# Fast $ZZ$-Free Entangling Gates for Superconducting Qubits Assisted by a
Driven Resonator

Ziwen Huang<EMAIL_ADDRESS>Superconducting Quantum Materials and Systems
Center, Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA
Taeyoon Kim  Superconducting Quantum Materials and Systems Center, Fermi
National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  Department of
Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA  Tanay
Roy  Superconducting Quantum Materials and Systems Center, Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  Yao Lu  Superconducting
Quantum Materials and Systems Center, Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA  Alexander Romanenko  Superconducting Quantum
Materials and Systems Center, Fermi National Accelerator Laboratory (FNAL),
Batavia, IL 60510, USA  Shaojiang Zhu  Superconducting Quantum Materials and
Systems Center, Fermi National Accelerator Laboratory (FNAL), Batavia, IL
60510, USA  Anna Grassellino<EMAIL_ADDRESS>Superconducting Quantum Materials
and Systems Center, Fermi National Accelerator Laboratory (FNAL), Batavia, IL
60510, USA

###### Abstract

Engineering high-fidelity two-qubit gates is an indispensable step toward
practical quantum computing. For superconducting quantum platforms, one
important setback is the stray interaction between qubits, which causes
significant coherent errors. For transmon qubits, protocols for mitigating
such errors usually involve fine-tuning the hardware parameters or introducing
usually noisy flux-tunable couplers. In this work, we propose a simple scheme
to cancel these stray interactions. The coupler used for such cancellation is
a driven high-coherence resonator, where the amplitude and frequency of the
drive serve as control knobs. Through the resonator-induced-phase (RIP)
interaction, the static $ZZ$ coupling can be entirely neutralized. We
numerically show that such a scheme can enable short and high-fidelity
entangling gates, including cross-resonance CNOT gates within 40 ns and
adiabatic CZ gates within 140 ns. Our architecture is not only $ZZ$ free but
also contains no extra noisy components, such that it preserves the coherence
times of fixed-frequency transmon qubits. With the state-of-the-art coherence
times, the error of our cross-resonance CNOT gate can be reduced to below
$10^{-4}$.

##  I Introduction

Strong interactions are a typical requirement for faster entangling operations
between multiple qubits [1, 2]. However, increasing the coupling strength can
come with larger stray interactions, which limit the fidelities of both single
and two-qubit gates. For superconducting qubits, one solution is to introduce
flux-tunable couplers, which can mitigate such cross-talk and has led to
increased gate fidelities [3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. However, for the
widely used transmon processors [13], it has been observed that such tunable
elements also lead to extra decoherence errors due to the high loss rate of
the coupler [14, 15, 16, 2].

Alternative strategies for $ZZ$ cancellation include using multi-path couplers
[17] and ac Stark shifts [18, 19, 20, 21]. By either fine-tuning the
parameters of the bus coupler [22] or Stark driving the transmon qubits, one
can suppress the $ZZ$ interaction without any flux tunability. In this paper,
we propose a different non-tunable architecture that is even free of the
parameter fine-tuning and direct transmon drives for the cancellation. In our
scheme, a high-coherence microwave resonator is used to cancel unwanted $ZZ$
coupling via the resonator-induced-phase (RIP) interaction [23, 24, 25, 26].
For such cancellation, we apply a constant off-resonant drive that displaces
the resonator state ($\approx\\!10$ photons in the steady state). Due to the
dispersive shifts, the displacement of the resonator state depends on the
states of the two transmon qubits, which in turn introduces a dynamical $ZZ$
interaction between the qubits. The strength of this coupling is highly
tunable and can be employed to cancel the static $ZZ$ coupling. With
superconducting microwave cavities with high coherence times [27, 28, 29], the
resonator photon loss only negligibly affects gate fidelities.

We show by simulations that our architecture offers a family of fast
entangling gates between qubits, including the cross-resonance (CR) [30, 31,
17, 32, 33] CNOT gates ($\lesssim\\!40$ ns) and adiabatic CZ gates [6, 12]
($\lesssim\\!140$ ns). Particularly, the CR-CNOT gates can be significantly
accelerated since the cancellation of the static $ZZ$ coupling ($\approx$ 6
MHz in our simulation) allows stronger qubit coupling without introducing
coherent errors caused by stray interactions [18]. Due to the shorter gate
times, the infidelity of the CR gates can reach below $10^{-4}$ according to
our simulation. Besides, the stronger coupling also allows a much larger qubit
frequency separation compared to those on typical CR architectures [34, 35],
which can alleviate the issue of frequency crowding among the transmon qubits.

Figure 1: Schematic of the lumped-element model of the composite transmon-
resonator-transmon system. In this model, both transmon qubits are
capacitively coupled to a high-coherence resonator, which is modeled as a
lumped-element resonator. The excitation energies of the left ($L$), right
($R$) qubit and the coupling resonator ($C$) are denoted by $\omega_{L,R,C}$,
respectively. The frequencies of the drives on the three elements are denoted
by $\omega^{\mathrm{d}}_{L,R,C}$.

The paper is structured as follows. In Sec. II, we introduce the lumped-
element model of this architecture and demonstrate both analytically and
numerically the $ZZ$ cancellation mechanism. In Sec. III, we discuss the
implementation of the fast cross-resonance gates and present simulations of
their gate fidelities. In Sec. IV, we show simulations of the adiabatic CZ
gates via the adiabatic tuning of resonator drives. In Sec. V, we briefly
discuss other potential error channels and their magnitudes. In Sec. VI, we
outline a plan to scale up this $ZZ$-free architecture. In Sec. VII, we append
our conclusions.

##  II Model and $ZZ$ Cancellation

###  II.1 Hamiltonian and dispersive transformation

The circuit we study comprises two superconducting qubits both capacitively
coupled to a central linear resonator. The lumped-element model of this system
is shown in Fig. 1. The Hamiltonian of this system is

$\displaystyle\hat{H}(t)=\sum_{\upsilon=L,R}\hat{H}_{\upsilon}+\hat{H}_{C}+\hat{H}_{\mathrm{int}}+\hat{H}_{\mathcal{D}}(t),$
(1)

where $\hat{H}_{L(R)}$ denotes the Hamiltonian of the left (right) qubit,
$\hat{H}_{C}$ is the resonator Hamiltonian, and $\hat{H}_{\mathrm{int}}$ is
their interaction. Besides the static terms, the time-dependent Hamiltonian
$\hat{H}_{\mathrm{d}}(t)$ describes the drives on the three components.

For this study, we consider the two qubits as fixed-frequency transmon qubits
[13, 36, 37, 38, 39, 40], which are among the most coherent superconducting
quantum elements and have been demonstrated to possess coherence times
approaching one millisecond [37, 39, 38, 41]. The Hamiltonian of the left
(right) transmon qubit is given by
$\hat{H}_{L(R)}=4E_{C_{L(R)}}\hat{n}^{2}_{L(R)}-E_{J_{L(R)}}\cos\hat{\varphi}_{L(R)}$,
where $\hat{n}_{L(R)}$ and $\hat{\varphi}_{L(R)}$ denote the charge and phase
operators of the left (right) qubit. The central lumped-element resonator is
assumed to be physically implemented as a 2D or 3D high-coherence resonator.
The Hamiltonian of this resonator is given by
$\hat{H}_{C}=\bar{\omega}_{C}\hat{a}^{\dagger}\hat{a}$, where $\hat{a}$
$(\hat{a}^{\dagger})$ denotes the annihilation (creation) operator, and
$\bar{\omega}_{C}$ is the bare resonator frequency. The interaction term is
specified by
$\hat{H}_{\mathrm{int}}=(\hat{a}+\hat{a}^{\dagger})(g_{L}\hat{n}_{L}+g_{R}\hat{n}_{R})$,
where $g_{L,R}$ denotes the qubit-resonator coupling strength. For these three
components, the drive term $\hat{H}_{\mathcal{D}}(t)$ is specified as the
charge drives that address each element. In the bare basis, it takes on the
form

$\displaystyle\hat{H}_{\mathcal{D}}(t)=\sum_{\upsilon=L,R}\mathcal{D}_{\upsilon}(t)\hat{n}_{\upsilon}+\mathcal{D}(t)(\hat{a}^{\dagger}+\hat{a}),$
(2)

where $\mathcal{D}_{L,R}(t)$ and $\mathcal{D}(t)$ denote the time-dependent
drive strengths.

Due to the coupling between the three components described by
$\hat{H}_{\mathrm{int}}$, the true eigenstates of this composite system are
hybridizations of the bare eigenstates of the Hamiltonians $\hat{H}_{L(R)}$
and $\hat{H}_{C}$. In terms of these true (dressed) eigenstates, the
Hamiltonian in Eq. (1) can be expressed as

$\displaystyle\hat{H}^{\prime}(t)\approx$
$\displaystyle\\!\\!\\!\sum_{\upsilon=L,R}\omega_{\upsilon}\hat{b}_{\upsilon}^{\dagger}\hat{b}_{\upsilon}+\frac{\eta_{\upsilon}}{2}\hat{b}_{\upsilon}^{\dagger}\hat{b}_{\upsilon}^{\dagger}\hat{b}_{\upsilon}\hat{b}_{\upsilon}+\chi_{\upsilon}\hat{b}_{\upsilon}^{\dagger}\hat{b}_{\upsilon}\hat{a}^{\dagger}\hat{a}$
(3)
$\displaystyle+{\chi^{\prime}_{LR}}\hat{b}_{L}^{\dagger}\hat{b}_{L}\hat{b}_{R}^{\dagger}\hat{b}_{R}+{\omega_{C}}\hat{a}^{\dagger}\hat{a}+\frac{\eta_{C}}{2}\hat{a}^{\dagger}\hat{a}^{\dagger}\hat{a}\hat{a}+\hat{H}^{\prime}_{\mathcal{D}}(t).$

Above, $\hat{b}_{L(R)}$ is the annihilation operator for the left (right)
transmon qubit, $\omega_{L,R,C}$ denote the dressed frequencies of the left,
right qubit, and the resonator, respectively. As common in experiments on
fixed-frequency transmons [24, 42], we consider the transmon frequencies to be
around 4.5 GHz (detuning between qubit frequencies is specified later), and
the frequency of the resonator is designed to be significantly detuned from
those of the transmons $\omega_{L,R}$ to serve only as a passive coupler. (We
consider this detuning $\approx 5$ GHz in this work.) The anharmonicity of the
left (right) transmon qubit is denoted by $\eta_{L(R)}$. Because of the
hybridization with the two non-linear transmon qubits, the resonator also
acquires a small self-Kerr term in its Hamiltonian, whose strength is denoted
by $\eta_{C}$. The full dispersive shift between the left (right) qubit and
the resonator is denoted by $\chi_{L(R)}$. Due to their mutual coupling to the
resonator and relatively closer frequencies, the two qubits also share a $ZZ$
interaction, whose magnitude is denoted by $\chi^{\prime}_{LR}$. Finally, the
drive Hamiltonian $\hat{H}^{\prime}_{\mathcal{D}}(t)$ in Eq. (3) is
transformed from $\hat{H}_{\mathcal{D}}(t)$ in Eq. (1) after the dispersive
treatment. We will specify it later in the following sections for different
types of operations.

In our simulations, we specify the dressed-basis Hamiltonian (3) by
numerically diagonalizing the full Hamiltonian (1) via SCQUBITS [43]. The
analytical expressions of the parameters in Eq. (3) have been extensively
explored in the literature, e.g., Refs. [35, 25, 34, 44].

###  II.2 Adiabatic elimination and $ZZ$ cancellation

According to Eq. (3), the interaction between each qubit and the coupler
resonator is described by the dispersive-shift terms such as
$\chi_{L}\hat{b}^{\dagger}_{L}\hat{b}_{L}\hat{a}^{\dagger}\hat{a}$. If the
passive resonator mode is kept in its ground state [31, 17], these
interactions can be safely neglected. However, if we apply a drive on the
resonator, such as 111In the dressed basis, the charge operator of the
resonator yields other small terms, but for a constant drive amplitude, those
additional terms are far off-resonant.

$\displaystyle\hat{H}^{\prime}_{\mathcal{D,C}}(t)=2\mathcal{D}\cos\big{(}\omega^{\mathrm{d}}_{C}t\big{)}(\hat{a}^{\dagger}+\hat{a}),$
(4)

this usually neglected interaction can induce entanglement between the two
qubits [23, 24]. Such interaction is the key element in this $ZZ$-free
architecture.

To see the $ZZ$ cancellation, we next derive an effective Hamiltonian with the
resonator degree of freedom eliminated. After a frame transformation according
to the unitary
$\hat{U}_{C}(t)=\exp(-i\omega^{\mathrm{d}}_{C}t\hat{a}^{\dagger}\hat{a})$, the
Hamiltonian in the rotating frame (with fast-rotating terms neglected) is
given by

$\displaystyle\tilde{H}=$
$\displaystyle\,\hat{U}^{\dagger}_{C}(t)\hat{H}(t)\hat{U}_{C}(t)-i{\hat{U}}^{\dagger}_{C}(t)\dot{\hat{U}}_{C}(t)$
$\displaystyle\approx$
$\displaystyle\,\hat{H}^{\prime}_{q}-\Delta_{d}\hat{a}^{\dagger}\hat{a}+\frac{\eta_{C}}{2}\hat{a}^{\dagger}\hat{a}^{\dagger}\hat{a}\hat{a}+\mathcal{D}(\hat{a}^{\dagger}+\hat{a})$
$\displaystyle+\chi_{L}\hat{b}_{L}^{\dagger}\hat{b}_{L}\hat{a}^{\dagger}\hat{a}+\chi_{R}\hat{b}_{R}^{\dagger}\hat{b}_{R}\hat{a}^{\dagger}\hat{a},$
(5)

where we define $\Delta_{d}\equiv\omega^{\mathrm{d}}_{C}-\omega_{C}$, and
$\hat{H}^{\prime}_{q}$ is the Hamiltonian for two transmons [first line in Eq.
(3)]. Note that since the self-Kerr strength $\eta_{C}$ is much smaller than
the other coefficients in the parameter regime we study, we neglect it for the
following analytical derivation for simplicity but include it in our numerical
simulation.

The drive term $\mathcal{D}(\hat{a}^{\dagger}+\hat{a})$ displaces the
resonator state away from the zero-photon state. Therefore, we can no longer
assume that the resonator only stays in its ground state. However, one can
perform a transformation to adiabatically eliminate the drive term and further
the resonator degree of freedom. The unitary useful for such elimination is
given by

$\displaystyle\hat{U}_{\mathrm{dis}}=\exp\Bigg{[}\frac{\mathcal{D}(\hat{a}^{\dagger}-\hat{a})}{\Delta_{d}-\chi_{L}\hat{b}^{\dagger}_{L}\hat{b}_{L}-\chi_{R}\hat{b}^{\dagger}_{R}\hat{b}_{R}}\Bigg{]},$
(6)

which further transforms Eq. (5) into

$\displaystyle\tilde{H}_{\mathrm{dis}}=$
$\displaystyle\,\hat{U}^{\dagger}_{\mathrm{dis}}\tilde{H}\hat{U}_{\mathrm{dis}}$
$\displaystyle\approx$
$\displaystyle\,\hat{H}^{\prime}_{q}+\big{(}-\Delta_{d}+\chi_{L}\hat{b}_{L}^{\dagger}\hat{b}_{L}+\chi_{R}\hat{b}_{R}^{\dagger}\hat{b}_{R}\big{)}\hat{a}^{\dagger}\hat{a}$
$\displaystyle+\frac{\mathcal{D}^{2}}{\Delta_{d}-\chi_{L}\hat{b}^{\dagger}_{L}\hat{b}_{L}-\chi_{R}\hat{b}^{\dagger}_{R}\hat{b}_{R}}.$
(7)

The second line in the equation above contains both the transmon Hamiltonians
and the remaining coupling between the transmon qubit and the resonator in
this displaced frame. The latter interaction can lead to shifts of the qubit
frequencies if the resonator is not in the displaced vacuum state. However, if
the resonator initially has zero photons and the ramp-up of the drive strength
is sufficiently slow, during this ramp-up, the state of the resonator will
remain in the vacuum defined in the displaced frame. Given that the resonator
has a sufficiently small decoherence rate, we can safely assume that the
resonator remains in this vacuum during our gate operations. (We discuss this
effect in more detail in Sec. V.)

Figure 2: Numerical demonstration of $ZZ$ cancellation. Panel (a) plots the
two-qubit controlled phase $\phi_{\mathrm{Cph}}$ as a function of the driving
strength $\mathcal{D}$ for three different evolution times $\tau$. We confirm
that $\phi_{\mathrm{Cph}}$ vanishes for all times for a certain $\mathcal{D}$,
which is marked by the red solid line. Visibly, this value is close to the
prediction by Eq. (11), but considerably differs from that by Eq. (9). In (b),
the energy shifts of different transition frequencies are plotted as functions
of the driving strength. Dashed curves are predictions by Eq. (8). The device
parameters used for simulation are as follows. The detuning of the two qubits
is $\Delta_{LR}/2\pi\approx-660$ MHz. The resonator drive frequency is detuned
from the resonator frequency by $\Delta_{d}/2\pi=100$ MHz. The anharmonicities
of the three components are $\eta_{L}/2\pi\approx\eta_{R}/2\pi\approx-320$
MHz, and $\eta_{C}/2\pi\approx-90$ kHz. The three dispersive shifts are given
by $\chi_{L}/2\pi\approx-6.0$ MHz, $\chi_{R}/2\pi\approx-8.4$ MHz, and
$\chi^{\prime}_{LR}/2\pi=-5.7$ MHz. The effective $ZZ$ coupling between the
qubits vanishes at $\mathcal{D}_{0}/2\pi\approx$ 0.27 GHz.

The third line in Eq. (7) describes the RIP interaction between the two qubits
introduced by the driven resonator. Approximately, in the limit of
$|\chi_{L}|,|\chi_{R}|\ll|\Delta_{d}|$, the strength of the dynamical $ZZ$
coupling can be evaluated by expanding the last line of Eq. (7) by

$\displaystyle\,\frac{\mathcal{D}^{2}}{\Delta_{d}-\chi_{L}\hat{b}_{L}^{\dagger}\hat{b}_{L}-\chi_{R}\hat{b}_{R}^{\dagger}\hat{b}_{R}}$
(8) $\displaystyle\approx$
$\displaystyle\,\frac{\mathcal{D}^{2}}{\Delta_{d}}+\frac{\mathcal{D}^{2}}{\Delta_{d}^{2}}\Big{(}\chi_{L}\hat{b}_{L}^{\dagger}\hat{b}_{L}+\chi_{R}\hat{b}_{R}^{\dagger}\hat{b}_{R}\Big{)}$
$\displaystyle-\frac{\mathcal{D}^{2}}{\Delta_{d}^{3}}\Big{(}\chi_{L}\hat{b}_{L}^{\dagger}\hat{b}_{L}+\chi_{R}\hat{b}_{R}^{\dagger}\hat{b}_{R}\Big{)}^{2}.$
(9)

Eq. (9) results in a two-qubit coupling term
$-2\mathcal{D}^{2}\chi_{L}\chi_{R}\hat{b}^{\dagger}_{L}\hat{b}_{L}\hat{b}^{\dagger}_{R}\hat{b}_{R}/\Delta_{d}^{3}$
[23], which provides a sign and amplitude-tunable $ZZ$ interaction. The
condition

$\displaystyle
2\chi_{L}\chi_{R}\mathcal{D}^{2}/\Delta_{d}^{3}=\chi^{\prime}_{LR}$ (10)

then approximately gives the $ZZ$-free operating point. However, this
approximation only holds in the limit of $|\chi_{L,R}|\ll|\Delta_{d}|$. In
Appendix B, we show another approach to derive the expression of the magnitude
of this dynamical coupling.

Beyond this limit, we find that using Eq. (8) to calculate the entangling
strength is more accurate. Specifically, the $ZZ$ cancellation requires

$\displaystyle\Delta E_{\mathrm{zp}}(1,1)\\!+\Delta
E_{\mathrm{zp}}(0,0)\\!-\Delta E_{\mathrm{zp}}(1,0)\\!-\Delta
E_{\mathrm{zp}}(0,1)\\!+\chi^{\prime}_{LR}=0,$ (11)

where we define

$\displaystyle\Delta
E_{\mathrm{zp}}(j_{L},j_{R})\equiv\frac{\mathcal{D}^{2}}{\Delta_{d}-j_{L}\chi_{L}-j_{R}\chi_{R}}.$
(12)

To numerically confirm such a prediction, we perform a simulation [46] of the
controlled phase in a transmon-resonator-transmon system described by
Hamiltonian (3), where the previously neglected resonator self-Kerr term is
included. We present our results in Fig. 2 (a). For this simulation, we choose
realistic parameters motivated by previous experiments [27, 28, 24], which are
listed in the caption. The controlled phase is obtained by evaluating
$\phi_{\mathrm{Cph}}=\phi_{10}+\phi_{01}-\phi_{00}-\phi_{11}$ for a given
evolution time $\tau$ and resonator driving strength $\mathcal{D}$, where
$\phi_{j_{L},j_{R}}$ denotes the system’s accumulated phase if the number of
photons in the left (right) transmon qubit is maintained at $j_{L}(j_{R})$. By
varying $\mathcal{D}$ and $\tau$, we find that $\phi_{\mathrm{Cph}}$ vanishes
at all times for a certain $\mathcal{D}$, implying the desired $ZZ$
cancellation. Such a value considerably differs from the leading-order
approximation (9), but can be closely predicted by solving Eq. (11) for
$\mathcal{D}$. The remaining deviation between the prediction by Eq. (11) and
the numerical result is related to the neglection of the weak resonator self-
Kerr described by
$\eta_{C}\hat{a}^{\dagger}\hat{a}^{\dagger}\hat{a}\hat{a}/2$.

Besides canceling the $ZZ$ interaction, turning on a resonator drive also
shifts the 0-1 excitation frequencies of the two qubits. In Fig. 2 (b), we
also plot the shift of the 0-1 transition frequencies. These shifts are also
well approximated by Eq. (8). If the strength of the drive on the resonator is
sufficiently stable, these frequency shifts can be easily calibrated
experimentally.

##  III Fast Cross-Resonance Gate

For such a fixed-frequency architecture, one of the most popular and
convenient gates to apply is the cross-resonance gate [30, 47, 34, 48].
Generally, this type of entangling gate takes longer times than parametric
gates on frequency tunable architectures [6, 7, 49, 50, 14, 12]. Impressively,
we find our $ZZ$-free scheme can significantly reduce the required gate time,
which can be comparable to those obtained for the parametric gates.

###  III.1 Traditional CR gates and their errors

To see how the $ZZ$ cancellation can help with improving the fidelities of the
CR gates, we first revisit the traditional CR gates, and understand the
current limitations on gate fidelities. It has been pointed out that three
factors especially contribute to gate errors [34, 44, 17, 18, 44], which are
summarized and discussed below.

Decoherence.– We assume that the strength of the coupling between the two
transmon qubits is $J$, the detuning between the frequencies of the two qubits
is $\Delta_{LR}\equiv\omega_{L}-\omega_{R}$, and their anharmonicities satisfy
$\eta_{L}\approx\eta_{R}\approx\eta$. For the resonator mediated coupling
assumed in our architecture, the coupling strength $J$ is approximately [25,
35]

$\displaystyle J\approx\,\tilde{g}_{L}\tilde{g}_{R}\Bigg{[}$
$\displaystyle\frac{\bar{\omega}_{L}+\bar{\omega}_{R}-2\bar{\omega}_{C}}{2(\bar{\omega}_{L}-\bar{\omega}_{C})(\bar{\omega}_{R}-\bar{\omega}_{C})}$
$\displaystyle-$
$\displaystyle\frac{\bar{\omega}_{L}+\bar{\omega}_{R}+2\bar{\omega}_{C}}{2(\bar{\omega}_{L}+\bar{\omega}_{C})(\bar{\omega}_{R}+\bar{\omega}_{C})}\Bigg{]},$
(13)

where $\bar{\omega}_{L,R,C}$ are the bare frequencies of the two qubits and
the resonator, and $\tilde{g}_{L(R)}$ are the normalized coupling strength
defined by
$\tilde{g}_{L(R)}\approx[E_{J_{L(R)}}/32E_{C_{L(R)}}]^{\frac{1}{4}}g_{L(R)}$.
To activate the cross-resonance gate, we consider a drive on the control qubit
(chosen as $L$) whose frequency is close to that of the target qubit ($R$).
The strength of this drive is denoted by $\epsilon_{\mathrm{CR}}$.

Generally speaking, a longer gate time implies more decoherence loss [51, 52].
To reduce the CR gate duration, a stronger effective CR driving strength is
needed. The effective $CX$ rotation rate induced by the CR drive is
approximately $\epsilon_{CX}\approx A_{CX}\epsilon_{\mathrm{CR}}$, where
$\epsilon_{CR}$ is the amplitude of the drive on the control qubit, and the
coefficient is approximately [34, 1]

$\displaystyle
A_{CX}\approx\Bigg{(}\frac{E_{J_{L}}}{32E_{C_{L}}}\Bigg{)}^{\frac{1}{4}}\frac{2J\eta}{\Delta_{LR}(\eta+\Delta_{LR})}.$
(14)

Roughly, if the transmon loss rate is characterized by $\gamma$, the
decoherence error scales as

$\displaystyle\mathrm{Err}_{\mathrm{noise}}\approx$
$\displaystyle\,\frac{4\pi\gamma}{5|\epsilon_{CX}|}$ $\displaystyle\propto$
$\displaystyle\begin{cases}{|\Delta_{LR}}|\cdot|J|^{-1}\cdot|\epsilon_{\mathrm{CR}}|^{-1}\cdot\gamma,&|\Delta_{LR}|\ll|\eta|;\\\
|\Delta_{LR}|^{2}\cdot|J|^{-1}\cdot|\epsilon_{\mathrm{CR}}|^{-1}\cdot|\eta|^{-1}\cdot\gamma,&|\Delta_{LR}|\gg|\eta|.\end{cases}$
(15)

Leakage.– The second major error source is the leakage due to off-resonant
transitions. Ideally, the CR drive only induces rotations in the target qubit;
however, to ensure a sufficiently fast gate, the control qubit is strongly
driven, as shown in Eq. (14). Such drive causes off-resonant errors [34, 44].
An accurate estimation of such error requires the knowledge of specific
parameter regimes of the device and the pulse shapes [34, 44], but generally
speaking, this off-resonant error $\mathrm{Err}_{\mathrm{leak}}$ increases
with larger driving strength $\epsilon_{\mathrm{CR}}$ but decreases with
larger detuning $\Delta_{LR}$.

Imperfect rotation.– Due to the unwanted interactions, especially the $ZZ$
coupling, the conditional Rabi rotation usually cannot be made fully resonant.
To quantify this type of error, we first evaluate the $ZZ$ rate, which is
approximated by [17, 34]

$\displaystyle\chi^{\prime}_{LR}\approx\frac{4J^{2}\eta}{(\Delta_{LR}+\eta)(\Delta_{LR}-\eta)}.$
(16)

Such coupling introduces a coherent error roughly proportional to [17, 34]

$\displaystyle\mathrm{Err}_{ZZ}\propto$
$\displaystyle\,\Big{|}\frac{\chi^{\prime}_{LR}}{\epsilon_{CX}}\Big{|}^{2}$
$\displaystyle\propto$
$\displaystyle\begin{cases}|\Delta_{LR}|^{2}\cdot|J|^{2}\cdot|\epsilon_{\mathrm{CR}}|^{-2}\cdot|\eta|^{-2},&|\Delta_{LR}|\ll|\eta|;\\\
|J|^{2}\cdot|\epsilon_{\mathrm{CR}}|^{-2},&|\Delta_{LR}|\gg|\eta|.\end{cases}$
(17)

By inspecting the scaling laws of the magnitude of the three types of errors,
one can find that there is no clear parameter regime where all three can be
suppressed simultaneously [18]. Specifically, reducing
$\mathrm{Err}_{\mathrm{noise}}$ requires a smaller $\Delta_{LR}$, reducing
$\mathrm{Err}_{\mathrm{leak}}$ requires a smaller $\epsilon_{\mathrm{CR}}$,
and reducing $\mathrm{Err}_{ZZ}$ requires a smaller $J$. However, if all of
them are small, neither type of error can be efficiently suppressed. We
summarize this situation in a diagram in Fig. 3.

Figure 3: A diagram illustrating how traditional cross-resonance gates are
affected by the three error channels, namely the decoherence loss, leakage,
and imperfect rotations due to $ZZ$ interactions. The contributions of these
channels are determined by three parameters, i.e., the two-qubit detuning
$\Delta_{LR}$, the driving strength $\epsilon_{\mathrm{CR}}$, and the two-
qubit effective coupling strength $J$. The dependence is roughly characterized
by positive (red arrows) and negative (blue arrows) correlations, as sketched
in the diagram above. Using the architecture presented in this work, the $ZZ$
error can be removed. Such mitigation strategy is indicated by the dashed
lines.

To overcome this apparent “trilemma”, one promising route is to cancel or
suppress the $ZZ$ coupling strength. This has been achieved by introducing
extra coupling elements [17, 8] or additional drives [18]. These cancellation
approaches open up new parameter space for error reduction. For example, if we
maintain a reasonable ratio $\epsilon_{\mathrm{CR}}/\Delta_{LR}$ to suppress
leakage, we can increase the coupling strength $J$ between the two qubits to
further reduce the gate infidelity [18].

In our $ZZ$-free regime, the third error source is eliminated by the RIP
interaction, which does not introduce additional loss channels, or require the
fine-tuning of hardware parameters. The drive is applied to the resonator,
which can avoid directly affecting the coherence times of the transmon qubits
[53, 54]. The highly tunable dynamical $ZZ$ interaction provides a tool to
cancel a relatively stronger static $ZZ$ coupling. In the next subsection, we
show our simulation where we implement CR gates with the static $ZZ$ coupling
at $\chi^{\prime}_{LR}/2\pi\approx-5.7$ MHz dynamically cancelled by the
resonator drive.

###  III.2 $ZZ$-free CR gate

Motivated by the consideration above, we choose a stronger two-qubit coupling
strength (effectively $|J/2\pi|\approx 42$ MHz) compared to the values used in
the literature [17, 48, 18]. Such strong coupling allows us to explore beyond
the straddling regime, where the problem of frequency crowding between qubits
can be alleviated. For example, we choose the detuning as
$\Delta_{LR}/2\pi=-660$ MHz. These choices result in a static $ZZ$ coupling
strength $\chi^{\prime}_{LR}/2\pi=-5.7$ MHz. Other parameters are listed in
the caption of Fig. 2. To neutralize the stray coupling, we operate the
resonator at the $ZZ$ free point $\mathcal{D}=\mathcal{D}_{0}$ as shown in
Fig. 2 (a).

Figure 4: Numerical simulation of the cross-resonance gates with $ZZ$
cancellation. Two types of gates are chosen for such simulation: the left
panels (a)-(d) show the results for the 0-Controlled NOT gate and the right
panels (e)-(h) for the 1-Controlled NOT, respectively. Among these panels, (a)
and (e) show the envelopes of the CR drive on the control qubit (red solid)
and the cancellation drive (purple dashed) on the target qubit, and their
insets contain circuit diagrams for the two types of gates. The envelopes are
chosen as truncated Gaussian functions ($2\sigma$ on each side). Panels (b),
(c), (f), and (g) show evolutions of the four computational states during
40-ns gates. (d) and (h) present the minimized coherent gate error for the two
types of entangling gates versus given gate times. For this optimization, we
fix $\omega^{\mathrm{d}}_{R}=\omega^{\mathrm{d}}_{L}$, and vary the driving
frequency $\omega^{\mathrm{d}}_{L}$, the maximal amplitude of
$\epsilon_{\,\mathrm{CR}}(t)$, and the maximal amplitude of
$\epsilon_{\,\mathrm{cancel}}(t)$ to search for the minimal average gate
error. As shown in stars in (d) and (h), the coherent errors (shown in blue
and red stars) are below $10^{-4}$ for most gate durations simulated.
Consequently, the decoherence contribution dominates the total gate error.
Even for an optimistic estimation of the transmon coherence times
($T_{1}=T_{2}=500\,\mu$s), the decoherence errors still predominate the
coherent ones for gate durations longer than 40 ns, which is indicated by the
shaded areas. (We assume that both the depolarization and dephasing noise are
Markovian.) For all the simulations, the strength of the drive on the
resonator is fixed at the $ZZ$-free point, i.e.,
$\mathcal{D}=\mathcal{D}_{0}$.

To verify the viability of the CR gates, we use the model described in Eq. (3)
to numerically simulate the evolution of the qubit states under CR gate
drives. For this simulation, we choose the $L$ qubit as the control and $R$ as
the target, and drive the $L$ qubit via the charge operator $\hat{n}_{L}$
according to

$\displaystyle\hat{H}_{\mathcal{D},L}(t)=2\epsilon_{\mathrm{CR}}(t)\cos\big{(}\omega^{\mathrm{d}}_{L}t\big{)}\hat{n}_{L}.$
(18)

Importantly, the bare charge operator is transformed into a sum of
contributions in the dressed basis, which is expressed as

$\displaystyle\hat{n}_{L}\rightarrow$
$\displaystyle\,A_{L}\hat{b}_{L}+A_{R}\hat{b}_{R}+A_{CX}\hat{b}^{\dagger}_{L}\hat{b}_{L}\hat{b}_{R}$
$\displaystyle+A^{\prime}_{R}\hat{b}^{\dagger}_{R}\hat{b}_{R}\hat{b}_{R}+A^{\prime}_{CX}\hat{b}^{\dagger}_{L}\hat{b}_{L}\hat{b}^{\dagger}_{R}\hat{b}_{R}\hat{b}_{R}+\cdots$
$\displaystyle+\text{H.c.}$ (19)

Again, we obtain the coefficients in this expansion numerically using
SCQUBITS. Among the terms in this expansion, the following are important for
understanding and designing the CR gate. Specifically, $A_{L}\hat{b}_{L}$
describes the drive on the control qubit, and $A_{R}\hat{b}_{R}$ induces
single-qubit rotations on the target qubit. Their coefficients can be
approximated as
$A_{L}\approx\big{(}E_{J_{L}}/32E_{C_{L}}\big{)}^{\frac{1}{4}}$, and
$A_{R}\approx\big{(}E_{J_{L}}/32E_{C_{L}}\big{)}^{\frac{1}{4}}(-J/\Delta_{LR})$.
The term $A_{CX}\hat{b}^{\dagger}_{L}\hat{b}_{L}\hat{b}_{R}$ induces
conditioned rotation depending on the state of the control qubit. [See
approximation of $A_{CX}$ in Eq. (14).] This term can be put on resonance if
the drive frequency $\omega^{\mathrm{d}}_{L}$ is chosen to be close to the
frequency of the target qubit $\omega_{R}$. The other two terms shown in Eq.
(19) are related to the possible leakage in this gate, which we aim to
minimize. They are also included in our following simulation.

Besides the drive on the resonator and on the control qubit, we also apply a
cancellation pulse to the target qubit [17] to directly correct the rotations
in the target qubit induced by the term $A_{R}\hat{b}_{R}$. This drive is
denoted by

$\displaystyle\hat{H}_{\mathcal{D},R}(t)=2\epsilon_{\mathrm{cancel}}(t)\cos\big{(}\omega^{\mathrm{d}}_{R}t\big{)}\hat{n}_{R}.$
(20)

By tuning and the strength and phase of $\epsilon_{\mathrm{cancel}}(t)$, this
cancellation tone can activate either a 0-CNOT gate (target qubit flipped when
the control is in the ground state) or a 1-CNOT gate [see circuit diagrams in
the insets of Fig. 4 (a) and (e)]. They are equivalent to each other up to
single-qubit rotations. The envelopes of these two pulses
$\epsilon_{\mathrm{CR}}(t)$ and $\epsilon_{\mathrm{cancel}}(t)$ are shown in
Fig. 4 (a) and (e). Besides these two, we also add DRAG pulses [55, 56] in our
simulations to mitigate leakage to the second excited state of the target
qubit.

By fine-tuning the pulse parameters, we numerically realize the two types of
entangling gates with gate times ranging from $30$ to $60$ ns. For a clearer
demonstration of this gate, we plot the evolution of the populations of the
four computational states during a 40-ns gate in Fig. 4 (b) and (c) for the
0-controlled CNOT gate, and in (f) and (g) for the 1-controlled CNOT gate.
Clearly, the target qubit ends up in different states for different initial
states of the control qubit. In Fig. 4 (d) and (h), we show the optimized gate
fidelites for different gate times. Impressively, the coherent error is
negligible ($\approx 10^{-5}$) for gate times as short as $40$ ns.

For such low coherence errors, the two-qubit gates are still limited by the
transmon decoherence. For the state-of-the-art coherence times of the transmon
qubits ($T_{1}=T_{2}=500\,\mu s$) [37, 38, 39], the gate error of this gate is
below $10^{-4}$ for a 40-ns duration, which is shown in Fig. 4 (d) and (h).
Even for more moderate coherence times such as $T_{1}=T_{2}=100\,\mu s$, this
error can still reach as low as $5\times 10^{-4}$. Such error can be further
reduced once the coherence times of single transmon qubits are further
improved, and the absence of flux-tunable elements spares the gates from
suffering from other prominent decoherence channels.

##  IV Adiabatic CZ gates

Besides the CR gates, this architecture supports other entangling operations,
which can complement the gates introduced above. For example, we can engineer
CZ gates via the RIP interaction by adiabatic tuning the resonator driving
strength. This strategy is analogous to that used in gates enabled by tunable
couplers [12].

Specifically, we adiabatically move $\mathcal{D}$ in Eq. (4) away from the
cancellation point $\mathcal{D}_{0}$ over a certain duration $T_{g}$ and
return it back. The $ZZ$ coupling is recovered during this process, which can
be used to engineer CZ gates. To maximally accelerate the entangling gate, we
adiabatically tune $\mathcal{D}$ to 0 in our simulation [see Fig. 5 (a)]. To
ensure high-fidelity state transfers, we choose an analytically designed pulse
envelope [57]

$\displaystyle\mathcal{D}(t)=2^{n}\mathcal{D}_{0}\Bigg{[}10\left(\frac{t}{T_{g}}\right)^{3}-15\left(\frac{t}{T_{g}}\right)^{4}+6\left(\frac{t}{T_{g}}\right)^{5}-\frac{1}{2}\Bigg{]}^{n},$
(21)

whose first and second-order derivatives vanish at $t=0$ and $t=T$ for $n=2m$,
$m\in\mathbb{Z}^{+}$. Such a feature is useful for preserving adiabaticity in
the state evolution of a harmonic oscillator.

By increasing $n$, one can shorten the ramp time of the pulse as shown in Fig.
5 (a), which leads to faster CZ gates at the cost of more diabatic leakage. To
verify this, we increase $n$ from 2 to 32 and find optimal gate durations for
the target CZ gates. As shown in Fig. 5 (b), the gate duration decreases from
160 ns to 110 ns, while the error due to diabatic resonator evolution
increases from a negligible value to $4\times 10^{-4}$, approaching the
decoherence limit. In addition, shorter pulse duration eventually results in a
broader bandwidth of the pulse, which increases the risk of photon exchanges
between the qubits and the resonator and leads to more errors [25, 58]. In
principle, this problem can be alleviated by carefully choosing the hardware
and driving parameters. We leave the analysis and mitigation of these error
channels in the current scheme for future investigation.

Although the durations of such CZ gates are relatively longer than those of
the CR gates studied previously, they require simpler control protocols, which
are especially convenient for experiments where strong transmon drives are
unavailable.

Figure 5: Numerical simulations of the adiabatic CZ gates. (a) shows the
envelopes of the resonator drives that enable such phase entangling. The
different curves correspond to different parameters $n$ in Eq. (21). (b) shows
the optimized gate durations and corresponding resonator diabatic errors as a
function of $n$. The dashed curve indicates the lower bound of gate duration
set by $T_{\mathrm{min}}=\pi/{\chi^{\prime}_{LR}}$. The shaded region
indicates the decoherence contribution for transmon qubits with
$T_{1}=T_{2}=500\,\mu$s.

##  V Other error channels

###  V.1 Resonator Decoherence

In the previous sections, we have neglected the decoherence of the resonator
by assuming that they have exceedingly longer coherence times than the
transmon qubits [27, 28, 29]. However, such a condition may not always apply
in the experiments. In this subsection, we quantify the error caused by the
decoherence of the resonator.

For a fixed resonator drive, the resonator-induced dephasing for the $L$
transmon qubit is given by [59]

$\displaystyle\Gamma_{m,L}\approx\frac{2\bar{n}\kappa_{C}\chi^{2}_{L}}{\kappa_{C}^{2}+\chi^{2}_{L}+4\Delta_{d}^{2}},$
(22)

where $\kappa_{C}$ denotes the resonator loss rate and $\bar{n}$ denotes the
average photon number kept in the resonator. For the control protocols we
consider in this paper, the detuning $\Delta_{d}$ is much larger than
$\chi_{L}$ and $\kappa_{C}$. Under the limit
$\Delta_{d}\gg\chi_{L},\kappa_{C}$, Eq. (22) can be further approximated as
$\Gamma_{m,L}\approx(\bar{n}\chi^{2}_{L}/2\Delta_{d}^{2})\kappa_{C}$. For a
simple estimation, we take the parameters used in the previous simulations,
i.e., $\Delta_{d}/2\pi=100$ MHz, $\chi_{L}/2\pi=6$ MHz, and $\bar{n}\approx
10$, and find the coefficient $\bar{n}\chi_{L}^{2}/2\Delta_{d}^{2}\approx
0.02$. Therefore, for the resonator lifetime $1/\kappa_{C}=100\,\mu$s, the
coherence limit on the transmon qubit is approximately 5 ms, which is still
much higher than its typical dephasing time.

For non-steady drives applied to the resonator such as those proposed in Sec.
IV, the simple estimation (22) should be modified by a more detailed formula
provided in Ref. [23]. However, we deem that such modification does not
qualitatively change the estimation given above. Besides the inherent loss of
the resonator, the stability of the microwave drives on the resonator can also
be an error source that reduces the transmon coherence times. Since the
magnitude of such instability has not been well established, we choose to
leave such a contribution for future research.

###  V.2 Four-wave mixing

For the traditional CR gates [35], the charge drives on either transmon qubit
usually only affect the qubits, since the frequencies of these drives are far
detuned from that of the resonator excitation. In our scheme, however, if a
displacement drive Eq. (4) is applied to the resonator, the transmon drive Eq.
(18) can introduce an intriguing four-wave-mixing interaction between the
qubit and resonator.

We can make this interaction more evident by inspecting the transformation of
$\hat{b}_{L(R)}$ according to the unitary $\hat{U}_{\mathrm{dis}}$. Toward
this expansion, we first write $\hat{U}_{\mathrm{dis}}$ as a Taylor series

$\displaystyle\hat{U}_{\mathrm{dis}}=\exp\Bigg{[}\frac{\mathcal{D}(\hat{a}^{\dagger}-\hat{a})}{\Delta_{d}}\sum_{k=0}^{\infty}\Bigg{(}\frac{\chi_{L}\hat{b}_{L}^{\dagger}\hat{b}_{L}+\chi_{R}\hat{b}_{R}^{\dagger}\hat{b}_{R}}{\Delta_{d}}\Bigg{)}^{k}\Bigg{]}.$
(23)

Then, to the leading order of $\chi_{L,R}/\Delta_{d}$, we find

$\displaystyle\hat{b}_{L(R)}\rightarrow$
$\displaystyle\,\hat{U}^{\dagger}_{\mathrm{dis}}\hat{b}_{L(R)}\hat{U}_{\mathrm{dis}}$
$\displaystyle\approx$
$\displaystyle\,\hat{b}_{L(R)}+\frac{{\mathcal{D}\chi_{L(R)}}}{\Delta^{2}_{d}}(\hat{a}^{\dagger}-\hat{a})\hat{b}_{L(R)}.$
(24)

Note that in the displaced frame where the Hamiltonian (7) is defined, the
resonator frequency is only $\Delta_{d}$. Therefore, if we apply a drive on
the qubits close to $\omega_{L,R}$, the four-wave-mixing interaction is only
detuned by $\Delta_{d}$, which may cause the resonator to leave its displaced
vacuum once the coefficient $\mathcal{D}\chi_{L(R)}/\Delta_{d}^{2}$ is not
sufficiently small.

For the hardware and drive parameters we choose for the $ZZ$ cancellation and
the CR gates, this four-wave-mixing interaction has not introduced additional
gate errors, which we have checked by numerical simulation. [Note that our
simulation is based on Eq. (3), where this type of interaction is fully
included.] For the experimental implementation of this scheme, one can
suppress or mitigate this potential error channel by carefully choosing the
detuning $\Delta_{d}$.

##  VI $ZZ$-free architecture for a larger qubit network

In this section, we outline a plan to scale up this $ZZ$-free architecture
toward a larger qubit network. We consider a transmon-resonator chain that
consists of $N$ qubits and $N-1$ cavities. The schematic of such a chain is
shown in Fig. 6. The adjacent qubit pair indexed by $(j,j+1)$ in this chain
are both coupled by the $j$th resonator. The Hamiltonian of such a chain is
given by

$\displaystyle\hat{H}_{\mathrm{chain}}=$
$\displaystyle\,\sum^{N}_{j=1}\Big{(}\omega_{q,j}\hat{b}^{\dagger}_{j}\hat{b}_{j}+\frac{\eta_{q,j}}{2}\hat{b}^{\dagger}_{j}\hat{b}^{\dagger}_{j}\hat{b}_{j}\hat{b}_{j}\Big{)}+\sum_{j=1}^{N-1}\omega_{C,j}\hat{a}^{\dagger}_{j}\hat{a}_{j}$
$\displaystyle+\sum^{N-1}_{j=1}\chi^{\prime}_{j,j+1}\hat{b}^{\dagger}_{j+1}\hat{b}_{j+1}\hat{b}^{\dagger}_{j}\hat{b}_{j}$
$\displaystyle+\sum_{j=1}^{N-1}\Big{(}\chi_{j,j+1}\hat{b}^{\dagger}_{j+1}\hat{b}_{j+1}+\chi_{j,j}\hat{b}^{\dagger}_{j}\hat{b}_{j}\Big{)}\hat{a}^{\dagger}_{j}\hat{a}_{j},$
(25)

where $\hat{a}_{j}$ ($\hat{b}_{j}$) is annihilation operators for the $j$th
resonator (qubit), and $\omega_{C,j}$ ($\omega_{q,j}$) denotes its frequency.
Other parameters are similarly defined as in the main text. To cancel the
stray $ZZ$ interactions described by terms such as
$\chi^{\prime}_{j,j+1}\hat{b}^{\dagger}_{j+1}\hat{b}_{j+1}\hat{b}^{\dagger}_{j}\hat{b}_{j}$,
we can set the driving strength $\mathcal{D}_{j}$ as the solution of the
equation

$\displaystyle\mathcal{D}^{2}_{j}\Bigg{(}$
$\displaystyle\frac{1}{\Delta_{d,j}-\chi_{j,j}-\chi_{j,j+1}}+\frac{1}{\Delta_{d,j}}$
$\displaystyle-\frac{1}{\Delta_{d,j}-\chi_{j,j}}-\frac{1}{\Delta_{d,j}-\chi_{j,j+1}}\Bigg{)}+\chi^{\prime}_{j,j+1}=0.$
(26)

Note that there is freedom in choosing the detuning $\Delta_{d,j}$, which
provides a way to avoid crosstalks in resonator drives.

In realistic devices, however, a transmon qubit in such a chain can have small
but non-zero $ZZ$ coupling with a distant resonator or qubit. In Appendix A,
we numerically simulate a three-transmon chain as a minimal example to discuss
the residual $ZZ$ interactions and potential mitigation solutions.

Figure 6: Schematic of a multi-transmon chain connected by cavities. The drive
on each resonator can cancel the static $ZZ$ interaction between adjacent
qubits, according to Eq. (26).

##  VII Conclusion

In conclusion, we propose a simple architecture where the $ZZ$ coupling
between transmon qubits can be dynamically canceled via the RIP interaction.
The coupler considered here is a high-coherence resonator controlled by a
microwave drive, which preserves the high coherence times of the fixed-
frequency transmon qubits. The $ZZ$ cancellation enables strong coupling
between qubits without introducing stray inter-qubit interaction, which
significantly accelerates two-qubit entangling gates. Using realistic
parameters, we numerically demonstrate two types of short and high-fidelity
entangling gates, namely, cross-resonance CNOT gates within 40 ns and
adiabatic CZ gates within 140 ns. As a showcase, the error of the cross-
resonance gate on this architecture can reach below $10^{-4}$ based on our
numerical simulation. Besides enabling short gate times and high fidelities,
this scheme also allows a relatively larger separation between the qubit
frequencies, which can ease the precise requirement of the transmon
parameters. We also show that such an architecture is straightforward to scale
up.

These advantages make our scheme competitive toward realizing error-
correctable quantum computing, especially with the rapid advances in the
material improvement of the fixed-frequency transmon qubits. In the future,
one can also explore similar $ZZ$-free schemes with other novel
superconducting qubits [60].

###### Acknowledgements.

This material is based upon work supported by the U.S. Department of Energy,
Office of Science, National Quantum Information Science Research Centers,
Superconducting Quantum Materials and Systems Center (SQMS) under contract
number DE-AC02-07CH11359. We thank Jens Koch and Hanhee Paik for the helpful
discussion.

##  Appendix A $ZZ$ cancellation for a three-transmon chain

Figure 7: Residue stray couplings in a three-transmon chain after RIP $ZZ$
suppression. The frequencies of the three transmons are
$\omega_{q,1}/2\pi=4.24$ GHz, $\omega_{q,2}/2\pi=4.89$ GHz, and
$\omega_{q,3}/2\pi=5.42$ GHz; their anharmonicies are $\eta_{q,1}/2\pi=-317$
MHz, $\eta_{q,2}/2\pi=-341$ MHz, and $\eta_{q,3}/2\pi=-299$ MHz. The strengths
of the dispersive coupling between the transmon qubits and the cavities are:
$\chi_{11}/2\pi=-3.66$ MHz, $\chi_{12}/2\pi=-4.05$ kHz, $\chi_{21}/2\pi=-4.96$
MHz, $\chi_{22}/2\pi=-3.61$ MHz, $\chi_{31}/2\pi=-13.8$ kHz,
$\chi_{32}/2\pi=-4.44$ MHz. The static $ZZ$ coupling strengths are
$\chi^{\prime}_{12}/2\pi=-2.258$ MHz, $\chi^{\prime}_{23}/2\pi=-3.45$ MHz and
$\chi^{\prime}_{31}/2\pi=-0.1$ kHz. The detunings of the drive on the
resonators are both set by $\Delta_{d,1}/2\pi=\Delta_{d,2}/2\pi=100$ MHz.

In this appendix, we show the $ZZ$ suppression in a chain of three transmon
qubits (labeled as qubit 1, 2 and 3) as an example of the architecture shown
in Fig. 6. The three transmon qubits are coupled by two resonators (labeled as
resonator 1 and 2). We list the relevant parameters of the qubits and
resonators in the caption of Fig. 7. The magnitudes of the static $ZZ$
coupling between adjacent qubits ($\chi^{\prime}_{12}$ and
$\chi^{\prime}_{23}$) are a few MHz, which we aim to cancel via the RIP
interaction.

By fine-tuning the drive strengths around the theoretical predictions via Eq.
(26), we find that the static $ZZ$ interactions can be mostly canceled. In
Fig. 7, we plot the strengths of the residue $ZZ$ couplings between both the
adjacent qubits ($\tilde{\chi}^{\prime}_{12}$ and
$\tilde{\chi}^{\prime}_{23}$) in Fig. 7, both of which vanish for the drive
strengths indicated by the white cross. However, due to the dispersive
coupling between distant qubits and resonators, the RIP drives also lead to
two additional interactions: the first is the $ZZ$ interaction between distant
qubits, whose strength is denoted by $\tilde{\chi}^{\prime}_{31}$ [Fig. 7
(c)]; the second is a $ZZZ$ interaction among the three qubits, whose strength
is denoted by $\tilde{\chi}^{\prime}_{123}$ [Fig. 7 (d)]. Fortunately, the
magnitudes of them are only at kHz level. (At the white cross, we find
$\tilde{\chi}^{\prime}_{31}/2\pi\approx 5.1$ kHz and
$\tilde{\chi}^{\prime}_{123}/2\pi\approx 2.1$ kHz.)

Although the remaining kHz-level $ZZ$ couplings are already weak compared to
those in many state-of-the-art two-qubit gates, they can be further mitigated
by introducing extra coupler cavities. For example, we can further couple
transmon 1 and transmon 3 with an additional resonator and displace the state
of that resonator toward the cancellation of all stray interactions.

##  Appendix B Alternative approach to deriving the dynamical $ZZ$ coupling

In this appendix, we describe an alternative method to derive the strength of
the dynamical $ZZ$ interaction. The framework introduced here can also be
useful for more efficient numerical simulation of our scheme.

Different from the derivation of Eq. (3), here we start from Eq. (1), and
directly perform a displacement transformation
$\hat{U}^{\prime}_{\mathrm{dis}}(t)=\exp[\mathcal{D}(\hat{a}^{\dagger}-\hat{a})/\Delta_{d}]\exp(-i\omega^{\mathrm{d}}_{C}t\hat{a}^{\dagger}\hat{a})$
to adiabatically eliminate the drive term Eq. (4). After this step, the
displaced Hamiltonian is transformed to

$\displaystyle\tilde{H}^{\prime}_{\mathrm{dis}}=$
$\displaystyle\,\hat{H}_{L}+\hat{H}_{R}-\Delta_{d}\hat{a}^{\dagger}\hat{a}$
$\displaystyle+\big{(}\hat{a}^{\dagger}e^{i\omega^{\mathrm{d}}_{C}t}\\!+\\!\hat{a}e^{-i\omega^{\mathrm{d}}_{C}t}\big{)}\big{[}\tilde{g}_{L}(\hat{b}^{\dagger}_{L}\\!+\\!\hat{b}_{L})+\tilde{g}_{R}(\hat{b}^{\dagger}_{R}\\!+\\!\hat{b}_{R})\big{]}$
$\displaystyle+\frac{2\mathcal{D}}{\Delta_{d}}\cos\omega^{\mathrm{d}}_{C}t\big{[}\tilde{g}_{L}(\hat{b}^{\dagger}_{L}\\!+\\!\hat{b}_{L})+\tilde{g}_{R}(\hat{b}^{\dagger}_{R}\\!+\\!\hat{b}_{R})\big{]}.$
(27)

Above, the displacement drive on the resonator is converted to drives on the
two qubits. To further simplify this Hamiltonian, we approximate the transmon
Hamiltonian by that of a duffing oscillator, i.e.,

$\displaystyle\hat{H}_{L(R)}=\bar{\omega}_{L(R)}\hat{b}^{\dagger}_{L(R)}\hat{b}_{L(R)}+\frac{\eta_{L(R)}}{2}\hat{b}^{\dagger}_{L(R)}\hat{b}^{\dagger}_{L(R)}\hat{b}_{L(R)}\hat{b}_{L(R)},$
(28)

and perform an additional unitary transformation according to
$\hat{U}^{\prime\prime}(t)=\exp[-i\omega^{\mathrm{d}}_{C}t(\hat{b}^{\dagger}_{L}\hat{b}_{L}+\hat{b}_{R}^{\dagger}\hat{b}_{R})]$.
After discarding fast-rotating terms, we arrive at

$\displaystyle\tilde{H}^{\prime\prime}_{\mathrm{dis}}=$
$\displaystyle\,(\bar{\omega}_{L}-\omega_{C}^{\mathrm{d}})\hat{b}^{\dagger}_{L}\hat{b}_{L}+\frac{\eta_{L}}{2}\hat{b}_{L}^{\dagger}\hat{b}_{L}^{\dagger}\hat{b}_{L}\hat{b}_{L}$
$\displaystyle\,+\tilde{g}_{L}(\hat{a}^{\dagger}\hat{b}_{L}+\hat{a}\hat{b}^{\dagger}_{L})+{g_{L}\alpha}(\hat{b}^{\dagger}_{L}+\hat{b}_{L})+(L\xleftrightarrow{}R)$
$\displaystyle\,-\Delta_{d}\hat{a}^{\dagger}\hat{a}.$ (29)

where we define $\alpha=\mathcal{D}/\Delta_{d}$.

The transmon displacement terms $\hat{b}^{\dagger}_{L(R)}+\hat{b}_{L(R)}$
hybridizes the bare transmon states, which eventually leads to the dynamical
$ZZ$ interaction [18]. To derive such hybridization, we consider a third
transformation

$\displaystyle\hat{U}_{\mathrm{SW}}=\exp\Bigg{[}$
$\displaystyle\hat{b}^{\dagger}_{L}\frac{-\tilde{g}_{L}\alpha}{\bar{\omega}_{L}-\omega^{\mathrm{d}}_{C}+\eta_{L}\hat{b}_{L}^{\dagger}\hat{b}_{L}}$
$\displaystyle+\hat{b}^{\dagger}_{R}\frac{-\tilde{g}_{R}\alpha}{\bar{\omega}_{R}-\omega^{\mathrm{d}}_{C}+\eta_{R}\hat{b}_{R}^{\dagger}\hat{b}_{R}}-\mathrm{H.c.}\Bigg{]},$
(30)

which to the leading order of
$\tilde{g}_{L(R)}\alpha/[\bar{\omega}_{L(R)}-\omega_{C}^{\mathrm{d}}]$
eliminates the transmon displacement terms. Importantly, the transformation
described above also modifies the coupling term such as
$\tilde{g}_{L}(\hat{a}^{\dagger}\hat{b}_{L}+\hat{a}\hat{b}^{\dagger}_{L})$,
which causes the energy shifts of the four transmon states. To derive the
transformed qubit-resonator coupling, we first inspect the transformation of
$\hat{b}_{L(R)}$. Again, to the leading order, we find

$\displaystyle\hat{b}_{L}\rightarrow$
$\displaystyle\,\hat{U}^{\dagger}_{\mathrm{SW}}\hat{b}_{L}\hat{U}_{\mathrm{SW}}$
$\displaystyle\approx$
$\displaystyle\,\hat{b}_{L}+f(\hat{b}_{L}^{\dagger}\hat{b}_{L})$
$\displaystyle+\hat{b}^{\dagger}_{L}\hat{b}_{L}[f(\hat{b}^{\dagger}_{L}\hat{b}_{L})-f(\hat{b}^{\dagger}_{L}\hat{b}_{L}-1)]$
$\displaystyle+\hat{b}_{L}^{2}[f(b_{L}^{\dagger}\hat{b}_{L})-f(b_{L}^{\dagger}\hat{b}_{L}+1)],$
(31)

where we define
$f(x)\equiv-\tilde{g}_{L}\alpha/[2(\bar{\omega}_{L}-\omega^{\mathrm{d}}_{L}+\eta_{L}x)]$.

In the limit of $|\omega_{L(R)}-\omega_{C}^{\mathrm{d}}|\gg\eta_{L(R)}$, we
can further approximate this transformation to the leading order of
$\eta_{L(R)}/|\omega_{L(R)}-\omega_{C}^{\mathrm{d}}|$ by,

$\displaystyle\hat{b}_{L}\rightarrow$
$\displaystyle\,\hat{U}^{\dagger}_{\mathrm{SW}}\hat{b}_{L}\hat{U}_{\mathrm{SW}}$
$\displaystyle\approx$
$\displaystyle\,\hat{b}_{L}-\frac{\tilde{g}_{L}\alpha}{\big{(}\bar{\omega}_{L}-\omega^{\mathrm{d}}_{C}\big{)}}$
$\displaystyle+\frac{\tilde{g}_{L}\alpha\eta_{L}}{\big{(}\bar{\omega}_{L}-\omega^{\mathrm{d}}_{C}\big{)}^{2}}(2\hat{b}^{\dagger}_{L}\hat{b}_{L}-\hat{b}_{L}^{2}).$
(32)

The useful term in this expansion for the generation of $ZZ$ coupling is
$\hat{b}^{\dagger}_{L}\hat{b}_{L}$, while the squeezing term $\hat{b}^{2}_{L}$
is less important. If the two qubits are coupled by direct exchange
interaction
$J(\hat{b}^{\dagger}_{L}\hat{b}_{R}+\hat{b}^{\dagger}_{R}\hat{b}_{L})$,
inserting $\hat{b}^{\dagger}_{L}\hat{b}_{L}$ and similarly
$\hat{b}^{\dagger}_{R}\hat{b}_{R}$ into the previous $\hat{b}_{L}$ and
$\hat{b}_{R}$ yields the “siZZle” dynamical $ZZ$ coupling [18]. For our case,
the qubit-resonator coupling
$\tilde{g}_{L(R)}[\hat{a}\hat{b}^{\dagger}_{L(R)}+\hat{a}^{\dagger}\hat{b}_{L(R)}]$
instead yields a term

$\displaystyle\alpha(\chi_{L}\hat{b}^{\dagger}_{L}\hat{b}_{L}+\chi_{R}\hat{b}^{\dagger}_{R}\hat{b}_{R})(\hat{a}^{\dagger}+\hat{a}),$
(33)

where we have used the approximation $\chi_{L(R)}\approx
2\tilde{g}^{2}_{L(R)}\eta_{L(R)}/[\bar{\omega}_{L(R)}-\omega^{\mathrm{d}}_{C}]^{2}$
($\omega^{\mathrm{d}}_{C}$ is set to be detuned from but close to
$\omega_{C}$) 222More accurately, this coefficient is derived as
$\chi_{L(R)}=2\tilde{g}^{2}_{L(R)}\eta_{L(R)}/[(\bar{\omega}_{L(R)}-\bar{\omega}_{C})(\bar{\omega}_{L(R)}-\bar{\omega}_{C}+\eta_{L(R)})]$
[1].. Finally, we only need to integrate out the resonator degree of freedom,
which can be done by a Born-Oppenheimer approximation [62, 63]. This step
results in a two-qubit interaction term

$\displaystyle-\frac{\alpha^{2}}{\Delta_{d}}(\chi_{L}\hat{b}_{L}^{\dagger}\hat{b}_{L}+\chi_{R}\hat{b}^{\dagger}_{R}\hat{b}_{R})^{2},$
(34)

which reproduces Eq. (9).

Besides cross-checking the strength of the dynamical $ZZ$ interaction, the
framework detailed in Eq. (29) can also be useful for faster numerical
simulation. In this frame, the resonator does not possess a large number of
photons, which can reduce the dimension of the Hilbert space and further the
numerical cost.

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