# Long-distance transmon coupler with CZ gate fidelity above
$99.8\text{\,}\mathrm{\char 37\relax}$

Fabian Marxer  Antti Vepsäläinen<EMAIL_ADDRESS>Shan W. Jolin  Jani
Tuorila  Alessandro Landra  Caspar Ockeloen-Korppi  Wei Liu  Olli Ahonen  IQM
Quantum Computers, Espoo 02150, Finland  Adrian Auer  IQM Quantum Computers,
Munich 80636, Germany  Lucien Belzane  Ville Bergholm  Chun Fai Chan  Kok Wai
Chan  Tuukka Hiltunen  Juho Hotari  Eric Hyyppä  Joni Ikonen  David Janzso
Miikka Koistinen  Janne Kotilahti  Tianyi Li  Jyrgen Luus  IQM Quantum
Computers, Espoo 02150, Finland  Miha Papic  IQM Quantum Computers, Munich
80636, Germany  Matti Partanen  Jukka Räbinä  Jari Rosti  Mykhailo Savytskyi
Marko Seppälä  Vasilii Sevriuk  Eelis Takala  Brian Tarasinski  IQM Quantum
Computers, Espoo 02150, Finland  Manish J. Thapa  IQM Quantum Computers,
Munich 80636, Germany  Francesca Tosto  Natalia Vorobeva  Liuqi Yu  Kuan Yen
Tan  Juha Hassel  IQM Quantum Computers, Espoo 02150, Finland  Mikko Möttönen
IQM Quantum Computers, Espoo 02150, Finland  QCD Labs, QTF Centre of
Excellence, Department of Applied Physics, Aalto University, Aalto 00076,
Finland  VTT Technical Research Centre of Finland, QTF Center of Excellence,
VTT, VTT 02044, Finland  Johannes Heinsoo  IQM Quantum Computers, Espoo 02150,
Finland

###### Abstract

Tunable coupling of superconducting qubits has been widely studied due to its
importance for isolated gate operations in scalable quantum processor
architectures. Here, we demonstrate a tunable qubit-qubit coupler based on a
floating transmon device which allows us to place qubits at least
$2\text{\,}\mathrm{mm}$ apart from each other while maintaining over
$50\text{\,}\mathrm{MHz}$ coupling between the coupler and the qubits. In the
introduced tunable-coupler design, both the qubit-qubit and the qubit-coupler
couplings are mediated by two waveguides instead of relying on direct
capacitive couplings between the components, reducing the impact of the qubit-
qubit distance on the couplings. This leaves space for each qubit to have an
individual readout resonator and a Purcell filter needed for fast high-
fidelity readout. In addition, simulations show that the large qubit-qubit
distance significantly lowers unwanted non-nearest-neighbor coupling and
allows multiple control lines to cross over the structure with minimal
crosstalk. Using the proposed flexible and scalable architecture, we
demonstrate a controlled-$Z$ gate with $99.81\pm 0.02\text{\,}\mathrm{\char
37\relax}$ fidelity.

††thanks: These two authors contributed equally to this work††thanks: These
two authors contributed equally to this work

##  I Introduction

Figure 1: (a) Quasi-lumped-element circuit diagram of qubits (blue and orange)
coupled by a tunable coupling structure consisting of waveguide extenders
(turquoise) and a floating coupler qubit (red). The gray lines show the
effective lumped-element model for the coupling capacitances arising from the
black waveguide extenders C and F. (b) False-colored micrograph of the qubit-
coupler-qubit system. The qubits, coupler, and waveguide extenders follow the
color code from (a). The airbridges are colored non-transparently to protect
unpublished intellectual property. (c) Simulation of the effective coupling
strengths between qubit 1 and the coupler, $g_{\rm 1c}$, qubit 2 and the
coupler, $g_{\rm 2c}$, and qubit 1 and qubit 2, $g_{\rm 12}$, for different
qubit-qubit distances $d_{\rm qq}$ (solid lines). The measured coupling values
(dots with $68\text{\,}\mathrm{\char 37\relax}$ confidence intervals) are
shown for our device with $d_{\rm qq}=$1960\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$ (dashed line). (d) Simulated spurious next-nearest-neighbor coupling
$g_{13}$ between $\rm{Q}_{1}$ and $\rm{Q}_{3}$ in a square lattice
configuration (schematic), and the nearest-neighbor coupling $g_{12}$ as a
function of the distances $d_{\rm qq}$ between $\rm{Q}_{1}$ and $\rm{Q}_{2}$.
The gray shaded area indicates the region, where the coupling values fall
below the simulation accuracy. (e) Schematic of a simplified flip-chip
architecture, where the qubits and the coupler are connected via waveguide
extenders (light gray) on the bottom chip and a long perpendicular
transmission line (black) is crossing above the qubit-coupler structure on the
top chip. (f) Simulated capacitance coupling ratios (see main text) from each
component to the transmission line (color code same as in (e)) at various
crossing positions $x_{\rm cross}$ of the transmission line, total coupling
(black dashed line) and region for low-crosstalk crossings (blue shading).

The implementation of high-fidelity two-qubit gates is a key requirement for
scalable quantum processors Reagor  _et al._ (2018); Arute  _et al._ (2019);
Chen  _et al._ (2021). The performance of quantum gates relying only on the
static qubit-qubit coupling Majer  _et al._ (2007); DiCarlo  _et al._ (2009);
Kandala  _et al._ (2019) is typically limited by stray $ZZ$ interaction
between the qubits resulting in long gate times due to a poor on/off ratio for
the coupling. To address this problem, different tunable-coupler designs have
been proposed Hime  _et al._ (2006); Niskanen  _et al._ (2007); Chen  _et al._
(2014); Geller  _et al._ (2015); Chen  _et al._ (2022) with increasing
sophistication.

An important step toward tunable couplers with a high on/off ratio and with
minimal impact on qubit coherence was the observation that one can design a
coupler-mediated tunable interaction that cancels out the static qubit-qubit
coupling at a specific coupler off-frequency which is above the qubit
frequencies Yan  _et al._ (2018a). Since then, such couplers have been
successfully used in several experiments Arute  _et al._ (2019); Li  _et al._
(2020); Xu  _et al._ (2020); Collodo  _et al._ (2020); Sung  _et al._ (2021);
Ye  _et al._ (2021). In Ref. Sete  _et al._ (2021), a floating transmon
coupler that can also be operated below the qubit frequencies was proposed,
allowing two-qubit-gate operations near the flux insensitive coupler sweet
spot and thereby reducing the impact of coupler flux noise during the gate.
Moreover, the concept was used in Ref. Stehlik  _et al._ (2021) for
implementing high-fidelity CZ gates for fixed-frequency floating transmons.

Although the $ZZ$-interaction-free tunable-coupler designs have seen
significant success in enabling high-fidelity two-qubit gates, the static
qubit-qubit coupling needed for canceling the coupler-mediated interaction
arises from the direct coupling between the qubits, controlled mainly by the
qubit-qubit distance. Having no residual $ZZ$ coupling in such architectures
therefore requires small qubit-qubit distances. This restriction introduces
severe limitations for the qubit locations, leading to high spurious non-
nearest-neighbor coupling, and furthermore, limits the space in a square qubit
lattice, allowing only the most essential components to be placed in between
the lattice sites.

Here, we introduce and experimentally demonstrate an extended floating coupler
that allows us to increase the physical distances between qubits, thus
providing the space on the chip for readout resonators with individual Purcell
filters for high-fidelity readout Heinsoo  _et al._ (2018); Sete  _et al._
(2015). Furthermore, long qubit-qubit distances significantly lower the
parasitic non-nearest-neighbor coupling and enable low-crosstalk paths for
multiple control lines above the qubit-coupler-qubit structure if flip-chip
technology is utilized. Similar to Refs. Sete  _et al._ (2021); Stehlik  _et
al._ (2021), the tunable coupler is operated below the qubit frequencies,
allowing two-qubit-gate operations closer to its flux insensitive sweet spot,
and furthermore, preventing the coupler to cross any qubit readout-resonator
modes. The ability to have long qubit-qubit distances is achieved using two
waveguide extenders that mediate the direct qubit-qubit coupling, as well as
the couplings between the qubits and the coupler. These couplings
predominantly originate from interdigital capacitors between the two waveguide
extenders and the coupler, making it possible to maintain couplings with
similar magnitude as previous designs Sung  _et al._ (2021); Li  _et al._
(2020) across a broad range of qubit-qubit distances. To demonstrate that the
couplings between the qubits and the coupler are high enough for qubit-qubit
distances beyond $1\text{\,}\mathrm{mm}$, we implement a fast and high-
fidelity controlled-$Z$ (CZ) gate with a duration of $33\text{\,}\mathrm{ns}$
and $99.81\pm 0.02\text{\,}\mathrm{\char 37\relax}$ fidelity for a qubit-qubit
distance of $1.96\text{\,}\mathrm{mm}$, which is four times longer than in
typical tunable coupler designs Sung  _et al._ (2021); Collodo  _et al._
(2020). Our coupler design is readily extendable to a square qubit lattice,
making it an appealing building block for a scalable high-fidelity quantum
processor.

##  II Extended floating transmon coupler design

We propose a coupler design in which two grounded transmon qubits interact
through a floating transmon coupler. As shown in Fig. 1(a), the qubits are
connected to the coupler with two waveguide extenders that mediate effective
capacitances $C_{\rm 1c}$, $C_{\rm 2c}$, and $C_{12}$ between the qubits and
the coupler. The two waveguide extenders shown in Fig. 1(b) are implemented as
$0.9\text{\,}\mathrm{mm}$-long coplanar waveguides whose resonance frequencies
are an order of magnitude higher than the qubits and coupler frequencies, such
that the waveguide extender modes do not couple to the qubit and coupler
modes. At one end, the waveguide extenders are coupled to each other and to
the coupler via interdigital capacitors, and at the other end to the qubits
via gap capacitors. These gap capacitors are designed to accommodate four
waveguide extenders which are coupled to a single qubit to form a square
lattice. All the capacitance values between the extenders, the qubits, and the
coupler can be independently adjusted to realize the desired effective
capacitances, see Appendix A for the detailed model. Within the effective
capacitance model, the dynamics of the system can be described using the
coupling strengths between qubit 1 ($\rm{Q}_{1}$) and the coupler, $g_{\rm
1c}$, qubit 2 ($\rm{Q}_{2}$) and the coupler, $g_{\rm 2c}$, and the qubit-
qubit coupling $g_{\rm 12}$.

In contrast to direct capacitive couplings between the qubits and the coupler
studied in Ref. Sete  _et al._ (2021), the waveguide extenders facilitate
flexible positioning of the individual components by simply elongating the
waveguide extenders. To investigate the impact of qubit-qubit distance in our
system, we calculate the effective coupling strengths for waveguide extenders
of varying lengths using simulated capacitance values between all the
superconducting islands, A–H in Fig. 1(a). As shown in Fig. 1(c), changing the
distance $d_{\rm qq}$ between the qubits by up to
$1000\text{\,}\mathrm{\SIUnitSymbolMicro m}$ impacts the coupling strengths
between the qubits and the coupler by less than $50\text{\,}\mathrm{\char
37\relax}$, implying that the couplings are still large enough for fast two-
qubit gates over a wide range of qubit-qubit distances. In the simulation, we
use scikit-rf Arsenovic  _et al._ (2022) to model the qubits and the coupler
as lumped-element circuits, and apply the transmission line model for the
waveguide extenders, see Appendix B for details on the simulation. The slight
decrease in coupling strengths with increasing $d_{\rm qq}$ can be attributed
to the increasing extender capacitance to ground which starts to contribute to
the total capacitance of the system, ultimately limiting $d_{\rm qq}$.

Even though the test device studied in this work has only two qubits connected
using the coupler, the layout is designed such that identical couplings can be
simultaneously reached for up to four neighboring qubits, compatible with
scaling to large qubit lattices. In such a lattice, the resulting freedom of
qubit spacing allows the spurious couplings to non-nearest-neighbor qubits to
be reduced while maintaining strong nearest-neighbor (NN) coupling, which is
essential for a $ZZ$-interaction-free idling configuration of a multi-qubit
system. The dominating non-nearest-neighbor couplings in a square qubit
lattice architecture are the diagonal next-nearest-neighbor (NNN) qubit pairs.
As shown in Fig. 1(d), the simulated spurious NNN coupling $g_{13}$ between
two diagonally placed qubits with the typical qubit-qubit distance of $d_{\rm
qq}=$500\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ has the same order of
magnitude as the typical NN coupling $g_{12}$ Sung  _et al._ (2021); Li  _et
al._ (2020), potentially leading to large residual $ZZ$-couplings during
idling time. As $g_{13}$ decreases with increasing $d_{\rm qq}$, the NNN
coupling at our qubit-qubit distance of
$1960\text{\,}\mathrm{\SIUnitSymbolMicro m}$ is less than
$g_{13}<$30\text{\,}\mathrm{kHz}$$, which is at least two orders of magnitude
smaller than $g_{12}$. The upper bound of the estimate is limited by the
accuracy of the simulation. The NNN coupling $g_{13}$ was extracted from an
Ansys HFSS finite-element-method (FEM) simulation of the direct capacitance
between qubits ${\rm Q_{1}}$ and ${\rm Q_{3}}$, and did not include other
qubits or couplers. However, we verified that the inclusion of $\rm{Q}_{2}$
and the couplers connecting $\rm{Q}_{1}$, $\rm{Q}_{2}$, and $\rm{Q}_{3}$ in
the simulation did not show a significant difference. From these capacitances,
we calculate $g_{13}$ by modeling the two qubits as coupled harmonic
oscillators with a frequency of $4.3\text{\,}\mathrm{GHz}$ Yan  _et al._
(2018a). For the simulation of the mediated qubit-qubit coupling $g_{12}$ the
coupler was included, limiting the smallest simulation range of the qubit-
qubit distance to $d_{\rm qq}=$920\text{\,}\mathrm{\SIUnitSymbolMicro m}$$
corresponding to the width of the coupler.

In addition to suppressing the NNN coupling, the large qubit-qubit distance
gives us physical space for readout structures with individual Purcell filters
for each qubit Sete  _et al._ (2015); Heinsoo  _et al._ (2018), enabling fast
and high-fidelity readout Walter  _et al._ (2017). Implementing e.g.
$6\text{\,}\mathrm{GHz}$ readout resonators and Purcell filters as spiral
$\lambda/4$ coplanar resonators requires together at least
$1\text{\,}\mathrm{m}\mathrm{m}^{2}$ of space, which easily fits in a square
qubit lattice with our qubit-qubit distances of $d_{\rm
qq}=$1960\text{\,}\mathrm{\SIUnitSymbolMicro m}$$.

Figure 2: (a) The effective $ZZ$ interaction strength $\zeta$ between the
qubits is shown as a function of the coupler frequency $\omega_{\rm c}/(2\pi)$
and the frequency detuning between the two qubits, $\Delta/(2\pi)$. If
$\alpha_{2}<\Delta<-\alpha_{1}$, there exists a contour (black dotted line)
along which the $ZZ$ interaction vanishes. The qubit frequencies are shown
with orange dashed lines and the detuning values corresponding to their
anharmonicities are marked with white dashed lines. The solid black line
indicates the lowest coupler frequency at which a conditional phase of
$\phi_{11}=\pi$ can be accumulated in $25\text{\,}\mathrm{ns}$. The solid red
line illustrates the path along which the coupler is operated in the
experiments. (b) Experimentally measured interaction strength (dots) as a
function of the coupler frequency $\omega_{\rm c}/(2\pi)$ in units of flux
quanta $\Phi_{0}$ for the specific qubit-qubit detuning chosen to implement
the CZ gate. The numerical model (purple line) is fitted to the data to
extract the coupling strengths between the two qubits and between each qubit
and the coupler. (c) Experimentally measured eigenfrequencies (white to blue
spectrum) and fitted eigenfrequencies (lines) of the qubits and the coupler as
a function of the external flux $\Phi_{\rm ext}^{\rm c}$ threading through the
coupler SQUID loop. The coupler spectrum below $3.7\text{\,}\mathrm{GHz}$ is
measured using ${\rm Q_{2}}$ as an ancilla, see Appendix F. The red circles
show the idling and the operation configuration of the coupler.

An additional benefit of the waveguide extenders is that the electric field
density of the qubit and the coupler modes is reduced above the extenders.
This allows microwave control lines to cross the qubit-coupler structure with
low crosstalk. To support this claim, we conduct a FEM simulation to extract
the capacitance between the components on the chip and a transmission line
($\rm TL$) that crosses the structure $8\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ above the qubits (Fig. 1(e)), corresponding to a typical flip-chip
topology Kosen  _et al._ (2022), see Appendix C for further details on the
simulation. To put the capacitance values into context, we calculate the drive
crosstalk from the capacitance ratio $r_{k}=C_{{\rm
k}\leftrightarrow\rm{TL}}/C_{\rm Q\leftrightarrow\rm{DL}}$, where $C_{{\rm
k}\leftrightarrow\rm{TL}}$ is the capacitance between the center conductor of
the transmission line and component $k=\\{{\rm Q_{1}},{\rm Q_{2}},{\rm c}\\}$,
and $C_{\rm Q\leftrightarrow\rm{DL}}=$0.12\text{\,}\mathrm{fF}$$ is the
capacitance between each qubit and its corresponding drive line, see Fig.
1(f). For this geometry, crossing the tunable coupling structure above the
waveguide extenders reduces the parasitic coupling by a factor of 10 compared
to passing over the tunable coupler. Above the extenders, the crosstalk is
$r_{\rm Q_{i}}<$0.1$=$-20\text{\,}\mathrm{dB}$$, which is comparable to the
typical crosstalk between two microwave control lines and can be further
reduced by increasing the chip-to-chip distance. This level of crosstalk
enables the implementation of qubit lattices with crossing lines without
resorting to more sophisticated technologies such as through silicon vias,
significantly reducing the complexity of the scalable architecture.

##  III CZ gate implementation

In our design, the effective static $ZZ$ interaction strength
$\zeta=\omega_{11}-\omega_{10}-\omega_{01}+\omega_{00}$ between the qubits can
be eliminated if the detuning between the two qubits is smaller than their
anharmonicity Sete  _et al._ (2021),
$\Delta=\omega_{1}-\omega_{2}\in[\alpha_{2},-\alpha_{1}]$, where $\alpha_{i}$
and $\omega_{i}$ are the anharmonicity and the angular frequency of qubit $i$
in the laboratory frame, and $\omega_{ij}/(2\pi)$ is the eigenfrequency for
the computational state in which $\rm{Q}_{1}$ is in state $i$ and $\rm{Q}_{2}$
is in state $j$. If this condition is met, a contour with $\zeta=0$ for
coupler frequencies below the qubit frequencies can be identified, as depicted
by the simulation shown in Fig. 2(a). In the simulation, we numerically solve
for the $ZZ$ interaction strength in the dressed-state basis comprising the
three lowest-energy states of the coupler and the qubits, see Appendix A for
details.

The $ZZ$ interaction can be turned on by tuning the coupler frequency away
from the $\zeta=0$ contour. The highest coupling and therefore the fastest
gate can be reached when the coupler frequency is close to the qubit
frequencies, as shown in the $25\text{\,}\mathrm{ns}$-gate-time curve in Fig.
2(a). The most relevant region for a fast and high-fidelity CZ gate is close
to $\Delta=-\alpha_{1}$, where the qubit states evolve through a near-resonant
$\ket{11}\leftrightarrow\ket{20}$ transition if the coupling is turned on,
implementing the diabatic version of the CZ gate. In the experiment, we set
the qubit frequencies at $\omega_{1}=\omega_{2}-a\alpha_{1}$, where
$\omega_{1}/(2\pi)=$4.10\text{\,}\mathrm{GHz}$$,
$\omega_{2}/(2\pi)=$3.89\text{\,}\mathrm{GHz}$$,
$\alpha_{1}/(2\pi)=$-216\text{\,}\mathrm{MHz}$$ and $a=0.97$, see Appendix D
for the full experimental setup and Appendix E for details on sample
fabrication. Even though the fastest gate times can be reached when the
$\ket{11}$ and $\ket{20}$ transitions are resonant, i.e. $a=1$, we
deliberately operate the qubits slightly away from the resonance to mitigate
the effect of drive crosstalk during single-qubit gates. The drive crosstalk
could also be reduced by idling the qubits in a configuration where the states
$\ket{11}$ and $\ket{20}$ are initially far-detuned and tuning the qubits to
their operation point only during the CZ gate Sung  _et al._ (2021). However,
in order to reduce the number of required microwave control lines in the
system, we choose not to use fast flux control for the qubits.

Figure 3: Interleaved randomized benchmarking experiment for two qubits. (a)
Experimental data of the sequence fidelity for randomized benchmarking with
interleaved CZ gates (purple dots) and the reference Clifford sequence (blue
dots) as a function of the number of Clifford gates. The purple and blue
shaded regions indicate the standard deviation of 30 random sequences for the
interleaved randomized benchmarking experiment and the reference Clifford
sequence, respectively. The histograms in the inset show the error per
Clifford gate (blue) and the error per CZ gate (purple) for the 14 repeated
experiments, from which the average error per Clifford gate of
$\epsilon^{\rm{avg}}_{\rm{Clifford}}=$(1.39\pm 0.03)\text{\times}{10}^{-2}$$
and the average error per CZ gate of $\epsilon^{\rm{avg}}_{\rm{CZ}}=$(1.9\pm
0.2)\text{\times}{10}^{-3}$$ were estimated. (b) Experimentally obtained error
per CZ gate as a function of the number of interleaved CZ gates $n$ in the
randomized benchmarking sequence. The error bars indicate the standard
deviation extracted from the residuals of the best model fit of the sequence
decay for $n$ interleaved CZ gates. The gate sequence is depicted in the
inset: A random Clifford gate $\rm C$ followed by $n$ CZ gates, repeated $m$
times, after which a reversing Clifford gate $\rm C_{r}$ is applied.
Extracting the fitting error for the sequence fidelity for $n$ interleaved CZ
gates yields a reduction in the uncertainty of the error per CZ gate by a
factor of $1/n$.

We experimentally determine the coupler idling frequency by measuring the
conditional phase of $\rm{Q}_{1}$ when $\rm{Q}_{2}$ is initialized either in
the ground or in the excited state, and find
$\lvert\zeta\rvert<$2\text{\,}\mathrm{kHz}$$ at $\omega_{\rm
c}/(2\pi)=$3.195\text{\,}\mathrm{GHz}$$, see Fig. 2(b). The fit to the
measured $ZZ$ interaction strength as a function of the coupler frequency
gives an estimate of $g_{\rm 1c}/(2\pi)=$51.5\pm 0.7\text{\,}\mathrm{MHz}$$,
$g_{\rm 2c}/(2\pi)=$53.9\pm 0.6\text{\,}\mathrm{MHz}$$, and $g_{\rm
12}/(2\pi)=$3.7\pm 0.5\text{\,}\mathrm{MHz}$$ where the couplings are defined
at the idling configuration. The fitted coupling strengths deviate less than
$15\text{\,}\mathrm{\char 37\relax}$ from the design and provide the strong
$ZZ$ interaction strength needed to implement a CZ gate in less than
$25\text{\,}\mathrm{ns}$ interaction time. In this configuration, we measure
simultaneous single-qubit gate errors $\epsilon_{\rm Q1}^{\rm sim}=$(1.32\pm
0.05)\text{\times}{10}^{-3}$$ and $\epsilon_{\rm Q2}^{\rm sim}=$(8.9\pm
0.5)\text{\times}{10}^{-4}$$, which are only slightly higher than individual
single-qubit gate errors, $\epsilon_{\rm Q1}=$(7.8\pm
0.3)\text{\times}{10}^{-4}$$ and $\epsilon_{\rm Q2}=$(6.8\pm
0.4)\text{\times}{10}^{-4}$$, respectively.

To realize the diabatic version of the CZ gate Barends  _et al._ (2019), we
turn on the effective exchange interaction between the states $\ket{20}$ and
$\ket{11}$ of the qubits by applying a $22\text{\,}\mathrm{ns}$-long flux
pulse to the coupler, shifting its frequency from the idling point to the
operation point, see Fig. 2(c). We use a Slepian-shaped flux pulse to minimize
the dominating leakage processes during the gate from $\rm{Q}_{2}$ to the
coupler Martinis and Geller (2014); Sung  _et al._ (2021). In addition, to
account for the distortions in the flux pulse shape, we use two consecutive
infinite-impulse-response filter to correct for their shape in real-time Rol
_et al._ (2020). To mitigate the impact of the flux distortion at the
nanosecond time scale, we add a $5.5\text{\,}\mathrm{ns}$ idle time before and
after the flux pulse, resulting in a total flux pulse duration of
$33\text{\,}\mathrm{ns}$.

Figure 4: The impact of the coupler decoherence on the qubits. (a) Energy
relaxation times of the hybridized qubit-coupler modes measured directly
(dots), measured via an ancilla qubit (diamonds), and evaluated numerically
based on the hybridization with the coupler (solid lines) as a function of the
coupler frequency $\omega_{\rm c}/(2\pi)$. The shaded areas indicate the
coupler frequencies swept during the CZ gate (gray) and regions with unknown
sources of decoherence, partly attributed to parasitic two-level systems
(orange and blue shading for $\rm{Q}_{1}$ and $\rm{Q}_{2}$, respectively). (b)
Measured frequencies of hybridized modes during coherence measurements (dense
set of dots) and frequencies of the uncoupled system (black dashed lines). (c)
as in (a) but shows the Gaussian component of the pure dephasing time for the
hybridized qubit and coupler modes. (d) Coherence limit of the CZ gate (solid
black line) as a function of the interaction time $\tau$ calculated using Eq.
(3) with contributions from individual coherence times (dashed lines), and the
coherence limit for no decoherence contributions from the coupler (green
stars). When evaluating the gate error, an idling time of
$11\text{\,}\mathrm{ns}$ is added on top of the interaction time. The black
dot shows the experimental gate error with the $68\text{\,}\mathrm{\char
37\relax}$ confidence interval. The data for $T_{\phi,1,{\rm eff}}^{\rm
Q_{2}}$ is below the curve corresponding to $T_{1,{\rm eff}}^{\rm Q_{2}}$ and
not well visible in the plot.

We optimize the CZ gate by adjusting the amplitude and duration of the coupler
flux pulse to find an operation point that minimizes the occupation in the
state $\ket{20}$ while accumulating a conditional phase shift $\phi_{11}=\pi$
in state $\ket{11}$. We account for the additional single-qubit phase
accumulation caused by the dispersive interaction between the coupler and the
qubits during the gate by applying virtual $Z$ gates McKay  _et al._ (2017) to
both qubits after each CZ gate.

##  IV CZ gate characterization

To characterize the average gate fidelity, we employ randomized benchmarking
where a sequence of random two-qubit Clifford gates is interleaved with CZ
gates Magesan  _et al._ (2012); Barends  _et al._ (2014). The experiment was
repeated 14 times over a period of $15\text{\,}\mathrm{h}$, shown in Fig.
3(a). We extract an average error per CZ gate of $\epsilon^{\rm avg}_{\rm
CZ}=$(1.9\pm 0.2)\text{\times}{10}^{-3}$$ from the collected data, where the
uncertainty indicates the $68\text{\,}\mathrm{\char 37\relax}$ confidence
interval. For consistency, the average error per Clifford $\epsilon_{\rm
Clifford}=$(1.39\pm 0.03)\text{\times}{10}^{-2}$$ can be compared to the
errors of its constituent CZ and single-qubit gates. On average, each two-
qubit Clifford gate consists of 1.5 CZ gates and 8.25 single-qubit gates
Barends  _et al._ (2014), which by simple summation of errors results in the
estimate for the Clifford error $\epsilon_{\rm Clifford}^{\rm
estimate}=$(1.1\pm 0.4)\text{\times}{10}^{-2}$$, in agreement with the
measured value.

We can improve the estimate for the CZ gate error by interleaving multiple CZ
gates in the randomized benchmarking sequence and calculating the average
error per gate for different numbers $n$ of interleaved gates Sheldon  _et
al._ (2016). As shown in Fig. 3(b), the error per CZ gate converges from
$\epsilon_{\rm CZ}^{(1)}=$(2.0\pm 0.5)\text{\times}{10}^{-3}$$ for one
interleaved CZ gate towards $\epsilon_{\rm CZ}^{(20)}=$(2.35\pm
0.07)\text{\times}{10}^{-3}$$ for 20 interleaved CZ gates, demonstrating a
seven-fold reduction in the uncertainty and a minor increase for the error per
CZ gate, see Appendix G for further details. In contrast to conventional
randomized benchmarking, where coherent errors are suppressed due to
randomization of the basis between each application of the CZ gate,
interleaving the sequence with several CZ gates reveals coherent errors as
well since they tend to add up for the $n$ interleaved CZ gates. The minor
increase in the error rate when interleaving multiple CZ gates suggests that
the CZ gate fidelity is mostly limited by decoherence at its chosen operation
point, rather than by coherent errors. The low error rate per CZ gate in the
iterated randomized benchmarking experiment is important for real-world
applications due to the accumulation of coherent errors being particularly
detrimental to many quantum algorithms Kjaergaard  _et al._ (2022).

During the CZ gate, the coupler and the qubits become strongly hybridized
which affects the coherence of the computational states. To quantify the
impact of decoherence during the CZ gate, we first measure the energy
relaxation times of all the three hybridized qubit and coupler modes as a
function of the uncoupled coupler frequency, shown in Fig. 4(a). Even though
there are no individual drive and readout lines for the coupler, we can
measure the energy-relaxation time of the coupler-like mode using the qubit
readout and drive lines when the coupler and qubit states are hybridized. For
the coupler frequencies below $3.9\text{\,}\mathrm{GHz}$ the degree of
hybridization is low and we employ two-probe readout with $\rm{Q}_{2}$ as an
ancilla qubit instead, see Appendix F for details on the experiment. The
measured frequencies of the hybridized modes are shown in Fig. 4(b). In the
frequency range relevant for the CZ gate (gray shaded area), we observe that
the energy-relaxation times of the qubit-like modes are not significantly
affected by the hybridization with the coupler due to its high intrinsic
energy-relaxation time, $T_{1}^{\rm c}>$40\text{\,}\mathrm{\SIUnitSymbolMicro
s}$$ at the idling configuration.

To model the dependency of the energy relaxation times of the hybridized
states on the coupler frequency, we assume that the energy relaxation times of
the uncoupled modes stay constant as a function of the coupler frequency. We
then numerically evaluate the energy relaxation time of the coupled modes
based solely on the degree of hybridization with the coupler, shown with solid
lines in Fig 4(a). In the frequency range relevant for the CZ gate and at
coupler frequencies above $4.1\text{\,}\mathrm{GHz}$, we find a good match
between the measurements and the model. For coupler frequencies between
$3.9\text{\,}\mathrm{GHz}$ and $4.1\text{\,}\mathrm{GHz}$, we observe
unexpected dips in the relaxation times, which can potentially be attributed
to parasitic two-level systems (TLSs) near ${\rm Q_{1}}$ and ${\rm Q_{2}}$
Klimov  _et al._ (2018); Müller  _et al._ (2019); Krinner  _et al._ (2022).
However, these TLSs do not have a significant impact on the performance of the
gate as long as the coupler remains below $3.9\text{\,}\mathrm{GHz}$.

Next, we measure the dephasing times of the three hybridized modes as a
function of the coupler frequency, see Fig. 4(c). At each of the frequencies,
we perform a Ramsey experiment and extract the dephasing times from a fit to
the model

$\nu(t)=a(t)\mathrm{e}^{-\frac{t}{2T_{1}}-\frac{t}{T_{\phi,1}}-\left(\frac{t}{T_{\phi,2}}\right)^{2}},$
(1)

where $a(t)$ is the response of the Ramsey experiment excluding decoherence,
$T_{1}$ is the energy-relaxation time measured earlier, and $T_{\phi,1}$ and
$T_{\phi,2}$ are the exponential and Gaussian components of the dephasing
time. The exponential dephasing time originates from excess wide-band noise
such as thermal excitations in the readout resonators Yan  _et al._ (2018b)
and is expected to only moderately contribute to the total dephasing. Indeed,
we find $T_{\phi,1}^{\rm Q1}\approx$70\text{\,}\mathrm{\SIUnitSymbolMicro s}$$
and $T_{\phi,1}^{\rm Q2}\approx$45\text{\,}\mathrm{\SIUnitSymbolMicro s}$$ for
the coupler frequencies relevant for the CZ gate, see Appendix H for the full
data. The Gaussian component of the dephasing time is dominated by low-
frequency flux noise Bylander  _et al._ (2011) (see Appendix F for details),
which is a function of flux-dispersion of the uncoupled modes and the degree
of their hybridisation. We model the dephasing rate of the coupled mode $k$ as

$\Gamma_{\phi,2}^{k}=1/T_{\phi,2}^{k}=\sqrt{\sum_{i=\rm{Q_{1}},\rm{Q_{2}},\rm{c}}\left(p_{i,k}\tilde{\Gamma}_{\phi,2}^{i}\right)^{2}},$
(2)

where $p_{i,k}$ is the participation ratio of the uncoupled mode $i$ in the
hybridized mode $k$ and $\tilde{\Gamma}_{\phi,2}^{i}$ are the dephasing rates
of the uncoupled modes, see Fig. 4(c) for the comparison of the model and the
experimental data. Above we have assumed that flux noise affecting the
uncoupled modes originates from independent sources and can be approximated as
quasi-static noise, resulting in the dephasing rates of the uncoupled modes to
sum in square Ithier  _et al._ (2005). To account for the change in the flux
noise sensitivity of the uncoupled coupler mode as its frequency is varied, we
model the coupler dephasing rate as $\tilde{\Gamma}_{\phi,2}^{\rm
c}=B|\frac{\partial\omega_{\rm c}}{\partial\Phi_{\rm c}}|$ Bylander  _et al._
(2011), where $\omega_{\rm c}/2\pi$ is the frequency of the uncoupled coupler
mode, $\Phi_{\rm c}$ is the coupler flux,
$B\approx$53\text{\,}\Phi_{\mathrm{0}}$$ is the flux noise amplitude extracted
from a fit to the coupler dephasing rate when it is decoupled from the qubits,
and $\Phi_{0}$ is the flux quantum. With these assumptions, we observe an
excellent match between the experimental data and the model across the
measured coupler frequency range, and note that the dephasing times of the
qubit modes stay above $1\text{\,}\mathrm{\SIUnitSymbolMicro s}$ even when
maximally hybridized with the coupler due to having the coupler flux sweet
spot being located near the qubit frequencies.

In order to estimate the coherence limit of an arbitrarily long CZ gate, we
calculate effective relaxation times $T_{1,{\rm eff}}$ and effective dephasing
times $T_{\phi,1,{\rm eff}}$ and $T_{\phi,2,{\rm eff}}$ for both qubits by
weighting the measured coherence rates with the time spend on the
corresponding continuous adiabatic trajectories [Fig. 4(b)] during the CZ
gate, see Appendix H for additional information. The time spent on each
frequency is determined by the Slepian pulse shape, which we parameterize with
the interaction time $\tau$. For each $\tau$, we re-calculate the pulse shape
using the $ZZ$ interaction strength calibration data in Fig. 2(b) to ensure
that a conditional phase of $\phi_{11}=\pi$ is accumulated.

The coherence limit can then be calculated as Chu and Yan (2021); Abad  _et
al._ (2021)

$\epsilon_{\rm CZ}^{\rm
limit}(\tau)=\sum_{i=1,2}\frac{2}{5}\left[\frac{\tau}{T_{1,{\rm eff}}^{{\rm
Q}_{i}}(\tau)}+\frac{\tau}{T_{\phi,1,{\rm eff}}^{{\rm
Q}_{i}}(\tau)}+\left(\frac{\tau}{T_{\phi,2,{\rm eff}}^{{\rm
Q}_{i}}(\tau)}\right)^{2}\right],$ (3)

where the first term in the square brackets corresponds to the coherence limit
of the CZ gate due to the relaxation times of qubits and the second and third
terms yield the coherence limits due to exponential and Gaussian components of
dephasing, respectively. Figure 4(d) depicts the coherence limit for the gate
error $\epsilon_{\rm CZ}^{\rm limit}$ for various different gate durations
along with the contributions of the individual terms in Eq. (3).

For our CZ gate with the interaction time of $22\text{\,}\mathrm{ns}$ and an
additional $11\text{\,}\mathrm{ns}$ wait time around it, we obtain $T_{1,{\rm
eff}}^{{\rm Q}_{1}}=$14\text{\,}\mathrm{\SIUnitSymbolMicro s}$$, $T_{1,{\rm
eff}}^{{\rm Q}_{2}}=$43\text{\,}\mathrm{\SIUnitSymbolMicro s}$$,
$T_{\phi,1,{\rm eff}}^{{\rm Q}_{1}}=$67\text{\,}\mathrm{\SIUnitSymbolMicro
s}$$, $T_{\phi,1,{\rm eff}}^{{\rm
Q}_{2}}=$43\text{\,}\mathrm{\SIUnitSymbolMicro s}$$, $T_{\phi,2,{\rm
eff}}^{{\rm Q}_{1}}=$17\text{\,}\mathrm{\SIUnitSymbolMicro s}$$ and
$T_{\phi,2,{\rm eff}}^{{\rm Q}_{2}}=$6\text{\,}\mathrm{\SIUnitSymbolMicro
s}$$, yielding a coherence limit of $\epsilon_{\rm CZ}^{\rm
limit}=$1.7\text{\times}{10}^{-3}$$. This limit can be compared with the
experimentally measured gate error $\epsilon_{\rm CZ}=$(1.9\pm
0.2)\text{\times}{10}^{-3}$$ which is only slightly higher, indicating that
the gate is mostly coherence limited, as was already suggested by the
randomized benchmarking experiment with multiple interleaved CZ gates. The
most significant contribution to the errors comes from the energy relaxation
time $T_{1,{\rm eff}}^{{\rm Q}_{1}}$. Importantly, the fidelity is only
modestly affected by dephasing, mainly due to the possibility to operate the
coupler near its flux insensitive sweep spot during the CZ gate. To
demonstrate that the coupler coherence times at the two-qubit-gate operation
region are high enough not to have a detrimental impact on the gate errors, we
re-compute the coherence limit assuming that hybridization with the coupler
does not affect the qubit coherence times and use this as a reference, see
Fig. 4(d). Comparing the reference case with the coherence limit
$\epsilon_{\rm CZ}^{\rm limit}$, we find a negligible difference, confirming
the hypothesis.

Possible reasons for the remaining errors include system fluctuations due to
TLSs nearby the qubits, leading to rare uncontrollable jumps in the qubit
frequencies, as well as uncompensated flux pulse distortions in the nanosecond
range and leakage to non-computational states.

##  V Conclusions

We have proposed and demonstrated a tunable-coupler design based on a floating
transmon coupled to two computational qubits via waveguide extenders. This
design allows the qubits to be separated at least by $2\text{\,}\mathrm{mm}$,
enabling each qubit to have individual readout resonators and Purcell filters
for high-fidelity readout, low non-nearest-neighbor coupling, and reduced
crosstalk from passing control lines to the qubits in a flip-chip
architecture. Although there is a minor trade-off between qubit-qubit distance
and the coupling strengths due to the added capacitance of the waveguide
extenders to ground, we have shown that the interaction strength is large
enough for a qubit-qubit distance beyond $1\text{\,}\mathrm{mm}$ by
demonstrating a fast and high-fidelity CZ gate with $99.81\pm
0.02\text{\,}\mathrm{\char 37\relax}$ fidelity. We confirmed that the CZ gate
fidelity is not limited by the coupler coherence, but mostly determined by the
qubit energy-relaxation rate, making state-of-the-art two-qubit-gate
fidelities on large quantum processors based on this coupler design an
achievable goal.

##  VI Acknowledgement

We acknowledge Matthew Sarsby, Roope Kokkoniemi, Ali Yurtalan Jean-Luc
Orgiazzi, Lucas Ortega, Jorge Santos, Jaakko Jussila, Illari Kuronen, Jaakko
Salo, Tiina Naaranoja, Otto Koskinen, and Tero Somppi for supporting the
conceptualization, construction, and maintenance of the experimental setup,
Ferenc Dósa-Rácz, Janne Mäntylä, Sinan Inel, and Leon Wubben for additional
software support, and Olli-Pentti Saira for valuable discussions. We would
additionally like to thank the rest of the IQM team for creating the entire
infrastructure, laying the foundation of this work.

A.V. and J.HE. conceptualized the project, F.M., S.J., and A.V. planned and
executed the experiments and analyzed the experimental data, C.O.-K. and A.L.,
conducted and analyzed the microwave simulations, J.T., M.PAP., A.V., M.T.,
and A.A. provided theoretical modeling and simulations, C.O.-K., A.L., E.T.,
J.RÄ., and J.HE. designed and simulated the sample, F.M., J.I., T.H., M.K.,
J.RO., N.V., M.SE., L.B., O.A., B.T., V.B., J.K., D.J., E.H., C.F.C., M.SA.,
V.S., M.PAR., F.T., J.L., J.HE., A.V., and S.J. developed the experiment and
analysis software, W.L. and T.L. designed Josephson Junctions. W.L. developed
qubit and airbridge process, fabricated the qubit devices, W.L. and Y.L.
benchmarked the room temperature resistance, J.HO bonded and packaged the
device, F.M., A.V., S.J., J.T., A.L., C.O.-K., W.L, M.M., and J.HE. drafted or
revised the work, and J.HE., M.M., J.HA., and K.Y.T. supervised the work.
Experimental data are available upon reasonable request to the authors.

The work was partly supported by the European Innovation Council (EIC) under
Prometheus (grant No. 959521), Business Finland (grant No. 7547/31/2021), and
by the German Federal Ministry of Education and Research (BMBF) under Q-Exa
(grant No. 13N16062), QSolid (grant No. 13N16161), and MUNIQC-SC (grant No.
13N16185). M.M. is partly supported by the Academy of Finland through its
Centers of Excellence Program (project No. 336810) and by the European
Research Council under Advanced Grant ConceptQ (grant No. 101053801). Parts of
this work are included in patents applications filed by IQM Finland Oy. This
work used resources from the OtaNano Micronova cleanroom.

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Long-distance transmon coupler with CZ gate fidelity above
$99.8\text{\,}\mathrm{\char 37\relax}$

##  APPENDIX A EFFECTIVE COUPLING MODEL

Figure S1: Lumped-element circuit diagram of the floating-coupler scheme. The
drawn sizes of the capacitors roughly indicate the relative magnitudes between
the designed values of the capacitances. The crosses indicate the Josephson
junctions with the Josephson energy $E_{\rm J}$.

In this Appendix, we derive an effective Hamiltonian operator for the extended
floating-coupler setup. The essential physics can be described with the
lumped-element circuit shown in Fig. S1. Since the microwave wavelengths
considered here are much longer than the lengths of the waveguide extenders,
the extenders can be modeled as lumped capacitors for the derivation of the
effective Hamiltonian, as shown in Fig. S1. We also observe that the shunt
capacitances $C_{\rm sC}$, $C_{\rm sD}$, $C_{\rm sE}$, and $C_{\rm sF}$ to the
ground, and the coupling capacitances $C_{\rm CE}$ and $C_{\rm DF}$ to
opposite coupler islands are parasitic couplings, hence not needed to operate
the two-qubit system. However, such capacitances are unavoidably present in
our design, and thus, also included in the discussion below.

The derivation of the Hamiltonian follows the conventional circuit-
quantization procedure described in the sections below. However, we note that
the circuit has six voltage nodes, three of which can be eliminated. We
eliminate the two (inactive) purely capacitive nodes C and F using three
successive star-mesh transformations. Consequently, we write down the
Lagrangian of the circuit and show that the relative and the ”center-of-mass”
motion of the coupler can be separated. The center-of-mass coordinate is
cyclic, and thus can also be eliminated. The resulting Hamiltonian has three
degrees of freedom, one for each qubit and the coupler. Finally, we show how
the effective $ZZ$ interaction strength can be computed numerically using the
obtained effective Hamiltonian operator.

###  A.1 Elimination of the inactive nodes

Figure S2: Derivation of the effective model using star-mesh transformations.
The capacitances are defined in Eqs. (S3) and (S4).

Here, we observe that in the circuit shown in Fig. S1, the nodes C and F are
inactive because they are coupled to other nodes only capacitively. In Fig.
S2, we show that the direct coupling between nodes C and F can be eliminated
by first applying the mesh-star ($\Delta$-Y) transformation for the nodes C,
F, and ground, defined as

$\begin{split}C_{\rm CI}&=\frac{C_{\rm CF}C_{\rm sC}+C_{\rm CF}C_{\rm
sF}+C_{\rm sC}C_{\rm sF}}{C_{\rm sF}},\\\ C_{\rm FI}&=\frac{C_{\rm CF}C_{\rm
sC}+C_{\rm CF}C_{\rm sF}+C_{\rm sC}C_{\rm sF}}{C_{\rm sC}},\\\ C_{\rm
sI}&=\frac{C_{\rm CF}C_{\rm sC}+C_{\rm CF}C_{\rm sF}+C_{\rm sC}C_{\rm
sF}}{C_{\rm CF}}.\end{split}$ (S1)

Consequently, the coupling is mediated by an effective node I. Then, we apply
the star-mesh transformations to the nodes C and F, resulting in the desired
elimination of the nodes. Subsequently, we eliminate the node I with an of the
star-mesh transformation. The star-mesh transformation is defined in the
general case as

$C_{\rm XY}=\frac{C_{\rm X}C_{\rm Y}}{C_{||}},$ (S2)

where $C_{||}=\sum_{n}C_{n}$ is the sum of all capacitances $C_{n}$ coupled to
the central node in the star configuration, and $C_{\rm XY}$ is the effective
capacitance coupling nodes $X$ and $Y$ in the mesh configuration.

Figure S3: Effective-circuit schematic of the floating-coupler setup. The
capacitances are defined in Eq. (S3).

Consequently, we obtain the equivalent circuit shown in Fig. S3, in which the
inactive nodes C and F have been eliminated. The related effective
capacitances can be written in terms of the physical capacitances as

$\begin{split}C_{1}&=C_{\rm AB}+\frac{C_{\rm BI}C_{\rm sI}}{C_{||}^{\rm I}}\\\
&\approx C_{\rm AB}+C_{\rm sC}\frac{C_{\rm BC}}{C_{||}^{\rm
C}}\left[1+\frac{C_{\rm CF}}{C_{\rm||}^{\rm F}}\frac{C_{\rm FI}}{C_{\rm
CI}}+\frac{C_{\rm CF}}{C_{\rm||}^{\rm C}}\frac{C_{\rm CI}}{C_{\rm
FI}}\right],\\\ C_{2}&=C_{\rm GH}+\frac{C_{\rm GI}C_{\rm sI}}{C_{||}^{\rm
I}}\\\ &\approx C_{\rm GH}+C_{\rm sF}\frac{C_{\rm FG}}{C_{||}^{\rm
F}}\left[1+\frac{C_{\rm CF}}{C_{||}^{\rm F}}\frac{C_{\rm FI}}{C_{\rm
CI}}+\frac{C_{\rm CF}}{C_{\rm||}^{\rm C}}\frac{C_{\rm CI}}{C_{\rm
FI}}\right],\\\ C_{\rm c}&=C_{\rm DE}+\frac{C_{\rm CD}C_{\rm CE}}{C_{||}^{\rm
C}}+\frac{C_{\rm DF}C_{\rm EF}}{C_{||}^{\rm F}}+\frac{C_{\rm DI}C_{\rm
EI}}{C_{||}^{\rm I}}\\\ &\approx C_{\rm DE}+\frac{C_{\rm CD}C_{\rm
EF}\tilde{C}_{\rm CF}}{C_{||}^{\rm C}C_{||}^{\rm F}},\\\ \tilde{C}_{\rm
sD}&=C_{\rm sD}+\frac{C_{\rm DI}C_{\rm sI}}{C_{||}^{\rm I}}\\\ &\approx C_{\rm
sD}+C_{\rm sC}\frac{C_{\rm CD}}{C_{||}^{\rm C}}\left[1+\frac{C_{\rm
CF}}{C_{||}^{\rm F}}\frac{C_{\rm FI}}{C_{\rm CI}}+\frac{C_{\rm
CF}}{C_{\rm||}^{\rm C}}\frac{C_{\rm CI}}{C_{\rm FI}}\right],\\\ \tilde{C}_{\rm
sE}&=C_{\rm sE}+\frac{C_{\rm EI}C_{\rm sI}}{C_{||}^{\rm I}}\\\ &\approx C_{\rm
sE}+C_{\rm sF}\frac{C_{\rm EF}}{C_{||}^{\rm F}}\left[1+\frac{C_{\rm
CF}}{C_{||}^{\rm F}}\frac{C_{\rm FI}}{C_{\rm CI}}+\frac{C_{\rm
CF}}{C_{\rm||}^{\rm C}}\frac{C_{\rm CI}}{C_{\rm
FI}}\right].\end{split}\qquad\qquad\begin{split}C_{\rm 1c}^{||}&=\frac{C_{\rm
BC}C_{\rm CD}}{C_{||}^{\rm C}}+\frac{C_{\rm BI}C_{\rm DI}}{C_{||}^{\rm I}}\\\
&\approx\frac{C_{\rm BC}C_{\rm CD}}{C_{||}^{C}}\left[1+\frac{C_{\rm
CI}}{C_{\rm FI}}\frac{\tilde{C}_{\rm CF}}{C_{||}^{\rm C}}\right],\\\ C_{\rm
1c}^{\perp}&=\frac{C_{\rm BC}C_{\rm CE}}{C_{||}^{\rm C}}+\frac{C_{\rm
BI}C_{\rm EI}}{C_{||}^{\rm I}}\\\ &\approx\left[C_{\rm CE}+\frac{C_{\rm
EF}C_{\rm CF}}{C_{||}^{\rm F}}\right]\frac{C_{\rm BC}}{C_{||}^{\rm C}},\\\
C_{\rm 2c}^{||}&=\frac{C_{\rm EF}C_{\rm FG}}{C_{||}^{\rm F}}+\frac{C_{\rm
EI}C_{\rm GI}}{C_{||}^{\rm I}}\\\ &\approx\frac{C_{\rm EF}C_{\rm
FG}}{C_{||}^{\rm F}}\left[1+\frac{C_{\rm FI}}{C_{\rm CI}}\frac{\tilde{C}_{\rm
CF}}{C_{||}^{\rm F}}\right],\\\ C_{\rm 2c}^{\perp}&=\frac{C_{\rm DF}C_{\rm
FG}}{C_{||}^{\rm F}}+\frac{C_{\rm DI}C_{\rm GI}}{C_{||}^{\rm I}}\\\
&\approx\left[C_{\rm DF}+\frac{C_{\rm CD}C_{\rm CF}}{C_{||}^{\rm
C}}\right]\frac{C_{\rm FG}}{C_{||}^{\rm F}},\\\ C_{12}&=\frac{C_{\rm BI}C_{\rm
GI}}{C_{||}^{\rm I}}=\frac{C_{\rm BC}\tilde{C}_{\rm CF}C_{\rm FG}}{C_{||}^{\rm
C}C_{||}^{\rm F}},\end{split}$ (S3)

where the effective capacitances that arise from the mesh-star and star-mesh
transformations are defined as

$\begin{split}C_{\rm CI}&=C_{\rm CF}+C_{\rm sC}+\frac{C_{\rm CF}C_{\rm
sC}}{C_{\rm sF}},\\\ C_{\rm FI}&=C_{\rm CF}+C_{\rm sF}+\frac{C_{\rm CF}C_{\rm
sF}}{C_{\rm sC}},\\\ C_{\rm sI}&=C_{\rm sC}+C_{\rm sF}+\frac{C_{\rm sC}C_{\rm
sF}}{C_{\rm CF}},\\\ C_{\rm BD}&=\frac{C_{\rm BC}C_{\rm CD}}{C_{||}^{\rm
C}},\\\ C_{\rm BE}&=\frac{C_{\rm BC}C_{\rm CE}}{C_{||}^{\rm C}},\\\ C_{\rm
BI}&=\frac{C_{\rm BC}C_{\rm CI}}{C_{||}^{\rm C}},\\\ C_{\rm DI}&=\frac{C_{\rm
DF}C_{\rm FI}}{C_{||}^{\rm F}}+\frac{C_{\rm CD}C_{\rm CI}}{C_{||}^{\rm C}},\\\
C_{||}^{\rm C}&=C_{\rm BC}+C_{\rm CD}+C_{\rm CE}+C_{\rm CF}+C_{\rm
sC}+\frac{C_{\rm CF}C_{\rm sC}}{C_{\rm sF}},\\\
\end{split}\qquad\qquad\begin{split}\tilde{C}_{\rm DE}&=C_{\rm
DE}+\frac{C_{\rm CD}C_{\rm CE}}{C_{||}^{\rm C}}+\frac{C_{\rm DF}C_{\rm
EF}}{C_{||}^{\rm F}},\\\ C_{\rm DG}&=\frac{C_{\rm DF}C_{\rm FG}}{C_{||}^{\rm
F}},\\\ C_{\rm EG}&=\frac{C_{\rm EF}C_{\rm FG}}{C_{||}^{\rm F}},\\\ C_{\rm
EI}&=\frac{C_{\rm CI}C_{\rm CE}}{C_{||}^{\rm C}}+\frac{C_{\rm EF}C_{\rm
FI}}{C_{||}^{\rm F}},\\\ C_{\rm GI}&=\frac{C_{\rm FG}C_{\rm FI}}{C_{||}^{\rm
F}},\\\ C_{||}^{\rm F}&=C_{\rm DF}+C_{\rm EF}+C_{\rm FG}+C_{\rm CF}+C_{\rm
sF}+\frac{C_{\rm CF}C_{\rm sF}}{C_{\rm sC}},\\\ C_{||}^{\rm I}&=C_{\rm
BI}+C_{\rm DI}+C_{\rm EI}+C_{\rm GI}+C_{\rm sI},\\\ \tilde{C}_{\rm
CF}&=\frac{C_{\rm CI}C_{\rm FI}}{C_{||}^{\rm I}}=\frac{C_{\rm CF}}{1-C_{\rm
CF}\left[\frac{C_{\rm CI}}{C_{\rm FI}C_{||}^{\rm C}}+\frac{C_{\rm FI}}{C_{\rm
CI}C_{||}^{\rm F}}\right]}.\end{split}$ (S4)

Above, the approximations hold if $C_{\rm CE},C_{\rm CI}\ll C_{||}^{\rm C}$
and $C_{\rm DF},C_{\rm FI}\ll C_{||}^{\rm F}$.

An alternative approach to eliminate the inactive nodes C and F in Fig. S1 is
to calculate the admittance matrix $Y$ of a four-port network, which includes
the nodes B, D, E and G. We can model the network by connecting each node to
every other node with a lumped element with a certain admittance $y_{ij}$,
where $i,j\in\\{\rm{B,D,E,G}\\}$. Each admittance $y_{ij}$ can be calculated
at a certain frequency $\omega$ from the initial capacitances shown in Fig.
S1. The admittance matrix can then be calculated as

$Y_{ij}=\begin{cases}y_{i}+\sum_{k=\rm{B,D,E,G}\atop k\neq
i}{y_{ik}},&\mbox{if}\quad i=j\\\ -y_{ij},&\mbox{if}\quad i\neq
j\quad.\end{cases}$ (S5)

Assuming having only capacitive elements in the network, we can then calculate
the capacitance matrix as $C=\rm{Im}(Y(\omega))/\omega$ at a certain angular
frequency $\omega$, where $C_{ii}$ is the capacitance from node $i$ to ground
and $C_{ij}=C_{ji}$ is the capacitance from node $i$ to node $j$.

###  A.2 Elimination of the center of mass

Here, we write the Lagrangian of the system as $L=T-V$ in which

$\begin{split}T=&\frac{1}{2}C_{1}\dot{\phi}_{\rm B}^{2}+\frac{1}{2}C_{\rm
1c}^{||}(\dot{\phi}_{\rm B}-\dot{\phi}_{\rm D})^{2}+\frac{1}{2}C_{\rm
1c}^{\perp}(\dot{\phi}_{\rm B}-\dot{\phi}_{\rm E})^{2}+\frac{1}{2}C_{\rm
12}(\dot{\phi}_{\rm B}-\dot{\phi}_{\rm G})^{2}\\\
&+\frac{1}{2}C_{2}\dot{\phi}_{\rm G}^{2}+\frac{1}{2}C_{\rm
2c}^{||}(\dot{\phi}_{\rm E}-\dot{\phi}_{\rm G})^{2}+\frac{1}{2}C_{\rm
2c}^{\perp}(\dot{\phi}_{\rm D}-\dot{\phi}_{\rm G})^{2}+\frac{1}{2}C_{\rm
c}(\dot{\phi}_{\rm D}-\dot{\phi}_{\rm E})^{2}+\frac{1}{2}\tilde{C}_{\rm
sD}\dot{\phi}_{\rm D}^{2}+\frac{1}{2}\tilde{C}_{\rm sE}\dot{\phi}_{\rm
E}^{2},\end{split}$ (S6)

and

$V=E_{\rm J_{1}}\left(1-\cos\left[\frac{2\pi}{\Phi_{0}}\phi_{\rm
B}\right]\right)+E_{\rm J_{2}}\left(1-\cos\left[\frac{2\pi}{\Phi_{0}}\phi_{\rm
G}\right]\right)+E_{\rm J_{\rm
c}}\left(1-\cos\left[\frac{2\pi}{\Phi_{0}}(\phi_{\rm D}-\phi_{\rm
E})\right]\right),$ (S7)

where $\Phi_{0}=h/(2e)$ is the flux quantum and $\phi_{i}$ is the node flux
related to the node $i\in\\{\rm B,D,E,G\\}$. We emphasize that both qubits and
the coupler have a charging energy $E_{\rm C}\ll E_{\rm J}$, and thus operated
in the transmon regime. Consequently, we have neglected the offset charges
from the above expression for kinetic energy, as their detrimental effect to
quantum coherence is exponentially suppressed in this regime of parameters.

We observe that the potential energy depends only on the relative motion
$\phi_{\rm c}\equiv\phi_{\rm D}-\phi_{\rm E}$. This suggests that there exists
a constant of motion analogous to the center-of-mass in mechanical systems.
Therefore, we make a linear transformation $(\phi_{\rm D},\phi_{\rm
E})\rightarrow(\phi_{\rm c},\theta_{\rm c})$ to flux coordinates of relative
motion, defined as

$\left(\begin{array}[]{cc}\phi_{\rm c}\\\ \theta_{\rm
c}\end{array}\right)=\left(\begin{array}[]{cc}1&-1\\\
a&b\end{array}\right)\left(\begin{array}[]{cc}\phi_{\rm D}\\\ \phi_{\rm
E}\end{array}\right).$ (S8)

Below, we require that the determinant of the transformation matrix is equal
to $a+b=1$, such that there is no scaling involved. Moreover, we define the
transformation such that the fluxes $\phi_{\rm c}$ and $\theta_{\rm c}$ are
decoupled. Consequently, we obtain that

$\theta_{\rm c}\equiv\frac{(C_{\rm 1c}^{||}+C_{\rm 2c}^{\perp}+\tilde{C}_{\rm
sD})\phi_{\rm D}+(C_{\rm 1c}^{\perp}+C_{\rm 2c}^{||}+\tilde{C}_{\rm
sE})\phi_{\rm E}}{C_{\rm 1c}^{||}+C_{\rm 1c}^{\perp}+C_{\rm 2c}^{||}+C_{\rm
2c}^{\perp}+\tilde{C}_{\rm sD}+\tilde{C}_{\rm sE}},$ (S9)

which is a cyclic coordinate. We observe that the flux $\theta_{\rm c}$ can be
interpreted as the ”center of capacitance” of the two islands D and E, in
which the total (decoupled) island capacitances are given as sums of the
capacitances coupled to the island, i.e. $C_{\rm 1c}^{||}+C_{\rm
2c}^{\perp}+\tilde{C}_{\rm sD}$ and $C_{\rm 1c}^{\perp}+C_{\rm
2c}^{||}+\tilde{C}_{\rm sE}$, respectively.

Denoting $\phi_{1}=\phi_{\rm B}$ and $\phi_{2}=\phi_{\rm G}$, the kinetic
energy defined in Eq. (S6) can be written after the transformation as
$T=\frac{1}{2}\dot{\Phi}^{\rm T}C\dot{\Phi}$, where $\Phi^{\rm
T}=(\phi_{1},\phi_{\rm c},\theta_{\rm c},\phi_{2})$, and the corresponding
capacitance matrix is given as

$C=\left(\begin{array}[]{cccc}C_{\sigma_{1}}&-C_{\rm 1c}&-C_{\rm
1c}^{||}-C_{\rm 1c}^{\perp}&-C_{12}\\\ -C_{\rm 1c}&C_{\sigma_{\rm c}}&0&C_{\rm
2c}\\\ -C_{\rm 1c}^{||}-C_{\rm 1c}^{\perp}&0&C_{\sigma_{\theta}}&-C_{\rm
2c}^{||}-C_{\rm 2c}^{\perp}\\\ -C_{12}&C_{\rm 2c}&-C_{\rm 2c}^{||}-C_{\rm
2c}^{\perp}&C_{\sigma_{2}}\end{array}\right),$ (S10)

where

$\begin{split}C_{\sigma_{1}}&=C_{1}+C_{\rm 1c}^{||}+C_{\rm
1c}^{\perp}+C_{12},\\\ C_{\sigma_{\rm c}}&=C_{\rm c}+\left[\frac{1}{C_{\rm
1c}^{||}+C_{\rm 2c}^{\perp}+\tilde{C}_{\rm sD}}+\frac{1}{C_{\rm
2c}^{||}+C_{\rm 1c}^{\perp}+\tilde{C}_{\rm sE}}\right]^{-1}=C_{\rm
c}+\gamma_{1}\gamma_{2}C_{\sigma_{\theta}},\\\ C_{\sigma_{2}}&=C_{2}+C_{\rm
2c}^{||}+C_{\rm 2c}^{\perp}+C_{12},\\\ C_{\sigma_{\theta}}&=C_{\rm
1c}^{||}+C_{\rm 1c}^{\perp}+C_{\rm 2c}^{||}+C_{\rm 2c}^{\perp}+\tilde{C}_{\rm
sD}+\tilde{C}_{\rm sE},\\\ C_{\rm 1c}&=\gamma_{1}C_{\rm
1c}^{||}-\gamma_{2}C_{\rm 1c}^{\perp},\\\ C_{\rm 2c}&=\gamma_{2}C_{\rm
2c}^{||}-\gamma_{1}C_{\rm 2c}^{\perp},\end{split}$ (S11)

and $\gamma_{1}=(C_{\rm 2c}^{||}+C_{\rm 1c}^{\perp}+\tilde{C}_{\rm
sE})/C_{\sigma_{\theta}}$, $\gamma_{2}=(C_{\rm 1c}^{||}+C_{\rm
2c}^{\perp}+\tilde{C}_{\rm sD})/C_{\sigma_{\theta}}$. We point out that the
second term in the definition of $C_{\sigma_{\rm c}}$ is equivalent to the
reduced mass of the mechanical two-body problem. We observe that after the
transformation, the relative and center-of-mass motion are indeed decoupled.
However, there is a coupling between the center-of-mass and the qubit fluxes.
The potential energy does not depend on the center-of-mass flux which is,
thus, a cyclic coordinate. According to the Euler–Lagrange equation, the
corresponding conjugate momentum,

$q_{\theta}=\frac{\partial L}{\partial\dot{\theta}_{\rm c}}=-(C_{\rm
1c}^{||}+C_{\rm 1c}^{\perp})\dot{\phi}_{\rm B}-(C_{\rm 2c}^{||}+C_{\rm
2c}^{\perp})\dot{\phi}_{\rm G}+C_{\sigma_{\theta}}\dot{\theta}_{\rm c},$ (S12)

is a constant of motion, i.e. $q_{\theta}=$ constant. Consequently, we obtain

$\dot{\theta}_{\rm c}=\frac{1}{C_{\sigma_{\theta}}}\left[q_{\theta}+(C_{\rm
1c}^{||}+C_{\rm 1c}^{\perp})\dot{\phi}_{\rm B}+(C_{\rm 2c}^{||}+C_{\rm
2c}^{\perp})\dot{\phi}_{\rm G}\right].$ (S13)

Substituting this back to the Lagrangian, and neglecting the constant term
proportional to $q_{\theta}$, we obtain the effective capacitance matrix

$C^{\prime}=\left(\begin{array}[]{ccc}C_{\Sigma_{1}}&-C_{\rm
1c}&-C_{12}^{*}\\\ -C_{\rm 1c}&C_{\Sigma_{\rm c}}&C_{\rm 2c}\\\
-C_{12}^{*}&C_{\rm 2c}&C_{\Sigma_{2}}\end{array}\right),$ (S14)

where

$\begin{split}C_{\Sigma_{1}}&=C_{\sigma_{1}}-\frac{(C_{\rm 1c}^{||}+C_{\rm
1c}^{\perp})^{2}}{C_{\sigma_{\theta}}},\\\ C_{12}^{*}&=C_{12}+\frac{(C_{\rm
1c}^{||}+C_{\rm 1c}^{\perp})(C_{\rm 2c}^{||}+C_{\rm
2c}^{\perp})}{C_{\sigma_{\theta}}},\\\
C_{\Sigma_{2}}&=C_{\sigma_{2}}-\frac{(C_{\rm 2c}^{||}+C_{\rm
2c}^{\perp})^{2}}{C_{\sigma_{\theta}}}.\end{split}$ (S15)

We note that the effective coupling capacitance $C_{12}^{*}$ between the
qubits consists of the direct capacitance $C_{12}$ and that mediated by the
coupler structure. The relevant coordinate flux vector is defined as
$\Phi^{\prime}=(\phi_{1},\phi_{\rm c},\phi_{2})$, in which $\phi_{1}=\phi_{\rm
B}$, $\phi_{2}=\phi_{\rm G}$, and $\phi_{\rm c}=\phi_{\rm D}-\phi_{\rm E}$ is
the coordinate for the relative motion between the nodes D and E. Similar
discussions on mediated interactions in a chain of floating transmons have
been presented recently in Refs. Yanay  _et al._ , 2022; Zhao  _et al._ ,
2022.

###  A.3 Effective model without cross-island coupling

Let us consider the effective capacitance matrix in Eq. (S14). We observe that
the elements of the matrix depend on cross-island coupling capacitances
$C_{\rm 1c}^{\perp}$ and $C_{\rm 2c}^{\perp}$. Here, we show that an
equivalent effective capacitance matrix can be realized with a circuit in
which $C_{\rm 1c}^{\perp}=C_{\rm 2c}^{\perp}=0$, provided that the other
capacitances are adjusted such that the coupling capacitances in Eq. (S14)
remain unaltered. We find that this can be achieved by making a transformation
$C\rightarrow\bar{C}$ to Eq. (S10), in which the elements of $\bar{C}$ are
defined as

$\begin{split}\bar{C}_{\rm 1c}^{\perp}&=\bar{C}_{\rm 2c}^{\perp}=0,\\\
\bar{C}_{\rm 1c}^{||}&=C_{\rm 1c}^{||}-C_{\rm 1c}^{\perp}\frac{C_{\rm
1c}^{||}+C_{\rm 2c}^{\perp}+\tilde{C}_{\rm sD}}{C_{\rm 2c}^{||}+C_{\rm
1c}^{\perp}+\tilde{C}_{\rm sE}},\\\ \bar{C}_{\rm 2c}^{||}&=C_{\rm
2c}^{||}-C_{\rm 2c}^{\perp}\frac{C_{\rm 2c}^{||}+C_{\rm
1c}^{\perp}+\tilde{C}_{\rm sE}}{C_{\rm 1c}^{||}+C_{\rm
2c}^{\perp}+\tilde{C}_{\rm sD}},\\\ \bar{C}_{\rm
12}&=C_{12}^{*}-\frac{\bar{C}_{\rm 1c}^{||}\tilde{C}_{\rm
2c}^{||}}{\bar{C}_{\sigma_{\theta}}},\\\
\bar{C}_{\sigma_{\theta}}&=\bar{C}_{\rm 1c}+\bar{C}_{\rm 2c}+\tilde{C}_{\rm
sD}+\tilde{C}_{\rm sE},\\\
\bar{C}_{\sigma_{1}}&=C_{\Sigma_{1}}+\frac{\bar{C}_{\rm
1c}^{||2}}{\bar{C}_{\Sigma_{\theta}}},\\\
\bar{C}_{\sigma_{2}}&=C_{\Sigma_{2}}+\frac{\bar{C}_{\rm
2c}^{||2}}{\bar{C}_{\Sigma_{\theta}}},\\\ \bar{C}_{\rm c}&=C_{\Sigma_{\rm
c}}-\left[\frac{1}{\bar{C}_{\rm 1c}+\tilde{C}_{\rm sD}}+\frac{1}{\bar{C}_{\rm
2c}+\tilde{C}_{\rm sE}}\right]^{-1}.\end{split}$ (S16)

We note here that the transformed capacitance matrix $\bar{C}$ above is
equivalent to that shown in Eq. (S10) in the sense that it results into
formally identical effective capacitance matrix $C^{\prime}$ given in Eq.
(S14), and thus, an identical effective Hamiltonian derived in the following
section. Therefore, despite the actual values of the circuit capacitances, one
can always describe the system with an effective circuit in which $C_{\rm
1c}^{\perp}=C_{\rm 2c}^{\perp}=0$. This equivalent circuit diagram of the
device is shown in Fig. S4, and also in Fig. 1(a) of the main text. Note that
in the main text $C_{12}\mathrel{\widehat{=}}\bar{C}_{12}$, $C_{\rm
1c}\mathrel{\widehat{=}}\bar{C}_{\rm 1c}$ and $C_{\rm
2c}\mathrel{\widehat{=}}\bar{C}_{\rm 2c}$.

The capacitance values of the capacitances shown in Fig. S4 are listed in
Table 1. Comparing the capacitance of $\rm{Q}_{1}$ to ground,
$\bar{C}_{1}=$82\text{\,}\mathrm{fF}$$, to the combined coupling from
$\rm{Q}_{1}$ to the coupler and $\rm{Q}_{2}$,
$\bar{C}^{||}_{1c}+\bar{C}_{12}=$7.25\text{\,}\mathrm{fF}$$, we show that
coupling four waveguide extenders to $\rm{Q}_{1}$ is achievable, since
$(\bar{C}^{||}_{1c}+\bar{C}_{12})/\bar{C}_{1}\ll 1/4$.

Figure S4: Schematic of the reduced effective circuit of the floating-coupler
setup. The capacitances are defined in Eq. (S16).

###  A.4 Hamiltonian of the setup

The Hamiltonian of the system is obtained with the Legendre transformation and
can be written as

$H=\frac{1}{2}\mathbf{q}C^{\prime-1}\mathbf{q}^{\rm T}+V,$ (S17)

where $\mathbf{q}=(q_{1},q_{\rm c},q_{2})$. Here, the canonical charges are
defined as

$q_{\lambda}=\frac{\partial L}{\partial\dot{\phi}_{\lambda}}=\frac{\partial
T}{\partial\dot{\phi}_{\lambda}}.$ (S18)

After inverting the capacitance matrix, we obtain the Hamiltonian as

$\begin{split}H=&\frac{1}{2|C^{\prime}|}\left[A_{\rm 11}q_{\rm 1}^{2}+A_{\rm
cc}q_{\rm c}^{2}+A_{\rm 22}q_{\rm 2}^{2}+2A_{\rm 1c}q_{\rm 1}q_{\rm c}+2A_{\rm
12}q_{\rm 1}q_{\rm 2}+2A_{\rm 2c}q_{\rm 2}q_{\rm c}\right]+V.\end{split}$
(S19)

Above, we have defined the inverse capacitance matrix as

$C^{\prime-1}\equiv\frac{1}{|C^{\prime}|}\left(\begin{array}[]{ccc}A_{\rm
11}&A_{\rm 1c}&A_{\rm 12}\\\ A_{\rm 1c}&A_{\rm cc}&A_{\rm 2c}\\\ A_{\rm
12}&A_{\rm 2c}&A_{\rm 22}\\\ \end{array}\right),$ (S20)

where $|\cdot|$ indicates the matrix determinant. In the following, we denote
$C_{\Sigma_{ij}}^{*}\equiv|C^{\prime}|/A_{ij}$ and $C_{\sigma_{ii}}\equiv
C_{\sigma_{i}}$ for $i,j\in\\{1,\textrm{c},2\\}$.

Here, we provide the expressions for the matrix elements of the inverse
capacitance matrix in Eq. (S20) as

$\begin{split}A_{11}=&C_{\Sigma_{2}}C_{\Sigma_{\rm c}}-C_{\rm 2c}^{2}\approx
C_{\Sigma_{2}}C_{\Sigma_{\rm c}},\\\ A_{\rm
cc}=&C_{\Sigma_{1}}C_{\Sigma_{2}}-(C_{12}^{*})^{2}\approx
C_{\Sigma_{1}}C_{\Sigma_{2}},\\\ A_{22}=&C_{\Sigma_{1}}C_{\Sigma_{\rm
c}}-C_{\rm 1c}^{2}\approx C_{\Sigma_{1}}C_{\Sigma_{\rm c}},\\\ \\\ A_{\rm
1c}=&C_{\rm 1c}C_{\Sigma_{2}}-C_{\rm 2c}C_{12}^{*}\approx C_{\rm
1c}C_{\Sigma_{2}}\\\ A_{\rm 2c}=&-C_{\rm 2c}C_{\Sigma_{1}}+C_{\rm
1c}C_{12}^{*}\approx-C_{\rm 2c}C_{\Sigma_{1}},\\\ A_{\rm
12}=&C_{12}^{*}C_{\Sigma_{\rm c}}-C_{\rm 1c}C_{\rm 2c},\end{split}$ (S21)

and the determinant of the capacitance matrix is given by

$\begin{split}|C^{\prime}|=C_{\Sigma_{1}}C_{\Sigma_{2}}C_{\Sigma_{\rm
c}}-C_{\rm 1c}^{2}C_{\Sigma_{2}}-C_{\rm
2c}^{2}C_{\Sigma_{1}}-(C_{12}^{*})^{2}C_{\Sigma_{\rm c}}+2C_{\rm 1c}C_{\rm
2c}C_{12}^{*}\approx C_{\Sigma_{1}}C_{\Sigma_{2}}C_{\Sigma_{\rm
c}}.\end{split}$ (S22)

For the approximations above, we have assumed that $C_{12}\ll C_{\rm
1c},C_{\rm 2c}\ll C_{\Sigma_{1}},C_{\Sigma_{2}},C_{\Sigma_{\rm c}}$.

We define the effective capacitances

$\begin{split}C_{\rm\Sigma_{1}}^{*}&=\frac{|C^{\prime}|}{A_{11}}\approx
C_{\Sigma_{1}},\\\ C_{\rm\Sigma_{2}}^{*}&=\frac{|C^{\prime}|}{A_{22}}\approx
C_{\Sigma_{2}},\\\ C_{\rm\Sigma_{\rm c}}^{*}&=\frac{|C^{\prime}|}{A_{\rm
cc}}\approx C_{\Sigma_{\rm c}},\\\ C_{\Sigma_{\rm
1c}}^{*}&=\frac{|C^{\prime}|}{A_{\rm
1c}}\approx\frac{C_{\Sigma_{1}}C_{\Sigma_{\rm c}}}{C_{\rm 1c}},\\\
C_{\Sigma_{\rm 2c}}^{*}&=\frac{|C^{\prime}|}{A_{\rm
2c}}\approx-\frac{C_{\Sigma_{2}}C_{\Sigma_{\rm c}}}{C_{\rm 2c}},\\\
C_{\Sigma_{12}}^{*}&=\frac{|C^{\prime}|}{A_{\rm
12}}\approx\frac{C_{\Sigma_{1}}C_{\Sigma_{2}}}{C_{12}^{*}(1-\eta)},\end{split}$
(S23)

where

$\eta=\frac{C_{\rm 1c}C_{\rm 2c}}{C_{12}^{*}C_{\Sigma_{\rm c}}}.$ (S24)

Consequently, the classical Hamiltonian of the system can be expressed as

$\begin{split}H=&\frac{q_{1}^{2}}{2C_{\Sigma_{1}}^{*}}+\frac{q_{2}^{2}}{2C_{\Sigma_{2}}^{*}}+\frac{q_{\rm
c}^{2}}{2C_{\Sigma_{\rm c}}^{*}}+\frac{q_{1}q_{\rm c}}{C_{\Sigma_{\rm
1c}}^{*}}+\frac{q_{\rm 2}q_{\rm c}}{C_{\Sigma_{\rm 2c}}^{*}}+\frac{q_{\rm
1}q_{\rm 2}}{C_{\Sigma_{12}}^{*}}\\\ &+E_{\rm
J_{1}}\left(1-\cos\left[\frac{2\pi}{\Phi_{0}}\phi_{1}\right]\right)+E_{\rm
J_{2}}\left(1-\cos\left[\frac{2\pi}{\Phi_{0}}\phi_{2}\right]\right)+E_{\rm
J_{\rm c}}\left(1-\cos\left[\frac{2\pi}{\Phi_{0}}\phi_{\rm
c}\right]\right),\end{split}$ (S25)

We follow the canonical quantization procedure and replace the flux
coordinates $(\phi_{1},\phi_{\rm c},\phi_{2})$ and the corresponding
canonically conjugated charges $(q_{1},q_{\rm c},q_{2})$ with operators as
$\phi_{\lambda}\rightarrow\hat{\phi}_{\lambda}$ and
$q_{\lambda}\rightarrow\hat{q}_{\lambda}$ where $\lambda\in[1,\textrm{c},2]$.
The operators obey the commutation relation
$[\hat{\phi}_{\lambda},\hat{q}_{\lambda}]=\textrm{i}\hbar$. Here, we define
the charging energies as $E_{\rm
C_{\lambda}}=e^{2}/(2C_{\Sigma_{\lambda}}^{*})$. In the limit $E_{\rm
C_{\lambda}}\ll E_{\rm J_{\lambda}}$, the setup is equivalent to three
bilinearly coupled transmons. Since the transmon is a weakly anharmonic
oscillator, it is beneficial to express the flux and charge operators in terms
of the annihilation ($\hat{b}_{\lambda}$) and creation
($\hat{b}_{\lambda}^{{\dagger}}$) operators of harmonic oscillators
corresponding to the linearized potential of the circuit, defined as

$\begin{split}\hat{\phi}_{\lambda}=&\frac{\Phi_{0}}{2\pi}\left(\frac{8E_{\rm
C_{\lambda}}}{E_{\rm
J_{\lambda}}}\right)^{1/4}\frac{1}{\sqrt{2}}(\hat{b}^{{\dagger}}_{\lambda}+\hat{b}_{\lambda}),\\\
\hat{q}_{\lambda}=&\textrm{i}2e\left(\frac{E_{\rm J_{\lambda}}}{8E_{\rm
C_{\lambda}}}\right)^{1/4}\frac{1}{\sqrt{2}}(\hat{b}^{{\dagger}}_{\lambda}-\hat{b}_{\lambda}).\end{split}$
(S26)

Consequently, assuming that $E_{\rm J_{\lambda}}\gg E_{\rm C_{\lambda}}$, the
Hamiltonian operator of the system can be written as

$\hat{H}=\hbar\sum_{\lambda}\left(\omega_{\lambda}\hat{b}_{\lambda}^{{\dagger}}\hat{b}_{\lambda}+\frac{\alpha_{\lambda}}{2}\hat{b}_{\lambda}^{{\dagger}}\hat{b}_{\lambda}^{{\dagger}}\hat{b}_{\lambda}\hat{b}_{\lambda}\right)-\hbar
g_{\rm 1c}(\hat{b}_{1}^{{\dagger}}-\hat{b}_{1})(\hat{b}_{\rm
c}^{{\dagger}}-b_{\rm c})+\hbar g_{\rm
2c}(\hat{b}_{2}^{{\dagger}}-\hat{b}_{2})(\hat{b}_{\rm
c}^{{\dagger}}-\hat{b}_{\rm c})-\hbar
g_{12}(\hat{b}_{1}^{{\dagger}}-\hat{b}_{1})(\hat{b}_{2}^{{\dagger}}-\hat{b}_{2})$
(S27)

where we have defined the qubit angular frequencies, anharmonicities, and the
relevant coupling rates as

$\begin{split}\hbar\omega_{\lambda}=&\sqrt{8E_{\rm
J_{\lambda}}E_{C_{\lambda}}}-E_{C_{\lambda}},\\\
\hbar\alpha_{\lambda}=&-E_{\rm C_{\lambda}},\\\ \hbar g_{\rm
1c}=&\frac{e^{2}}{\sqrt{2}C_{\Sigma_{\rm 1c}}^{*}}\left(\frac{E_{\rm
J_{1}}E_{\rm J_{\rm c}}}{E_{\rm C_{1}}E_{\rm C_{\rm
c}}}\right)^{1/4}\approx\frac{\hbar}{2}\frac{C_{\rm
1c}}{\sqrt{C_{\Sigma_{1}}C_{\Sigma_{\rm c}}}}\sqrt{\omega_{1}\omega_{\rm
c}},\\\ \hbar g_{\rm 2c}=&-\frac{e^{2}}{\sqrt{2}C_{\Sigma_{\rm
2c}}^{*}}\left(\frac{E_{\rm J_{2}}E_{\rm J_{\rm c}}}{E_{\rm C_{2}}E_{\rm
C_{\rm c}}}\right)^{1/4}\approx\frac{\hbar}{2}\frac{C_{\rm
2c}}{\sqrt{C_{\Sigma_{2}}C_{\Sigma_{\rm c}}}}\sqrt{\omega_{2}\omega_{\rm
c}},\\\ \hbar
g_{12}=&\frac{e^{2}}{\sqrt{2}C_{\Sigma_{12}}^{*}}\left(\frac{E_{\rm
J_{1}}E_{\rm J_{2}}}{E_{\rm C_{1}}E_{\rm
C_{2}}}\right)^{1/4}\approx\frac{\hbar}{2}\frac{C_{12}^{*}(1-\eta)}{\sqrt{C_{\Sigma_{1}}C_{\Sigma_{\rm
2}}}}\sqrt{\omega_{1}\omega_{2}},\end{split}$ (S28)

respectively. Above, we have again assumed that $C_{12}^{*}\ll C_{\rm
1c},C_{\rm 2c}\ll C_{\Sigma_{1}},C_{\Sigma_{2}},C_{\Sigma_{\rm c}}$ and also
neglected the anharmonicities in the latter equalities. For later use, we
define the transmon-frequency independent ratio between the coupling rates as

$\xi=\frac{2g_{\rm 1c}g_{\rm 2c}}{g_{12}\omega_{\rm c}}=\frac{\eta}{1-\eta}.$
(S29)

We emphasize that the Hamiltonian in Eq. (S27) is similar to that for the
conventional tunable-coupler design derived in Ref. Yan  _et al._ (2018a).
However, the sign of the coupling term between $\rm{Q}_{2}$ and the coupler is
different, and consequently, the idling frequency of the coupler is located
below the qubit frequencies, as we show below.

We have verified the above analytical results independently using a quasi-
lumped-element model based on the full circuit schematic shown in Fig. S1. The
coupler capacitance matrix has been solved using FEM simulations and the full
network model has been synthesized from those components. This way we were
able to reproduce the theoretical result.

###  A.5 Idling frequency of the coupler

Similar to the conventional design in Ref. Yan  _et al._ (2018a), the
Hamiltonian in Eq. (S27) can be approximately diagonalized up to the second
order in the qubit-coupler couplings $g_{\rm 1c}$ and $g_{\rm 2c}$ using the
Schrieffer–Wolff transformation. Consequently, the effective transverse
coupling strength between the single-excitation qubit states can be expressed
as

$\tilde{g}=g_{12}-\frac{g_{\rm 1c}g_{\rm 2c}}{2}\left(\frac{1}{\Delta_{\rm
1c}}+\frac{1}{\Delta_{\rm 2c}}-\frac{1}{\Sigma_{\rm 1c}}-\frac{1}{\Sigma_{\rm
2c}}\right),$ (S30)

where $\Delta_{i\textrm{c}}=\omega_{i}-\omega_{\rm c}$ and
$\Sigma_{i\textrm{c}}=\omega_{i}+\omega_{\rm c}$ with $i\in\\{1,2\\}$. We note
that the effective coupling strength $\tilde{g}$ between the coupler-dressed
qubits consists of contributions arising from a direct capacitive interaction
and an indirect interaction mediated by the coupler. Moreover, the sign of the
second term is different compared to that of the conventional design. As a
consequence, the coupler frequency at which $\tilde{g}=0$ is located below
both qubit frequencies. For example, assuming the qubits are in resonance, we
obtain $\tilde{g}=0$ with the coupler frequency

$\omega_{\rm c}^{\rm off}=\frac{\omega_{1}}{\sqrt{1+\xi}}<\omega_{1}.$ (S31)

We emphasize that $\tilde{g}$ gives the coupling strength between the
hybridized qubit-coupler states which approximate the eigenstates of the setup
accurately only in the limit of $g_{12}=0$. Furthermore, biasing the coupler
at the frequency $\omega_{\rm c}=\omega_{\rm c}^{\rm off}$, does not guarantee
that the diagonal, Kerr-type couplings are also zero. Especially, the residual
$ZZ$ coupling is, in general, not zero at $\omega_{\rm c}=\omega_{\rm c}^{\rm
off}$ which is therefore not an optimal choice as the idling frequency.

The computational states of the effective two-qubit system are defined as
eigenstates, or dressed states, of the coupled system consisting of the two
transmons and the tunable floating coupler. In practice, the computational
two-qubit states $|00\rangle$, $|10\rangle$, $|01\rangle$, and $|11\rangle$
are the dressed states that have the largest overlap with the bare-basis
states $|\rm ggg\rangle$, $|\rm egg\rangle$, $|\rm gge\rangle$, and $|\rm
ege\rangle$ of the non-interacting part of the Hamiltonian (labeling defined
as $|\rm Q1,C,Q2\rangle$), respectively. This provides a unique definition of
the computational basis in the dispersive regime.

The computational states collect conditional phase through the interactions
between the bare-basis states in the two-excitation manifold $\\{|\rm
eeg\rangle,|ege\rangle,|gee\rangle,|fgg\rangle,|gfg\rangle,|ggf\rangle\\}$.
The conditional-phase rate, i.e., the effective $ZZ$ interaction strength, is
defined as

$\zeta=(\omega_{11}-\omega_{10})-(\omega_{01}-\omega_{00}),$ (S32)

where $\omega_{ij}/(2\pi)$ are the eigenfrequencies of the computational
states. Similar to the effective coupling $\tilde{g}$ in the single-excitation
manifold, the conditional-phase rate consists of competing contributions
arising from direct and indirect interactions between two-excitation states
with different signs. At suitable values for qubit and coupler frequencies,
these contributions can cancel each other out, resulting in a negligible
effective $ZZ$ interaction rate. We define the idling frequency as the coupler
frequency $\omega_{\rm c}=\omega_{\rm c}^{\rm idle}$ for which the effective
$ZZ$ interaction is minimized.

In the dispersive regime, one can compute the conditional-phase rate $\zeta$
analytically using the Schrieffer–Wolff approach, similar to the case of the
effective transverse coupling frequency. However, here one needs to do the
Schrieffer–Wolff expansion at least to the fourth order in the coupling
strengths in order to obtain accurate results. This has been carried out in
Refs. Sung  _et al._ (2021) and Chu and Yan (2021). In our case, however, we
often operate in the non-dispersive parameter regime, particularly, close to
the resonance between the computational state $|11\rangle$ and non-
computational state $|02\rangle$. Consequently, we compute the conditional-
phase rate by numerically solving the eigenproblem for the Hamiltonian in Eq.
(S27) using $N=3$ states for each transmon in the system.

We show the numerically obtained effective $ZZ$ interaction strength in Fig.
2(a) of the main text. Similar to the analytic expression Sung  _et al._
(2021); Chu and Yan (2021), the data show that for the qubit-qubit detunings
$\Delta=\omega_{1}-\omega_{2}\in[\alpha_{2},-\alpha_{1}]$, the system has two
coupler frequencies for which $\zeta=0$. In the plane spanned by the coupler
frequency and the detuning, the zero-coupling condition $\zeta=0$ forms a
characteristic oval-shaped contour, each point of which can be used as the
idling configuration for two-qubit gates.

##  APPENDIX B SIMULATION ON COUPLING STRENGTH FOR DIFFERENT QUBIT-QUBIT
DISTANCES

Fig. S1
---
Capacitance | Value (fF)
$C_{\rm AB}$ | 93
$C_{\rm BC}$ | 12
$C_{\rm CD}$ | 189
$C_{\rm CF}$ | 41
$C_{\rm CE}$ | 2
$C_{\rm sC}$ | 50
$C_{\rm DE}$ | 6
$C_{\rm DF}$ | 5
$C_{\rm sD}$ | 62
$C_{\rm EF}$ | 192
$C_{\rm sE}$ | 62
$C_{\rm FG}$ | 12
$C_{\rm sF}$ | 47
$C_{\rm GH}$ | 93

Fig. S3
---
Capacitance | Value (fF)
$C_{1}$ | 82
$C_{2}$ | 82
$C_{\rm c}$ | 28
$C_{\rm 1c}^{||}$ | 8
$C_{\rm 1c}^{\perp}$ | 1
$C_{\rm 2c}^{||}$ | 8
$C_{\rm 2c}^{\perp}$ | 1
$C_{12}$ | 0.07
$\tilde{C}_{\rm sD}$ | 100
$\tilde{C}_{\rm sE}$ | 98

Fig. S4
---
Capacitance | Value (fF)
$\bar{C}_{1}$ | 82
$\bar{C}_{2}$ | 82
$\bar{C}_{\rm c}$ | 29
$\bar{C}_{\rm 1c}^{||}$ | 7
$\bar{C}_{\rm 2c}^{||}$ | 7
$\bar{C}_{12}$ | 0.25
Eq. (S14)
Capacitance | Value (fF)
$C_{\Sigma_{1}}$ | 91
$C_{\Sigma_{2}}$ | 91
$C_{\Sigma_{\rm c}}$ | 82
$C_{\rm 1c}$ | 3
$C_{\rm 2c}$ | 3
$C_{\rm 12}^{*}$ | 0.5

Josephson energy | Value (GHz)
---|---
$E_{\rm J_{1}}$ | 12.1
$E_{\rm J_{2}}$ | 13.2
$E_{\rm J_{c}}$ | 11.8

Table 1: Lumped-element capacitance values for models defined in Figs. S1, S3,
and S4. We also show the elements of the effective capacitance matrix in Eq.
(S14), and the designed values for Josephson energies used in the models.

The sample is fully designed with KQCircuits Heinsoo  _et al._ (2021), our
open source Python package for superconducting circuit design, from which we
can export netlists. We model the two-qubit system as a quasi-lumped-element
network model using scikit-rf Arsenovic  _et al._ (2022). In this model, the
qubits and the coupler are represented by lumped-element circuits, whereas the
waveguide extenders are modeled as transmission lines with analytical
expressions for the effective dielectric constant and the characteristic
impedance. We fix the capacitances of the system as shown in Table 1 and use
Ansys Q3D Extractor to find the shapes of the qubits, the coupler, and the
waveguide extender capacitor pads which realize the required capacitance
values.

To study the coupling strength as a function of the qubit spacing, as given in
Fig. 1(b) of the main text, we change the length of the waveguides in the
quasi-lumped-element model, while keeping the geometry of the finger and gap
capacitors the same. We then reduce the network to the one given in Fig. S4
and calculate the effective capacitances from the admittance matrix of the
network at $5\text{\,}\mathrm{GHz}$. Lastly, we use Eq. (S28) to calculate the
coupling strengths shown in Fig. 1(b) in the main text. Here, we use the
Josephson energies shown in Table 1 to obtain the frequencies
$\omega_{1}/(2\pi)=$4.10\text{\,}\mathrm{GHz}$$,
$\omega_{2}/(2\pi)=$3.89\text{\,}\mathrm{GHz}$$, and $\omega_{\rm
c}/(2\pi)=$3.195\text{\,}\mathrm{GHz}$$ for $\rm{Q}_{1}$, $\rm{Q}_{2}$, and
the coupler, respectively.

##  APPENDIX C SIMULATION ON CROSSTALK OF PASSING TRANSMISSION LINE

In this section, we will elaborate on the details of the simulation results,
where we simulate the crosstalk between a transmission line running
$8\text{\,}\mathrm{\SIUnitSymbolMicro m}$ above the qubit-coupler-qubit
structure and each component, as shown in Fig. 1(d) of the main text. In this
simulation, a $3.5\text{\,}\mathrm{mm}$-long transmission line on the top chip
crosses the qubit-coupler-qubit line on the bottom chip almost
perpendicularly. The line is long compared to our qubit size of
$0.5\text{\,}\mathrm{mm}$ to account for all of the extra capacitance. We
simulate the sample with finite element simulation, where the other
capacitances are listed in Table 1. We place five ports on the simulated
sample: one at each qubit ($\rm{Q}_{1}$ and $\rm{Q}_{2}$), one at each island
of the coupler ($\rm{c}_{C}$ and $\rm{c}_{F}$) and one at the end of the
transmission line. We extract the capacitance to the crossing transmission
line from the obtained admittance matrix at $4.3\text{\,}\mathrm{GHz}$. This
frequency is chosen to represent the worst case, as it is at an identical
frequency to the designed qubit frequencies. Since the coupler is floating, we
are interested in the differential capacitance between the two coupler islands
$C_{\rm{TC}\leftrightarrow\rm{TL}}=|C_{\rm{c_{C}}\leftrightarrow\rm{TL}}-C_{\rm{c_{F}}\leftrightarrow\rm{TL}}|$.
As a consequence, there is a crossing position of the transmission line, where
$C_{\rm{c_{C}}\leftrightarrow\rm{TL}}=C_{\rm{c_{F}}\leftrightarrow\rm{TL}}$.
Since our coupler structure is symmetric around its center, a dip in the
coupling ratio $r_{\rm c}$ of the coupler at $x_{\rm cross}=0$ appears in Fig.
1(d) in the main text.

To confirm that the excitation of $4.3\text{\,}\mathrm{GHz}$ on the
transmission line does not excite any particular mode, we repeat the
simulations for different frequencies in the range between
$1\text{\,}\mathrm{GHz}$ to $10\text{\,}\mathrm{GHz}$ and find no quantitative
deviations from the simulations with a $4.3\text{\,}\mathrm{GHz}$ excitation.

##  APPENDIX D EXPERIMENTAL SETUP

The used experimental setup including a schematic of the sample is shown in
Fig. S5. Both qubits used in the experiment are flux-tunable transmon qubits.
Single-qubit gates are implemented using capacitively coupled drive lines
attenuated by a total of $60\text{\,}\mathrm{dB}$ throughout the different
temperature stages. Each of the $20\text{\,}\mathrm{dB}$ attenuators are
thermalized at the corresponding temperature stages, as shown in Fig. S5. The
pulse envelopes for the qubit drives are generated by a Zurich Instrument
arbitrary-waveform generator HDAWG and the carrier for the in-phase and
quadrature (IQ) mixing to the qubit frequency via the Analog Devices mixer
HMC8193 is provided by a Rohde&Schwarz microwave generator SGS100A. To control
the frequency of each qubit, a voltage source at room temperature is connected
to a twisted-pair cable which reaches down to the $10\text{\,}\mathrm{mK}$
stage. One conductor of the twisted pair is grounded near the sample and the
other conductor leads through the sample as an on-chip flux line and is then
grounded on the sample near the SQUID loop. To filter the noise from the
voltage source and the noise picked up by the twisted pair, we employ a
$25\text{\,}\mathrm{kHz}$ second-order differential low-pass filter
thermalized at the $4\text{\,}\mathrm{K}$ stage.

|  | Measured State
---|---|---
|  |  $\ket{\rm 00}$ |  $\ket{\rm 01}$ |  $\ket{\rm 10}$ |  $\ket{\rm 11}$
Prepared state | $\ket{\rm 00}$ |  0.902 |  0.055 |  0.040 | 0.003
$\ket{\rm 01}$ |  0.052 |  0.905 |  0.003 | 0.040
$\ket{\rm 10}$ |  0.065 |  0.004 |  0.880 | 0.050
$\ket{\rm 11}$ |  0.004 |  0.061 |  0.110 | 0.824
Table 2: Readout correlation matrix for the two-qubit system which is prepared
$2\times 10^{4}$ times in each state in the computational basis $\\{\ket{\rm
00},\ket{\rm 01},\ket{\rm 10},\ket{\rm 11}\\}$. The single-shot measurement
probabilities are then obtained for each of the four computational states
using previously determined discrimination lines in the in-phase–quadrature-
phase plane of the measurement signal of each qubit.

Each qubit has its own readout structure which consists of a readout resonator
and a Purcell filter Sete  _et al._ (2015); Heinsoo  _et al._ (2018). Both
readout structures are coupled to a single transmission line. The signal
traveling to the output line is amplified with a low-noise high-electron-
mobility transistor (HEMT) at the $4\text{\,}\mathrm{K}$ stage. Despite not
having a quantum-limited amplifier in the setup, readout fidelities around
$88\text{\,}\mathrm{\char 37\relax}$ are reached due to strongly coupled
readout resonators designed for high fidelity readout, see Table 2 for
details. The probe pulse at the intermediate frequency is generated and the
response is digitized using a Zurich Instrument Quantum Analyzer UHFQA. For up
and down conversion to the readout frequency, a single Rohde&Schwarz microwave
generator is used as a local oscillator.

The tunable-coupler structure includes a flux-tunable floating transmon qubit.
Since the coupler state does not necessarily need to be read out, no separate
readout structure was designed for the coupler. However, by swapping the
coupler state to the qubit or by using coupler-state-dependent dispersive
shift of a qubit, it is still possible to read out the state of the coupler
for characterization experiments, see Appendix F. On this device, the tunable
coupler is the only component which needs fast flux control. For improved
coupler coherence, a $1\text{\,}\mathrm{GHz}$ low-pass filter is installed to
the flux line at the $10\text{\,}\mathrm{mK}$ stage of the cryostat. The
direct current (DC) bias of the coupler flux line and the fast pulses are
generated by a single channel of a Zurich Instrument arbitrary-waveform
generator.

Figure S5: The diagram of the used experimental setup. The background colors
depict the different temperature stages of the dilution refrigerator. The
coaxial microwave cables (straight black lines) and twisted-pair cables
(braided lines) are used to control and read out the qubit and coupler states.
The schematic of the sample in the white box consists of two flux-tunable
qubits (blue and orange), a readout structure for each qubit (light blue and
light orange), and a flux-tunable coupler (red).

The summary of the device parameters is provided in Table 3. Here, the
frequencies and the coherence times are given both at the flux insensitive
point (sweet spot) and at the two-qubit-gate idling configuration. The two-
qubit-gate idling point is chosen such that the frequency of the $\ket{11}$
state is close to the $\ket{02}$ state to enable fast CZ gates. However, since
we chose to not have fast flux control for the qubits, the smaller the
detuning between the $\ket{11}$ and $\ket{02}$ states, the larger the effect
of drive crosstalk. In particular, when driving the
$\ket{0}\leftrightarrow\ket{1}$ transition of $\rm{Q}_{1}$, we also spuriously
drive the $\ket{1}\leftrightarrow\ket{2}$ transition of $\rm{Q}_{2}$.
Therefore, we tune the qubit frequencies to the values shown in Table 3 to
have an ideal trade-off between fast two-qubit gates and weak drive crosstalk.

Quantity, symbol (unit) |  Qubit 1 |  Qubit 2 |  Coupler
---|---|---|---
Readout resonator frequency, $\omega_{\mathrm{R}}/2\pi$ (GHz) |  4.950 |  6.134 |  -
Readout Purcell filter frequency, $\omega_{\mathrm{Rpf}}/2\pi$ (GHz) |  5.037 |  6.125 |  -
Effective readout resonator bandwidth, $\kappa_{\mathrm{eff}}/2\pi$ (MHz) |  13 |  22 |  -
Readout circuit dispersive shift, $\chi_{\mathrm{R}}/2\pi$ (MHz) |  5.3 |  4.7 |  -
Qubit frequency at sweet spot, $\omega_{\mathrm{ge}}/2\pi$ (GHz) |  4.102 |  3.972 |  4.210
Qubit frequency at 2QG idling point, $\omega_{\mathrm{ge}}/2\pi$ (GHz) |  4.102 |  3.892 |  3.195
Transmon anharmonicity, $\alpha$ (MHz) |  $-215$ |  $-217$ |  $-250$
Energy relaxation time at sweet spot, $T_{1}$ (µs) |  13 |  39 |  30
Energy relaxation time at 2QG idling point, $T_{1}$ (µs) |  13 |  42 |  50
Transverse relaxation time at Sweet spot, $T_{2}$ (µs) |  14 |  14 |  1.4
Transverse relaxation time at 2QG idling point, $T_{2}$ (µs) |  14 |  8 |  0.35
Table 3: Summary of device parameters. Parameters at the flux insensitive
point (sweet spot) and at the two-qubit gate (2QG) idling configurations are
listed.

##  APPENDIX E DEVICE FABRICATION

The sample used in the experiments is fabricated at the OtaNano Micronova
cleanroom. First, a high-purity $200\text{\,}\mathrm{nm}$-thick niobium layer
is deposited on a high-resistivity
($\rho>$10\text{\,}\mathrm{k\SIUnitSymbolOhm}\text{\,}\mathrm{cm}$$) non-
oxidized $n$-type undoped (100) six-inch silicon wafer by sputtering. The
coplanar waveguides and capacitive structures are then formed by
photolithography with subsequent reactive ion etching. After etching, the
photoresist residuals are cleaned in an ultrasonic bath with acetone and
isopropanol. Next, electron beam lithography is used with subsequent electron
beam shadow evaporation and lift-off processes to form the Josephson junctions
of the qubits. The qubit junctions are formed by two
$20\text{\,}\mathrm{nm}$-thick aluminum layers. Before the evaporation of
these structures, natural oxides are removed from the surface by argon ion
milling. Finally, $800\text{\,}\mathrm{nm}$-thick aluminum airbridges are
fabricated by sputtering and subsequent lift off. After the device has been
fabricated, we measure the room temperature resistance of the qubit junctions,
producing the scratches on the qubit islands seen in Fig. 1(b) in the main
text.

##  APPENDIX F COUPLER CHARACTERIZATION

###  F.1 Three-tone spectroscopy

In the absence of a dedicated readout circuit for the coupler, we use three-
tone spectroscopy with $\rm{Q}_{2}$ as an ancilla qubit to find the coupler
frequency Li  _et al._ (2020). We begin by applying a weak probe tone to
$\rm{Q}_{2}$ at its transition frequency (see the experimental setup in Fig.
S5 and the qubit properties in Table 3) and monitor its state by sending a
continuous wave to its readout resonator. At the same time, we strongly drive
the coupler through the drive line of $\rm{Q}_{1}$. When the coupler drive
tone is in resonance with the coupler frequency, the coupler gets partially
excited, and $\rm{Q}_{2}$ frequency shifts down due to dispersive interaction
between the qubit and the coupler. As a result, the weak probe tone driving
$\rm{Q}_{2}$ no longer excites $\rm{Q}_{2}$, leading to a change in the
readout signal of $\rm{Q}_{2}$. This effectively maps the coupler state to the
readout signal of $\rm{Q}_{2}$.

To illustrate the effect of the qubit probe frequency and the coupler drive
frequency, we sweep both of the parameters and plot the readout voltage, see
Fig. S6(a). When the coupler drive frequency is far away from the coupler
transition frequency of $\omega_{\rm c}/(2\pi)=$3.192\text{\,}\mathrm{GHz}$$,
the experiment resembles a normal qubit spectroscopy experiment with the qubit
frequency at $4.1016\text{\,}\mathrm{GHz}$. As the coupler drive frequency
gets close to the coupler transition frequency, the frequency of $\rm{Q}_{2}$
decreases down to $4.100\text{\,}\mathrm{GHz}$ due to dispersive shift. By
fixing the probe tone at the frequency of $\rm{Q}_{2}$ and sweeping the
coupler drive frequency, we implement the coupler frequency spectroscopy
experiment, see Fig. S6(b). In Fig. 2(c) of the main text, this method is used
for measuring coupler frequencies below $3.8\text{\,}\mathrm{GHz}$, where the
coupling between the hybridized coupler and qubit mode to the readout
resonators is too weak for standard dispersive readout.

Although this method works with either $\rm{Q}_{1}$ or $\rm{Q}_{2}$ as the
ancilla, the transition frequency difference between
$\ket{000}\leftrightarrow\ket{001}$ and $\ket{010}\leftrightarrow\ket{011}$
with $\rm{Q}_{2}$ as an ancilla is larger than with $\rm{Q}_{1}$ as an
ancilla.

Figure S6: Results from a three-tone spectroscopy experiment. (a) Qubit
spectroscopy with an additional frequency sweep of the coupler drive. The
horizontal black dashed line indicates the one-dimensional sweep for a three-
tone experiment shown in (b). The coupler frequency $\omega_{\rm c}/(2\pi)$ is
extracted by a minimum search of the one-dimensional sweep (vertical black
dashed line).

###  F.2 Coupler coherence and flux noise

Above, we discussed the three-tone spectroscopy measurement in the context of
continuous waves and observed that it allows us to read out the coupler state.
Here, we use two-probe readout in standard time-dependent experiments to
characterize the coupler coherence as a function of the coupler frequency, see
Fig. S7(a) for the results. Instead of a continuous drive, the ancilla qubit
and the readout resonator are now pulsed consecutively. The length of the
$\pi$-pulse on the ancilla qubit is chosen to have the spectral width narrower
than the coupler state-dependent frequency shift. The dephasing rates for the
echo and Ramsey experiments are calculated from the fit to Eq. (1) in the main
text, where $T_{1}$ is acquired from the energy relaxation measurements of the
coupler.

We can utilize the extracted data for $T_{\phi,2}^{\rm echo}$ to characterize
the $1/f$ flux noise of the coupler Ithier  _et al._ (2005); Yoshihara  _et
al._ (2006); Braumüller  _et al._ (2020); Bylander  _et al._ (2011). In the
presence of $1/f$ flux noise, the pure dephasing rate is

$\displaystyle\Gamma_{\phi,2}^{\rm echo}=1/T^{\rm echo}_{\phi,2}=\sqrt{A\ln
2}\left|\frac{\partial\omega}{\partial\Phi}\right|,$ (S33)

where $\left|\partial\omega/\partial\Phi\right|$ is the slope of the coupler
flux dispersion curve and $\sqrt{A}$ is the flux noise amplitude. The power
spectral density can then be extracted as $S(\omega)=A/|\omega|$, assuming
$1/f$ flux noise Ithier  _et al._ (2005); Yoshihara  _et al._ (2006); Bylander
_et al._ (2011); Braumüller  _et al._ (2020). In Fig. S7(b), the pure
dephasing rate $\Gamma_{\phi,2}^{\rm echo}$ is shown as a function of the flux
dispersion slope $\left|\partial
f/\partial\Phi\right|=(2\pi)^{-1}\left|\partial\omega/\partial\Phi\right|$.
Each flux dispersion slope point is extracted by fitting a cubic polynomial to
the measured flux dependency of the coupler frequency and then extracting the
derivative. We fit Eq. (S33) together with an offset parameter to the data,
resulting in a flux noise amplitude of $\sqrt{A}=$21.4\pm
0.2\text{\,}\mathrm{\SIUnitSymbolMicro}\Phi_{\mathrm{0}}$$. From the fit, we
extract an offset of $-0.132\pm
0.006\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}^{-1}$, where the deviation
from zero is possibly caused by other high-frequency dephasing processes
Braumüller  _et al._ (2020).

Figure S7: (a) Experimentally obtained relaxation and coherence times of the
coupler as functions of its frequency. (b) Pure dephasing rate from echo
measurements as a function of the flux dispersion slope. A line fit yields the
flux noise amplitude $\sqrt{A}$.

We compare the extracted flux noise amplitude to the theoretically expected
value given the geometry of the coupler SQUID loop following the analysis of
Ref. Braumüller  _et al._ (2020). From the geometry, we estimate
$\sqrt{A}=$7.2\text{\,}\mathrm{\SIUnitSymbolMicro}\Phi_{\mathrm{0}}$$, which
is significantly smaller than the measured value. The source of the excess
flux noise unexplained by the SQUID geometry is unknown, but might be
originating from the flux noise of the voltage source. Furthermore, we can
compare the coupler flux noise amplitude to the measured qubit flux noise
amplitudes which are $4.12\pm
0.04\text{\,}\mathrm{\SIUnitSymbolMicro}\Phi_{\mathrm{0}}$ and $4.29\pm
0.05\text{\,}\mathrm{\SIUnitSymbolMicro}\Phi_{\mathrm{0}}$ for $\rm Q_{1}$ and
$\rm Q_{2}$, respectively. These values correspond well to the expected value
from the SQUID geometry alone
($4.9\text{\,}\mathrm{\SIUnitSymbolMicro}\Phi_{\mathrm{0}}$). Despite the
coupler suffering from excess flux noise, the dephasing time of the coupler is
not expected to limit our gate fidelity, as we have shown in Fig. 4(d) in the
main text.

##  APPENDIX G RANDOMIZED BENCHMARKING WITH MULTIPLE INTERLEAVED CZ GATES

To further illustrate the results obtained in Fig. 3(b), we show in Fig. S8
the sequence fidelity of the randomized benchmarking for each number $n$ of
interleaved CZ gates. We fit the sequence fidelity of the reference trace
$\mathcal{F}_{\rm ref}$ with the exponential model $\mathcal{F}_{\rm
ref}=A{p_{\rm ref}}^{m}+B$, where $p_{\rm ref}$ is the sequence decay, $m$ is
the number of Clifford gates, and $A$ and $B$ are parameters to capture the
state preparation and measurement errors Barends  _et al._ (2014). The error
per Clifford is given by $\epsilon_{\rm Clifford}=(1-p_{\rm ref})(d-1)/d$,
where $d=2^{N_{\rm q}}=4$ is the dimensionality of the hilbert space for
$N_{\rm q}$ qubits. This model can be applied to fit the sequence fidelity
$\mathcal{F}_{n\rm CZ}$ for $n$ interleaved CZ gates and extract the sequence
decay $p_{n\rm CZ}$. The errors per $n$ CZ gates can then be calculated as
$\epsilon_{n\rm CZ}=(1-p_{n\rm CZ}/p_{\rm ref})(d-1)/d$, where the
experimental results are shown in the legend of Fig. S8. The error per CZ gate
$\epsilon_{\rm CZ}^{(n)}$ can then be calculated as $\epsilon_{\rm
CZ}^{(n)}=1-\sqrt[n]{1-\epsilon_{n\rm CZ}}$ to reach the results shown in Fig.
3(b) of the main text. For small $\epsilon_{n\rm CZ}$, this expression reduces
to $\epsilon_{\rm CZ}^{(n)}\approx\epsilon_{n\rm CZ}/n$. Here we assume that
each of the $n$ CZ gates contributes with an equal amount to the total error
$\epsilon_{n\rm CZ}$. This assumption becomes invalid as soon as the
distortion of one flux pulse affects one or more of its subsequent flux
pulses. Therefore, it is important to compensate for the flux pulse
distortions in order for this assumption to hold.

Figure S8: Experimentally obtained sequence fidelity as a function of the
number of Clifford gates in the randomized benchmarking sequence with $n$
interleaved CZ gates, where $n$ ranges from 1 to 20. Each data point (symbols)
represents the average result of 30 random sequences. Each sequence fidelity
trace is fitted using an exponential model (solid lines) to extract the error
$\epsilon_{n\rm CZ}$ per $n$ CZ gates.

##  APPENDIX H ESTIMATING COHERENCE LIMIT OF AVERAGE GATE ERRORS

We estimate the coherence limit of the gate error measured with randomized
benchmarking (RB) using the theoretical results derived in Refs. Chu and Yan
(2021); Abad  _et al._ (2021). We start by writing the coherence limit for two
isolated qubits, and later generalize the result for the qubit-coupler-qubit
system using the effective coherence time method presented in Refs. Xu  _et
al._ (2020); Chu and Yan (2021).

###  H.1 Coherence limit for CZ gate

For simplicity, we assume that the noise sources acting on the qubits are
independent, which is mostly accurate for the dominating noise sources in our
system: longitudinal noise and flux noise from the oxygen impurities in the
SQUID loops Braumüller  _et al._ (2020). We also assume that the environment
of the qubits is cold, which allows us to neglect the excitation rate of the
qubits. Then, in the limit of small errors and Markovian noise, the error rate
from energy-relaxation can be written as Abad  _et al._ (2021)

$\epsilon_{T_{1}}=\sum_{i}^{N_{\rm q}}\frac{N_{\rm
q}d}{2(d+1)}\frac{\tau}{T_{1}^{{\rm Q}_{i}}},$ (S34)

where $d=2^{N_{\rm q}}$ is the dimensionality of the Hilbert space for $N_{\rm
q}$ qubits.

As opposed to energy relaxation, which is typically Markovian, the impact of
phase noise significantly depends on the temporal correlations in the phase
fluctuations, often invalidating the Markov approximation. Nevertheless, the
amplitude of phase noise is typically assumed to be Gaussian distributed, with
a notable counter-example being if the qubit is operated at or near its flux
sweet spot Sung  _et al._ (2019). However, the non-Gaussian component is often
relatively small compared to the dominating noise sources, and thus it can be
typically neglected for practical applications. A broad review of the impact
of phase noise on the decay of qubit coherence can be found in Ref. Ithier
_et al._ (2005). For the sake of simplicity, we only discuss two extreme cases
here, quasi-static noise and white noise, and divide all the noise sources
into either one of these two categories.

White noise has a uniform noise spectral density over a wide band of
frequencies and, therefore, does not have any temporal correlations, enabling
Markov approximation. This results in exponential decay in the off-diagonal
elements in the qubit density matrix, characterized by the decay envelope

$\chi(t)=\mathrm{e}^{-\frac{t}{T_{\phi,1}}},$ (S35)

similar to the result obtained with the Lindblad master equation.

Quasi-static noise arises from any noise source whose noise power originates
from frequencies below the smallest relevant rates in the system, typically
the repetition rate of the experiment. On the other hand, noise at frequencies
below the calibration rate of the system can be considered static and fully
eliminated by the calibration. Quasi-static noise therefore covers all the
noise power roughly between the repetition rate of the experiment and the
calibration rate of the experiment, $P_{\rm q-s}=\int_{1/t_{\rm
repetition}}^{1/t_{\rm calibration}}S_{\rm ff}(f)\mathrm{d}f$, where $S_{\rm
ff}(f)$ is the single-sided noise power spectral density of the qubit
frequency fluctuations. Quasi-static noise results in Gaussian decay envelope
for the qubit coherence,
$\chi(t)=\mathrm{e}^{-\left(t/T_{\phi,2}\right)^{2}}$, where
$T_{\phi,2}=1/P_{\rm q-s}$ is the dephasing time Ithier  _et al._ (2005).

Noise with a significant spectral weight in between these two extreme cases
results in decay envelopes that complicate the analytical calculations. Most
importantly, the dephasing time for $1/f^{\alpha}$-noise can no longer be
simply defined as a single coefficient. For simplicity, here we approximate
$1/f^{\alpha}$ as quasi-static since its decay envelope only significantly
deviates from Gaussian when $t$ approaches the root for
$\chi(t)=1/\mathrm{e}$. The most important difference between $1/f^{\alpha}$
noise and quasi-static noise is observed when noise-decoupling sequences are
considered, for example, spin echo completely eliminates quasi-static noise
but not $1/f^{\alpha}$-noise Ithier  _et al._ (2005); Bylander  _et al._
(2011).

With these assumptions, the contributions from white noise and quasi-static
noise to the gate error rate can be modeled as Abad  _et al._ (2021)

$\epsilon_{T_{\phi,1}}=\sum_{i}^{d}\frac{N_{\rm
q}d}{2(d+1)}\frac{\tau}{T_{\phi,1}^{{\rm Q}_{i}}},$ (S36)

and for quasi-static noise as Chu and Yan (2021)

$\epsilon_{T_{\phi,2}}=\sum_{i}^{d}\frac{N_{\rm
q}d}{2(d+1)}\left(\frac{\tau}{T_{\phi,2}^{{\rm Q}_{i}}}\right)^{2}.$ (S37)

###  H.2 Coherence limit for CZ gate with tunable coupler

As the coupler hybridizes with the qubits during the gate, the coherence of
the computational states is affected by the coupler coherence. In Ref. Xu  _et
al._ (2020), a method to estimate the CZ gate fidelity based on effective
qubit coherence times was proposed. The qubit coherence times are measured at
each of the coupler frequency relevant for the gate, and an effective
coherence time is calculated as an average over the coherence times weighted
by the time coupler spends at each of those frequencies.

This yields

$T_{1}^{\rm eff}=\tau/\int_{0}^{\tau}1/T_{1}[\omega_{\rm c}(t)]\mathrm{d}t,$
(S38)

$T_{\phi,{\rm 1}}^{\rm eff}=\tau/\int_{0}^{\tau}1/T_{\phi,{\rm 1}}[\omega_{\rm
c}(t)]\mathrm{d}t,$ (S39)

$T_{\phi,{\rm 2}}^{\rm
eff}=1/\sqrt{\frac{1}{\tau}\int_{0}^{\tau}\left\\{1/T_{\phi,{\rm
2}}[\omega_{\rm c}(t)]\right\\}^{2}\mathrm{d}t},$ (S40)

where the weighting is carried out in terms of decoherence rates, and
$\omega_{c}(t)$ is the coupler pulse shape. Additionally, the quasi-static
dephasing rates need to be averaged under a square root due to their quadratic
contribution to the error rates. In the above analysis, we have assumed that
both qubits contribute equally to the CZ gate error rate. However, during the
gate the state $\ket{11}$ evolves to $\ket{02}$ and back, reducing the
probability of a relaxation error for $Q_{1}$ and correspondingly increasing
it for $Q_{2}$.

Figure S9: Figure shows the exponential part of the dephasing times as a
function of the coupler frequency for the hybridized modes. For the coupler
frequencies relevant for the CZ gate, the exponential part of dephasing mostly
stays constant, except for a sharp drop at $3.8\text{\,}\mathrm{GHz}$ for
qubit 1. We observed the feature to move to different frequencies over time
indicating that it originates from a spectrally active TLS. The solid lines
show the median of the measured exponential dephasing rates, evaluated at the
frequencies relevant for the CZ gate, and used for calculating $T_{\phi,1,{\rm
eff}}^{k}$.

To model the effective decoherence rates based on the decoherence rates of the
uncoupled modes, we calculate the participation ratios of the uncoupled modes
in the computational states for various coupler frequencies, see Appendix A
for details. The energy-relaxation rate of the coupled mode is then

$\Gamma_{1}^{k}=1/T^{k}_{1}=\sum_{i={\rm Q_{1}},{\rm Q_{2}},{\rm
c}}p_{i,k}\tilde{\Gamma}^{i}_{1},$ (S41)

where $\tilde{\Gamma}^{i}_{1}$ is the energy-relaxation rate of the uncoupled
mode $i$ and $p_{i,k}$ is the participation ratio of uncoupled mode $i$ in the
coupled mode $k$. We note that due to the hybridization of the modes, the
energy-relaxation events become correlated with each other even though the
energy-relaxation events of the individual components were assumed to be
independent. Nevertheless, for first-order Markovian noise processes resulting
from a linear coupling to a bath, the error rate is identical for correlated
and non-correlated noise Abad  _et al._ (2021), allowing us to use Eq. (S34)
to calculate the expected error. Similarly, we can calculate the dephasing
rate corresponding to the Markovian part of the dephasing noise as

$\Gamma_{\phi,1}^{k}=1/T^{k}_{\phi,1}=\sum_{i={\rm Q_{1}},{\rm Q_{2}},{\rm
c}}p_{i,k}\tilde{\Gamma}^{i}_{\phi,1},$ (S42)

where, as before, $\tilde{\Gamma}^{i}_{\phi,1}$ is the exponential dephasing
rate of the uncoupled mode $i$ and $p_{i,k}$ is the participation ratio. Fig.
S9 shows the measured values for $T^{i}_{\phi,1}$ and the solid line which
indicates the value for $T^{i}_{\phi,1,{\rm eff}}$ used in the main text.

To evaluate the Gaussian part of the dephasing noise for the coupled mode, we
need to separately account for the independent low-frequency fluctuations in
the SQUID loops of the qubits and the couplers. The combined effect of the
fluctuations on the dephasing of the coupled mode is Chu and Yan (2021);
Ithier  _et al._ (2005)

$\Gamma_{\phi,2}^{k}=1/T_{\phi,2}^{k}=\sqrt{\sum_{i=\rm{Q_{1}},\rm{Q_{2}},\rm{c}}\left(p_{i,k}\tilde{\Gamma}_{\phi,2}^{i}\right)^{2}},$
(S43)

where $p_{i,k}$ is the participation ratio of the uncoupled mode $i$ in the
hybridized mode $k$ and $\tilde{\Gamma}_{\phi,2}^{i}$ are the dephasing rates
of the uncoupled modes. Strictly speaking, Eq. (S36) is no longer valid for
the non-Markovian correlated noise described by Eq. (S43), but for simplicity,
we decide to rely on it for the estimate of the coherent error limit.