# Quantum crosstalk cancellation for fast entangling gates and improved multi-
qubit performance

K. X. Wei<EMAIL_ADDRESS>E. Magesan  I. Lauer  S. Srinivasan  D. F. Bogorin
S. Carnevale  G. A. Keefe  Y. Kim  D. Klaus  W. Landers  N. Sundaresan  C.
Wang  E. J. Zhang  M. Steffen  O. E. Dial  D. C. McKay<EMAIL_ADDRESS>A.
Kandala<EMAIL_ADDRESS>IBM Quantum, IBM T.J. Watson Research Center,
Yorktown Heights, NY 10598, USA

###### Abstract

Quantum computers built with superconducting artificial atoms already stretch
the limits of their classical counterparts. While the lowest energy states of
these artificial atoms serve as the qubit basis, the higher levels are
responsible for both a host of attractive gate schemes as well as generating
undesired interactions. In particular, when coupling these atoms to generate
entanglement, the higher levels cause shifts in the computational levels that
leads to unwanted $ZZ$ quantum crosstalk. Here, we present a novel technique
to manipulate the energy levels and mitigate this crosstalk via a simultaneous
AC Stark effect on coupled qubits. This breaks a fundamental deadlock between
qubit-qubit coupling and crosstalk, leading to a 90ns CNOT with a gate error
of (0.19 $\pm$ 0.02) $\%$ and the demonstration of a novel CZ gate with fixed-
coupling single-junction transmon qubits. Furthermore, we show a definitive
improvement in circuit performance with crosstalk cancellation over seven
qubits, demonstrating the scalability of the technique. This work paves the
way for superconducting hardware with faster gates and greatly improved multi-
qubit circuit fidelities.

Existing quantum processors Zhang  _et al._ (2020); Arute  _et al._ (2019)
based on superconducting transmon qubits are pushing the limits of classical
simulability. However, the realization of quantum advantage requires these
processors to scale up in both size and operational fidelity. Reaching a
suitable threshold on both counts would further enable quantum error
correction and the realization of a fault tolerant quantum computer. These
objectives require overcoming several technical challenges, notably, two-qubit
gate fidelity, crosstalk, system stability and qubit coherence. One common
architecture, based on fixed-frequency transmon qubits with fixed couplings,
has a distinct advantage in terms of stability and coherence, but has
limitations on gate speed and minimizing crosstalk due to always on
interactions, and their relation to the exchange coupling strength, $J$. While
a larger $J$ enables a faster entangling gate, the coupling leads to state
dependent frequency shifts of neighboring coupled qubits, which is a source of
quantum crosstalk that takes the form of a $ZZ$ interaction in the system
Hamiltonian. This is formally seen from the standard cQED Hamiltonian for a
pair of coupled transmons ($i=\\{0,1\\}$), modelled as Duffing oscillators,

$\displaystyle H_{0}/h$ $\displaystyle=$
$\displaystyle\sum_{i=\\{0,1\\}}\left(\nu_{i}\hat{a}_{i}^{\dagger}\hat{a}_{i}+\frac{\alpha_{i}}{2}\hat{a}_{i}^{\dagger}\hat{a}_{i}\left(\hat{a}_{i}^{\dagger}\hat{a}_{i}-1\right)\right)$
(1)
$\displaystyle+J(\hat{a}_{0}^{\dagger}+\hat{a}_{0})(\hat{a}_{1}^{\dagger}+\hat{a}_{1}),$

with bare qubit frequencies $\nu_{i}$, bare anharmonicities $\alpha_{i}$ and
coupling strength $J$. The coupling dresses the energy levels, and the
crosstalk arising from state dependent frequency shifts is expressed as,

$\displaystyle\nu_{ZZ}$ $\displaystyle=$
$\displaystyle(\nu_{11}-\nu_{10})-(\nu_{01}-\nu_{00}).$ (2)

For fixed couplings, this is an always-on source of crosstalk, referred to as
a static $ZZ$ interaction, with the following perturbative form,

$\displaystyle\nu_{ZZ,s}$ $\displaystyle=$
$\displaystyle-\frac{2J^{2}(\alpha_{0}+\alpha_{1})}{(\alpha_{1}-\Delta_{0,1})(\alpha_{0}+\Delta_{0,1})},$
(3)

where $\Delta_{0,1}$ represents the qubit-qubit detuning. This crosstalk has
been seen to be an important limitation to multi-qubit circuit performance in
tests of quantum volume Jurcevic  _et al._ (2020), randomized benchmarking
McKay  _et al._ (2019), and error correction codes Takita  _et al._ (2016),
and may prevent device scaling Berke  _et al._ (2020). Several hardware
strategies have been employed to mitigate this crosstalk. The simplest
approach, as seen from Eq. (3), is to lower $J$, however, this comes at the
expense of gate speed and lowers the overall gate fidelity due to finite qubit
coherence. More involved strategies include the the introduction of tunable
$J$ coupling Chen  _et al._ (2014); Arute  _et al._ (2019); Stehlik  _et al._
(2021); coupling different flavors of qubits with opposite signs of
anharmonicity Zhao  _et al._ (2020a); Ku  _et al._ (2020); Xu and Ansari
(2020) (see Eq. (3)); and the use of engineered multi-path coupling elements
Mundada  _et al._ (2019); Yan  _et al._ (2018); Kandala  _et al._ (2020); Zhao
_et al._ (2020b); Xu and Ansari (2020). An alternative approach is a quantum
control strategy to $ZZ$ cancellation via the AC Stark effect, using off-
resonant radiation to selectively tune the energy levels, and modulate $ZZ$,
as seen from Eq. (2). This has been demonstrated with a single near-resonant,
continuous wave (CW) drive in flux-tunable superconducting qubit architectures
Noguchi  _et al._ (2020); Xiong  _et al._ (2021). However, this requires being
close to a resonant transition outside the computational space, and is
susceptible to charge noise in transmon qubits.

In this work we show that the $ZZ$ interaction for a pair of coupled transmon
qubits can be tuned over several orders of magnitude by far-off resonant
driving on both qubits. We develop an analytical model of the effect for
transmons, building off previous theoretical work studying the case of coupled
spins Li  _et al._ (2008). We then demonstrate that the effect, dubbed siZZle
- Stark induced $ZZ$ by level excursions - can be employed for both static
$ZZ$ cancellation as well as implementing $ZX$ and $ZZ$ entangling gates in
all-transmon processors with simple direct capacitive coupling. The ability to
cancel the static $ZZ$ interaction enables us to employ stronger qubit-qubit
coupling, leading to a state-of-the-art cross-resonance gate with over a
factor of 2 improvement in gate time from previous reports Kandala  _et al._
(2020). Furthermore, we demonstrate a novel high-fidelity CZ gate based on
siZZle which adds to the toolkit of microwave-only Chow  _et al._ (2011);
Poletto  _et al._ (2012); Chow  _et al._ (2013); Paik  _et al._ (2016);
Krinner  _et al._ (2020) two qubit gates. In contrast to previous approaches
Noguchi  _et al._ (2020); Xiong  _et al._ (2021), our approach with Stark
tones on both qubits introduces an additional control parameter, the phase
difference between the two tones, that is particularly useful for extending to
larger devices. We demonstrate $ZZ$ cancellation on a line of 7 qubits
combining siZZle with the hardware approach of multi-path couplers, and
demonstrate improvements in the performance of Quantum Volume (QV) circuits
Cross  _et al._ (2019).

To describe the physics of siZZle, we consider the Hamiltonian of Eqn. (1) and
add off-resonant drives on both qubits,

$\displaystyle H_{\textrm{siZZle}}/h=H_{0}/h+$
$\displaystyle\sum_{i=\\{0,1\\}}\Omega_{i}\cos{(2\pi\nu_{d}t+\phi_{i})}(\hat{a}_{i}^{\dagger}+\hat{a}_{i}),$
(4)

with amplitudes $\Omega_{i}$, phases $\phi_{i}$, and a common frequency
$\nu_{d}$ . The device schematic in Fig. 1(a) depicts a simple direct
capacitive coupling between the qubits that produces the Hamiltonian model of
Eq. 4. In the limit of ${\Omega_{i}}/{|\nu_{d}-\nu_{i}|}\ll 1$, we can write
the dressed RWA Hamiltonian as,

$H_{\textrm{eff}}/h=\tilde{\nu}_{ZI}{ZI}/4+\tilde{\nu}_{IZ}{IZ}/4+\tilde{\nu}_{ZZ}{ZZ}/4,$
(5)

where the tilde notation refers to being in the doubly-dressed frame with
respect to the exchange coupling and Stark tones. To second order in
$\Omega_{i}$ and first order in $J$, the $ZZ$ coefficient is,

$\displaystyle\tilde{\nu}_{ZZ}$ $\displaystyle=$ $\displaystyle\nu_{ZZ,s}+$
(6)
$\displaystyle\frac{2J\alpha_{0}\alpha_{1}\Omega_{0}\Omega_{1}\cos{(\phi_{0}-\phi_{1})}}{\Delta_{0,d}\Delta_{1,d}(\Delta_{0,d}+\alpha_{0})(\Delta_{1,d}+\alpha_{1})},$

where the static term is given by Eqn. (3). In the above equations,
$\Delta_{i,j}=(\nu_{i}-\nu_{j})$ denotes detunings where $i,j\in\\{0,1,d\\}$.
The most significant contribution to the Stark shifts comes from the term
associated with a single, isolated drive

$\displaystyle\tilde{\nu}_{ZI,\textrm{single}}=-\frac{\Omega_{0}^{2}\alpha_{0}}{\Delta_{0,d}(\Delta_{0,d}+\alpha_{0})},$
(7)

which will be of significance in later discussions for the impact of the Stark
tones on qubit coherence. A formal derivation of these expressions is
discussed in the Supplementary Information. Eq. (6) reveals the various
control knobs to manipulate the strength of the Stark induced $ZZ$
interaction: the amplitudes of the two tones, the drive-qubit detunings, the
anharmonicities, and the phase differences between the two drive tones.

Figure 1: Physics of siZZle (a) Modulation of the $ZZ$ interaction strength
$\tilde{\nu}_{ZZ}$ as the Rabi amplitude of the Stark tones is swept (ratio
$\Omega_{1}/\Omega_{0}=0.5$) for fixed frequency $\nu_{d}=5.075$ GHz and phase
difference $\phi=\pi$. Experimental data (black circles) is compared to
numerical (blue line) and perturbative (red line) calculations using the
device parameters of Table 1. The inset shows a circuit representation of the
primary two-qubit device discussed in this work. (b) The corresponding
excursions of the computational levels, calculated numerically, to generate
the $\tilde{\nu}_{ZZ}$ shown in (a).(c) Modulation of the $ZZ$ interaction
strength $\tilde{\nu}_{ZZ}$ as the phase difference between the Stark tones is
swept, for fixed frequency $\nu_{d}=5.075$ GHz and and drive amplitudes
$\Omega_{1}=0.5\Omega_{0}=20$ MHz. Experimental data (black circles) is
compared to numerical (blue line) and perturbative (red line) calculations
using the device parameters of Table 1 (d) The corresponding excursions of the
computational levels, calculated numerically, to generate the
$\tilde{\nu}_{ZZ}$ shown in (c).  Figure 2:  Mapping the siZZle parameter
space (a) Experimental sweep of $\tilde{\nu}_{ZZ}$ versus Stark amplitudes for
fixed Stark frequency $\nu_{d}=$ 5.065 GHz and calibrated phase $\phi=\pi$.
The red dotted line highlights the
$\tilde{\nu}_{ZZ}\propto\Omega_{0}\Omega_{1}$ dependence that is expected from
the perturbative expression of (4). (b) Experimental sweep of
$\tilde{\nu}_{ZZ}$ versus Stark amplitude and Stark frequency for a fixed
ratio of Stark amplitudes $\Omega_{1}/\Omega_{0}=0.4$ and calibrated phase
$\phi=\pi$. (c) Experimental sweep of $\tilde{\nu}_{ZZ}$ versus the phase
difference $\phi$ and Stark frequency for a fixed Stark amplitudes
$\Omega_{0}=37.5$ MHz, $\Omega_{1}=15$ MHz. The + and - symbols in the 3 sub-
figures refer to the sign of $\tilde{\nu}_{ZZ}$.

Fig. 1 reveals the physics of siZZle, employing the parameters of the primary
two-qubit device studied in this work, device A. The parameters are given in
Table 1. We perform numerical diagonalization of Eq. (4) after moving into the
frame of the drive. Fig. 1 depicts how the excursions of the computational
levels leads to a modulation of $\tilde{\nu}_{ZZ}$, as the Stark tone
amplitudes ((a), (b)) and phase difference ((c), (d)) are swept. We also see
good agreement between the numerical calculations and the derived analytical
expression of Eq. 6 in the perturbative limit. Experimentally, we measure
$\tilde{\nu}_{ZZ}$ by employing standard Ramsey sequences on Q0 while Q1 is in
$|0\rangle$ or $|1\rangle$. The experimentally measured values show very good
agreement with numerics in Fig. 1(a), (c). A wider parameter space is
experimentally mapped in the 2D sweeps of Fig. 2 and further depicts the
physics of siZZle. Fig. 2(a) maps $\tilde{\nu}_{ZZ}$ versus the Rabi
amplitudes of the Stark tones on both qubits, and the region of
$\tilde{\nu}_{ZZ}\sim 0$ kHz clearly highlights the
$\tilde{\nu}_{ZZ}\propto\Omega_{0}\Omega_{1}$ dependence expected from Eq.
(6). Fig. 2 (b) shows that modulation of $\tilde{\nu}_{ZZ}$ versus siZZle
frequency and the Rabi amplitudes, and shows that sizeable $ZZ$ modulation can
be obtained over a wide range of frequencies. As can be seen qualitatively
from Eq. (6), placing the Stark tone away from the qubit frequency can be
compensated by increasing the drive amplitude, for the same
$\tilde{\nu}_{ZZ}$. Fig. 2 (c) demonstrates the sinusoidal phase dependence of
$\tilde{\nu}_{ZZ}$, over a range of frequencies. The experimental data of
Figures 1 and 2 reveal two interesting regimes of operation. At fairly modest
drives, we observe see that we can cancel the $ZZ$ interaction to operate at
$\tilde{\nu}_{ZZ}\sim 0$. At stronger drive amplitudes, one can generate large
$ZZ$ rates for two qubit entangling gates. These regimes of operation are
discussed in Fig. 3 and 4 and in the next two sections.

| $\tilde{\nu}_{0}$ | $\tilde{\nu}_{1}$ | $\tilde{\alpha}_{0}$ | $\tilde{\alpha}_{1}$
---|---|---|---|---
No siZZle | 4.960 | 5.016 | -0.283 | -0.287
siZZle | 4.953 | 5.014 | -0.276 | -0.286
Table 1: Qubit frequencies for device A depicted in Fig. 1(a) before and after
$ZZ$ cancellation. All the numbers are in units of GHz. We note that these
numbers represent the experimentally measured frequencies, dressed by the
coupling $J=7.745$ MHz.  Figure 3: Fast cross-resonance with static ZZ
cancellation (a) Simultaneous randomized benchmarking (RB) of 50 ns single
qubit gates in the absence of static $ZZ$ cancellation (blue) leads to an
average error per gate (EPG) of 6.6e-3. After static $ZZ$ cancellation with a
pair of CW Stark tones at $\nu_{d}=$ 5.1 GHz, the EPG dramatically improves to
7.1e-4 (red). Bold symbols represent mean of the individual seeds (represented
by light symbols), and dotted lines are exponential fits to the decay of the
excited state probability $P_{1}$. (b) Phase calibration of the CW Stark tones
to $\tilde{\nu}_{ZZ}\sim 0$ for $\Omega_{0}=59$ MHz and $\Omega_{1}=22$ MHz.
(c) Strength of $ZX$ interaction $\tilde{\nu}_{ZX}$ versus cross-resonance
drive amplitude $\Omega_{\textrm{CR}}$ with (red) and without (blue) static
$ZZ$ cancellation. Here, Q1 is the control qubit and Q0 is the target qubit.
Bold circles represent experimentally measured rates, using Hamiltonian
tomography. Dotted lines are perturbative estimates, using Eq. 8. (d) EPG
measured by interleaved RB, for direct CNOT gates constructed from cross-
resonance, after $ZZ$ cancellation, as a function of CNOT gate time. The blue
dotted line represent the coherence limit to gate error from estimated using
standard $T_{1}$ and $T_{2}$ measurements before every RB run. (e) Post-$ZZ$
cancellation interleaved RB of a 90 ns direct CNOT gate reveals a best EPG of
1.86e-3, with an upper bound on the EPG of 4.0e-3.

In the first regime of operation, siZZle is used to cancel $ZZ$, which can be
utilized to increase the speed of entangling gates, such as cross-resonance
(CR) Paraoanu (2006); Chow  _et al._ (2011), which are set by the coupling
strength $J$. As discussed previously in Eq. 3, increasing $J$ typically leads
to large values of static $ZZ$ crosstalk. Recent work Kandala  _et al._ (2020)
with multi-path couplers demonstrated a way to break the standard relationship
between $J$ and $\nu_{ZZ,\textrm{static}}$ (operating at
$J/\nu_{ZZ,\textrm{static}}\sim 130$), leading to state-of-the art CR gate
fidelities. A drawback of the multi-path coupler approach is that
$\nu_{ZZ,\textrm{static}}$ depends strongly on the qubit frequencies, so that
attempting to achieve $\nu_{ZZ,\textrm{static}}\sim 0$ is non-trivial in fixed
frequency architectures given current precision over qubit frequency
allocation Zhang  _et al._ (2020). Our quantum control approach to $ZZ$
cancellation introduced here enables tuning to $\tilde{\nu}_{ZZ}\sim 0$ over a
range of parameters since we have several degrees of freedom in our control
space. Importantly, this allows for a decoupling of $J$ and $\tilde{\nu}_{ZZ}$
so that fast, high-fidelity entangling gates are possible with minimal static
crosstalk in an architecture consisting of standard single path couplers and
nominally fixed-frequency qubits.

To test this, our device, described in Table 1 has a large coupling strength
of $J\sim 7.745$ MHz, leading to a very large static $ZZ$ interaction of
$\nu_{ZZ,\textrm{static}}=875$ kHz. Without any further mitigation of $ZZ$,
this prevents high-fidelity simultaneous single qubit operation due to
strongly state-dependent qubit frequencies. This is seen in the decay and
variance of simultaneous single qubit randomized benchmarking sequences shown
in Fig. 3(a) with an estimated average error per gate (EPG) of 6.6e-3. In
order to mitigate this crosstalk, we add continuous wave (CW) Stark drives to
cancel $ZZ$ and operate in a basis dressed by these off-resonant drives. The
system Hamiltonian builds off Eq. (4) to now include additional drives for
gate operation:

$\displaystyle H/h$ $\displaystyle=$ $\displaystyle H_{\textrm{siZZle}}/h+$
$\displaystyle\sum_{i=\\{0,1\\}}\Omega_{i,\textrm{gate}}(t)\cos{(2\pi\nu_{i,\textrm{gate}}t+\phi_{i})}(\hat{a}_{i}^{\dagger}+\hat{a}_{i})$

where $\Omega_{i,\textrm{gate}}(t)$ and $\nu_{i,\textrm{gate}}$ are the time-
dependent amplitude and frequency of the single/two-qubit gate drive on qubit
$i$ respectively.

The large choice of operating parameters for the $ZZ$ cancellation tones makes
identifying an optimal set of working parameters a complex task. First, we
limit leakage out of the computational subspace by placing the $ZZ$
cancellation tone above both qubits. Next, we optimize the detuning of the
cancellation tone. Smaller detuning reduces the drive amplitude required for
$ZZ$ cancellation. There is a practical limit to the amount of amplitude that
can delivered to the qubits before there is heating of system components.
However, if the detuning is too small then the cancellation tone may start to
interfere with the gate drive and time-dependent terms in the effective
Hamiltonian in the frame of the drive can no longer be ignored. For these
reasons, we select $\nu_{d}=$ 5.1 GHz, for device A. The CW amplitudes are
chosen to be sufficient to just approach $\tilde{\nu}_{ZZ}\sim 0$ after phase
calibration (i.e at $\phi=\pi$), see Fig. 3(b). We estimate the CW amplitudes
from the independent qubit Stark shifts to be $\Omega_{0}=59$ MHz and
$\Omega_{1}=22$ MHz. After tuning to $\tilde{\nu}_{ZZ}\sim 0$, the single
qubit gates are re-calibrated with the cancellation drives on. The new
operating frequencies of the qubits are $\tilde{\nu_{0}}=4.953$ GHz and
$\tilde{\nu_{1}}=5.014$ GHz, and so, the qubits have modest Stark shifts of
-7.8 MHz and -1.7 MHz respectively. Reducing the $ZZ$ in this way results in
remarkable improvements in simultaneous single qubit operation for 50 ns
gates, with an estimated gate error of 7.1e-4 from randomized benchmarking,
see Fig. 3(a). We note that there are several operating points for achieving
$\nu_{ZZ}\sim 0$, but operating at stronger CW amplitudes with larger Stark
shifts can to lead to additional dephasing.

With $ZZ$ cancelled and single-qubit gates calibration, we now calibrate a
two-qubit gate with cross-resonance. This entails additional drives on the
control qubit (Q1) at the dressed target qubit (Q0) frequency. In Fig. 3c, we
measure the generated $ZX$ rates versus CR drive amplitude from tomography of
the CR drive Hamiltonian, with and without $ZZ$ cancellation. The $ZX$ rate is
modified due to the presence of the cancellation tones, however, as a
consequence of the large $J$ coupling, one can access fairly large $ZX$ rates
at modest CR drive amplitudes. A perturbative model for the $ZX$ rate is
derived that includes the contribution from the cancellation tones. Assuming a
CR tone on transmon 0 (for the experiment of this paper the CR tone is on
transmon 1 so the labels will be swapped) we have,

$\displaystyle\tilde{\nu}_{ZX}$ $\displaystyle\sim
J\Omega_{0,\text{gate}}\left(A+B\Omega_{0}^{2}+C\Omega_{1}^{2}\right),$ (8)

where

$\displaystyle A$
$\displaystyle=-\frac{\delta}{\Delta_{0,1}(\delta+\Delta_{0,1})},$ (9)

and $B$, $C$ are given in the supplement. We see that the $ZX$ rate has
contributions that are quadratic in the cancellation tone amplitudes. The
zero-point slope for the $ZX$ rate is modified by the Stark tones and when
$\Omega_{0}=\Omega_{1}=0$ the usual first order expression for
$\tilde{\nu}_{ZX}$ is obtained. Fig. 3c contains the $ZX$ rates with the Stark
tones both off and on, and we see good agreement between the perturbative
model and experiment at low CR amplitudes.

The large $J$ coupling is also of consequence for the reduced control qubit
Stark shift, discussed previously in Kandala  _et al._ (2020), and the
resulting stability of unechoed direct CNOT gates constructed using CR. We
construct and calibrate direct CNOT gates, similar to Kandala  _et al._
(2020), and study the gate error obtained from interleaved RB as a function of
CNOT gate time in Fig. 3d. The calibration sequences and pulse shapes are
detailed in the supplement. At the optimal gate time of 90 ns, we depict
results from interleaved RB sequences in Fig. 3e, that we use to estimate an
error per gate (EPG) of 1.86e-3, with an error per Clifford (EPC) of 6.0e-3
from standard RB. Our decomposition has 1.5 CNOT gates per Clifford on average
and this places an upper bound on the EPG of EPC/1.5 $\sim$ 4.0e-3. The ratio
of EPG/EPC can be compared to analysis in Epstein  _et al._ (2014) for
confidence in the interleaved RB estimates. We also note that our gate errors
fluctuate with changes in coherence and the defect environment Carroll  _et
al._ (2021) in the vicinity of the qubit frequencies. At the time of the
displayed benchmarking, our measured coherence times for Q0(Q1) were $T_{1}=$
66 (66) $\mu$s and $T_{2}=$ 49(84) $\mu$s.

Figure 4:  All $ZZ$ SiZZle gate (a) Post ZZ cancellation 2D sweep of
$\nu_{ZZ}$ with pulsed Stark frequency $\nu_{\textrm{gate}}$ and amplitude,
with the ratio of the two amplitudes fixed to $\Omega_{0}=\Omega_{1}$, and
phase calibrated to maximum contrast. The CW tones to cancel $ZZ$ use the same
parameters discussed in Fig 3, with $\nu_{d}=5.1$ GHz. (b) Interleaved RB of a
calibrated CZ gate based on siZZle reveals an error per gate of 5e-3, with an
upper bound on that figure of 7.6e-3.  Figure 5:  Dependence of multi-qubit
circuit fidelity on $ZZ$ interaction (Top) A device schematic of the line of 7
qubits, with a combination of hardware and control approaches to $ZZ$
modulation. The device employs multi-path couplers composed of a direct
capacitive coupling and a $\lambda$/4 bus resonator. (Bottom) Average heavy
output probability (HOP) for the same set of 200 random quantum volume (QV)
circuits, at different levels of $\tilde{\nu}_{ZZ}$. Error bars represent
standard error of the mean. The maximum and minimum $\tilde{\nu}_{ZZ}$ data
points are tuned by setting the pair wise phase difference between the siZZle
tones to $\phi\sim 0$ and $\phi\sim\pi$ respectively. The middle data point is
measured in the absence of siZZle. (Inset) Scatter of individual circuit HOPs
for the native (bare) device versus post-$ZZ$ cancellation.

In the second regime of operation, siZZle can be used as a standalone method
for performing a two-qubit gate due to the large $ZZ$ rates that can be
generated as shown in Figs. 1 and 2. In order to mitigate the static $ZZ$, we
continue to use CW tones at $\nu_{d}=$ 5.1 GHz, but, additionally pulse a
second set of off-resonant tones at a different frequency
$\nu_{\textrm{gate}}$ to generate large $\tilde{\nu}_{ZZ}$. This is shown in
Figure 4a, where we sweep the pulsed tone frequency and amplitudes
($\Omega_{0,\textrm{gate}}=\Omega_{1,\textrm{gate}}$) to generate
$\tilde{\nu}_{ZZ}$ exceeding a few MHz. We note that $ZZ$ gate operation can
also be achieved with a single frequency, using amplitude or phase modulation
to switch between low and high $ZZ$ rates. Once again, the operating parameter
space is very large, and finding a parameter set that is optimized for gate
fidelity, speed and leakage is a challenging task that is left for future
study. Here, we provide a proof-of-concept example of siZZle gate operation at
$\nu_{\textrm{gate}}=4.9$ GHz, with maximum amplitudes
$\Omega_{0,\textrm{gate}},\Omega_{1,\textrm{gate}}\sim 26$ MHz. We calibrate
the phase difference between the phase tones for maximum $\tilde{\nu}_{ZZ}$,
and employ frame changes on the control and target qubits to construct a novel
direct CZ gate of length 200 ns. Interleaved RB, shown in Fig. 4b reveals a
gate error of 5e-3, with an error per gate upper bound of 7.6e-3.

Finally, we study the impact of siZZle on multi-qubit circuit fidelity, using
a line of 7 qubits from a 27 qubit device with a heavy-hex architecture
Jurcevic  _et al._ (2020), that we shall refer to as Device B. In order to
reduce the impact to qubit coherence from the Stark tones, our experiment
combines the quantum control approach to static $ZZ$ cancellation introduced
here with the hardware approach of multi-path couplers Kandala  _et al._
(2020). The multi-path couplers already suppress the $\tilde{\nu}_{ZZ}$
compared to an equivalent direct coupler with the same effective $J$. This
reduces the amplitude of the siZZle tones required to then tune to
$\tilde{\nu}_{ZZ}\sim 0$, and consequently, the magnitude of the individual
qubit Stark shifts (see Eq. 7), and the impact to qubit coherence, if any.
This discussion and the device properties are detailed in the Supplementary
Information. As a reminder, we have three knobs to manipulate the $ZZ$
interaction in the device: the amplitude of the off-resonant tones, their
detuning from the qubit frequencies, and the pair wise phase difference. This
makes it particularly attractive for device-wide $ZZ$ cancellation on even
more complex topologies. For the considered line of qubits, we choose a common
Stark frequency set to 5.1 GHz, above all the qubit frequencies, leaving the
individual amplitudes and phases as the free control parameters. Placing the
Stark frequency above all the qubits reduces the possibility of undesired
frequency collisions. We then adjust the Stark amplitude on one of the qubits
to induce a Stark shift of $\sim$ 1 MHz. The amplitudes of the CW tones on the
subsequent qubits are then adjusted sequentially such that it is just
sufficient to tune to $\tilde{\nu}_{ZZ}\sim 0$ for every pair (i.e.
$\phi_{i}-\phi_{j}\sim\pi$). We then re-calibrate the single and two qubit
gates at the new dressed frequencies. We see that we can tune to
$\tilde{\nu}_{ZZ}\sim 0$ with very modest Stark shifts ($\sim$ 1 MHz), which
is important for reducing the impact to qubit coherence, as discussed above.
We then use cross-resonance to calibrate an echo CNOT with rotary target
drives, as in Sundaresan  _et al._ (2020). We emphasize that we observe no
large changes in CNOT gate fidelity for all the pairs, at the different
$\tilde{\nu}_{ZZ}$ levels, which highlights the need for circuit-level
benchmarks such as quantum volume (QV) Cross  _et al._ (2019) that are
sensitive to accumulated $ZZ$ errors from qubit idle times. In order to
benchmark multi-qubit performance, we employ seven-qubit QV circuits and
observe an improvement in the heavy output probability (HOP) from 0.5810 $\pm$
0.0027 to 0.5996 $\pm$ 0.0023 as the average $\tilde{\nu}_{ZZ}$ is tuned from
the bare value $\sim$ 40 kHz to $\sim$ 0 kHz. We employ 200 random circuits,
with a mean circuit time of $\sim$ 14.1 $\mu s$, and $83$ CNOT gates on
average. The improvement in the distribution of the individual circuit HOPs
with $ZZ$ cancellation is also depicted in Fig. 5. For the purpose of this
demonstration, we do not employ the circuit optimization and improved readout
techniques discussed in Jurcevic  _et al._ (2020). Our control knobs also
enable us to systematically study the impact of $\tilde{\nu}_{ZZ}$ on circuit
performance. We modulate the average $\tilde{\nu}_{ZZ}$ in the device merely
by adjusting the pair-wise phase differences, and re-calibrate all the gates
at every step. Fig. 5 depicts the systematic decrease in HOP with increase in
average $\tilde{\nu}_{ZZ}$, and highlights why $ZZ$ cancellation will be
crucial for improving the performance of superconducting quantum processors.
The technique also opens up the path to more targeted studies of the impact of
the $ZZ$ interaction on spectator interactions and parallel gate operation,
all in a single device.

In conclusion, we demonstrate an all microwave technique - siZZle - for
arbitrary control of the $ZZ$ interaction rate in coupled transmon devices. We
use siZZle to demonstrate a novel high-fidelity CZ gate that could enable
hardware-efficient implementations of near-term algorithms on existing fixed-
frequency quantum processors. Furthermore, static $ZZ$ cancellation with
siZZle enables us to take cross-resonance past the 100 ns milestone for two-
qubit gate time, with state-of-the-art fidelity. This gives us a clear path to
increasing the fixed $J$ coupling in devices and also serves as a platform to
explore the physics of well-controlled strong coupling interactions. Finally,
combining siZZle with hardware approaches to $ZZ$ cancellation is leveraged to
definitively improve multi-qubit circuit fidelity, and highlights the
scalability of our technique. These results reveal quantum control with multi-
color drive tones to be an attractive approach to extend the reach of fixed
frequency superconducting quantum architectures.

We note recent independent work Mitchell  _et al._ (2021) reporting siZZle and
a CZ gate based on the effect.

###### Acknowledgements.

We acknowledge Malcolm Carroll, Antonio Corcoles, Pat Gumann, Micheal Gordon,
Shawn Hall, Sean Hart, Muir Kumph, Jim Rozen, Maika Takita for experimental
contributions and Doug McClure, Petar Jurcevic for helpful discussions. The
device bring-up, gate calibration and characterization work was supported by
IARPA under LogiQ (contract W911NF-16-1-0114).

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Supplementary Information: Quantum crosstalk cancellation for fast entangling
gates and improved multi-qubit performance

##  I SiZZle theory

We first provide some intuition for the dual-drive Stark induced $ZZ$ effect.
We consider a simple two-level model for the qubits, dressed by monochromatic
drives, with the $J$ coupling introduced as a perturbation. In the absence of
coupling, each qubit can be described independently by

$H_{0}/h=\nu_{Q0}|1\rangle\langle
1|+\Omega_{0}\cos(2\pi\nu_{d}t+\phi_{0})(|0\rangle\langle 1|+|1\rangle\langle
0|).$ (S1)

The off-resonant drive dresses the eigenstates and shifts the eigenvalues,

$\displaystyle|\overline{0}\rangle$ $\displaystyle\approx$
$\displaystyle|0\rangle-\frac{\Omega_{0}}{2\Delta_{0}}e^{-i\phi_{0}}|1\rangle,$
(S2) $\displaystyle|\overline{1}\rangle$ $\displaystyle\approx$
$\displaystyle|1\rangle+\frac{\Omega_{0}}{2\Delta_{0}}e^{i\phi_{0}}|0\rangle,$
(S3) $\displaystyle\overline{E}_{0}/h$ $\displaystyle\approx$
$\displaystyle-\frac{\Omega^{2}_{0}}{4\Delta_{0}},$ (S4)
$\displaystyle\overline{E}_{1}/h$ $\displaystyle\approx$
$\displaystyle\nu_{Q0}+\frac{\Omega^{2}_{0}}{4\Delta_{0}},$ (S5)

where $\Delta=\nu_{Q0}-\nu_{d}$ is the detuning. Therefore, in the two qubit
basis, the dressed states are,

$\displaystyle|\overline{00}\rangle$ $\displaystyle\approx$
$\displaystyle|00\rangle-\frac{\Omega_{0}}{2\Delta_{0}}e^{-i\phi_{0}}|10\rangle-\frac{\Omega_{1}}{2\Delta_{1}}e^{-i\phi_{1}}|01\rangle+\frac{\Omega_{0}\Omega_{1}}{4\Delta_{0}\Delta_{1}}e^{-i(\phi_{0}+\phi_{1})}|11\rangle,$
(S6) $\displaystyle|\overline{10}\rangle$ $\displaystyle\approx$
$\displaystyle|10\rangle+\frac{\Omega_{0}}{2\Delta_{0}}e^{i\phi_{0}}|00\rangle-\frac{\Omega_{1}}{2\Delta_{1}}e^{-i\phi_{1}}|11\rangle-\frac{\Omega_{0}\Omega_{1}}{4\Delta_{0}\Delta_{1}}e^{i(\phi_{0}-\phi_{1})}|01\rangle,$
(S7) $\displaystyle|\overline{01}\rangle$ $\displaystyle\approx$
$\displaystyle|01\rangle-\frac{\Omega_{0}}{2\Delta_{0}}e^{-i\phi_{0}}|11\rangle+\frac{\Omega_{1}}{2\Delta_{1}}e^{i\phi_{1}}|00\rangle-\frac{\Omega_{0}\Omega_{1}}{4\Delta_{0}\Delta_{1}}e^{-i(\phi_{0}-\phi_{1})}|10\rangle,$
(S8) $\displaystyle|\overline{11}\rangle$ $\displaystyle\approx$
$\displaystyle|11\rangle+\frac{\Omega_{0}}{2\Delta_{0}}e^{i\phi_{0}}|01\rangle+\frac{\Omega_{1}}{2\Delta_{1}}e^{i\phi_{1}}|10\rangle+\frac{\Omega_{0}\Omega_{1}}{4\Delta_{0}\Delta_{1}}e^{i(\phi_{0}+\phi_{1})}|00\rangle.$
(S9)

The dressing of the $|00\rangle$ and $|11\rangle$ with $|01\rangle$ and
$|10\rangle$ allows exchange interactions to directly couple them, leading to
a $ZZ$ interaction. We explicitly show this by calculating the energy shifts
due to a $J$ coupling, $H_{J}/h=J(|01\rangle\langle 10|+|10\rangle\langle
01|)$ ,

$\displaystyle\langle\overline{01}|H_{J}/h|\overline{01}\rangle$
$\displaystyle\approx$
$\displaystyle-J\frac{\Omega_{0}\Omega_{1}}{4\Delta_{0}\Delta_{1}}e^{-i(\phi_{0}-\phi_{1})},$
(S10) $\displaystyle\langle\overline{10}|H_{J}/h|\overline{10}\rangle$
$\displaystyle\approx$
$\displaystyle-J\frac{\Omega_{0}\Omega_{1}}{4\Delta_{0}\Delta_{1}}e^{i(\phi_{0}-\phi_{1})},$
(S11) $\displaystyle\langle\overline{00}|H_{J}/h|\overline{00}\rangle$
$\displaystyle\approx$ $\displaystyle
J\frac{\Omega_{0}\Omega_{1}}{2\Delta_{0}\Delta_{1}}\cos(\phi_{0}-\phi_{1}),$
(S12) $\displaystyle\langle\overline{11}|H_{J}/h|\overline{11}\rangle$
$\displaystyle\approx$ $\displaystyle
J\frac{\Omega_{0}\Omega_{1}}{2\Delta_{0}\Delta_{1}}\cos(\phi_{0}-\phi_{1}).$
(S13)

From Eq. 2, we thus obtain for the doubly dressed frame,

$\displaystyle\tilde{\nu}_{ZZ}$ $\displaystyle\approx$ $\displaystyle
2J\frac{\Omega_{0}\Omega_{1}}{\Delta_{0}\Delta_{1}}\cos(\phi_{0}-\phi_{1}).$
(S14)

For transmons we perform a similar calculation and including the $|2\rangle$
state leads to the expression in Eq. 6 of the main text.

More generally, we start from Eq. 4 of the main text,

$\displaystyle H_{\textrm{siZZle}}/h$ $\displaystyle=$ $\displaystyle
H_{0}/h+\sum_{i=\\{0,1\\}}\Omega_{i}\cos{(2\pi\nu_{d}t+\phi_{i})}(\hat{a}_{i}^{\dagger}+\hat{a}_{i}),$
(S15)

with amplitudes $\Omega_{i}$, phases $\phi_{i}$, and a common frequency
$\nu_{d}$. First, we move into the frame rotating at $\nu_{d}$ via the unitary
operator,

$\displaystyle R_{d}$
$\displaystyle=e^{-i2\pi\nu_{d}t\left(\hat{a}_{0}^{\dagger}\hat{a}_{0}+\hat{a}_{1}^{\dagger}\hat{a}_{1}\right)}.$
(S16)

The RWA is made on the drive tones by ignoring fast rotating terms. The result
is a time-independent Hamiltonian and diagonalizing followed by restoring
$\nu_{d}$ via $R_{d}^{\dagger}$ gives the effective Hamiltonian describing the
dynamics under the exchange coupling and Stark tones,

$\displaystyle H_{\text{eff}}/h$
$\displaystyle=\tilde{\nu}_{IZ}IZ/4+\tilde{\nu}_{ZI}ZI/4+\tilde{\nu}_{ZZ}ZZ/4,$
(S17)

where
$\tilde{\nu}_{ZZ}=(\tilde{E}_{00}+\tilde{E}_{11}-\tilde{E}_{01}-\tilde{E}_{10})/h$.
For the case $\alpha_{0}\approx\alpha_{1}$, $\tilde{\nu}_{ZZ}$ is given in Eq.
6 of the main text and $\tilde{\nu}_{IZ}$, $\tilde{\nu}_{ZI}$ are given by,

$\displaystyle\tilde{\nu}_{IZ}$
$\displaystyle=(\tilde{E}_{01}-\tilde{E}_{00}+\tilde{E}_{11}-\tilde{E}_{10})/h\approx\nu_{IZ,J}+\nu_{1,s}+\frac{J(\alpha_{0}+\alpha_{1})\Omega_{0}\Omega_{1}\cos(\phi_{0}-\phi_{1})}{\Delta_{1,d}(\alpha_{0}+\Delta_{0,d})(\alpha_{1}+\Delta_{1,d})},$
(S18) $\displaystyle\tilde{\nu}_{ZI}$
$\displaystyle=(\tilde{E}_{10}-\tilde{E}_{00}+\tilde{E}_{11}-\tilde{E}_{01})/h\approx\nu_{ZI,J}+\nu_{0,s}+\frac{J(\alpha_{0}+\alpha_{1})\Omega_{0}\Omega_{1}\cos(\phi_{0}-\phi_{1})}{\Delta_{0,d}(\alpha_{0}+\Delta_{0,d})(\alpha_{1}+\Delta_{1,d})},$
(S19)

where

$\displaystyle\tilde{\nu}_{IZ,J}$
$\displaystyle=2\left(-\nu_{1}+J^{2}\left(\frac{1}{\Delta_{01}}+\frac{\alpha_{0}+\alpha_{1}}{(\Delta_{01}+\alpha_{0})(\Delta_{01}-\alpha_{1})}\right)\right),$
$\displaystyle\tilde{\nu}_{ZI,J}$
$\displaystyle=2\left(-\nu_{0}+J^{2}\left(-\frac{1}{\Delta_{01}}+\frac{\alpha_{0}+\alpha_{1}}{(\Delta_{01}+\alpha_{0})(\Delta_{01}-\alpha_{1})}\right)\right),$
(S20) $\displaystyle\nu_{0,s}$
$\displaystyle=-\frac{\Omega_{0}^{2}\alpha_{0}}{\Delta_{0,d}(\alpha_{0}+\Delta_{0,d})},$
$\displaystyle\nu_{1,s}$
$\displaystyle=-\frac{\Omega_{1}^{2}\alpha_{1}}{\Delta_{1,d}(\alpha_{1}+\Delta_{1,d})}.$
(S21)

##  II Cross-resonance with $ZZ$ cancellation tones

The starting Hamiltonian is given by,

$\displaystyle H/h$ $\displaystyle=$
$\displaystyle\sum_{i\in\\{0,1\\}}\left(\nu_{i}\hat{a}_{i}^{\dagger}\hat{a}_{i}+\frac{\alpha_{i}}{2}\hat{a}_{i}^{\dagger}\hat{a}_{i}\left(\hat{a}_{i}^{\dagger}\hat{a}_{i}-1\right)\right)+J(\hat{a}_{0}^{\dagger}+\hat{a}_{0})(\hat{a}_{1}^{\dagger}+\hat{a}_{1})$
(S22)
$\displaystyle+\sum_{i\in\\{0,1\\}}\Omega_{i}\cos{(2\pi\nu_{d}t+\phi_{i})}(\hat{a}_{i}^{\dagger}+\hat{a}_{i})+\Omega_{0,\textrm{gate}}(t)\cos{(2\pi\nu_{0,\textrm{gate}}t+\phi_{0,\textrm{gate}})}(\hat{a}_{0}^{\dagger}+\hat{a}_{0}).$

In order to find the effective Hamiltonian describing the system including the
cross-resonance tone, we first find the effective Hamiltonian describing the
dynamics of the exchange coupling and Stark tones. The series of
transformations are also applied to the CR drive tone
$\Omega_{0,\textrm{gate}}(t)\cos{(2\pi\nu_{0,\textrm{gate}}t+\phi_{0,\textrm{gate}})}(\hat{a}_{0}^{\dagger}+\hat{a}_{0})$
so we obtain,

$\displaystyle H$ $\displaystyle\rightarrow
H_{\text{eff}}+\Omega_{0,\textrm{gate}}(t)\cos{(2\pi\nu_{0,\textrm{gate}}t+\phi_{0,\textrm{gate}})}D_{\textrm{CR}}(t),$
(S23)

where $D_{\textrm{CR}}(t)$ is the transformed drive operator. We set
$\nu_{0,\textrm{gate}}=\tilde{\nu}_{1}$ and $\phi_{0,\textrm{gate}}=0$. Moving
into the frame rotating at $\tilde{\nu}_{1}$ and making the RWA gives to
first-order in the cross-resonance tone amplitude, first order in $J$, second
order in the Stark tone amplitudes, and assuming
$\alpha_{0}=\alpha_{1}=\alpha$ for simplicity,

$\displaystyle\tilde{\nu}_{ZX}$
$\displaystyle=\text{tr}\left(H_{\text{eff,CR}}\frac{ZX}{2}\right)=J\Omega_{0,\text{gate}}\left(A+B\Omega_{0}^{2}+C\Omega_{1}^{2}\right),$
(S24)

where

$\displaystyle A$
$\displaystyle=-\frac{\alpha}{\Delta_{0,1}(\alpha+\Delta_{0,1})},$ (S25)
$\displaystyle B$
$\displaystyle=-\frac{\alpha}{4\Delta_{0,1}(\alpha+\Delta_{0,1})^{2}\Delta_{0,d}}+\frac{(2\alpha+\Delta_{0,1})}{8(\alpha+\Delta_{0,1})\Delta_{0,d}\Delta_{0,1}^{2}}-\frac{\alpha}{4(\alpha+\Delta_{0,d})(\alpha+\Delta_{0,1})\Delta_{0,1}^{2}}$
$\displaystyle-\frac{\alpha}{4(\alpha+\Delta_{0,1}+\Delta_{0,d})(\alpha+\Delta_{0,1})\Delta_{0,1}^{2}}+\frac{(2\alpha+\Delta_{0,1})}{8\Delta_{1,d}(\alpha+\Delta_{0,1})\Delta^{2}}+\frac{\Delta_{0,1}(\alpha+\Delta_{0,d}+\Delta_{1,d})}{8(\alpha+\Delta_{0,1})^{2}(2\alpha+\Delta_{0,1})(\alpha+\Delta_{0,d})\Delta_{1,d}}$
$\displaystyle+\frac{1}{16(\alpha+\Delta_{0,1})^{2}}\Bigg{(}-\frac{2}{\Delta_{0,d}}-\frac{2}{\alpha+\Delta_{0,d}}-\frac{2\alpha}{(2\alpha+\Delta_{0,1})(\alpha+\Delta_{0,d})}+\frac{6\alpha}{(2\alpha+\Delta_{0,1})(2\alpha+\Delta_{0,d})}$
$\displaystyle+\frac{2\alpha}{(\alpha+\Delta_{0,d})(\alpha+\Delta_{0,1}+\Delta_{0,d})}+\frac{6\alpha}{(2\alpha+\Delta_{0,1})(3\alpha+\Delta_{0,1}+\Delta_{0,d})}-\frac{10\alpha+4\Delta_{0,1}}{\Delta_{1,d}(2\alpha+\Delta_{0,1})}\Bigg{)},$
(S26)

and

$\displaystyle
C=\frac{\alpha}{4\Delta_{0,1}^{2}}\Bigg{(}\frac{1}{(\Delta_{0,1}-\alpha)\Delta_{0,d}}-\frac{\Delta_{0,1}}{(\alpha+\Delta_{0,1})^{2}(\alpha+\Delta_{0,d})}+\frac{\alpha(\alpha+3\Delta_{0,1})}{(\Delta_{0,1}-\alpha)(\alpha+\Delta_{0,1})^{2}\Delta_{1,d}}$
$\displaystyle-\frac{\alpha(\alpha+3\Delta_{0,1})}{(\Delta_{0,1}-\alpha)(\alpha+\Delta_{0,1})^{2}(\alpha+\Delta_{1,d})}+\frac{\Delta_{0,1}}{(\alpha+\Delta_{0,1})^{2}(\Delta_{1,d}-\Delta_{0,1})}+\frac{1}{(\alpha-\Delta_{0,1})(\alpha-\Delta_{0,1}+\Delta_{1,d})}\Bigg{)}.$
(S27)

##  III Gate Calibration: Device A

The single qubit gates are 4$\sigma$ derivative Gaussian quadrature corrected
(DRAG) pulses Chow  _et al._ (2010) of duration 50 ns. The CNOT gate consists
of two flat-topped Gaussian pulses applied simultaneously on the control and
target qubits at the target frequency, followed by Z rotations on both qubits
implemented by frame changes McKay  _et al._ (2017). The target pulse envelope
is given by

$\displaystyle\Omega(t)=\Omega_{x}(t)\cos(2\pi\nu_{tg}t)+(\beta\dot{\Omega}_{x}(t)+\gamma|\dot{\Omega}_{x}(t)|)\sin(2\pi\nu_{tg}t)$

where $\Omega_{x}$ is the flat-topped Gaussian pulse, $\nu_{tg}$ is the target
frequency, $\beta$ and $\gamma$ are the DRAG and skew corrections
respectively. The control pulse does not have DRAG or skew correction.

To begin with the CNOT gate calibration, we do a rough amplitude calibration
on the control pulse such that the $ZX$ rotation on the target is $\pi/2$,
then we apply a pulsed version of Hamiltonian tomography Sheldon  _et al._
(2016) on the control pulse to align the $ZX$ interaction along the $-x$ axis.
Next we turn on the target pulse and do a fine calibration by simultaneously
varying the control amplitude, target amplitude, control and target phases,
target drag, target skew, and target frame change to tune the gate unitary to
be $\outerproduct{0}{0}\otimes I+e^{-i\varphi}\outerproduct{1}{1}\otimes X$.
Finally, the control frame change is calibrated to cancel $\varphi$, which
brings the unitary to a CNOT gate. The fine calibration sequences are shown in
Fig. S1 A-F, which measures the target dynamics when the control qubit is in
either $\ket{0}$ or $\ket{1}$ state. The control amplitude and target
amplitude are updated according to Fig. S1 A and B, the goal to simultaneously
satisfy $\theta_{ZX}+\theta_{IX}=0$ and $-\theta_{ZX}+\theta_{IX}=\pi$, where
$\theta_{ZX}$ and $\theta_{IX}$ are the rotations due to cross-resonance and
target pulses respectively. The target drag ($\beta$) and the gate angle are
updated according to Fig. S1 D and F, these calibrations make sure the target
rotation is along the $x$-axis when the control is in $\ket{1}$. When
calibrating the gate angle, the control and target phases are updated
together. Finally, the target skew ($\gamma$) and target frame change (fc) are
calibrated according to Fig. S1 E and C, which ensures the target dynamics is
identity when control is in $\ket{0}$. The control frame change (FC) is
calibrated at the very end, using the sequence in Fig. S1 G. The final
calibrated pulse envelope is shown in FIG. S1 H, the rise and fall for the
flat-topped Gaussian pulses are $2\sigma$ long where $\sigma=10$ns.

Figure S1: Direct CNOT gate calibration sequences Sequences A-F are
implemented simultaneously. Target and control amplitudes are updated
according to the outputs of A and B. The target frame change, target drag,
target skew, and control/target phase are updated according to the outputs of
C-F respectively. Sequence G is used to calibrate the control frame change.
The calibrated pulse envelope for the CNOT gate is shown in H. The sequence in
bracket is repeated $n$ times. For A-F the target qubit population is
measured, for G the control qubit population is measured. The sequence in
bracket is repeated n times

The calibration of the CZ gate is similar to that of the CNOT gate. Here, two
flat-topped Gaussian pulses are applied simultaneously to the control and
target qubits at the siZZle frequency, followed by two frame changes on the
control and target qubits. We fix the target amplitude and the relative phase
between the two siZZle pulses, then calibrate the control amplitude and target
frame change simultaneously to satisfy $\theta_{ZZ}+\theta_{IZ}=0$ and
$-\theta_{ZZ}+\theta_{IZ}=\pi$. Finally we calibrate the control frame change
to cancel the control Stark shift and bring the unitary to a CZ gate. The
calibration sequence are shown in Fig. S2 A-C, and the final CZ pulse envelope
are shown in Fig. S2 D, where the rise and fall times are $3\sigma$ with
$\sigma=10$ns. Unlike for CNOT gate, drag and skew are not used in the CZ gate
calibration.

Figure S2: Direct CZ gate calibration sequences Sequence A and B are
implemented simultaneously, and the outputs are used to update the control
amplitude and target frame change (fc). Sequence C is used to calibrate the
control frame change (FC). The calibrated pulse envelope for the CZ gate is
shown in D.

We show the calibration data for both the CNOT and CZ gates. The fine
calibration routine is an iterative process Sheldon  _et al._ (2015), which
terminates when the absolute difference between the calibrated rotation angles
and the desired rotation angles becomes less than $0.01$. In FIG S3 A-G. we
show the converged data for sequence used in the CNOT gate calibration, and in
FIG S3 H-J the final converged data for sequence used in the CZ gate
calibration.

Figure S3: Output of the calibration sequences used for CNOT and CZ gates. A-G
correspond to the output of the CNOT calibration sequences respectively. Where
as H-J correspond to the output of the CZ calibration sequences respectively.
The blue points are experimental data, and the red dashed lines are the fits
to experimental data. We extract the calibrated rotation angles from the fits.
The y-axis in each plot corresponds to either the control or target
population, and the x-axis is number of repetitions (n) shown in the
calibration sequence. Figure S4: Device A coherence Scatter plots of $T_{1}$
(left), $T_{2}$ (middle), and $T_{2}^{*}$ (right) times for Q0 (top, red) and
Q1 (bottom ,blue), with and without CW siZZle tones. All the measurements were
interleaved and taken at 30 minute intervals. The Stark drive on Q0 for $ZZ$
cancellation is larger than Q1 at the chosen operating point, as well as the
corresponding Stark shift, resulting in a clear reduction $T_{2}$ and
$T_{2}^{*}$.

##  IV SiZZle with multi-path ZZ cancellation couplers

As seen in the coherence data of Device A, discussed in Fig. S4, there can be
a degradation of coherence with $ZZ$ cancellation. Particularly, at large $J$
couplings with standard couplers, one requires large siZZle amplitudes
$\Omega$ to achieve $ZZ$ cancellation. Since the qubit Stark shifts are
proportional to $\Omega^{2}$, this makes the qubits more susceptible to
amplitude noise, and consequently can lead to additional dephasing. In this
section, we numerically show that using multi-path couplers, for the same
effective $J$, one can achieve full $ZZ$ cancellation at smaller siZZle
amplitudes due to requiring smaller qubit Stark shifts. We start with the
following form of the Hamiltonian with a direct qubit coupling and an
additional coupling path via a bus resonator.

$\displaystyle H/h$ $\displaystyle=$
$\displaystyle\sum_{i=\\{0,1\\}}\left(\nu_{i}\hat{a}_{i}^{\dagger}\hat{a}_{i}+\frac{\alpha_{i}}{2}\hat{a}_{i}^{\dagger}\hat{a}_{i}\left(\hat{a}_{i}^{\dagger}\hat{a}_{i}-1\right)\right)+J(\hat{a}_{0}^{\dagger}+\hat{a}_{0})(\hat{a}_{1}^{\dagger}+\hat{a}_{1})+\sum_{j}\nu_{j}\hat{b}_{j}^{\dagger}\hat{b}_{j}$
(S28)
$\displaystyle+\sum_{i=\\{0,1\\},j}g_{i,j}(\hat{a}_{i}^{\dagger}+\hat{a}_{i})(\hat{b}_{j}^{\dagger}+\hat{b}_{j})+\sum_{i=\\{0,1\\}}\Omega_{i}\cos{(2\pi\nu_{d}t+\phi_{i})}(\hat{a}_{i}^{\dagger}+\hat{a}_{i}).$

Most terms here have already been defined in Eq. 4 of the main text. The
additional terms arise from the bus coupling, where $g_{i,j}$ is the coupling
from qubit $i$ to the $j$’th harmonic mode of the bus at frequency $\nu_{j}$.
For simplicity, we drop the counter-rotating terms and consider a single bus
mode. Eq. S28 is then transformed into a time independent form by moving into
a frame rotating at the drive frequency $\nu_{d}$ via the rotation operator
$\hat{R}/h=e^{-i2\pi\nu_{d}t(\hat{a}_{0}^{\dagger}\hat{a}_{0}+\hat{a}_{1}^{\dagger}\hat{a}_{1}+\hat{b}^{\dagger}\hat{b})}$
and applying the RWA. One can then obtain the Stark shifts
$\tilde{\nu}_{ZI},\tilde{\nu}_{IZ}$ and the $ZZ$ interaction
$\tilde{\nu}_{ZZ}$ by diagonalizing the time independent Hamiltonian.

We consider the following parameters, that are similar to pairs on device B:
$\nu_{0}=4.85$ GHz, $\nu_{1}=4.95$ GHz, $\alpha_{0}=\alpha_{1}=-290$ MHz,
$g_{0}=g_{1}=135$ MHz, $J=10.6$ MHz, $\nu_{\textrm{bus}}=6.35$ GHz,
$\nu_{d}=5.1$ GHz and $\Omega_{0}=\Omega_{1}$. From the low amplitude
dependence of $\tilde{\nu}_{ZZ}$, we estimate an effective $J$ coupling for
the multi-path coupler(mpc) using the form of Eq. 6 to be
$J_{\textrm{eff}}=3.28$ MHz. We then compare the Stark tone amplitude
dependence of $\tilde{\nu}_{ZI},\tilde{\nu}_{IZ}$ and $\tilde{\nu}_{ZZ}$ for
this mpc device with a single path coupler (spc) of the same
$J_{\textrm{eff}}$. This is depicted in Fig. S5. For the mpc, while $ZZ$
cancellation only requires Stark tone amplitudes $\sim 15$ MHz, the spc
requires amplitudes $\sim 55$ MHz, seen in Fig. S5a. Consequently, the qubit
Stark shifts are much smaller at $ZZ$ cancellation for the mpc device, seen in
Fig. S5b and c, thereby reducing the sensitivity to Stark tone amplitude
noise.

Figure S5: SiZZle with multi-path $ZZ$ cancellation couplers Numerical
simulations of the Stark tone amplitude dependence of (a) $\tilde{\nu}_{ZZ}$,
(b) $\tilde{\nu}_{ZI}$ and (c) $\tilde{\nu}_{IZ}$ for multi-path coupler
(blue) and a single-path coupler (red) with the same $J_{\textrm{eff}}=3.2$
MHz, defined in the text. We consider the following parameters for the mpc
device: $\nu_{0}=4.85$ GHz, $\nu_{1}=4.95$ GHz, $\alpha_{0}=\alpha_{1}=-290$
MHz, $g_{0}=g_{1}=135$ MHz, $J=10.6$ MHz, $\nu_{\textrm{bus}}=6.35$ GHz,
$\nu_{d}=5.1$ GHz and $\Omega_{0}=\Omega_{1}$. The blue (red) dotted line
represents the operating point for ZZ cancellation for the mpc (spc) device.
For the mpc, the $ZZ$ cancellation is achieved at smaller Stark tone
amplitudes, and the smaller $\tilde{\nu}_{ZI}$ and $\tilde{\nu}_{IZ}$ are less
sensitive to amplitude noise on the Stark tones.

##  V Device B: Seven-qubit device with multi-path couplers for $ZZ$
cancellation

In this section, we detail device B from the main text. The seven qubits
represent a sub section of a larger lattice of 27 qubits in the heavy-hex
architecture. In order to reduce the static $ZZ$ interaction compared to
standard single path couplers, the device employs coupling elements composed
of a direct capacitive coupler and a $\lambda/4$ bus resonator Kandala  _et
al._ (2020). The bus resonator frequencies are in the range 6.35-6.55 GHz. For
$ZZ$ cancellation, a common frequency $\nu_{d}=5.1$ GHz was chosen for the CW
tones, above all the qubit transitions. Most of the qubit parameters and gate
fidelities are detailed in Fig. S6. The qubit anharmonicities are in the range
-288 to -295 MHz, and the average readout fidelity is 98.2 $\%$. As seen in
Fig. S6, the qubit frequencies are shifted by at most 1.2 MHz, by the CW tones
for full $ZZ$ cancellation. This helps retain good coherence times for the
device, even after $ZZ$ cancellation, depicted in Fig. S7. Some of the qubits
show a modest decrease in $T_{2}$, while the $T_{1}$ times are within typical
fluctuations. From the single drive Stark shifts of the qubits, we estimate
the amplitude of the CW tones driving Q0/1/2/3/4/5/6 for $ZZ$ cancellation to
be 17.6/16.8/20.4/19.5/21.1/13.0/21.3 MHz respectively.

Figure S6: Device B sub-system metrics Qubit frequencies, single and two-qubit
gate times and their respective error rates, and the strength of the pair wise
static $ZZ$ interaction $\tilde{\nu}_{ZZ}$ for (a) the native device without
siZZle tones (b) with siZZle tones at $\nu_{d}=5.1$ GHz, and pair-wise phases
tuned to $ZZ$ cancellation $\phi\sim\pi$. (c) with siZZle tones at
$\nu_{d}=5.1$ GHz, and pair-wise phases tuned to $ZZ$ amplification $\phi\sim
0$. All gate errors are estimated by randomized benchmarking. The arrows
represent the direction of the CNOT gates employed in the QV circuits
discussed in the main text, and the reported error per gates (EPG) represent
the upper bound obtained from the error per Clifford (EPC/1.5). The CNOT gates
are composed of two cross-resonance pulses and two finite-time single qubit
pulses, and the gate times are optimized for operation in the absence of
siZZle. The single qubit EPG’s represent the errors for simultaneous single
qubit operation.  Figure S7: Device B coherence Scatter plots of $T_{1}$ (top,
red) and $T_{2}$ (bottom, blue) times of the 7 qubits, with and without siZZle
tones. All the measurements were interleaved and taken at 30 minute intervals.