, the use of VCMA in controlling the magnetization
of nanomagnets results in an energy efficient method for controlling qubits.
Another interesting candidate is strain-mediated 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 [33]. However, this does not
result in coherent rotations of the qubits. More recently, coherent rotation
of single spin qubits in a NV center [34] by spin-waves propagating adjacent
to it has been demonstrated. Nanoscale manipulation of silicon qubits [14, 9,
15] including of flying qubits [13] have been demonstrated.

Herein we demonstrate the feasibility of scalable, small footprint, high-
fidelity, energy-efficient quantum gates based on VMCA. Here, we use electron
spins with $g$-factor 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 intermediate-field 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 rotating-wave approximation. We also consider control of qubit
ensembles located in a finite-size 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 [35, 36, 37, 38, 39, 40, 41, 42, 43] 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.

##  II Voltage control of nanomagnets to apply control pulses to the Qubit

The magnetization dynamics of the nanomagnets — as simulated by solving the
Landau-Lifshitz-Gilbert (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 $\times$ 5 nm $\times$ 1 nm) with a nanomagnet in each side is shown in
Fig. 1(a) and Fig. 1(b). 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 near-field as it is very close to the
nanomagnets (distance is $\sim$10 nm, which is a fraction of the wavelength
$\sim$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.

Figure 1: (a) Schematic diagram of simulation setup in MuMax3. (b) Schematics
of nanomagnet and qubit layers. (Left) Ferromagnet/antiferromagnet interface
to create exchange bias field in $+x$ direction in the nanomagnet, so it
deterministically oscillates from $+z$ to $+x$ when VCMA is applied in an
external bias field of 0.3 T. (Right) FM/Oxide interface to create PMA in the
film (magnetized to point along $-z$) below the qubit volume that cancels part
of the external magnetic bias field so the effective $+z$ field seen by the
qubit produces Larmor precession at 0.5 GHz (or 2 GHz) simulated in this work.
(Images are not to scale.)

The nanomagnets and the qubit volume are assumed to be placed in a uniform
external magnetic field pointing along the direction of the $z$-axis. Due to
perpendicular magnetic anisotropy (PMA) as well as the global bias magnetic
field (along $+z$) the magnetization of the nanomagnets are out-of-plane (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 in-plane direction easy, the shape
anisotropy due to elliptical nanomagnet shape drives the magnetization to the
easy (either $\pm x$) axis of the nanomagnet with equal probability [44]. 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, $B_{\text{bias}\\_x}^{\text{ext}}$ 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. 1(b).
The rotation of the magnetization to the $+x$-direction 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. 1(a). The magnetization is restored to the $z$-direction 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 $x$-axis 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 $z$-direction (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 spin-transfer-torque (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 $z$-direction and the field along the $z$-direction
due to PMA.

Table 1: List of parameters used in the simulation. Parameters | Value
---|---
Saturation magnetization ($M_{s}$) | 0.8$\times$106 A/m
Gilbert damping constant ($\alpha$) | 0.1
Exchange stiffness ($A_{\text{ex}}$) | 10$\times$10-12 J/m
Maximum PMA constant ($K_{u1,\text{max}}$) | 1.0$\times$106 J/m3
Exchange bias field ($B_{\text{bias}\\_x}^{\text{ext}}$) | 100 mT
Nanomagnet thickness ($t$) | 1 nm
VCMA frequency ($\nu$) | 500 MHz and 2 GHz
VCMA coefficient ($\eta$) | 500 fJ/Vm
Figure 2: Field gradient histogram plots in the qubit volume (a) Maximum
magnetic field of each spin along $x$-axis and (b) along $z$-axis with a
single nanomagnet. (c) Maximum magnetic field of each spin along $x$-axis and
(d) along $z$-axis with two nanomagnets.

##  III 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 Figs. 2(a) and (b), the maximum induced magnetic field along the $x$-axis
$B_{\text{max},x}$ and the $z$-axis $B_{\text{max},z}$ 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 $x$-direction. 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 for two nanomagnets. The maximum
amplitude $B_{\text{max},x}$ 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.

Figure 3: Sinusoidal (500 MHz) perpendicular magnetic anisotropy in (a) time
and (b) frequency domain. Magnetization ($m_{x}$, $m_{y}$, $m_{z}$) of a
nanomagnet in (c) time and (d) frequency domain. Induced magnetic field
($B_{x}$, $B_{y}$, $B_{z}$) in the qubit volume in (e) time and (f) frequency
domain.

The sinusoidal variation of PMA applied to the nanomagnets through VCMA, VCMA-
induced sinusoidal variation of PMA leading to magnetization dynamics and
inducing a time varying magnetic field in the qubit volume are shown in Fig. 3
along with corresponding frequency domain plots. A sinusoidal VCMA of 500 MHz
is shown in time domain and frequency domain in Fig. 3(a) and Fig. 3(b). The
magnetization in the nanomagnet (Fig. 3(c)) and the induced magnetic field
(Fig. 3(e)) contains higher harmonics (1 GHz, 2 GHz etc.) due to the nonlinear
response of the nanomagnet to VCMA as shown in Fig. 3(d) and Fig. 3(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 $x$-component ($B_{x}$)
contains 2 GHz as well as higher order harmonics such as 4 GHz, 6 GHz etc.

Figure 4: Induced magnetic field ($B_{x}$, $B_{y}$, $B_{z}$) in the qubit
volume in (a) time and (b) frequency domain.

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, Fig. 6
panels a, b, and c, and for 2 GHz, in panels d, e, and f. The first 20 Fourier
coefficients of each field are plotted in the inset. 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.

##  IV Spin dynamics

We show how to use the induced field of a nanomagnet for implementation of
single-qubit gates on electron-spin 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, $\mathcal{H}(t)=-\gamma_{e}\vec{B}(t)\cdot\vec{S}$, for spin
interaction with a magnetic field $\vec{B}(t)=\vec{B}_{0}(t)+\vec{B}_{1}(t)$
is

$\displaystyle\mathcal{H}(t)=$
$\displaystyle-\omega_{1x}(t)\hat{S}_{x}-\omega_{1y}(t)\hat{S}_{y}-(\omega_{0}+\omega_{1z}(t))\hat{S}_{z}$
$\displaystyle=$
$\displaystyle-(\omega_{1x}^{\text{st}}+\omega_{1x}^{\text{var}}(t))\hat{S}_{x}-(\omega_{1y}^{\text{st}}+\omega_{1y}^{\text{var}}(t))\hat{S}_{y}$
$\displaystyle-(\omega_{0}+\omega_{1z}^{\text{st}}+\omega_{1z}^{\text{var}}(t))\hat{S}_{z}$
$\displaystyle=$ $\displaystyle\mathcal{H}_{0}+\mathcal{H}_{1}(t)$

where $\omega_{0}=\gamma_{e}B_{0}$ is the angular velocity of electron spins
when subjected to the external static field $B_{0}$, or the Larmor frequency,
$\gamma_{e}$ is the gyromagnetic ratio of the electron and
$\omega_{1\alpha}(t)=\gamma_{e}\vec{B}_{1}(t)\cdot\hat{\alpha}$ for
$\hat{\alpha}\in\\{\hat{x},\hat{y},\hat{z}\\}$, is proportional to the
strength of the control field in each direction. The time-independent portion
of the Hamiltonian
$\mathcal{H}_{0}=-\omega_{1x}^{\text{st}}\hat{S}_{x}-\omega_{1y}^{\text{st}}\hat{S}_{y}-(\omega_{0}+\omega_{1z}^{\text{st}})\hat{S}_{z}$
includes the external field $B_{0}$ and the time-independent part of the
nanomagnet induced field $B_{1}^{\text{st}}$. This is a result of the bias
field applied to fix the rotation direction of the magnetization vector in the
nanomagnet.
$\mathcal{H}_{1}(t)=-\omega_{1x}^{\text{var}}(t)\hat{S}_{x}-\omega_{1y}^{\text{var}}(t)\hat{S}_{y}-\omega_{1z}^{\text{var}}(t)\hat{S}_{z}$
represents the time-dependent part of the induced magnetic field, which is
used to control qubits, in place of RF or microwave pulses.

Figure 5: Evolution of spins in the lab frame for nanomagnet induced field
with 500 MHz drive (Left) and 2 GHz drive (Right). Average evolution of 25
spins initialized along the $x$, $y$, and $z$ directions are projected on $x$,
$z$, $y$ axes, respectively. The effective $\pi/2)_{x}$ rotation happens at
3.871$\pm 0.001$ ns and 4.498$\pm 0.001$ ns in 500 MHz and 2 GHz drives,
respectively. Spins rotate a $\pi$ Radian along the x-axis at 7.882$\pm 0.001$
ns and 8.998$\pm 0.001$ ns in 500 MHz and 2 GHz frequencies, respectively.

Spin dynamics in the lab frame, is described with the Liouville von-Neumann
equation, Eq. (4), with a unitary propagator defined as

$U_{1}=\tau\exp\left\\{-i\int_{0}^{t}\left[\mathcal{H}_{0}+\mathcal{H}_{1}(t^{\prime})\right]dt^{\prime}\right\\}$
(1)

where $\tau$ is the Dyson time ordering operator.

The induced magnetic field of nanomagnet has a pronounced static field along
$x$ and $z$ directions. These time-independent 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 $z$-axis. The angular velocity for this precession is

$\omega_{r}=\sqrt{(\omega_{1x}^{\text{st}})^{2}+(\omega_{1z}^{\text{st}}+\omega_{0})^{2}}.$

Time-independent components of the induced field, $\omega_{1x}^{\text{st}}$
and $\omega_{1z}^{\text{st}}$ are evaluated using the time average of field
components. Considering that $\omega_{r}$ should be in resonance with the
drive frequency of the nanomagnet, the amplitude of the static external field
is chosen such that $\omega_{0}$ satisfies this equation.

The unitary propagator is evaluated for the continuous application of drive
voltage using Eq. (1). Spin evolution shows the step-wise rotation of spins
modulated with the rotation along the effective field with angular velocity
$\omega_{r}$. Depiction of spin dynamics 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. Fig. 5 shows the average observed signal for both
drive frequencies at 500 MHz and 2 GHz. Since the largest time-dependent 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 $x$-axis 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 single-
qubit gate implementation. The $X/2$ gate which is a $\pi/2$ rotation along
the $x$-axis can be achieved by stopping the drive when $\rho_{y}$ rotates to
-$z$, or equivalently when $\rho_{z}$ rotates to $y$. This rotation happens at
3.872$\pm 0.001$ ns for 500 MHz case and at 4.498$\pm 0.001$ 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 $y$-axis, 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$\pm 0.001$ ns for 500 MHz and at
8.998$\pm 0.001$ ns for 2 GHz. Table 2 shows the field profile of two
rotations necessary for the implementation of Clifford gates, for the 2 GHz
drive case.

Table 2: Implementation of quantum gates using nanomagnets at 2GHz drive Gate | Induced Field Profile |  Fidelity Map Volume Average 25 spins
---|---|---
X/2 |  |
Ideal Hamiltonian $\mathcal{H}=\exp\left(\frac{i}{2}\frac{\pi}{2}\sigma_{\text{x}}\right)$ |  |
Central spin fidelity: 0.9997 $\pm 10^{-4}$ |  |
Fidelity averaged over lattice sites: 0.9990$\pm 10^{-4}$ |  |
X |  |
Ideal Hamiltonian $\mathcal{H}=\exp\left(\frac{i}{2}\pi\sigma_{\text{x}}\right)$ |  |
Central spin fidelity: 0.9987 $\pm 10^{-4}$ |  |
Fidelity averaged over lattice sites: 0.9952$\pm 10^{-4}$ |  |
X Knill CP |  |
Ideal Hamiltonian $\mathcal{H}=\exp\left(\frac{i}{2}\pi\sigma_{\text{x}}\right)$ |  |
Central spin fidelity: 0.9993 $\pm 10^{-4}$ |  |
Fidelity averaged over lattice sites: 0.9979$\pm 10^{-4}$ |  |

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 $3\times 10^{-3}$ (see Refs.
[45, 46, 47, 9]). The typical value used as a threshold for experimental
implementation of quantum computers is the EPG of less than $10^{-4}$ (see
Ref. [48]). In the case of two-level systems, average gate fidelity can be
computed by comparing the ideal and noisy implementation of unitary maps [49].
For a general, linear, and trace-preserving map $\mathcal{M}$, and its ideal
counterpart unitary $U$, the gate fidelity averaged over initial states is
defined in terms of the Hilbert-Schmidt inner product

$\bar{F}=\frac{1}{2}+\frac{1}{3}\sum_{j=x,y,z}\mbox{Tr}\left[U\frac{\sigma_{j}}{2}U^{\dagger}\mathcal{M}\left(\frac{\sigma_{j}}{2}\right)\right].$
(2)

As indicated in Table 2, the fidelity of the $X/2$ gate implemented by
nanomagnet, is well above the required EPG threshold at $99.97\pm 0.01\%$.
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.
$\bar{F}\equiv\bar{F}(\mathbf{r})$) and average over the lattice sites:

$\langle\bar{F}\rangle_{\Omega}=\frac{1}{N}\sum_{\mathbf{r}_{i}\in\Omega}^{N}\bar{F}(\mathbf{r}_{i})$

where $\Omega$ is the set of lattice points, $\bar{F}(\mathbf{r}_{i})$ is the
fidelity of the gate at lattice site $\mathbf{r}_{i}\in\Omega$ and
$N=|\Omega|$ is the number of lattice sites. For the $X/2$ gate the volume
averaged fidelity drops to $99.90\pm 0.01\%$. The fidelity for the longer $X$
gate ($99.87\pm 0.01\%$ at the central spin), on the other hand, falls
slightly 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 well-established method in nuclear magnetic resonance for
the design of robust pulses [50, 51]. The so-called Knill pulse is a composite
$\pi$ pulse designed specifically to be robust against frequency offset and
pulse amplitude fluctuation errors. It is the symmetric combination of the
following five pulses

$\pi_{0}^{\text{ideal}}(\frac{\pi}{3})_{z}=(\pi)_{\frac{\pi}{6}}(\pi)_{0}(\pi)_{\frac{\pi}{2}}(\pi)_{0}(\pi)_{\frac{\pi}{6}}.$

An $X$ gate implemented with Knill composite pulse shows significant
improvement and reaches an average gate fidelity of 0.9993$\pm 10^{-4}$. The
lattice-averaged fidelity of the composite pulse, 0.9979 $\pm 10^{-4}$, is
only slightly below threshold.

Lattice averaging was done on a 2D planar lattice $\Omega$ consisting of
$N=25$ spins separated 1 nm apart. As expected, performance degrades compared
to the case of 1 spin. This merely reflects the well-known 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 threshold for acceptable error rates, they
remain within $0.1\%$ of it. A map of average gate fidelity for spin-lattice,
with frequency detuning of $\pm 0.3\%$ and pulse amplitude errors of $\pm 1\%$
is plotted in table 2. The results show high fidelity regions that depict the
robustness of our gates.

##  V Conclusion

In conclusion, the use of nanoscale magnets allows the production of highly
localized AC magnetic fields to implement single qubit gates. 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 no attempt was made here to optimize the results. 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 (Table 2). 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.

##  VI Methods

###  VI.1 Micromagnetics

The simulations of the magnetization dynamics in the nanomagnets are performed
by solving the LLG equation using a micromagnetic framework (MuMax3 [52])

$\frac{d\vec{m}}{dt}=-\frac{\gamma\vec{m}\times\vec{H}_{\text{eff}}}{(1+\alpha^{2})}-\frac{\alpha\gamma\vec{m}\times(\vec{m}\times\vec{H}_{\text{eff}})}{(1+\alpha^{2})}$
(3)

Here, $\alpha$ is the Gilbert damping coefficient, $\gamma$ is the
gyromagnetic ratio, $\vec{m}=\frac{\vec{M}}{M_{s}}$ is the normalized
magnetization, where $\vec{M}$ is the magnetization and $M_{s}$ is the
saturation magnetization. The effective magnetic field, $\vec{H}_{\text{eff}}$
in this case consists of the fields due to the exchange interaction, uniaxial
anisotropy of the nanomagnets, and the demagnetizing field.

$\vec{H}_{\text{eff}}=\vec{H}_{\text{an}}+\vec{H}_{\text{ex}}+\vec{H}_{\text{d}}$

where $\vec{H}_{\text{an}}$ is the effective field due to the uniaxial
perpendicular magnetic anisotropy (PMA) which can be modulated using voltage
control of magnetic anisotropy (VCMA), $\vec{H}_{\text{ex}}$ is the effective
field due to Heisenberg exchange coupling and $\vec{H}_{\text{d}}$ is the
field due to the demagnetization energy (shape anisotropy).

The effective field due to the perpendicular magnetic anisotropy,
$\vec{H}_{\text{an}}$ is given as:

$\vec{H}_{\text{an}}=\frac{2K_{u1}}{\mu_{0}M_{s}}(\vec{z}\cdot\vec{m})\vec{z}.$

Here, the first order uniaxial anisotropy constant is $K_{u1}$, the magnetic
permeability of free space is $\mu_{0}$, and $\vec{z}$ 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 [53],
by the application of voltage pulse, the interface electron density as well as
perpendicular anisotropy can be changed [54]. This phenomenon is called VCMA
[55, 56, 57].

The cell sizes are chosen to be 1 nm3, so that all dimensions are well within
the limit of ferromagnetic exchange length calculated by
$\sqrt{2A_{\text{ex}}/\mu_{0}M_{s}^{2}}\approx 4.99$ nm.

###  VI.2 Quantum control with periodic, polychromatic, inhomogeneous field

Spin dynamics in magnetic resonance experiments is governed by the time-
dependant part of the Hamiltonian resulting from the application of r.f.
pulses. Consider the Zeeman interaction between spin $\vec{S}$ and external
static field ($\vec{B}_{0}=B_{0}\hat{z}=(\omega_{0}/\gamma)\hat{z}$, where
$\gamma$: gyromagnetic ratio, $g\mu_{B}/\hbar$ for electrons or
$g_{n}\mu_{N}/\hbar$ for nuclei) and also time-dependent r.f. fields

$\vec{B}_{1}(t)=(\omega_{x}(t)/\gamma)\hat{x}+(\omega_{y}(t)/\gamma)\hat{y}+(\omega_{z}(t)/\gamma)\hat{z}$

is:

$\begin{split}\mathcal{H}(t)&=-\gamma\left(\vec{B}_{0}+\vec{B}_{1}(t)\right)\cdot\vec{S}\\\
&=-\omega_{x}(t)\hat{S}_{x}-\omega_{y}(t)\hat{S}_{y}-(\omega_{0}+\omega_{z}(t))\hat{S}_{z}\\\
&=\mathcal{H}_{0}+\mathcal{H}_{1}(t)\end{split}$

where

$\mathcal{H}_{0}=-\omega_{0}\hat{S}_{z},\quad\mathcal{H}_{1}(t)=-(\omega_{x}(t)\hat{S}_{x}+\omega_{y}(t)\hat{S}_{y}+\omega_{z}(t)\hat{S}_{z}).$

Denoting operators transformed to the rotating frame by a tilde, we write:

$\tilde{\mathcal{H}}(t)=e^{i\omega_{r}\hat{S}_{z}t}\mathcal{H}(t)e^{-i\omega_{r}\hat{S}_{z}t}$
$\tilde{\rho}(t)=e^{i\omega_{r}\hat{S}_{z}t}\rho(t)e^{-i\omega_{r}\hat{S}_{z}t}$

By differentiating the latter expression with respect to time, we find the
evolution of density matrix in the interaction representation, a.k.a.
Liouville von-Neumann equation:

$\frac{\partial\tilde{\rho}}{\partial
t}=-i[\tilde{\mathcal{H}}(t),\tilde{\rho}(t)]\qquad$ (4)

where

$\displaystyle\tilde{\mathcal{H}}(t)=$
$\displaystyle\tilde{\mathcal{H}}_{0}+\tilde{\mathcal{H}}_{1}(t)-\omega_{r}\hat{S}_{z}$
$\displaystyle=$ $\displaystyle
e^{i\omega_{r}\hat{S}_{z}t}\left[-\omega_{x}(t)\hat{S}_{x}-\omega_{y}(t)\hat{S}_{y}\right]e^{-i\omega_{r}\hat{S}_{z}t}+$
(5) $\displaystyle\qquad(-\omega_{0}-\omega_{z}(t)+\omega_{r})\hat{S}_{z}$ (6)

The solution to Eq. (4) is given in terms of time-ordered exponentials:

$\tilde{\rho}(t)=\mathcal{T}e^{-i\int_{0}^{t}\tilde{\mathcal{H}}(t)dt}\tilde{\rho}(0)\mathcal{T}e^{i\int_{0}^{t}\tilde{\mathcal{H}}(t)dt}.$

Consider a single frequency r.f. pulse

$\omega_{x}(t)=w_{1}\cos(\omega_{r}t+\phi(t)),\quad\omega_{y}(t)=w_{1}\sin(\omega_{r}t+\phi(t))$

and let $\omega_{z}(t)=0$ for simplicity. At the resonance condition
$\omega_{0}=\omega_{r}$, the spin evolution in the interaction representation
is described with

$\tilde{\mathcal{H}}(t)=e^{i\omega_{0}\hat{S}_{z}t}w_{1}[\cos(\omega_{0}t+\phi(t))\hat{S}_{x}\\\
-\sin(\omega_{0}t+\phi(t))\hat{S}_{y}]e^{-i\omega_{0}\hat{S}_{z}t}.$ (7)

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

$\omega_{x}(t)=\sum_{n_{x}=-N}^{N}c_{x}[n_{x}]e^{i2\pi n_{x}t/T}.$

Here $f_{0}=1/T$ is the fundamental frequency and the coefficient in the
Fourier space are defined as

$c[n]=\frac{1}{T}\int_{0}^{T}\omega(t)e^{-i2\pi nt/T}dt.$

Similar expressions also exist for $y$ and $z$. Substitution into Eq. (6)
gives:

$\displaystyle\tilde{\mathcal{H}}(t)=$ $\displaystyle
e^{i\omega_{r}\hat{S}_{z}t}\left[\omega_{x}(t)\hat{S}_{x}+\omega_{y}(t)\hat{S}_{y}\right]e^{-i\omega_{r}\hat{S}_{z}t}$
$\displaystyle\quad+(\omega_{0}+\omega_{z}(t)-\omega_{r})\hat{S}_{z}$
$\displaystyle=$
$\displaystyle\sum_{n=-N}^{N}e^{i\omega_{r}\hat{S}_{z}t}\left(c_{x}[n]\hat{S}_{x}+c_{y}[n]\hat{S}_{y}\right)e^{i2\pi
nt/T}e^{-i\omega_{r}\hat{S}_{z}t}$ $\displaystyle+$
$\displaystyle(\omega_{0}+c_{z}[n]e^{i2\pi nt/T}-\omega_{r})\hat{S}_{z}$

By setting the fundamental frequency $2\pi f_{0}=\omega_{r}=\omega_{0}$ the
control Hamiltonian becomes:

$\displaystyle\tilde{\mathcal{H}}(t)=\sum_{n=-N}^{N}$ $\displaystyle
e^{i\omega_{0}\hat{S}_{z}t}\left(c_{x}[n]\hat{S}_{x}+c_{y}[n]\hat{S}_{y}\right)e^{i\omega_{0}nt}e^{-i\omega_{0}\hat{S}_{z}t}$
$\displaystyle+$ $\displaystyle c_{z}[n]e^{i\omega_{0}nt}\hat{S}_{z}$

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

$e^{i\omega_{0}\hat{S}_{z}t}\hat{S}_{\pm}e^{-i\omega_{0}\hat{S}_{z}t}=\hat{S}_{\pm}+[\hat{S}_{z},\hat{S}_{\pm}]i\omega_{0}t+\dots=\hat{S}_{\pm}e^{\pm
i\omega_{0}t},$

and the Hamiltonian becomes:

$\displaystyle\tilde{\mathcal{H}}(t)=\sum_{n=-N}^{N}$
$\displaystyle\hat{S}_{+}e^{i\omega_{0}(n+1)t}c_{+}[n]+\hat{S}_{-}e^{i\omega_{0}(n-1)t}c_{-}[n]$
$\displaystyle+$ $\displaystyle c_{z}[n]e^{i\omega_{0}nt}\hat{S}_{z}$ (8)

where we used the shorthand notation:

$c_{+}[n]:=\frac{c_{x}[n]}{2}+\frac{c_{y}[n]}{2i},\qquad
c_{-}[n]:=\frac{c_{x}[n]}{2}-\frac{c_{y}[n]}{2i}.$

The real parameters $c_{\alpha}[n]$ are the components of the control field
oscillating at the Larmor frequency $\omega_{0}$. The component $c_{z}[0]$ is
the time-average of the field ($z$ component). If the $z$-component 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 $\omega_{0}$ by the amount $c_{z}[0]$.

Figure 6: $x$, $y$, and $z$ component of the induced magnetic field with 500
MHz drive, are indicated in panels a, b, and c respectively. Panels d, e, and
f are showing the results for 2 GHz drive. In each case Fourier components
with $N=5,10,20$ are used to reconstruct the original field profile.

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. 6. For
each direction $\alpha$, Fourier components
$\omega_{\alpha}[n]=c_{\alpha}\text{e}^{i2\pi n_{\alpha}t/T}$, are added to
reconstruct the original time-domain 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. 6) 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.

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##  VII Acknowledgements

JA, MFC and MMR were supported in part by National Science Foundation (NSF)
grants 1815033 and 1909030. The research at UCLA was partially supported by
NSF awards CHE-2002313 and 1936375.

##  VIII Competing interests

The authors declare no competing interests.

##  IX Author contributions

JA, KLW, RNS and LSB conceived the idea. All authors discussed the results and
commented on the paper. JA defined the nanomagnet magnetization control and
field inhomogeneity problem and MFC performed the micromagnetic simulations
with help from MMR and WAM. LSB defined the spin evolution problem and MN
performed the spin dynamics simulations and quantum gate calculations. JA,
MFC, LSB and MN wrote the paper. †These authors (MN, MFC) contributed equally.