††thanks: These authors contributed equally.††thanks: These authors
contributed equally.††thanks: These authors contributed equally.

#  Focusing of quantum gate interactions using dynamical decoupling

M. C. Smith  Clarendon Laboratory, Department of Physics, University of
Oxford, Parks Road, Oxford OX1 3PU, U.K.  A. D. Leu  Clarendon Laboratory,
Department of Physics, University of Oxford, Parks Road, Oxford OX1 3PU, U.K.
M. F. Gely  Clarendon Laboratory, Department of Physics, University of Oxford,
Parks Road, Oxford OX1 3PU, U.K.  D. M. Lucas  Clarendon Laboratory,
Department of Physics, University of Oxford, Parks Road, Oxford OX1 3PU, U.K.

###### Abstract

In 1995, Cirac and Zoller proposed the first concrete implementation of a
small-scale quantum computer, using laser beams focused to micron spot sizes
to address individual trapped ions in a linear crystal. Here we propose a
method to focus entangling gate interactions, but driven by microwave fields,
to micron-sized zones, corresponding to $10^{-5}$ microwave wavelengths. We
demonstrate the ability to suppress the spin-dependent force using a single
ion, and find the required interaction introduces $3.7(4)\times 10^{-4}$ error
per emulated gate in a single-qubit benchmarking sequence. We model the scheme
for a 17-qubit ion crystal, and find that any pair of ions should be
addressable with an average crosstalk error of $\sim 10^{-5}$.

Dynamical decoupling (DD) is commonly used across many quantum computing
platforms to reduce the loss of quantum information caused by decoherence [1,
2, 3, 4, 5, 6, 7]. DD aims to suppress the interaction between a quantum
system and its environment by imposing a time-dependence on the interaction,
and averaging out the net effect on the system. This technique is used for
enhancing quantum memories [2, 3, 4, 4, 5, 6, 7] and reducing errors during
logical gate operations [8, 9, 10, 11]. In this article however, we propose an
extension of DD where qubits are not decoupled from the environment, but
rather selectively decoupled from gate interactions driven on other qubits,
with individual qubit addressing or crosstalk mitigation in mind.

We demonstrate this idea using trapped ion qubits driven by near-field
microwaves [12, 13, 9, 14, 15, 16]. More specifically, we show how DD can
enable the focusing of gate interactions onto individual ions, addressing a
key challenge with microwave-driven trapped ions over the laser-driven
alternative. Microwave technology offers attractive features for ion trap
scalability: robustness, cost and size, straightforward amplitude and phase
control, and easy integration of waveguides onto surface traps. However,
whilst laser beams can be focused onto individual ions [17], the centimeter-
wavelength of microwaves requires alternate techniques to address qubits [18,
19, 20, 21, 22, 23, 16], currently nonexistent for two-qubit gates on ions
confined to the same potential well.

We first show how DD can effectively suppress the interaction which drives
two-qubit gates, the spin-dependent force (SDF) [24]. The error associated
with suppression is measured to be much lower than typical two-qubit gate
errors: $3.7(4)\times 10^{-4}$. Secondly, we implement a spatially varying DD
phase to selectively suppress or enable the SDF with $\sim 1~{}\upmu$m spatial
resolution, as schematically illustrated in Fig. 1. We extrapolate our results
to a larger register of ions, showing how this technique could enable all-to-
all connectivity in a chain of ions solely by varying the amplitude of
microwave currents in three electrodes.

Figure 1:  Two-qubit gate addressing through spatially varying dynamical
decoupling. (a) A laser is commonly used to address ions by focusing beams to
small spot sizes [17]. To drive the two-qubit gate interaction – spin-
dependent force (SDF) – two sideband (SB) laser tones are applied,
symmetrically detuned from the qubit frequency by the motional frequency
$\omega_{m}$. (b) An SDF can also be driven using microwave currents, tuned to
SB frequencies, and injected through electrodes (light grey) passing under the
ions. Due to the long wavelength of microwaves, this force will affect all
ions trapped within the same potential well. Our proposed method for targeting
a single ion requires an additional dynamical decoupling tone (DD) which
drives the qubit resonantly. By spatially varying the phase difference between
DD and SB drives – through interference of the field generated by the
different electrodes – the SDF is either enabled (DD and SB in-phase) or
suppressed (DD and SB $\pm\pi/2$ out of phase).

For this demonstration, a single ion is displaced to different positions to
emulate different ions in a chain. Our experiments are carried out at room
temperature on a segmented electrode surface “chip” trap characterised in Ref.
[25]. The trap features an on-chip microwave (MW) resonator with a single ion
trapped $40~{}\upmu\text{m}$ above the chip surface. Our qubit is defined by
the hyperfine levels $|1\rangle=|F=4,M_{F}=1\rangle$ and
$|0\rangle=|F=3,M_{F}=1\rangle$ of the ground state manifold 4S1/2 of 43Ca+,
which form a clock transition at our static magnetic field strength of 28.8
mT. These states are connected by a magnetic dipole transition with frequency
splitting $\omega_{q}=2\pi\times 3.1$ GHz. To drive this transition, we use
near-field radiation generated by the currents propagating in the MW
resonator. The field gradient is used to generate the SDF enabling two-qubit
gates.

Figure 2:  Spin-dependent force suppression through dynamical decoupling. (a)
Position-momentum (x-p) diagram illustrating the effect of the spin-dependent
force (SDF) on the ion’s motional state. Starting with qubit state $\ket{0}$,
a measurement in the qubit basis ($\ket{0}$, $\ket{1}$), would show the qubit
evolve from state $\ket{0}$ to $(\ket{0}\pm\ket{1})/\sqrt{2}$. (b) Measured
(blue) and fitted (black) probability $P_{\ket{0}}$ of measuring the initial
state $\ket{0}$ after driving an SDF using two microwave sideband (SB) tones.
(c) To suppress motional entanglement, a dynamical decoupling (DD) tone drives
the qubit with a variable phase $\phi_{\text{DD}}$ (relative to the average
phase of the two sidebands). We measure $P_{\ket{0}}$ for a fixed pulse
duration (red star), such that suppression of the SDF corresponds to
$P_{\ket{0}}>0.6$. Measurements were taken at a fixed ion position (null of
the RF trapping field). The SDF is unaffected ($P_{\ket{0}}\approx 0.6$) or
suppressed ($P_{\ket{0}}\approx 1$) for $\phi_{\text{DD}}=0$ and
$\phi_{\text{DD}}=\pi/2$ respectively. All error bars indicate 68$\%$
confidence intervals.

An SDF is generated by driving the red and blue motional sidebands of the
qubit transition at frequencies $\omega_{q}\pm\omega_{m}$ (see Sec. S1) where
$\omega_{m}$ corresponds to the frequency of one of the motional modes of the
ion – here we use the in-plane radial mode. The SDF displaces the state of
this mode in position-momentum phase space where the sign of the displacement
depends on the qubit state [26]. By slightly detuning from the red and blue
sidebands, the motion describes loops in phase space – the central mechanism
of Mølmer-Sørensen (MS) two-qubit gates [24] – but we will mostly focus on the
resonant case for simplicity, and without loss of applicability to the detuned
case. Under a rotating wave approximation and ignoring negligible off-resonant
carrier driving, the SDF is described by the Hamiltonian (see Sec. S1)

$\hat{H}_{\text{SDF}}=\frac{\hbar}{2}\Omega_{\text{SB}}\hat{\sigma}_{x}\left(\hat{a}+\hat{a}^{\dagger}\right)\
,$ (1)

here written in the interaction frame where $\Omega_{\text{SB}}$ is the
sideband interaction strength, proportional to the gradient of the field
driving the ion. Re-written as

$\hat{H}_{\text{SDF}}=\frac{\hbar}{2}\Omega_{\text{SB}}\ket{+}\bra{+}\left(\hat{a}+\hat{a}^{\dagger}\right)-\frac{\hbar}{2}\Omega_{\text{SB}}\ket{-}\bra{-}\left(\hat{a}+\hat{a}^{\dagger}\right)$
(2)

the spin-dependent force is made explicit: the motion experiences a force
which is positive if the qubit is in state
$\ket{+}=(\ket{0}+\ket{1})/\sqrt{2}$ and negative if the qubit state is
$\ket{-}=(\ket{0}-\ket{1})/\sqrt{2}$.

We propose to selectively suppress the SDF through the choice of the phase of
an additional dynamical decoupling (DD) tone driving the qubit transition. For
the MW electrode geometry used in this experiment, the dynamical decoupling
(DD) drive phase changes with ion position, localising the effect to a chosen
region of space. The DD tone drives the interaction

$\hat{H}_{\text{DD}}=\frac{\hbar}{2}\Omega_{\text{DD}}e^{i\phi_{\text{DD}}}\hat{\sigma}_{+}+\text{h.c.}$
(3)

where $\hat{\sigma}_{+}=|1\rangle\langle 0|$, $\Omega_{\text{DD}}$ is the
strength of the DD drive, and the phase $\phi_{\text{DD}}$ depends on the ion
position and the phase of the injected microwave current. Note that if the DD
and SDF are in-phase, $\phi_{\text{DD}}=k\pi$ ($k\in\mathbb{Z}$), the
Hamiltonians commute, $[\hat{H}_{\text{SDF}},\hat{H}_{\text{DD}}]=0$, and the
dynamical decoupling does not alter the SDF dynamics. Whilst this has already
been demonstrated in two-qubit gates [9], here we also make use of the out-of-
phase case.

When the DD and SDF are out of phase $\phi_{\text{DD}}=\pi/2+k\pi$, DD driving
can suppress qubit-motion interaction. The DD tone will drive Rabi
oscillations between the states on which the SDF depends
$\ket{+}\leftrightarrow\ket{-}$, and, if
$\Omega_{\text{DD}}\gg\Omega_{\text{SB}}$, the rapid changes of SDF direction
will stop the motion from gaining significant amplitude in phase space. We
propose to combine this technique with a microwave electrode geometry that
enforces $\phi_{\text{DD}}=k\pi$ for ion positions where the SDF is desired,
and $\phi_{\text{DD}}=\pi/2+k\pi$ for ion positions where the SDF is
undesired. First, however, we experimentally demonstrate SDF suppression at a
fixed ion position.

Choosing to resonantly drive the sidebands, and subsequently measuring the
qubit state, provides a straightforward measurement of the SDF and its
suppression. We first demonstrate the effect of the SDF with no DD driving, as
shown in Fig. 2(a). Each experimental cycle starts by preparing the state
$\ket{0}\ket{0_{m}}$, where $\ket{0_{m}}$ designates the motional ground state
through microwave-enhanced optical pumping [13] and Raman-laser driven
sideband cooling [25]. A small thermal population does remain in the motional
mode (see Sec. S3), which is included in all our simulations, but omitted here
for simplicity. After driving the motional sidebands for a duration $t$, the
initial state

$\ket{0}\ket{0_{m}}=\frac{1}{\sqrt{2}}\left(\ket{+}+\ket{-}\right)\ket{0_{m}}$
(4)

will evolve under the SDF to

$\displaystyle\ket{\psi}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\ket{+}\ket{+\alpha}+\ket{-}\ket{-\alpha}\right)$
(5)
$\displaystyle=\frac{1}{2}\ket{0}\left(\ket{\alpha}+\ket{-\alpha}\right)+\frac{1}{2}\ket{1}\left(\ket{\alpha}-\ket{-\alpha}\right)\
,$

schematically shown in Fig. 2(a), where $\ket{\pm\alpha}$ designates a
coherent state with amplitude $\pm\alpha=\pm\Omega_{\text{SB}}t/2$. The
probability $P_{\ket{0}}$ of measuring the initial qubit state $\ket{0}$ then
decays to $1/2$ as $\alpha$ increases, following

$\displaystyle P_{\ket{0}}$
$\displaystyle=\frac{1}{4}\left(\bra{\alpha}+\bra{-\alpha}\right)\mathds{1}\left(\ket{\alpha}+\ket{-\alpha}\right)$
(6) $\displaystyle=\frac{1}{2}\left(1+e^{-2|\alpha|^{2}}\right)\ ,$

providing a measurement of the SDF (and later its suppression). For
simplicity, we describe the state $\ket{\psi}$ as pure, but in addition to
initial thermal population, coherence is degraded by the motional mode heating
rate of $\approx$370 quanta/s, comparable to the sideband interaction strength
$\Omega_{\text{SB}}/2\pi=380$ Hz. Loss of coherence increases the rate at
which $P_{\ket{0}}$ decays, but the physical intuition presented here remains
valid, and the measurement still provides a measurement of $\alpha$ once the
independently measured heating rate and initial motional mode occupation are
considered (see Sec. S3). The qubit state is read out by transferring
$\ket{0}$ to the “dark” 3D${}_{\text{5/2}}$ manifold and measuring the
probability of the ion fluorescing [27]. A measurement of the decay of
$P_{\ket{0}}$ to 0.5, as a consequence of the SDF, is shown in Fig. 2(b).

Driving the system at both DD and sideband frequencies, and varying their
relative phases, allows us to suppress the SDF. To demonstrate this, the qubit
state is measured after a pulse duration which, at most, will reduce
$P_{\ket{0}}$ to 0.6 when the DD drive commutes with the SDF. To ensure that
Rabi oscillations induced by DD driving do not have an impact on the final
qubit state (independently of the SDF), we switch the phase of the DD drive in
a Walsh-3 pattern [28] (see Sec. S4). The resulting data, shown in Fig. 2(c),
clearly shows that the SDF can be either undisturbed or suppressed by
selecting DD phases 0 or $\pi/2$ respectively. For this demonstration, we fix
the DD drive power injected into the trap such that the DD drive amplitude
$\Omega_{\text{DD}}=60~{}\Omega_{\text{SB}}$, which fulfills the requirement
for suppression $\Omega_{\text{DD}}\gg\Omega_{\text{SB}}$ (see Sec. S4). Note
that the microwave power of the dynamical decoupling tone, required to obtain
$\Omega_{\text{DD}}=60~{}\Omega_{\text{SB}}$, is two orders of magnitude
smaller than the power injected at sideband frequencies due to the small
effective Lamb-Dicke parameter ($\eta=1.25\times 10^{-3}$).

Figure 3:  Surface trap design giving a spatially-varying phase. (a) Schematic
of our surface trap. Two microwave electrodes produce interfering microwave
fields. These fields combine to produce $\text{B}_{\text{MW}}$ (solid arrow).
The component $\text{B}_{\text{x}}$ of this field (dashed arrow) – in the
direction of the quantisation axis, defined by a static field $\text{B}_{0}$ –
drives qubit transitions. The two microwave electrodes are connected 750
$\upmu$m away from the ion (grey dashed line) and their currents are
approximately $\pi$ radians out-of-phase (with a small phase offset
$2\epsilon$). (b) $\text{B}_{\text{x}}$ phasor in the complex plane (dots)
where the imaginary-axis has been magnified $\times 20$. Neighboring dots
correspond to ion positions $2~{}\upmu\text{m}$ apart in the x-direction. As
the ion is moved along the x-axis, the phase of the microwave field, dictating
the phase $\phi_{\text{DD}}$, changes by $\approx\pi$ radians. The phase of
the field gradient, however, remains constant and thus so does that of the
spin-dependent force. The change in phase of the field is used to obtain
suppression of the SDF away from the symmetry point of the microwave field.
Figure 4:  Spatially-selective ion-motion interaction. (a) Measured phase of
the microwave field (blue dots) and its gradient (grey dots), with fits (solid
lines) to a Biot-Savart model of the trap (see Sec. S2). Here $x=0$
corresponds to the null of the RF trapping field. The two shaded regions
correspond to the microwave field and its gradient being $\pi/2$ radians out-
of-phase (grey) or in-phase (blue). The gradient of the microwave field
features a slight change in its phase caused by a tilt in the direction of the
ion’s harmonic motion, consistent with simulated values (black dashed line).
(b) After preparing state $\ket{0}$ at position $x=0$, the ion is displaced to
position $x$, subjected to SB and DD driving, and returned to position $x=0$
where we measure the probability $P_{\ket{0}}$ of finding the qubit in its
initial state. The phase of the DD drive injected into the trap is kept
constant, but the the MW electrode geometry offsets the DD phase at the ion
position as shown in panel (a). This leads to a memory zone (grey) where the
SDF is suppressed and an interaction zone (blue), where it isn’t. Here, the
interaction zone is defined as where $P_{\ket{0}}$ is below 0.75. All error
bars indicate 68$\%$ confidence intervals.  Figure 5:  Simulated surface trap
with three microwave electrodes. (a) Schematic of the proposed trap design
using three microwave electrodes. The ion-chain lies perpendicular to the
microwave electrodes. A pair of ions (dark blue) have been selected to
demonstrate this scheme. Bar plots on each electrode show the required
sideband (red) and dynamical decoupling (blue) amplitudes to address this ion
pair. (b) Maximum crosstalk when driving a spin-dependent force on both ions
in a pair simultaneously. The DD phase (corresponding to the microwave field
phase) is shown for the ion pair selected in (a).

To use this technique to address individual ions, the phase difference between
the DD driving and SDF at different ion positions must vary – in our system
this arises from the microwave electrode geometry, schematically shown in Fig.
3(a). The ion is driven by near-field radiation generated by MW currents
flowing in a U-shaped electrode (see Ref. [25] for more details). Currents
propagate in opposite directions on each side of the ion (approximately $\pi$
radians out-of-phase), leading to destructive interference of the magnetic
field component $B_{x}$ aligned with the quantisation axis $B_{0}$ (see Fig.
3(a) for coordinate system), which is the component of the field that couples
to our “$\pi$-polarised” qubit transition. The interference leads to a change
in the phase of $B_{x}$ as a function of $x$, since the field switches from
being dominated by the field of one electrode to the other. This is
illustrated by plotting $B_{x}$ in the complex plane, see Fig. 3(b). The phase
of the gradient of the B-field however, which determines the phase of the SDF
(see Sec. S1), stays constant as the ion is moved across the trap.

The phase of the DD drive therefore changes with position, undergoing a phase
shift of $\approx\pi$ radians, whereas the phase of the SDF remains constant.
As a result, the DD and SDF can be in-phase at the symmetry of the field
whilst being $\approx\pm\pi/2$ out of phase away from it. An experimental
verification of this is shown in Fig. 4(a). The change in DD drive phase is
measured by preparing (and measuring) the $\ket{+}$ state at $x=0$ before (and
after) displacing the ion to position $x$, where it is subject to a MW pulse
with varying phase $\phi_{p}$. When averaged over multiple random pulse
amplitudes, sweeping the phase $\phi_{p}$ to maximise the probability of
measuring $\ket{+}$ at the end of the sequence constitutes a measurement of
the field’s position-dependent phase shift (see Sec. S2). The phase of the SDF
is measured relative to the DD by maximising $P_{\ket{0}}$ in a DD phase scan,
as described in Fig. 2(c), for different ion positions. Ion displacement is
achieved by varying the voltage delivered to the trap DC electrodes following
an analytical model of the trapping field (see Ref. [29]). Notably, we measure
a slight change in the SDF phase – rather than the desired constant phase –
which arises from a change in the direction of the ion’s harmonic motion,
which samples the microwave gradient in the $y$ direction. This results from
the limited control offered by our DC trap electrode geometry when displacing
the ion from the trap RF null (see Sec. S3).

The spatial variation in MW phase will “enable” the SDF for an ion displaced
by $x\approx 5~{}\upmu$m from the trap RF null but the SDF will be
increasingly suppressed by the DD drive when the ion is displaced out of this
“interaction zone”, creating a much broader “memory zone”. Experimental
verification of this is carried out as in Fig.2(c), but where the phase of the
DD tone injected in the trap is kept constant, and the ion position is varied
instead. To disentangle the change in relative phase of the DD from changes in
motional mode properties, we vary the pulse duration and sideband frequency
with position. (We refer here to changes in motional mode frequency, thermal
occupation or heating rate, as well as the change in direction of motion,
which are measured and presented in Sec. S3 but which would not be present in
applications of this technique where ions are not displaced, see Fig. 5.) The
pulse duration is chosen such that, starting in state $\ket{0}$ and in the
absence of DD driving, the SDF causes a decay of $P_{\ket{0}}$ to 0.51. The
measurements, shown in Fig. 4(b), reveal the variation in spin-dependent force
as an ion is moved across the surface trap, and the resulting “interaction”
and “memory” zones.

This technique, extended to two interaction zones, could be used to drive two-
qubit gates between arbitrary pairs of ions arranged in a chain, without
resorting to ion shuttling. We propose a different trap design which would
facilitate this, shown in Fig. 5(a), where the chain of ions is now
perpendicular to the microwave electrodes (e.g. by using a multi-layer surface
trap [30]). Here the additional interaction zone is enabled by a third
microwave electrode creating a second DD phase-flip (at a different position
to the first). The three electrodes are assumed to be fed by independent
microwave currents, where the central MW current is close to $\pi$ radians out
of phase with the other two. For our simulations, we set a phase difference
$(\pi-\epsilon)$ radians, where $\epsilon=3\times 10^{-5}$ (based on a typical
figure for the demonstrated accuracy in MW phase control [31]). The positions
of the two phase-flips, and corresponding interaction zones, can be set
through the amplitudes of the DD currents through each electrode. The SDF
strength at both zones can also be set independently by varying the SB
currents through each electrode.

We simulate the expected crosstalk when performing two-qubit gates between all
pairs in a chain of 17 ions, with a uniform ion spacing of $5~{}\upmu\text{m}$
and an ion height of $40~{}\upmu\text{m}$. We assume that the leading source
of crosstalk would be imperfect control over the spatial variation in DD
phase, i.e. $\phi_{\text{DD}}-\phi_{\text{SB}}\neq\pm\pi/2$ (see Sec. S5).
Crosstalk here measures the impurity of the partial trace of non-addressed
ions after a gate. In Fig. 5(b), we consider the worst affected non-addressed
ion for a given pair of addressed ions and so, since we are not showing the
average crosstalk across the chain, we refer to this as “maximum crosstalk”.
Alternatively, if we consider the mean crosstalk across the chain, we find a
crosstalk error of $1.04(9)\times 10^{-5}$ (averaging over all addressed ion
pairs). This shows that all-to-all connectivity is possible with error rates
far below typical error correction thresholds. Errors are lowest between the
electrodes – the natural location for creating field interference – but
increase as ion pairs are moved further out. Adding more microwave electrodes
or bringing the phase difference between currents closer to $\pi$ radians
reduces these errors and would enable even larger registers of ions.

For this design, we have considered the chain of ions to be perpendicular to
the MW electrodes, for an implementation of gates on axial motional modes.
However, changing the angle between the ion chain and MW electrodes would also
enable the use of radial motional modes. As the ion chain gets closer to being
parallel with the MW electrodes, the radial MW gradient increases, and
narrower interaction zones are required. Alternatively, one could use separate
electrodes for dynamical decoupling (perpendicular to the ion chain) and
sideband driving (parallel to the ion chain, maximising the radial MW
gradient). Neither electrode direction should however be perpendicular to the
quantisation axis.

Figure 6:  Single-qubit randomised benchmarking of the gate suppression error.
(a) The residual error after suppressing a gate is experimentally estimated by
embedding Mølmer-Sørensen (MS) gate pulses, applied to a single ion and
suppressed through a DD drive ($\phi_{\text{car}}=\pi/2$), in a single-qubit
randomised benchmarking (RB) sequence. (b) An error of $3.7(4)\times 10^{-4}$
is demonstrated (blue), indistinguishable from the error induced by DD driving
alone (red dashed line) indicating a complete suppression of the ion-motion
interaction. The DD driving error is partially accounted for by laser leakage
and qubit decoherence, measured by replacing the MS pulse by a delay of equal
duration (grey dashed line) (see Sec. S6). Error bars indicate 68$\%$
confidence intervals of the measurements and the shaded region indicates a
68$\%$ confidence interval of the fit.

We measure to what extent this crosstalk would be dominated by other effects
in our system by driving a suppressed Mølmer-Sørensen (MS) interaction in a
randomised-benchmarking type measurement [32] (see Sec. S6). We detune the
sidebands by $2\pi\times 770$ Hz in a 1.30 ms pulse to emulate – on a single
ion radial mode – a 1-loop MS gate for a Rabi frequency of
$\Omega_{\text{SB}}=2\pi\times 380$ Hz, and apply a DD drive with phase
$\phi_{\text{DD}}=\pi/2$ to suppress this interaction. The “suppressed MS
gate” is applied after every Clifford gate of a single-qubit randomised
benchmarking sequence. We obtain an error per “suppressed MS gate” of
$3.7(4)\times 10^{-4}$, shown in Fig. 6. DD driving alone produces the same
error, $4.0(6)\times 10^{-4}$, suggesting that residual qubit-motion
entanglement is negligible compared to this measured error, which is probably
of technical origin. This error is still far below typical error-correcting
thresholds, suggesting strong potential for scaling to multi-ion registers.
Finally, we measure the error arising if the MS-pulse is replaced by an
equally long time delay, revealing that about half of the error, $1.7(2)\times
10^{-4}$, is not related to the microwave driving at all.

In conclusion, we have demonstrated a method to focus a microwave-driven spin-
dependent force (SDF) to a sub-ion-spacing region. To do so, we make use of
(1) SDF suppression using dynamical decoupling (DD), and (2) a microwave
electrode geometry which generates interference, bringing the DD drive in- and
out-of-phase with the SDF. Building upon these concepts, we have proposed a
surface trap design to create and move two interaction zones over a chain of
ions. We predict that this will enable all-to-all connectivity between ions
with crosstalk levels far below error-correction thresholds. The crosstalk
error of $3.7(4)\times 10^{-4}$, emulated in a single-ion benchmarking
experiment, supports this prediction.

This technique could make microwave-driven logic a more practical approach to
constructing a large-scale universal quantum processor by reducing its
reliance on shuttling ions. Once two-qubit gates implemented on a chain of
ions become limited by multi-mode effects [33, 34], one could then rely on
shuttling between different logical zones [35] or optical networking between
different zones [36]. The proposed microwave electrode geometry can also
generate a microwave amplitude gradient enabling addressed single-qubit gates
with very low ($<10^{-4}$) errors [16], which, combined with the method for
addressed two-qubit gates presented in this report, forms a universal,
addressed, gate set. Finally, we believe this method could be applied to other
QC platforms where dynamical decoupling is used [2, 4, 5, 6, 7] to mitigate
qubit addressing or crosstalk problems.

Acknowledgments This work was supported by the U.S. Army Research Office (ref.
W911NF-18-1-0340) and the U.K. EPSRC Quantum Computing and Simulation Hub.
M.C.S. acknowledges support from Balliol College. A.D.L. acknowledges support
from Oxford Ionics Ltd. M.F.G. acknowledges support from the Netherlands
Organization for Scientific Research (NWO) through a Rubicon Grant.

Author Contributions M.C.S., A.D.L. and M.F.G. contributed equally to this
work. A.D.L. carried out simulations of the scheme for this experiment and
larger ion registers. M.C.S. and M.F.G. acquired and analysed the data. M.C.S.
and M.F.G. wrote the manuscript with contributions from all authors. M.C.S.,
A.D.L. and M.F.G. upgraded and maintained the experiment. D.M.L. and M.F.G.
supervised the project.

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Supplementary information

##  S1 Hamiltonian derivation and numerical approach

We first make the assumption that motional modes (other than the in-plane
rocking mode) and internal ion states (other than the qubit states) play
“spectator” roles and are not involved in the observed physics (justification
in Sec. S4.4). Since we make use of a “$\pi$” qubit which is only sensitive to
linearly polarized magnetic fields, in the direction of the quantization axis
“$x$”, we are interested in the interaction between the magnetic field
component $B_{x}$ (expressed here as a phasor) and the ion. Coupling to the
ion’s harmonic motion (oriented along a direction
$\underline{u}=q_{x}\underline{x}+q_{y}\underline{y}$) arises from the
changing field amplitude and phase as the ion moves by a distance $u$ with
respect to its equilibrium position $u=0$

$\displaystyle B_{x}(u)$ $\displaystyle=B_{x}(0)+u[\partial_{u}B_{x}]|_{u=0}$
(S1)
$\displaystyle=B_{x}(0)+u\left[q_{x}\partial_{x}B_{x}+q_{y}\partial_{y}B_{x}\right]|_{u=0}\
.$

Under these assumptions, the interaction-picture Hamiltonian (in the rotating
wave approximation) describing the interplay of microwave field, qubit states
and ion motion reads [12]

$\displaystyle\hat{H}=$
$\displaystyle\frac{\hbar\Omega}{2}e^{-i\left(\omega-\omega_{q}\right)t}\hat{\sigma}_{+}$
(S2) $\displaystyle\otimes\left(e^{i\phi_{\text{field}}}+\eta
e^{i\phi_{\text{grad}}}\left(e^{-i\omega_{m}t}\hat{a}+e^{i\omega_{m}t}\hat{a}^{\dagger}\right)\right)+\text{h.c.}$

where:

  * •

$\omega$, $\omega_{q}$, $\omega_{m}$ are the frequencies of the microwave
field, qubit transition, and ion motion respectively

  * •

$\hat{\sigma}_{+}=\ket{1}\bra{0}$ acts on the qubit states $\ket{0},\ket{1}$

  * •

$\hat{a}$ is the annihilation operator for the ion’s harmonic motion, related
to the ion’s displacement operator
$\hat{u}=u_{\text{zpf}}(\hat{a}+\hat{a}^{\dagger})$ through the zero-point
fluctuations in motion $u_{\text{zpf}}=\sqrt{\hbar/(2M\omega_{m})}$ ($M$ is
the mass of the ion)

  * •

$\phi_{\text{field}}$, $\phi_{\text{grad}}$ are the phase of the field $B_{x}$
and its gradient $\partial_{u}B_{x}$ respectively

  * •

$\Omega$ is the Rabi-frequency, quantifying the strength of the magnetic field
and the magnetic dipole moment $\mu_{x}$ of the qubit transition
$\hbar\Omega=\mu_{x}|B_{x}|$

  * •

$\eta$ is the effective Lamb-Dicke parameter, quantifying the relative change
in field resulting from zero-point fluctuations
$\eta=u_{\text{zpf}}|\partial_{u}B_{x}|/|B_{x}|$.

For a single microwave tone on resonance with the qubit ($\omega=\omega_{q}$)
we can neglect the motional coupling under the assumption
$\eta\Omega\ll\omega_{m}$ and write the dynamical decoupling (DD) driving
Hamiltonian shown in Eq. (3). The SDF arises in the case where the red and
blue motional sidebands are driven with two microwave tones at frequencies
$\omega_{q}-(\omega_{m}+\delta)$ and $\omega_{q}+(\omega_{m}+\delta)$
respectively. Under the assumption $|\delta|\ll\omega_{m}\ll\omega_{q}$, we
first discard terms rotating at the qubit frequency to obtain

$\displaystyle\hat{H}_{\text{SB}}$
$\displaystyle=\hat{H}_{\text{off}}+\hat{H}_{\text{SDF}}$ (S3)
$\displaystyle\hat{H}_{\text{SDF}}$
$\displaystyle=\frac{\hbar\eta\Omega}{2}\hat{\sigma}_{+}\left(e^{i\delta
t}\hat{a}+e^{-i\delta t}\hat{a}^{\dagger}\right)+\text{h.c.}$
$\displaystyle\hat{H}_{\text{off}}$
$\displaystyle=\frac{\hbar\Omega}{2}e^{i(\phi_{\text{field}}-\phi_{\text{grad}})}\hat{\sigma}_{+}\left(e^{i\left(\omega_{m}+\delta\right)t}+e^{-i\left(\omega_{m}+\delta\right)t}\right)+\text{h.c.}\
$

The resulting spin-dependent force Hamiltonian $\hat{H}_{\text{SDF}}$ was
presented in the main text (Eq. 1) using the sideband interaction strength
$\Omega_{\text{SB}}=\eta\Omega$ and $\delta=0$. The second part of the
Hamiltonian, $\hat{H}_{\text{off}}$, representing off-resonant carrier
driving, is better rewritten as

$\displaystyle\hat{H}_{\text{off}}$ $\displaystyle=\hbar\Omega
e^{i(\phi_{\text{field}}-\phi_{\text{grad}})}\hat{\sigma}_{+}\cos\left(\left(\omega_{m}+\delta\right)t\right)+\text{h.c.}\
.$ (S4)

It drives small oscillations on the Bloch sphere (as long as
$\Omega\ll\omega_{m}+\delta_{m}$) and has a negligible effect (see Sec. S4.3).

In simulations, we take into account the interaction of the motional mode’s
heating rate $\dot{\bar{n}}$ and initial thermal occupation $\bar{n}$. The
quantum dynamics of this open quantum system is captured by the Lindblad
master equation

$\displaystyle\dot{\hat{\rho}}=$
$\displaystyle-\frac{i}{\hbar}\left[\hat{H}_{\text{SDF}}+\hat{H}_{\text{off}}+\hat{H}_{\text{DD}},\hat{\rho}\right]$
(S5)
$\displaystyle+\frac{\dot{\bar{n}}}{2}\left(2\hat{a}\hat{\rho}\hat{a}^{\dagger}-\hat{a}^{\dagger}\hat{a}\hat{\rho}-\hat{\rho}\hat{a}^{\dagger}\hat{a}\right)$
$\displaystyle+\frac{\dot{\bar{n}}}{2}\left(2\hat{a}^{\dagger}\hat{\rho}\hat{a}-\hat{a}\hat{a}^{\dagger}\hat{\rho}-\hat{\rho}\hat{a}\hat{a}^{\dagger}\right)\
,$

where the initial state of the motional mode is a thermal state with average
occupation $\langle\hat{n}\rangle=\bar{n}$. Numerical simulation of this
master equation is carried out using the Julia QuantumOptics package [37].
Most of the parameters used in the simulations are dependent on the
displacement of the ion with respect to the RF-null of the trap (see Sec. S3).

##  S2 Microwave field and Biot-Savart model

The microwave fields generated by the electrodes in the measured trap and in
the proposed three-electrode trap of Fig. 5 are calculated using the Biot-
Savart law. The underlying assumption is that, since the ion-electrode
distance ($40\ \upmu$m) exceeds the dimensions of the electrode ($5\ \upmu$m
height and $8\ \upmu$m width), the electrode is well approximated by a 1D
wire. The total field is computed by adding the magnetic fields produced by
two of these effective wires, corresponding to currents traversing the
microwave electrode on each side of the ion. We fit this model to the measured
change in microwave-field amplitude and phase shown in Fig. 4. The free
parameters in this fit are the relative phase and amplitude in the currents
traversing the two wires. These parameters are determined (in an idealised
microwave electrode) by the losses and boundary condition of the electrode
respectively [25]. In our experiment, we find that we cannot predict the
fitted values from the length and qualify factor of the microwave electrode
(resonator). As is covered in detail in Ref. [25], coupling of the microwaves
to the RF electrode plays an important role. Regardless of these issues, as
shown in Fig. S1, the Biot-Savart model with two free parameters leads to
excellent fits to the measured microwave field. We determine the relative
phase to be 0.0480 radians and the relative amplitude to be 0.856. We
therefore take the same approach in computing the field for the three
electrode case of Fig. 5.

Figure S1:  Microwave field amplitude and phase. (a) Measured microwave field
amplitude (blue) compared to a fitted Biot-Savart model (red), relative to the
amplitude at 0 displacement. The amplitude is measured by preparing (and
measuring) the $\ket{0}$ state at x = 0 before (and after) displacing the ion
to position x, where a MW pulse of fixed amplitude and duration is applied.
(b) Microwave field phase. The phase is measured by preparing (and measuring)
the $\ket{+}$ state at $x=0$ before (and after) displacing the ion to position
$x$, where it is subject to a MW pulse with varying phase $\phi_{p}$. When
averaged over multiple random pulse amplitudes, sweeping the phase $\phi_{p}$
to maximise the probability of measuring $\ket{+}$ at the end of the sequence
constitutes a measurement of the field’s position dependent phase shift.
Figure S2:  Calibrations at displaced ion positions. (a) Measured in-plane
mode secular frequency (light blue) compared to simulated values for the in-
plane mode (red) and out-of-plane mode (orange). Measured motional mode
occupation (b) and heating rate (c) after sideband cooling at x=0 and
displacing the ion to position x.

##  S3 Displaced-ion calibrations

The ion is displaced away from the RF null in the x direction by changing the
voltage applied to the DC electrodes, which has the undesired effects of
changing the radial motional mode frequency $\omega_{m}$, the axis of motion
(characterized by the tilt angle $\theta$), and the heating rate
$\dot{\bar{n}}$, as well as introducing an additional heating mechanism (we
move the ion after its sideband cooling sequence) which modifies the initial
thermal population $\bar{n}$. Here we explain how we disentangle these effects
from the change in microwave field to best illustrate the ideas presented in
Figs. 1,5.

We first perform calibrations of the mode frequency, heating and thermal
occupations at different ion displacements from the RF null. The resulting
calibrations are shown in Fig. S2. The motional mode tilt angle $\theta$
(which we cannot straightforwardly measure) is then the only free parameter in
fitting the decay of $P_{\ket{0}}$, assuming that the microwave field is well
described by the Biot-Savart model. Indeed, a change in $\theta$ changes the
axis $\underline{u}$ along which the microwave gradient should be computed
(Eq. S1), in turn changing the sideband interaction strength. The angle
$\theta$ is thus determined by fitting a pulse duration scan, as shown for
example in Fig. S3(b), with a simulation of the Lindblad master equation S5
using a least-squares minimization routine [38]. The resulting field gradient
extracted at different positions is shown Fig. S3(a).

This approach relies on the accuracy of the “Biot-Savart” model. One way in
which we test this assumption is to check whether the fitted change in tilt
angle is consistent with the expected pseudopoential at the displaced ion
positions. By making the assumption that these electrodes have infinitely
small gaps between them, we can analytically estimate the pseudopoential
governing the ion’s harmonic motion [29]. The RF contribution to the
pseudopotential is shown in Fig. S4(a). The direction of motion is then given
by the local curvature of this pseudopotential, which shows good agreement
with the value independently extracted through measurements of the SDF, see
Fig. S4(b). Note that there is an intentional $15^{\circ}$ tilt imposed at
zero displacement, in order for the out-of-plane radial motion of the ion to
have a small projection on the horizontally-delivered Doppler cooling laser
beams [25]. The second way in which we verify the extracted tilt angles is by
comparing the measured phase of the microwave gradient – which would be
constant if $\theta=0$ – to values predicted by the extracted tilt angle
$\theta$. The comparison is shown in Fig. S5 and again demonstrates a good
understanding of the change in motional tilt angle.

Figure S3:  Amplitude of the gradient of the microwave field. (a) Measured
amplitude of the gradient of the microwave field (blue) relative to the
amplitude at zero displacement. This is compared to the predicted amplitude
from the Biot-Savart model, not accounting for motional mode tilting (red).
(b) Fitted simulation of a sideband pulse duration scan used to determine the
microwave gradient, based on independently-determined mode properties (heating
rate, thermal occupation, frequency).  Figure S4:  Estimated motional mode
tilt. (a) Simulated pseudopotential generated by RF trapping fields in our
surface trap. At zero displacement, the DC electrode voltages are chosen such
that the motional mode tilt angle, $\theta$, is 15∘. As the ion is displaced,
the curvature of the potential changes, causing the direction of the radial
motional modes to tilt. (b) Tilt angle of the in-plane mode (light blue)
extracted from the measured MW field gradient of Fig. S3(a) and compared to
the expected tilt angle predicted from the simulated pseudopoential (red).
Figure S5:  Phase of the gradient of the microwave field. Measured phase of
the gradient of the microwave field (blue) compared to a Biot-Savart model
when neglecting (red dashed line) or accounting for (orange solid line) the
motional mode tilt. The phase of the gradient is measured relative to the
field by maximising P${}_{\ket{0}}$ in a DD phase scan, as described in Fig.
2(c). The phase of the gradient alone is then extracted using the
independently measured phase of the field [Fig. S1(b)].

##  S4 Microwave pulse sequences

###  S4.1 Accounting for motional mode changes with displacement

Keeping in mind our intention to demonstrate the ideas of Figs. 1,5, we vary
the microwave drive parameters to simulate the driving of an ion chain with
three electrodes. In an ion chain, the motional mode is shared by all the
ions, and therefore mode characteristics are independent of ion position. We
simulate this by changing the microwave drive frequencies as the ion is
displaced from the RF null such that resonance with the motional sidebands is
maintained. Additionally, changes in motional occupation, heating rate, zero-
point motion and motional tilt angle are disentangled from the effective SDF
measurement shown in Fig. 4 by adjusting the microwave pulse duration. The
appropriate pulse duration for a given position is determined through a pulse
duration scan. After fitting the decay of $P_{\ket{0}}$ to a Gaussian (Eq. 6),
we determined the pulse duration required to reach an excited state population
of 0.51. These values are shown for different ion positions in Fig. S6. By
using this pulse duration, we maximise the change in $P_{\ket{0}}$ to best
reveal the SDF suppression, as shown in Fig. 4(b).

Figure S6:  Required pulse duration to maximally entangle spin and motion.
Sideband pulse duration required to reach P${}_{\ket{0}}=$ 0.51 when driving
the SDF at displaced ion positions. This is determined by fitting a Gaussian
decay to a SB pulse duration scan (as in Fig. 2(b)).

###  S4.2 Dynamical decoupling drive amplitude and Walsh sequence selection

The DD drive should fulfill two conditions: (1) its strength should exceed
that of the sideband interaction to suppress the SDF and (2) any rotations
that it induces should be reversed by the end of the sequence. In this section
we explain how these two requirements are achieved through selection of an
appropriate Walsh scheme [28].

Figure S7:  Dynamical decoupling amplitude required to suppress the SDF. For a
Walsh-3 scheme, the DD strength $\Omega_{\text{DD}}$ is varied relative to a
fixed sideband interaction strength $\Omega_{\text{SB}}$, illustrating that
suppression occurs when the DD driving dominates
$\Omega_{\text{DD}}\gg\Omega_{\text{SB}}$. For DD strengths where
$P_{\ket{0}}=1$ was measured for all 1000 measurement shots, we plot
$P_{\ket{0}}=0.9995$. This is compared to a simulation (red), where the
maximum suppression error follows the empirically determined scaling of Eq.
(S6) (black dashed line).  Figure S8:  Comparison of different Walsh
sequences. (a/b) Measured excited state population for varying DD strengths,
when using a Walsh-1 (dark blue) or Walsh-3 (red) pulse scheme to cancel qubit
rotations induced by DD. At lower DD strengths, the maximum measured
suppression of the motional interaction follows the empirically determined
scaling of Eq. (S6) (black dashed line), as shown in Fig. S7. At larger DD
strengths, the measured suppression of the motional interaction is limited by
incomplete cancellation of the effect of the DD. This was confirmed by
detuning the sidebands 100 kHz from the secular frequency such that no
motional interaction is driven (data: light blue/orange points, fits: blue/red
dashed lines). The grey area shows the state preparation and measurement
(SPAM) error. (c) Measured DD strength compared to sideband strength for
displaced ion positions (deduced from the data shown in Fig. S1 and Fig. S3).
The light grey dashed lines indicate the minimum and maximum strengths in the
range of ion displacements used for these experiments. This shows that we can
sufficiently suppress the SDF for this entire range of DD/sideband strengths
when using a Walsh-3 sequence.

The requirement for suppression, $\Omega_{\text{DD}}\gg\Omega_{\text{SB}}$, is
illustrated in Fig. S7 for a Walsh-3 pulse sequence. The suppression error
oscillates with DD strength, presumably due to the frequency of the Rabi
oscillations driven by the DD increasing with DD strength, while the pulse
duration remains constant, such that the suppression is increased at some
integer number of $\Omega_{\text{DD}}/\Omega_{\text{SB}}$. Therefore, a fairer
estimate of the expected suppression error is the maximum error, for which,
through simulations, we determine an empirical scaling

$1-P_{\ket{0}}=8\left(\frac{\Omega_{\text{DD}}}{\Omega_{\text{SB}}}\right)^{-2}.$
(S6)

Mismatch between simulated and measured outcomes is expected due to an unknown
transfer function distorting the DD and sideband amplitudes at the ion’s
position, relative to the amplitudes at the signal generator. We note that the
strength of the sidebands, $\Omega_{\text{SB}}$, includes the effective Lamb-
Dicke parameter $\eta$, which for our experiment is $1.25\times 10^{-3}$. This
means that we can fulfill the condition
$\Omega_{\text{DD}}\gg\Omega_{\text{SB}}$ using a lower power microwave
current for the DD than the sidebands.

For stronger DD driving, we encounter the limits of the Walsh-3 pulse
sequence, which we will now investigate in more detail. A Walsh pulse scheme
is used to ensure that no net rotation is driven by the DD by the end of the
pulse sequence. In this scheme each MW pulse is split into $2^{N}$ pulses of
equal duration and the phases of half of these pulses are shifted by $\pi$
radians. The choice of pulses which are $\pi$-shifted follows a Walsh sequence
of order $2^{N}-1$. A Walsh sequence of order 0 is the trivial case of a
single pulse. A Walsh sequence of order 1 (schematically shown in Fig. S8(a))
corresponds to two $\pi$-shifted pulses. In this case any rotation accumulated
in the first pulse will be cancelled in the second, assuming the DD drive
strength remains constant. A Walsh sequence of order 3 follows a more complex
phase shifting scheme featuring four seperate pulses (schematically shown in
Fig. S8(b)). The advantage gained from this more complex pulse sequence is
cancellation of rotations even if the DD drive strength varies linearly
throughout the sequence. A Walsh sequence of order 7 will cancel rotations in
the presence of quadratic changes, order 15 cancels cubic changes, and so on.
Here, we compare the Walsh-1 scheme to the Walsh-3 scheme used in most other
experiments, except the benchmarking experiment described in Sec. S6, which
uses Walsh-15.

For varying amplitudes of DD driving, and different Walsh orders, we measured
the change in initial state following a “suppressed SDF” pulse as described in
Fig. 2(d). In Fig. S8, we show that the Walsh-1 sequence is less effective at
cancelling the net effect of the DD at high DD drive amplitudes than the
Walsh-3 scheme. This would suggest that the drive strength varies linearly
with time throughout the sequence, for example due to thermal transients,
making Walsh-3 more effective. This is verified in Fig. S8 by performing the
same experiment but with the sideband frequencies symmetrically far-detuned
from the motional sideband (by $2\pi\times$100 kHz), hence solely observing
the consequences of DD driving.

As a result of the microwave field interference underpinning the DD phase
shift, the amplitude of the DD drive will decrease with field amplitude, as
shown in Fig. S1. We chose however to keep the microwave power driving the DD
transition constant, to best represent the situation that would be present in
Fig. 5. Furthermore, we select a DD drive amplitude which does not break the
suppression condition $\Omega_{\text{DD}}\gg\Omega_{\text{SB}}$ as the ion is
displaced away from the RF null. Note that this decision makes our experiment
a “worst possible” case, as a lower DD drive could be chosen to relax this
condition at the interaction zone, minimising interference of the DD and gate
interaction. The range of relative DD amplitudes sampled as the ion is
displaced is shown in Fig. S8(c).

###  S4.3 Impact of off-resonant carrier excitation

Here we give support to the claim that off-resonant carrier excitations driven
by sidebands $\hat{H}_{\text{off}}$ have negligible consequences. The first
impact of this term to consider is the net rotation it may drive on the qubit
over the course of a pulse. The second issue to discuss is that this term does
not commute with the SDF in the interaction zones, and does not commute with
the DD in the memory zones.

From Eq. S4, ignoring phases, and for resonant sideband driving, the
interaction Hamiltonian of interest is

$\hat{H}_{\text{off}}=\Omega\cos\left(\omega_{m}t\right)\hat{\sigma}_{x}$ (S7)

where $\Omega=\Omega_{\text{SB}}/\eta$ is the strength of the drive and
$\omega_{m}$ is the detuning of the sidebands from the qubit frequency. At the
RF null, we have used $\Omega_{\text{SB}}=2\pi\times$ 380 Hz and
$\omega_{m}=2\pi\times$ 5.7 MHz. Disregarding its commutation relation with
other terms of the Hamiltonian, and assuming an initial qubit state $\ket{0}$,
$\hat{H}_{\text{off}}$ will drive small oscillations of
$\langle\hat{\sigma}\rangle$ with period $2\pi/\omega_{m}$ and amplitude
$2(\Omega/\omega_{m})^{2}$ when $\Omega\ll\omega_{m}$. We ramp the sideband
microwave amplitude following a $\sin^{2}$ shape over a duration of $2.4\
\upmu$s $\gg 2\pi/\omega_{m}$ to ensure that no net rotation is driven by the
end of the gate.

We study the impact of the non-commutativity of $\hat{H}_{\text{off}}$ with
other terms through simulation. We restrict our analysis to the parameter
regime used in this demonstration at the RF null. For the memory zone,
$\hat{H}_{\text{off}}$ does not commute with $\hat{H}_{\text{DD}}$, and
therefore may limit its effectiveness. This is already taken into account in
the DD strength simulation of Fig. S7, demonstrating that for sufficiently
large DD drive strengths, there is no determined effect due to off-resonant
(and out-of-phase) carrier driving. For the interaction zone, we simulate the
fidelity of a hypothetical single-loop two-qubit gate using the same sideband
interaction and DD drive strengths as in our single-ion experiment at the RF-
null. The gate duration of such a gate would be $1.27$ ms, consistent with
typical microwave gate durations [9]. The Walsh-3 scheme is applied to the DD
drive in the simulated gate. The impact on (Bell-state preparation) fidelity,
even for off-resonant driving rates up to ten times larger than present in the
current experiment, is negligible ($\lesssim 10^{-6}$). Note that the strength
of the off-resonant carrier excitation, for a given gate duration, will be
inversely proportional to the effective Lamb-Dicke parameter $\eta$. The fact
that, in the scheme described in Fig. 5, gates are driven where the microwave
field is suppressed by interference will increase $\eta$ and decrease the
relevance of off-resonantly driven carrier excitations in interaction zones.

###  S4.4 Microwave pulse ramping scheme

The microwave amplitude of both sideband and DD drive should be ramped up and
down slowly to adiabatically transition in and out of hybridised states of:

  1. 1.

The qubit and out-of-plane motional spectator mode, driven with strength
$\Omega/2\pi<380$ Hz by the sidebands detuned by $\Delta/2\pi\gtrsim$300 kHz

  2. 2.

The two qubit states, driven off-resonantly with strength $\Omega/2\pi\lesssim
640$ kHz by the sidebands detuned by $\Delta=\omega_{m}\sim 2\pi\times$3–6 MHz

  3. 3.

The qubit and spectator hyperfine states, driven by both DD and sidebands
detuned by $\Delta/2\pi\sim$100 MHz

The impact of these off-resonant interactions on qubit state populations will
be, following first order perturbation theory, of order
$\sim(\Omega/\Delta)^{2}$. This renders the impact of the out-of-plane mode
negligible. For the other two interactions, the DD and sideband ramping rate
should far exceed the period of oscillations being driven ($\sim 1/\Delta$):
$10$ ns and $300$ ns respectively. Otherwise, these interactions would have an
impact of order $10^{-4}$ and $10^{-2}$ respectively. To satisfy this
condition, each pulse of the Walsh modulation sequence starts and ends with a
sin2 shaped ramping period of durations $120$ ns and $2.4\ \upmu$s for the DD
and sidebands respectively. For effective suppression of the SDF, we always
ramp up/down the DD driving before/after ramping up the sideband drive.

###  S4.5 Compensating for AC Zeeman shifts

The microwave sidebands off-resonantly drive the qubit, as well as “spectator”
hyperfine transitions, hybridising the qubit states and causing AC Zeeman
shifts.

The first contribution to these shifts $\Delta_{\pi}$, comes from
$\pi$-polarised fields (see Fig. S9(a)). A $\pi$-polarised (linearly polarized
in the $x$ direction) magnetic field drives transitions where $\Delta M=0$,
such as the qubit transition. Since the red and blue sidebands are oppositely
detuned from the qubit frequency
($\omega_{\text{r}}=\omega_{q}-\delta_{\text{r}}$ and
$\omega_{\text{b}}=\omega_{q}+\delta_{\text{b}}$) the red sideband will
increase the qubit frequency whilst the blue sideband will decrease the qubit
frequency. These contributions cancel with sidebands of equal amplitude, but a
small imbalance in the sidebands can be used to compensate the other AC Zeeman
shift.

A $\sigma$-polarised magnetic field will cause frequency shifts of transitions
where $\Delta M=\pm 1$, connecting the qubit states to states with $M=0,2$
(see Fig. S9(b)). Contrary to the $\pi$-shift, both sidebands have a similar
detuning to each off-resonantly driven transition, and will cause shifts in
the same direction. However, $\sigma_{-}$/$\sigma_{+}$ transitions connecting
the qubit to lower and higher $M$ states are negatively/positively detuned
from the sideband frequencies, creating shifts
$\Delta_{\sigma_{-}}$/$\Delta_{\sigma_{+}}$ of opposite signs.

The $\sigma$-shifts do not completely cancel, resulting in a microwave-power
dependent shift $\Delta_{\sigma}=\Delta_{\sigma_{-}}+\Delta_{\sigma_{+}}$.
This shift can be cancelled (for all amplitudes), by imposing a small sideband
amplitude imbalance, which has a much larger impact on the magnitude of
$\Delta_{\pi}$ than $\Delta_{\sigma}$, as shown in Fig. S9(c). This scheme
loses effectiveness around x $=5\ \upmu$m, requiring a much larger imbalance,
as the $\pi$-polarized field gets smaller (Fig. S1(a)) whereas the
$\sigma^{-}$field undergoes little change. Whilst we use this approach to
cancel AC Zeeman shifts, detuning microwave tones is an alternative solution,
although it has the added complexity of amplitude dependence in the shift.

We calibrate the imbalance required to cancel all AC Zeeman shifts by applying
the sidebands together with a weak tone at the qubit frequency, calibrated to
map the initial state $\ket{0}$ to $\ket{1}$ when on resonance. The required
imbalance is determined by minimising $P_{\ket{0}}$ for different values of
the imbalance. The change in required imbalance is shown in Fig. S9(d), and
compared to the expected imbalance based on the change in MW field (Fig.
S1(a)). Mismatch to the theoretical value is expected due to an unknown and
frequency-dependent transfer function distorting the RSB/BSB amplitude at the
ion’s position ($\Omega_{\text{RSB,BSB}}$), relative to the amplitude
($A_{\text{RSB,BSB}}$) at the signal generator.

Figure S9:  Compensating for the AC Zeeman shift. (a) Red and blue sidebands
are oppositely detuned from the $\pi$-transition, creating AC Zeeman shifts in
opposite directions. (b) Red and blue sidebands are both detuned from the
$\sigma$-transitions in the same direction. However, both sidebands create
$\sigma_{-}$ and $\sigma_{+}$ AC Zeeman shifts, which cancel themselves if the
manifold is symmetric. The schematic shows the shift caused a single sideband
on one energy level, but an identical shift will be produced both by the other
sideband and by both sidebands on the ground state of the qubit. (c) Total
shifts created by both sidebands for the $\sigma$ (light blue) and $\pi$ (dark
blue) transitions, at ion displacements of x $=0~{}\upmu\text{m}$ and x
$=5~{}\upmu\text{m}$. The amplitude of the blue sideband tone can be adjusted
such that the two shifts cancel each other out (vertical dashed line). (d)
Measured (blue) and expected (red) sideband amplitude imbalance required to
compensate for the net AC Zeeman shift experienced by the ion. The expected
imbalance is determined from the change in MW field amplitude (Fig. S1(a)).

##  S5 Ion chain crosstalk estimation

In this section, we discuss the assumptions and limitations of the crosstalk
simulation presented in Fig. 5. We consider a chain of 17 equally-spaced ions,
with an ion-ion distance of $5~{}\upmu$m, located $40~{}\upmu$m above a
surface trap. These values were chosen to match the ion height and typical
inter-ion-spacing used in our trap. We envisage a geometry where, unlike our
present trap, the MW electrodes are perpendicular to the ion string; this
could be created using a multi-layer surface trap [30]. The microwave
electrodes, $40~{}\upmu$m apart from each other, were simplified to wires to
model the magnetic field using the Biot-Savart law; this was shown to provide
reliable predictions for our electrode geometry (see Sec. S2). The three
electrodes are assumed to be fed by independent microwave currents, where the
central MW current is out-of-phase with the other two, but offset by $3\times
10^{-5}$ radians. This value is based on experiments of Ref. [31], where phase
control was sufficient to suppress the MW amplitude by a factor 40 with
respect to an ion position 350 $\upmu$m away from the point of maximum
interference. Using the Biot-Savart model of Sec. S2, we estimate the phase
uncertainty demonstrated in Ref. [31] to be $\sim 3\times 10^{-5}$ radians.

For every pair of ions, we use an optimisation routine (least-squares method)
to determine the amplitudes of the sideband and DD currents required to
address the selected ion-pair. Firstly, the amplitudes of the sideband
currents are chosen to minimise the difference in field gradient at the two
addressed ion positions. This ensures the spin-dependent force will be equal
at both positions during a two-qubit gate. Secondly, the DD currents are
chosen to minimise the phase-difference between the field and the gradient at
the two addressed ion positions.

Using the optimum current configuration, we then compute the field and its
gradient at all ion positions to determine the residual spin-dependent force.
An example of the amplitudes and phases of the fields is shown in Fig. S10(a)
and (b).

Using Eq. S3, we can write the Hamiltonian for the interaction at a given
position as

$\hat{H}=\frac{1}{2}\Omega_{\text{DD}}\hat{\sigma}_{+}+\frac{1}{2}\Omega_{\text{SB}}e^{\phi_{\text{grad}}-\phi_{\text{DD}}}\hat{\sigma}_{+}\left(\hat{a}+\hat{a}^{\dagger}\right)+\text{h.c.}\
$ (S8)

where we have neglected off-resonant carrier driving and used the DD phase as
a reference. This choice of reference phase makes it possible to write

$\displaystyle\hat{H}$
$\displaystyle=\frac{1}{2}\Omega_{\text{DD}}\hat{\sigma}_{x}$ (S9)
$\displaystyle-\frac{1}{2}\Omega_{\text{SB}}\sin\left(\phi_{\text{grad}}-\phi_{\text{DD}}\right)\hat{\sigma}_{y}\left(\hat{a}+\hat{a}^{\dagger}\right)$
$\displaystyle+\frac{1}{2}\Omega_{\text{SB}}\cos\left(\phi_{\text{grad}}-\phi_{\text{DD}}\right)\hat{\sigma}_{x}\left(\hat{a}+\hat{a}^{\dagger}\right)$

making the two parts of the SDF apparent: one which commutes with the DD and
one which does not. For a sufficiently strong DD drive with an ideal Walsh
cancellation scheme, the residual sideband interaction strength,
$\Omega_{\text{res}}$, is then given by

$\Omega_{\text{res}}=\Omega_{\text{SB}}\cos{(\phi_{\text{grad}}-\phi_{\text{DD}})}\\\
.$ (S10)

To estimate the crosstalk based on this residual sideband interaction
strength, we rely on the fact that phase loop-closure – a key component of
high-fidelity two-qubit gates – is ensured by choosing the correct gate
duration and detuning. Therefore, loop-closure is also enforced on non-
addressed ions with a residual sideband interaction. If an ion experiences an
imperfectly cancelled SDF, errors are induced by the phase loop it performs,
entangling it with other ions in the chain. The dominant entanglement occurs
with the addressed ions. We therefore define the parameter $\zeta$ as a
measure of the relative strength:

$\zeta=\frac{\Omega_{\text{res, non-addressed}}}{\Omega_{\text{res,
addressed}}}\\\ .$ (S11)

We estimate the magnitude of this error by considering a two-qubit system of
one addressed ion, which is driven by an SDF of amplitude $\Omega_{\text{res,
addressed}}$, and a decoupled ion, which experiences a residual SDF of
amplitude $\Omega_{\text{res, non-addressed}}$. The unitary evolution of such
a two-ion system can be calculated analytically with the Magnus expansion [39]
and is given by

$U_{\text{MS}}=\text{exp}\left[-i\frac{\pi}{8}\left(\sigma_{x}\otimes\mathds{1}-\zeta\mathds{1}\otimes\sigma_{x}\right)^{2}\right].$
(S12)

This drives the unitary evolution

$\displaystyle\ket{++}$ $\displaystyle\rightarrow$ $\displaystyle
e^{-i\frac{\pi}{4}\zeta}$ $\displaystyle\ket{++}$ (S13)
$\displaystyle\ket{+-}$ $\displaystyle\rightarrow$ $\displaystyle
e^{i\frac{\pi}{4}\zeta}$ $\displaystyle\ket{+-}$ $\displaystyle\ket{-+}$
$\displaystyle\rightarrow$ $\displaystyle e^{i\frac{\pi}{4}\zeta}$
$\displaystyle\ket{-+}$ $\displaystyle\ket{--}$ $\displaystyle\rightarrow$
$\displaystyle e^{-i\frac{\pi}{4}\zeta}$ $\displaystyle\ket{--}.$

To quantify the crosstalk given by the residual entanglement, we use the
impurity of the partial trace of the non-addressed qubit. Averaging over the
three bases ($\hat{\sigma}_{x/y/z}$) gives us the following crosstalk formula

$\frac{1}{6}\left(1-\cos{\left(\pi\zeta\right)}\right)\ .$ (S14)

We compute this crosstalk at all positions, as shown in Fig. S10(c), and take
the maximum value as the measure of “maximum crosstalk” shown in Fig. 5.

Figure S10:  Crosstalk simulation for a long ion chain. (a) Using a simple
three wire model, we use the Biot-Savart law to calculate the optimal fields
and phases to address the ions at positions $-20~{}\upmu$m and $-5~{}\upmu$m.
The phases of the DD (blue) and the sideband (red) are out of phase at all ion
positions (light grey dashed lines) except at the addressed ions (dark grey
dashed lines) (b) The interaction strength of the sideband (red) changes
across the ion chain, and is optimised to have the same amplitude at the two
addressed ion positions. (c) The estimated crosstalk calculated as a purity
loss arising from the residual entangling interaction.  Figure S11:
Randomised benchmarking of the gate suppression error. (a) The residual error
after suppressing a gate is experimentally estimated by embedding Mølmer-
Sørensen (MS) gate pulses in a single-qubit randomised benchmarking (RB)
sequence. The MS interaction is suppressed through a DD drive structured in a
Walsh-15 sequence. (b) Driving scheme: to closely emulate a (1-loop) MS gate,
the detuning from the single-ion motional sidebands $\delta/2\pi=770$ Hz is
consistent with both pulse time (1.30 ms) and sideband strength
($\Omega_{\text{sb}}/2\pi=380$ Hz). When the DD is in-phase with the SDF
driven by the sidebands ($\phi_{\text{car}}=0$), the MS pulses entangles the
qubit with the rapidly decohering motion, producing the maximum RB sequence
error (0.5). (c) With the DD out of phase with the SDF
($\phi_{\text{car}}=\pi/2$), a low error is demonstrated (left panel), on par
with the error induced by DD driving alone (center panel) indicating a
suppression of the ion-motion interaction. The DD driving error is partially
accounted for by leakage and qubit decoherence, measured by replacing the MS
pulse by a delay of equal duration (right panel).

##  S6 Suppression error

In this section, we provide further details of the single-qubit randomised
benchmarking (RB) [32] measurement, used to estimate the Mølmer-Sørensen (MS)
gate [24] suppression error, which is presented in Fig. 6. With a sideband
strength $\Omega_{\text{SB}}/2\pi=380$ Hz, an MS gate would be driven by a
$1.30$ ms pulse, with sidebands detuned by $\delta/2\pi=770$ Hz. Driving such
a pulse on a single qubit – with perfectly coherent motion – would cause
$\ket{\pm}$ states to undergo closed loops in phase space and induce no net
change in the qubit state. However, since in this case the motional heating
rate (370 quanta/s) is commensurate with the sideband strength, an MS pulse
will entangle the qubit with the rapidly decohering motion which in turn
decoheres the qubit. For two representative basis states, the MS pulse
(theoretically) transforms the state density matrix following

$\displaystyle\begin{pmatrix}1&0\\\ 0&0\end{pmatrix}$
$\displaystyle\rightarrow\begin{pmatrix}0.61&0\\\ 0&0.39\end{pmatrix}\ ,$
(S15) $\displaystyle\begin{pmatrix}1/2&i/2\\\ -i/2&1/2\end{pmatrix}$
$\displaystyle\rightarrow\begin{pmatrix}1/2&0.11\times i\\\ -0.11\times i&1/2\
.\end{pmatrix}$

This will serve as a proxy for the undesired motional and inter-qubit
entanglement which would occur in a multi-qubit register as protrayed in Fig.
5. We aim to gauge how effectively the effect is suppressed upon adding a DD
drive with a $\pi/2$ phase shift with respect to the SDF.

To obtain an error averaged over all possible starting states, we embed the MS
pulse in a single-qubit randomised benchmarking (RB) sequence [32] as shown in
Fig. S11(a). After each single-qubit Clifford gate, we drive an MS pulse and
measure the sequence error as a function of the number of Clifford/MS-pulse
pairs subjected to the qubit, as typically done in RB. Given our extremely low
Clifford gate error ($1.5\times 10^{-6}$) [16], the sequence error is
dominated by the MS-pulses.

Beyond the detuning and pulse timing, there are a few differences with respect
to microwave driving in the other presented experiments. First, we found that
the lowest errors are obtained by further increasing the Walsh modulation
order from 3 to 15. Secondly, in order for our control system to implement
this more complex pulsing scheme, we had to reduce the pulse ramping duration
from $2.4\ \upmu$s to $1.6\ \upmu$s. For the same reason, the delay between
individual pulses of the Walsh sequence is set to $24\ \upmu$s, and the delay
between two Clifford/MS-pulse sequences is $100\ \upmu$s. In total, an
individual Clifford/MS-pulse sequence has a duration of 1.78 ms, which will be
relevant when considering decoherence and leakage errors below. Also, the
change in motional mode occupation (due to heating) will be significant for
long benchmarking sequences, and so we restrict our experiments to 20
Clifford/MS-pulse pairs. We restrict our study to a single value of the DD
drive strength $\Omega_{\text{DD}}/2\pi=152$ kHz.

We first run a test measurement with $\phi_{\text{DD}}=0$, to verify the
effectiveness of the RB sequences in revealing errors arising from the MS-
pulse (Fig. S11(b)). We find that, as expected, the error signal rises to its
maximum value (0.5) indicating that the MS-pulses lead to a complete loss of
coherence.

Error source | Error ($/10^{-4}$ )
---|---
Decoherence $T_{\text{2}}$ | 1.3(2)
Leakage (from $\ket{0}$) | 1.5(1)
Leakage (from $\ket{1}$) | 0.36(8)
Detuning | 0.19
Expected error | 3.4(3)
Measured error | 1.7(3)
Table S1:  Time-delay error budget. Expected and measured error budget for the
RB sequence where MS pulses are replaced by delays of equal duration (Fig.
S11(c) right panel). Mismatch to measurement is attributed to a poor
understanding of the leakage error channel, and the difference in time-scales
probed by the RB sequence ($\sim$10 ms) and leakage/$T_{2}$ measurements
($\sim$1 s).

We then measure the case $\phi_{\text{DD}}=\pi/2$, obtaining an error per
Clifford/MS-pulse pair of $3.7(4)\times 10^{-4}$. This error is comparable to
that obtained with only DD driving ($4.0(6)\times 10^{-4}$), indicating that
the limiting factor is not related to motional coupling. Finally, we measure
the error arising if the MS-pulse is replaced by an equally long time delay,
revealing that about half of the error, $1.7(2)\times 10^{-4}$, is not related
to the microwave driving at all. These results are shown in Fig. S11(c). The
expected error from ion-motion coupling is simulated to be between $2.1\times
10^{-5}$ (when performing a single gate) and $3.0\times 10^{-4}$ (accounting
for heating during 20 sequential gates). The leading error we were able to
identify for DD or sideband driving is the coupling to spectator hyperfine
states, detuned by $\sim 100$ MHz and featuring low decoherence time scales
$\sim 20$ ms. The error arising from hybridisation of the qubit states with
these spectator states and subsequent decoherence is relatively small ($\sim
10^{-6}$). Whilst we cannot explain the origin of the error induced by DD
driving, we are reassured by the fact that we cannot identify fundamental
limitations to the Walsh sequence, indicating that the dominant error is
probably technical.

The error associated with the duration of the pulse, measured when replacing
the MS pulse with a delay, is roughly consistent with independently measured
error channels (see Table S1). Leakage was measured before and after the RB
data acquisition by preparing states $\ket{0}$ or $\ket{1}$, waiting for a
variable delay and subsequently reading out the qubit state. The measured
error is asymmetric with respect to the starting state, with leakage occurring
on time scales $\tau=$ 6.1(6) and 25(5) seconds for starting states $\ket{0}$
and $\ket{1}$ respectively (where $P(\ket{0,1})\approx e^{-t/\tau}$). Assuming
that during the RB sequence $P(\ket{0,1})\sim 1/2$ on average, this
corresponds to errors $1.5(1)\times 10^{-4}$ and $0.36(8)\times 10^{-4}$
respectively. A Ramsey measurement (with varying qubit detuning), performed
after the RB sequences, identifies the miscalibration or drift in qubit
frequency to be $2\pi\times 1.3$ Hz – which should only contribute to the
error measured without microwave driving, as the DD drive decouples the qubit
from detuning offsets. The impact of this detuning error is estimated through
simulating RB sequences to be $0.19\times 10^{-4}$. A Hahn-echo measurement
performed before and after the RB data acquisition reveals a $T_{2}$-echo time
of $4.4(6)$ seconds, consistent with values measured at small time-scales
through memory benchmarking [16]. This leads to an error of $1.3(2)\times
10^{-4}$. Mismatch of the total estimated error with the measured value is
associated with a poor understanding of the leakage error channel, and the
difference in time-scales probed by the RB measurement ($\sim$10 ms) and
leakage/$T_{2}$ measurements ($\sim$1 s).