[1]Aashish Sah

1]QCD Labs, QTF Centre of Excellence, Department of Applied Physics, Aalto
University, Tietotie 3, Espoo, 02150, Finland

2]VTT Technical Research Centre of Finland Ltd, Tietotie 3, Espoo, 02150,
Finland

# Decay-protected superconducting qubit with fast control enabled by
integrated on-chip filters

<EMAIL_ADDRESS>Suman Kundu  Heikki Suominen  Qiming Chen  Mikko Möttönen
[  [

###### Abstract

Subharmonic control of superconducting qubits simultaneously facilitates fast
single-qubit gates, long coherence times, and reduced thermal load. However,
its implementation requires a very strong coupling of the drive line to the
qubit or an excessively strong microwave signal. To address this challenge, we
introduce on-chip filters of the qubit drive that exhibit a stopband at the
resonance frequency and ultra-strong coupling at the subharmonic frequency.
The on-chip filters are designed to demonstrate qubit $T_{1}$ in the
millisecond range while enabling fast gates of tens of nanoseconds with
subharmonic control. For a resonant drive, we experimentally show a maximum of
200-fold improvement in the measured $T_{1}$ and a factor of 220 suppression
in the measured Rabi frequency at the stopband. Furthermore, we successfully
implemented subharmonic driving of Rabi oscillations with a $\pi$-pulse
duration of 12 ns. Our demonstration of on-chip filters and the resulting
introduction of efficient subharmonic driving in a two-dimensional quantum
processor paves the way for a scalable qubit architecture with reduced thermal
load and noise from the control line.

## Introduction

The physical realization of a universal useful quantum computer comes with
very stringent technical requirements, including those summarized in the
DiVincenzo criteria [1]. One such requirement is a high coherence-time-to-
gate-time (CT2GT) ratio, ensuring that the physical system representing a
quantum bit, qubit, stores the encoded information with high fidelity until
the execution of the assigned task. This property is reflected in the measured
energy relaxation time of the qubit, $T_{1}$, which is the time scale for the
qubit to exponentially lose an excitation and return to its ground state.
Various qubit modalities such as trapped ions [2, 3, 4, 5], spin qubits [6, 7,
8, 9], photonic qubits [10, 11, 12, 13], and superconducting qubits [14, 15,
16, 17] are being explored for longer relaxation times to enhance the CT2GT
ratio.

Superconducting qubits, in particular, have exhibited remarkable progress,
with relaxation times extending from a few nanoseconds [14, 18] to nearly a
few milliseconds [19, 20, 21, 22] over the past two decades. The quest for
further improvement of the coherence time is still ongoing. Alternatively,
gates can be sped up to further improve the CT2GT ratio [23, 24]. However,
this requires strong external coupling of the qubit to the drive line, which
inadvertently decreases the coherence of the qubit, thereby posing a
significant trade-off.

The coherence times of a qubit are influenced not only by the external
coupling but also by intrinsic losses which in properly designed qubits,
represent the primary loss mechanism, thus limiting the qubit performance.
These losses arise from various sources, such as the presence of two-level-
system defects near the qubit [25, 26, 27, 28, 29, 30, 31, 32] associated with
dielectric losses and quasiparticle tunneling [33, 34, 35, 36]. Remarkable
progress has been made in understanding [37, 38, 39, 40, 41, 42, 43, 44, 45]
and mitigating these internal losses through the implementation of high-
coherence materials [46, 47, 48, 49, 21, 22], fabrication recipes [50, 19, 20,
51], and improved qubit geometry [52, 53], leading to a situation where the
external coupling should be made very weak not to limit qubit coherence, thus
calling for high drive power at the qubit.

While engineering very weak coupling to the qubit can increase the CT2GT
ratio, it will impose a significant heat load on the dilution refrigerators
that house the qubit, and especially on the attenuators that provide an
electromagnetic environment for the qubit. To minimize the corresponding
thermal noise reaching the qubit, attenuators are placed at multiple stages of
the dilution refrigerators [54]. Additionally, this will also limit the number
of gate operations that can be carried out before the cooling capacity of the
dilution refrigerator is exceeded.

Recent progress [55, 56] has shown promise in addressing the above-discussed
challenge of simultaneously achieving fast control and long coherence time.
However, the approach of Ref. [55] raises concerns of quasiparticle generation
due to the saturation of the Josephson junction employed in the qubit drive
line at high power levels. Another important work [56] utilizes a novel way to
drive a three-dimensional (3D) transmon qubit by subharmonics of the qubit
resonance. Namely, the non-linearity of the qubit up-converts the drive at
one-third of the qubit frequency to the qubit frequency in the spirit of
three-wave mixing [56]. Inconveniently for scaling, the previous work employs
bulky low-pass filters in the control line to suppress the spontaneous qubit
decay to the strongly coupled drive line. Furthermore, to enable subharmonic
control, only 43 dB of attenuation is used in the control line, which leads to
a significant thermal noise photon of around 0.145 at qubit frequency of 5
GHz.

In this work, we introduce designs and provide implementations of on-chip
filters for subharmonic driving in two-dimensional qubit schemes. The on-chip
filters are devised to fully isolate the qubit from the drive line at its
resonance frequency while establishing two orders of magnitude stronger
coupling at the subharmonic frequency compared to standard drive lines. This
result enables fast single-qubit gates via subharmonic drive while
demonstrating the $T_{1}$ limit owing to the external coupling of
milliseconds, thereby substantially improving the corresponding CT2GT ratio.
Encouragingly, the introduced on-chip filters facilitate the use of over 60 dB
of attenuation in the control lines to ensure a thermal noise photon below
$2.5\times 10^{-3}$. Furthermore, we show that the use of a bulky low-pass
filter creates unwanted resonances in the qubit drive line, modulating the
measured $T_{1}$ in extreme ranges. Thus our first demonstration of
subharmonic driving of a qubit in two dimensions, integrated with an on-chip
filter, seems a promising pathway for scalable superconducting quantum
processing units.

## Results

### Design and implementation of on-chip filters

We have developed two distinct versions of on-chip filters:
$\mathrm{\lambda/4}$ and $\mathrm{\lambda/2}$ filters as shown in Fig. 1a, b.
Essentially, these filters are coplanar-waveguide (CPW) transmission lines
that connect capacitively to the qubit at the position $x=0$ and end in an
open circuit at $x=L_{\mathrm{f}}$, where $L_{\mathrm{f}}$ is the length of
the filter. In the case of the $\mathrm{\lambda/4}$ filter,
$L_{\mathrm{f}}=\lambda_{\mathrm{q}}/4$, where $\lambda_{\mathrm{q}}$ is the
wavelength of the mode in the drive line at the qubit frequency. Similarly,
for the $\mathrm{\lambda/2}$ filter, $L_{\mathrm{f}}=\lambda_{\mathrm{q}}/2$.
We also note that $\mathrm{\lambda/2}$ filter couples to the qubit at two
separate positions as opposed to the single-point coupling in the case of the
$\mathrm{\lambda/4}$ filter.

Figure 1: Device layout with integrated on-chip filters. a,b Schematic
representation of qubits (white crosses) strongly coupled to a transmission
line (turquoise) implemented as (a) $\mathrm{\lambda/4}$ and (b)
$\mathrm{\lambda/2}$ filters at the qubit frequency $\mathrm{f_{q}}$ which
corresponds to the full wavelength $\lambda_{\mathrm{q}}$. c,d Voltage
amplitude of the transmission line mode as a function of position $x$ along
the transmission line for the (c) $\mathrm{\lambda/4}$ and (d)
$\mathrm{\lambda/2}$ filters. At the resonance frequency $f=\mathrm{f_{q}}$
(red line), both on-chip filters effectively decouple from the qubit due to
net zero voltage at the qubit, whereas at the subharmonic frequency
$f=\mathrm{f_{q}/3}$ (blue line), the on-chip filters couple strongly to the
qubit. The pink shading highlights the region where the modes couple to the
qubit. e False-colour optical-microscope image of the device. The device
consists of three flux-tunable transmon qubits (violet) labeled Q1, Q2, and
Q3. Each qubit is individually coupled to its own readout resonator (red) and
flux lines (magenta). The readout resonators are further coupled to a common
Purcell filter (yellow) for qubit readout. Additionally, qubits Q2 and Q3 are
coupled to $\mathrm{\lambda/4}$ and $\mathrm{\lambda/2}$ filters respectively,
whereas qubit Q1 is weakly coupled to a standard transmission line for control
(turquoise).

The working principle of the on-chip filters is illustrated in Fig. 1c, d. In
the $\mathrm{\lambda/4}$ filter, the open-end creates a boundary condition for
the voltage along the transmission line, resulting in a voltage anti-node at
the open-end and a node at the qubit location when operating at the resonance
frequency of the qubit as depicted in Fig. 1c. In contrast, the
$\mathrm{\lambda/2}$ filter exhibits a voltage anti-node at both the $x=0$ and
$x=\lambda/2$ locations, but with opposite polarities as shown in Fig. 1d.
Both configurations enforce net-zero voltage amplitude for the qubit,
effectively decoupling the qubit from the drive line at the qubit resonance
frequency. However, at the subharmonic frequency, the voltage profile at the
qubit is close to its maximum, thereby establishing a strong coupling between
the qubit and the drive line. The length, over which the mode voltage is
integrated to obtain the effective coupling strength between the mode and the
qubit, is highlighted in Fig. 1a-d.

Our realized device is shown in Fig. 1e. The on-chip filters are integrated as
a part of the qubit drive line. The device features three flux-tunable Xmon-
style transmon qubits, labeled Q1, Q2, and Q3 [57]. Qubit Q1 is connected to a
standard, weakly coupled CPW transmission line. In contrast, qubits Q2 and Q3
are connected to $\mathrm{\lambda/4}$ and $\mathrm{\lambda/2}$ filters,
respectively. Additionally, the device includes flux lines to tune the
frequencies of the qubits. Each qubit is coupled to its respective quarter-
wavelength resonator for readout. These readout resonators are inductively and
capacitively connected to a common quarter-wavelength Purcell filter [58] to
suppress qubit decay through the resonator.

We target the length of the on-chip filters such that their stopband is
centered around 5 GHz. The results of our numerical simulations of these
filters are shown in Fig. 2. For a detailed description of the electromagnetic
(EM) simulations, we refer to the Supplementary Information S1.

Figure 2: Simulated transmission and relaxation time of the qubit from drive
lines. a Schematic of a 3-port network labeled P1, P2, and P3. Port P1 is
connected to a qubit drive line through a filter which is a standard
transmission line for Q1, $\lambda/4$ filter for Q2, and $\lambda/2$ filter
for Q3. b Equivalent lumped-element model, where $\mathrm{C_{d}}$ is the
capacitance between the qubit and the drive line, $\mathrm{C_{g}}$ is the
capacitance between the qubit and the readout resonator, and $\mathrm{C_{q}}$
is the shunt capacitance of the qubit to the ground. The coupling capacitances
$\mathrm{C_{d}}$ are 80 aF, 4.5 fF, and 9.2 fF for qubits Q1, Q2, and Q3,
respectively. c Transmission amplitude from the qubit port to the drive line,
$S_{\mathrm{21}}$, as a function of frequency for the different filter
configurations as indicated. The $\lambda/4$ and $\lambda/2$ filters are
designed to have a stopband at 5 GHz. The sharp features around 7 GHz
correspond to the individual readout resonator frequencies for each qubit. The
inset highlights the region in the vicinity of the filter stopband for
convenient comparison. d Estimated $T_{\mathrm{1}}$ of the qubit owing to its
coupling just to the drive line as a function of qubit frequency. This
estimation is based on the total capacitance of the qubit and the input
admittance $Y_{\mathrm{in}}$ parallel to the qubit port P2. At the stopband,
the estimated relaxation times of the qubit for the standard drive, the
$\lambda/4$ filter, and for the $\lambda/2$ filter are approximately 0.3 ms,
7161 ms, and 2421 ms, respectively.

Figures 2a, b depict a schematic and its equivalent lumped-element model of a
qubit coupled to an on-chip filter and to a readout resonator. In the model,
50-$\Omega$ lumped ports, labeled P1 for the drive line, P2 for the qubit, and
P3 for the readout resonator, are introduced for EM simulations. We focus on
the scattering parameter $S_{\mathrm{21}}$, which relates to the power
transmission from the drive line to the qubit port, and on the input
admittance $Y_{\mathrm{in}}$ parallel to the qubit. Here, $Y_{\mathrm{in}}$ is
calculated when port P1 is matched to an impedance of 50-$\Omega$, while port
P3 is grounded. With this procedure, one can estimate the external coupling
rate just from the drive line to be
$\gamma_{\mathrm{q}}(\omega)=\mathrm{Re}[Y_{\mathrm{in}}(\omega)]/C_{\mathrm{q}}$
[59, 60]. In the low-temperature limit, the relaxation time can be simply
calculated using $T_{\mathrm{1}}(\omega)=1/\gamma_{\mathrm{q}}(\omega)$, where
we assume negligible internal losses of the qubit.

Figure 2c shows that within the filter stopband centered at 5 GHz, the
transmission from the qubit port to the transmission line lies below $-110$
dB, greatly suppressed from $-72$ dB obtained for the standard drive line.
Importantly, within the subharmonic frequency range of 1–2 GHz, we observe a
notable increase of over 30 dB in the transmission of the on-chip filters over
the standard case. The observed sharp features around 7 GHz correspond to the
resonances from the readout resonators.

In Fig. 2d, we examine the relaxation time $T_{\mathrm{1}}$ of the qubit
resulting from the external coupling to the drive line. Within the filter
stopband, we observe a substantial enhancement in $T_{\mathrm{1}}$ by a factor
of over 1000 with the $\lambda/4$ and $\lambda/2$ filters compared to the
standard drive line. The estimated $T_{\mathrm{1}}$ times at 5 GHz are as
follows: 0.3 ms for the standard drive, 7161 ms for $\lambda/4$ filter, and
2421 ms for $\lambda/2$ filter. The widths of the frequency ranges where qubit
$T_{1}$ exceeds 1 ms are 70 MHz and 450 MHz for $\lambda/4$ and $\lambda/2$
filter, respectively. We attribute this difference of feasible frequency
ranges to the fact that the $\lambda/2$ filter is of higher order than the
$\lambda/4$ filter since the coupling points of the qubit are located at the
voltage anti-nodes where the gradient of the voltage with respect to position
vanishes.

Assuming a coupling that results in $T_{1}=1$ ms for a qubit driven resonantly
through a standard transmission line, we estimate that a microwave pulse with
a peak power of $-11$ dBm at room temperature is necessary to implement 10-ns
single-qubit gates. With the above simulated parameters of the on-chip
filters, corresponding peak powers of $-16$ dBm and $-22$ dBm will be required
for $\lambda/4$ filter and $\lambda/2$ filter, respectively, using a
subharmonic drive approach. Thus the filters combined with subharmonic driving
may yield a significant reduction in power consumption for single-qubit gates.
These values are calculated considering 60 dB of attenuation in the drive line
and a simple rectangular pulse shape. Note that the peak powers may
significantly increase for complex pulse shapes which are necessary to
minimize the leakage out of the computational subspace for high-fidelity gates
[23].

Furthermore, with the above estimates, we obtain a heat dissipation of $-53$
dBm at the base plate resulting from a resonant drive with the standard
transmission line. In the case of a subharmonic drive with the on-chip
filters, we estimate heat dissipations of $-58$ dBm with $\lambda/4$ filter
and $-64$ dBm with $\lambda/2$ filters.

### Characterization of on-chip filters

Figure 3: Characterization of the on-chip filters. a,b Measured (dots) and
simulated (dashed line) Rabi frequencies of the qubit for (a) $\lambda/4$ and
(b) $\lambda/2$ filters as functions of the qubit frequency
$f_{\mathrm{q}}=\omega_{\mathrm{q}}/(2\pi)$. We swept qubit frequency in the
range of 4 to 6.5 GHz and carried out the Rabi measurement to characterize the
coupling strength of the drive lines to the qubit. The dashed line represents
the Rabi frequency estimated using
$f_{\mathrm{R}}=\Omega_{\mathrm{R}}/2\pi=2\sqrt{\gamma_{\mathrm{q}}}\beta$,
where $\gamma_{\mathrm{q}}=\mathrm{Re}[Y_{\mathrm{in}}]/C_{\mathrm{q}}$ and
$\beta\propto\omega_{\mathrm{q}}^{-1/2}V$, where V is the peak-to-peak voltage
amplitude at the chip level. Note that the measured Rabi frequency for the
$\lambda/2$ filter shows two distinct stopbands centered around 4.7 GHz and
5.7 GHz, the presence of which is attributed to the unequal coupling at
positions $x=0$ and $x=\mathrm{\lambda/2}$. c,d Measured (dots) and simulated
(dashed line) $T_{1}$ of the qubit for (c) $\lambda/4$ and (d) $\lambda/2$
filters as functions of the qubit frequency. In the simulation, the dielectric
loss tangent of $3\times 10^{-6}$ obtained from the fit is used to account for
dielectric losses limiting the $T_{1}$. The error bars represent $1\sigma$
fitting uncertainty.

We follow the typical steps to fabricate the device on a silicon wafer, which
includes a niobium metallization layer on top. The micron-sized features in
the niobium are created using maskless lithography and etching, while the
Manhattan-style Josephson junctions are patterned using electron-beam
lithography and thermal evaporation of aluminium. We cool down the device in a
commercial Bluefors XLD500 dilution refrigerator for cryogenic measurements.
For more detailed information on the fabrication process and the experimental
setup, see the Supplementary Information S2 & S3.

We carry out typical time-domain measurements to characterize the device
parameters summarized in Table 1 of the Supplementary Information S4. To
quantify the performance of the on-chip filters, we measured the Rabi
frequency and the $T_{1}$ of the qubit around the target filter frequency of 5
GHz. A comprehensive description of the measurement sequence can be found in
greater detail in the Supplementary Information S4.

Figures 3a, b show the measured Rabi frequency for qubits driven through the
$\mathrm{\lambda/4}$ and $\mathrm{\lambda/2}$ filters. We observe an
impressive 50-fold and 220-fold suppression in measured Rabi frequency at the
stopbands of $\lambda/4$ and $\lambda/2$ filters, respectively, corresponding
to 34 dB and 47 dB changes in $S_{21}$. The simulated data depicted in Fig.
3a, b correspond to the Rabi frequency derived from the input admittance
$Y_{\mathrm{in}}$ obtained through EM simulations, see Methods. Conveniently,
the measured and simulated Rabi frequencies exhibit a high level of agreement.
Some qualitative deviations observed can be attributed to parasitic resonance
and imperfections stemming from coupling with the control lines.

Note the significant discrepancy in the measured Rabi frequency for the
$\lambda/2$ filter shown in Fig. 3b in comparison to the simulated
transmission depicted in Fig. 2c. In contrast to a single stopband at 5 GHz,
we observe two stopbands centered around 4.7 GHz and 5.7 GHz. As detailed in
Supplementary Information S1, this splitting of the stop band mainly arises
from the asymmetry in the coupling capacitance between the drive line and the
qubit at the positions $x=0$ and $x=\lambda/2$, in addition to the fact that
the coupling region at $x=\lambda/2$ is of finite length. Consequently, we
show a good agreement between the measured and simulated results in Fig. 3b by
introducing an asymmetric coupling at the open-end of the $\mathrm{\lambda/2}$
filter in the EM simulation.

Figures 3c, d depict the measured $T_{1}$ for the $\mathrm{\lambda/4}$ and
$\mathrm{\lambda/2}$ filter. In our device, we consistently observe a $T_{1}$
ceiling of approximately 10 $\mu\mathrm{s}$ across all qubits. To investigate
the reason for this limited $T_{1}$, we fit the Purcell decay and dielectric-
loss models to the measured $T_{1}$ of qubit Q1 near its readout resonator
frequency, see Supplementary Information S5. From the fit, we obtain a
dielectric loss tangent of $3\times 10^{-6}$, confirming dielectric loss as
the primary loss mechanism in our device. Consequently, we used the dielectric
loss tangent of $3\times 10^{-6}$ in the EM simulations, yielding the
simulation results for qubits Q2 and Q3 in good agreement with the
experiments.

For the qubit $T_{1}$ measured at the sweet spot, we observed a remarkable
20-fold improvement for the $\mathrm{\lambda/4}$ filter and an even more
substantial 200-fold improvement for the $\mathrm{\lambda/2}$ filter in
comparison to the stadard case of no on-chip filter. This enhancement has the
potential for further improvements through the use of low-loss dielectric
materials and implemention of state-of-the-art fabrication processes for high
coherence.

### Resonant and subharmonic drive

Figures 4a,b present the flux spectroscopy results for qubit Q3 coupled to
$\lambda/2$ filter using both resonant and subharmonic driving techniques. By
sweeping the voltage across a resistor installed in the flux line at room
temperature, we tune the frequency of qubit Q3 in a range of 500 MHz near the
second stopband of the $\lambda/2$ filter centered around 5.7 GHz. With
resonant driving, we observe a major reduction in the qubit linewidth as it
approaches the stopband at 5.7 GHz. Note that also the two-photon process to
excite the qubit from its ground state to the second excited state vanishes at
different flux bias but frequency matching to 5.7 GHz, consistent with the
filter protecting the qubit at its stopband rather than the qubit itself being
substantially different at a certain flux bias. However, with subharmonic
driving, we observe a broadened qubit linewidth at around $f=5.7/3=1.9$ GHz.
This observation indicates that the $\lambda/2$ filter exhibits strong
coupling at the subharmonic frequency but very weak coupling at the resonance
frequency, which agrees with our intended design and purpose.

Figure 4: Spectroscopy and Rabi oscillations of qubit Q3 with resonant driving
and subharmonic driving through the $\mathrm{\lambda/2}$ filter. a,b
Normalized readout signal of the qubit as a function of the flux bias and the
frequency of the (a) resonant drive and (b) subharmonic drive. The top
diagonal feature indicates the lowest transition frequency of the qubit and
the bottom feature reveals the two-photon transition to the second excited
state of the qubit. c,d Normalized readout signal of the qubit as a function
of the driving-voltage amplitude at room temperature and the frequency of the
(c) resonant drive and (d) subharmonic drive for a flux bias of 0.32 V. Note
the power broadening and up to $-70$ MHz alternating-current (AC) Stark shift
in (d) with increasing pulse amplitude. e,f Normalized readout signal of the
qubit as a function of the pulse length and the frequency of the (e) resonant
drive and (f) subharmonic drive for a flux bias of 0.32 V. In (e), the pulse
amplitude 0.55 V yields a Rabi frequency of 17.57 MHz and in (f), we have 0.9
V and 31.12 MHz, respectively, at the Stark-shifted frequency of 1.809 GHz.

Qubit power spectroscopy with resonant driving in Fig. 4c demonstrates a
broadening of the qubit line with increasing pulse amplitude. The
spectroscopic line remains centered at the frequency of 5.6 GHz independent of
the pulse amplitude. In contrast, Fig. 4d exhibits a pronounced AC Stark shift
caused by the off-resonant nature of the subharmonic drive. The AC Stark shift
of the qubit frequency is typical for an off-resonant driving of the qubit
[45]. This shift increases with increasing pulse amplitude, reaching around
$-70$ MHz at the maximum amplitude used. The negative direction of the shift
arises from the negative anharmonicity of the transmon qubit.

Figures 4d,e illustrate Rabi oscillations of the qubit under resonant and
subharmonic driving. For room-temperature pulse amplitudes of 0.55 V and 0.9
V, the measured Rabi frequencies for resonant and subharmonic driving are
17.57 MHz and 31.12 MHz, respectively. In the case of subharmonic drive, the
Rabi frequency corresponding to the slowest oscillation occurs with an AC
Stark shift of $-57.67$ MHz.

### Rabi frequency and AC Stark shift

Figure 5: Stark shifts and Rabi frequencies in subharmonic driving of qubits
Q2 and Q3. a,b Measured (markers) and model-fitted (solid lines) (a) AC Stark
shift and (b) Rabi frequency for subharmonic driving through the $\lambda/4$
and $\lambda/2$ filters as indicated as functions of the voltage amplitude of
the drive pulse at room temperature. The error bars indicate $1\sigma$
uncertainty.

Figure 5a shows the AC Stark shift arising from the off-resonant nature of the
subharmonic drive. For a maximum pulse amplitude of 1 V, the measured shift is
approximately $-33$ MHz for the $\lambda/4$ filter and $-73$ MHz for the
$\lambda/2$ filter. We obtain a good agreement between the measured Stark
shift and the fit using a quadratic function in the pulse amplitude as
theoretically expected from equation (5) in the Method section.

In Fig. 5b, we conduct a performance comparison between the $\lambda/4$ and
$\lambda/2$ filters using the subharmonic drive. This comparison involves
varying the pulse amplitude to measure the Rabi frequency at the AC-Stark-
shifted frequency studied in Fig. 5a. We observe that the maximum measured
Rabi frequency is approximately 13 MHz and 43 MHz, resulting in $\pi$ pulses
of lengths 37 ns and 12 ns with the $\lambda/4$ and $\lambda/2$ filters,
respectively. We employ a cubic function to fit the measured Rabi frequencies
as a function of the pulse amplitude, showing an excellent agreement with the
experimental results as expected from the three-wave-mixing nature of
subharmonic driving, as can be seen from equation (5) in the Method section.
Importantly, we maintain over 70 dB of attenuation in the drive line to ensure
that the thermal occupation of the qubit remains below $0.5\%$, and limit heat
dissipation at the base plate to less than $-40$ dBm $=100$ nW.

## Discussion

In this work, we addressed the seemingly competing challenges of implementing
brief single-qubit gates and mitigating the decoherence of qubits due to the
strong coupling needed for such brief gates. These challenges are intimately
related to the power dissipation budget in the dilution refrigerator that
houses the qubits and cross-talk mitigation between the qubits.

A transmon qubit with a standard capacitive coupling to its broadband drive
line resulting in $T_{1}$ of 1 ms requires a peak power of $-11$ dBm at room
temperature to implement 10-ns-long single-qubit gates using a resonant drive.
This power requirement is estimated considering 60 dB attenuation in the drive
line to keep thermal qubit excitations low. In contrast, a subharmonic drive
requires roughly $25$ dBm in this scenario with no drive line filter. Such
drive power leads to heat dissipation of about $-17.5$ dBm at the base plate,
unfortunately matching its cooling capacity of $-17$ dBm.

Our proposed on-chip filters are designed to achieve a $T_{1}$ bound of 7161
ms for the $\lambda/4$ filter and 2421 ms for the $\lambda/2$ filter at the
stopband near the qubit transition frequency. We estimate peak powers of $-16$
dBm for the $\lambda/4$ filter and $-22$ dBm for the $\lambda/2$ filter at
room temperature to implement 10-ns-long gates using subharmonic driving. With
60 dB of attenuation in the drive lines, the heat dissipation at the base
plate is reduced to $-58$ dBm for the $\lambda/4$ filter and to $-64$ dBm for
the $\lambda/2$ filter, enabling tens of thousands of simultaneous gate
operations before reaching the cooling power of typical refrigerators.

Experimentally, we achieved an impressive 50-fold and 220-fold suppression in
the measured Rabi frequency and a factor of 50 and 80 improvements in the
measured $T_{1}$ at the stopbands of the $\lambda/4$ and $\lambda/2$ filters,
respectively. We measured a maximum $T_{1}$ of 10 $\mu\mathrm{s}$ across the
device, mainly limited by dielectric losses. The simulated and measured data
exhibited excellent agreement, from which we conclude that the on-chip filters
function as desired. We note that using state-of-the-art fabrication processes
for high coherence and low-loss dielectric materials may further enhance these
results and allow for more thorough study of the effects of the filters on the
qubit energy relaxation time.

In our subharmonic-drive experiments, we obtained qubit spectroscopy akin to
those with resonant driving using standard two-tone measurements. We measured
$\pi$-pulse lengths of 37 ns and 12 ns for the $\lambda/4$ and $\lambda/2$
filters, respectively, using Rabi measurements. This result was obtained with
62 dB of attenuation distributed across various dilution stages, combined with
an additional 12 dB of attenuation from filters and wires at room temperature
and inside the refrigerator, yielding approximately 74 dB in total in the
drive line. This configuration effectively reduced thermal noise photons to
below $2.5\times 10^{-3}$ at qubit frequency and around $0.03$ at subharmonic
frequency. This is a notable improvement compared to the work in Ref. [56],
where single-qubit gate of length 35 ns has been demonstrated with only 43 dB
attenuation in the qubit control line and without the use of bulky off-chip
low-pass filters.

The off-resonant nature of the subharmonic drive led to a substantial AC Stark
shift of around $-73$ MHz at a Rabi frequency of about 43 MHz with the
$\lambda/2$ filter, significantly more pronounced than in resonant driving. As
a result, the standard approach for implementing and characterizing single-
qubit gates through Randomized Benchmarking may not be feasible as such,
calling for special attention to phase correction between gates. In addition,
minimization of leakage errors with such brief gates is also privotal.
Although the drag scheme effectively reduces leakage errors in resonant
driving, a similar approach adapted for subharmonic driving needs to consider
the large AC Stark shift. However, addressing this issue falls outside the
scope of this study and will be addressed in future research.

## Methods

### Single-qubit control

The Hamiltonian describing the control of the qubit has the following form

$\displaystyle\hat{H}=\hat{H}_{\mathrm{qubit}}+\hat{H}_{\mathrm{drive}}(t),$
(1)

where the first term on the right is the Hamiltonian of the undriven qubits
and the the second term is the drive Hamiltonian. The Hamiltonian of the
transmon qubit is given by

$\displaystyle\hat{H}_{\mathrm{qubit}}=\hbar\omega_{\mathrm{q}}\hat{b}^{\dagger}\hat{b}+\frac{\hbar\alpha}{2}\hat{b}^{\dagger}\hat{b}^{\dagger}\hat{b}\hat{b},$
(2)

where $\omega_{\mathrm{q}}$ is the angular frequency of the lowest qubit
transition, $\alpha$ is the anharmonicity, and $\hat{b}$ and
$\hat{b}^{\dagger}$ are the annihilation and creation operators acting on the
qubit states, respectively. The driving Hamiltonian assumes the form

$\hat{H}_{\mathrm{drive}}(t)=\hbar\Omega_{\mathrm{R}}e^{-i\omega_{\mathrm{d}}t-\varphi_{\mathrm{d}}}\hat{b}^{\dagger}+\mathrm{h.c.},$
(3)

where $\Omega_{\mathrm{R}}$ is the Rabi angular frequency, and
$\omega_{\mathrm{d}}$ and $\varphi_{\mathrm{d}}$ are the frequency and the
phase of the microwave drive. To implement arbitrary single-qubit gates, the
Rabi frequency and the phase of the drive are temporally controlled.

In the case of resonant driving, we have
$\omega_{\mathrm{d}}\approx\omega_{\mathrm{q}}$. By going into a frame
rotating at $\omega_{\mathrm{d}}$, we obtain a simplified qubit-drive
Hamiltonian with the detuning
$\delta_{\mathrm{res}}=\omega_{\mathrm{q}}-\omega_{\mathrm{d}}$,

$\displaystyle\begin{split}\hat{H}_{\mathrm{res}}&=\hbar\delta_{\mathrm{res}}\hat{b}^{\dagger}\hat{b}+\frac{\hbar\alpha}{2}\hat{b}^{\dagger}\hat{b}^{\dagger}\hat{b}\hat{b}\\\
&+\hbar(\Omega_{\mathrm{R}}\hat{b}^{\dagger}+\mathrm{h.c.}).\end{split}$ (4)

For the subharmonic driving, instead we have
$\omega_{\mathrm{d}}=\omega_{\mathrm{q}}/3+\delta_{\mathrm{sub}}$ and
following the derivation in [56], we find

$\begin{split}\hat{H}_{\mathrm{sub}}&=\hbar(2\alpha|\eta|^{2}-3\delta_{\mathrm{sub}})\hat{b}^{\dagger}\hat{b}+\frac{\hbar\alpha}{2}\hat{b}^{\dagger}\hat{b}^{\dagger}\hat{b}\hat{b}\\\
&+\frac{\hbar\alpha}{3}(\eta^{3}\hat{b}^{\dagger}+\mathrm{h.c.}),\end{split}$
(5)

where
$\eta=\frac{\Omega_{\mathrm{R}}(\omega_{\mathrm{q}}-\alpha)}{\omega^{2}_{\mathrm{d}}-(\omega_{\mathrm{q}}-\alpha)^{2}}$
represents the strength of the subharmonic drive. From the Hamiltonian, we
obtain the AC Stark shift $\delta_{\mathrm{sub}}=2\alpha|\eta|^{2}/3$
resulting from the off-resonant drive and the subharmonic Rabi frequency
$\Omega^{\mathrm{sub}}_{\mathrm{R}}=2\alpha\eta^{3}/3$.

### Relation of the external coupling strength and Rabi frequency with
admittance

In our experiments, we capacitively couple a CPW transmission line to the
qubit for control. The interaction strength between the drive line and the
qubit is given by [61]

$\gamma_{\mathrm{q}}=2e^{2}\bigg{(}\frac{C_{\mathrm{d}}}{C_{\mathrm{q}}}\bigg{)}^{2}\Bigg{(}\frac{E_{\mathrm{J}}}{2E_{\mathrm{C}}}\Bigg{)}^{\frac{1}{2}}\frac{Z_{\mathrm{tml}}\omega_{\mathrm{q}}}{\hbar},$
(6)

where $e$ is the elementary charge, $C_{\mathrm{d}}$ is the capacitance
between the qubit and the drive line, $E_{\mathrm{J}}$ is the Josephson
energy, $E_{\mathrm{C}}$ is the charging energy of the qubit,
$Z_{\mathrm{tml}}$ is the characteristic impedance of the drive line, and
$C_{\mathrm{q}}$ is the capacitance of the qubit island, respectively.

We express the Rabi frequency in terms of the interaction strength as

$\Omega_{\mathrm{R}}=2\sqrt{\gamma_{\mathrm{q}}}\beta,$ (7)

where
$\beta=\big{(}\frac{1}{2\hbar\omega_{\mathrm{q}}Z_{\mathrm{tml}}}\big{)}^{\frac{1}{2}}V$
and $V$ is the voltage amplitude of the drive [61]. From equations (6) and
(7), we observe that for the resonant case, the Rabi frequency
$\Omega_{\mathrm{R}}\propto V$, whereas for subharmonic drive,
$\Omega^{\mathrm{sub}}_{\mathrm{R}}\propto V^{3}$. This cubic scaling with
respect to the voltage amplitude renders the subharmonic drive favourable over
resonant driving at high amplitudes or coupling strengths.

The above simplified analytical equation may be used to obtain
$\gamma_{\mathrm{q}}$ of the qubit at weak coupling to the transmission line.
In our case, we also consider the complete system of a qubit strongly coupled
to an on-chip filter and to the readout resonator. Here, we resort to
electromagnetic simulations to estimate the external coupling strength using
$\gamma_{\mathrm{q}}(\omega)=\mathrm{Re}[Y_{\mathrm{in}}(\omega)]/C_{\mathrm{q}}$,
where $Y_{\mathrm{in}}$ is the admittance in parallel with the qubit obtained
from the simulations [59, 60].

### Electromagnetic simulations

The device layout is created using KQCircuits, an open-source Python library,
within the KLayout editor. Subsequently, this layout is exported to the
electromagnetic-simulation environment of the Sonnet software. Owing to the
relatively large size of the qubit-drive geometry and the associated
complexity of the simulation, we simplify the model by replacing a part of the
transmission line by a lumped port, which allows for building a simulation of
the full device using a lumped-circuit simulator.

The scattering-parameter data is exported from Sonnet to Microwave Office AWR,
where we construct and simulate a full-circuit model. We extract and export
the input admittance $Y_{\mathrm{in}}$ for further analysis in Python. A
comprehensive description of the steps taken to achieve these simulation
results is presented in the Supplementary Information S1.

## Data Availability

All reasonable data requests should be currently addressed to the
corresponding author. Research data will be available on the Zenodo platform
in the future.

## References

  *   * [1] DiVincenzo, D. P.  The physical implementation of quantum computation.  _Fortschritte der Physik_ 48, 771–783 (2000).  URL https://onlinelibrary.wiley.com/doi/10.1002/1521-3978.
  * [2] Cirac, J. I. & Zoller, P.  Quantum computations with cold trapped ions.  _Physical Review Letters_ 74, 4091–4094 (1995).  URL https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.74.4091.
  * [3] Bernien, H. _et al._ Probing many-body dynamics on a 51-atom quantum simulator.  _Nature_ 551, 579–584 (2017).  URL https://www.nature.com/articles/nature24622.
  * [4] Bluvstein, D. _et al._ A quantum processor based on coherent transport of entangled atom arrays.  _Nature_ 604, 451–456 (2022).  URL https://www.nature.com/articles/s41586-022-04592-6.
  * [5] Bluvstein, D. _et al._ Logical quantum processor based on reconfigurable atom arrays.  _Nature_ 1–8 (2023).  URL https://www.nature.com/articles/s41586-023-06927-3.
  * [6] Loss, D. & DiVincenzo, D. P.  Quantum computation with quantum dots.  _Physical Review A - Atomic, Molecular, and Optical Physics_ 57, 120–126 (1998).  URL https://journals.aps.org/pra/abstract/10.1103/PhysRevA.57.120.
  * [7] Xue, X. _et al._ Quantum logic with spin qubits crossing the surface code threshold.  _Nature_ 601, 343–347 (2022).  URL https://www.nature.com/articles/s41586-021-04273-w.
  * [8] Weinstein, A. J. _et al._ Universal logic with encoded spin qubits in silicon.  _Nature_ 615, 817–822 (2023).  URL https://www.nature.com/articles/s41586-023-05777-3.
  * [9] Burkard, G., Ladd, T. D., Pan, A., Nichol, J. M. & Petta, J. R.  Semiconductor spin qubits.  _Reviews of Modern Physics_ 95, 025003 (2023).  URL https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.95.025003.
  * [10] Knill, E., Laflamme, R. & Milburn, G. J.  A scheme for efficient quantum computation with linear optics.  _Nature_ 409, 46–52 (2001).  URL https://www.nature.com/articles/35051009.
  * [11] Browne, D. E. & Rudolph, T.  Resource-efficient linear optical quantum computation.  _Physical Review Letters_ 95, 010501 (2005).  URL https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.95.010501.
  * [12] O’Brien, J. L.  Optical quantum computing.  _Science_ 318, 1567–1570 (2007).  URL https://www.science.org/doi/10.1126/science.1142892.
  * [13] Shi, S. _et al._ High-fidelity photonic quantum logic gate based on near-optimal Rydberg single-photon source.  _Nature Communications_ 13, 1–6 (2022).  URL https://www.nature.com/articles/s41467-022-32083-9.
  * [14] Nakamura, Y., Pashkin, Y. A. & Tsai, J. S.  Coherent control of macroscopic quantum states in a single-Cooper-pair box.  _Nature_ 398, 786–788 (1999).  URL https://www.nature.com/articles/19718.
  * [15] Koch, J. _et al._ Charge-insensitive qubit design derived from the Cooper pair box.  _Physical Review A - Atomic, Molecular, and Optical Physics_ 76, 042319 (2007).  URL https://journals.aps.org/pra/abstract/10.1103/PhysRevA.76.042319.
  * [16] Devoret, M. H. & Martinis, J. M.  Implementing qubits with superconducting integrated circuits.  _Experimental Aspects of Quantum Computing_ 3, 163–203 (2005).  URL https://link.springer.com/article/10.1007/s11128-004-3101-5.
  * [17] Kockum, A. F. & Nori, F.  Quantum Bits with Josephson Junctions.  _Springer Series in Materials Science_ 286, 703–741 (2019).  URL https://link.springer.com/chapter/10.1007/978-3-030-20726-7_17.
  * [18] Bladh, K., Duty, T., Gunnarsson, D. & Delsing, P.  The single Cooper-pair box as a charge qubit.  _New Journal of Physics_ 7, 180 (2005).  URL https://iopscience.iop.org/article/10.1088/1367-2630/7/1/180https://iopscience.iop.org/article/10.1088/1367-2630/7/1/180/meta.
  * [19] Wang, C. _et al._ Towards practical quantum computers: transmon qubit with a lifetime approaching 0.5 milliseconds.  _npj Quantum Information_ 8, 1–6 (2022).  URL https://www.nature.com/articles/s41534-021-00510-2.
  * [20] Verjauw, J. _et al._ Path toward manufacturable superconducting qubits with relaxation times exceeding 0.1 ms.  _npj Quantum Information_ 8, 1–7 (2022).  URL https://www.nature.com/articles/s41534-022-00600-9.
  * [21] Place, A. P. _et al._ New material platform for superconducting transmon qubits with coherence times exceeding 0.3 milliseconds.  _Nature Communications_ 12, 1–6 (2021).  URL https://www.nature.com/articles/s41467-021-22030-5.
  * [22] Somoroff, A. _et al._ Millisecond Coherence in a Superconducting Qubit.  _Physical Review Letters_ 130, 267001 (2023).  URL https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.130.267001.
  * [23] Werninghaus, M. _et al._ Leakage reduction in fast superconducting qubit gates via optimal control.  _npj Quantum Information_ 7, 1–6 (2021).  URL https://www.nature.com/articles/s41534-020-00346-2.
  * [24] Ding, L. _et al._ High-Fidelity, Frequency-Flexible Two-Qubit Fluxonium Gates with a Transmon Coupler.  _Physical Review X_ 13, 031035 (2023).  URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.13.031035.
  * [25] Martinis, J. M. _et al._ Decoherence in Josephson qubits from dielectric Loss.  _Physical Review Letters_ 95, 210503 (2005).  URL https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.95.210503.
  * [26] Gao, J. _et al._ A semiempirical model for two-level system noise in superconducting microresonators.  _Applied Physics Letters_ 92, 212504 (2008).  URL /aip/apl/article/92/21/212504/851718/A-semiempirical-model-for-two-level-system-noise.
  * [27] Lindström, T., Healey, J. E., Colclough, M. S., Muirhead, C. M. & Tzalenchuk, A. Y.  Properties of superconducting planar resonators at millikelvin temperatures.  _Physical Review B - Condensed Matter and Materials Physics_ 80, 132501 (2009).  URL https://journals.aps.org/prb/abstract/10.1103/PhysRevB.80.132501.
  * [28] Macha, P. _et al._ Losses in coplanar waveguide resonators at millikelvin temperatures.  _Applied Physics Letters_ 96, 62503 (2010).  URL /aip/apl/article/96/6/062503/167025/Losses-in-coplanar-waveguide-resonators-at.
  * [29] Pappas, D. P., Vissers, M. R., Wisbey, D. S., Kline, J. S. & Gao, J.  Two level system loss in superconducting microwave resonators.  _IEEE Transactions on Applied Superconductivity_ 21, 871–874 (2011).
  * [30] Neill, C. _et al._ Fluctuations from edge defects in superconducting resonators.  _Applied Physics Letters_ 103, 72601 (2013).  URL /aip/apl/article/103/7/072601/150361/Fluctuations-from-edge-defects-in-superconducting.
  * [31] Müller, C., Lisenfeld, J., Shnirman, A. & Poletto, S.  Interacting two-level defects as sources of fluctuating high-frequency noise in superconducting circuits.  _Physical Review B - Condensed Matter and Materials Physics_ 92, 035442 (2015).  URL https://journals.aps.org/prb/abstract/10.1103/PhysRevB.92.035442.
  * [32] Wang, C. _et al._ Surface participation and dielectric loss in superconducting qubits.  _Applied Physics Letters_ 107, 162601 (2015).  URL /aip/apl/article/107/16/162601/593971/Surface-participation-and-dielectric-loss-in.
  * [33] Lutchyn, R. M., Glazman, L. I. & Larkin, A. I.  Kinetics of the superconducting charge qubit in the presence of a quasiparticle.  _Physical Review B - Condensed Matter and Materials Physics_ 74, 064515 (2006).  URL https://journals.aps.org/prb/abstract/10.1103/PhysRevB.74.064515.
  * [34] Paik, H. _et al._ Observation of high coherence in Josephson junction qubits measured in a three-dimensional circuit QED architecture.  _Physical Review Letters_ 107, 240501 (2011).  URL https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.107.240501.
  * [35] Catelani, G. _et al._ Quasiparticle relaxation of superconducting qubits in the presence of flux.  _Physical Review Letters_ 106, 077002 (2011).  URL https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.106.077002.
  * [36] Catelani, G., Nigg, S. E., Girvin, S. M., Schoelkopf, R. J. & Glazman, L. I.  Decoherence of superconducting qubits caused by quasiparticle tunneling.  _Physical Review B - Condensed Matter and Materials Physics_ 86, 184514 (2012).  URL https://journals.aps.org/prb/abstract/10.1103/PhysRevB.86.184514.
  * [37] Paladino, E., Galperin, Y., Falci, G. & Altshuler, B. L.  1/ f noise: Implications for solid-state quantum information.  _Reviews of Modern Physics_ 86, 361–418 (2014).  URL https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.86.361.
  * [38] Burnett, J. _et al._ Evidence for interacting two-level systems from the 1/f noise of a superconducting resonator.  _Nature Communications_ 5, 1–6 (2014).  URL https://www.nature.com/articles/ncomms5119.
  * [39] Pop, I. M. _et al._ Coherent suppression of electromagnetic dissipation due to superconducting quasiparticles.  _Nature_ 508, 369–372 (2014).  URL https://www.nature.com/articles/nature13017.
  * [40] Lisenfeld, J. _et al._ Electric field spectroscopy of material defects in transmon qubits.  _npj Quantum Information_ 5, 1–6 (2019).  URL https://www.nature.com/articles/s41534-019-0224-1.
  * [41] Yan, F. _et al._ The flux qubit revisited to enhance coherence and reproducibility.  _Nature Communications_ 7, 1–9 (2016).  URL https://www.nature.com/articles/ncomms12964.
  * [42] Vepsäläinen, A. P. _et al._ Impact of ionizing radiation on superconducting qubit coherence.  _Nature_ 584, 551–556 (2020).  URL https://www.nature.com/articles/s41586-020-2619-8.
  * [43] Murray, C. E.  Material matters in superconducting qubits.  _Materials Science and Engineering R: Reports_ 146, 100646 (2021).
  * [44] Siddiqi, I.  Engineering high-coherence superconducting qubits.  _Nature Reviews Materials_ 6, 875–891 (2021).  URL https://www.nature.com/articles/s41578-021-00370-4.
  * [45] Carroll, M., Rosenblatt, S., Jurcevic, P., Lauer, I. & Kandala, A.  Dynamics of superconducting qubit relaxation times.  _npj Quantum Information_ 8, 1–7 (2022).  URL https://www.nature.com/articles/s41534-022-00643-y.
  * [46] Wang, H. _et al._ Improving the coherence time of superconducting coplanar resonators.  _Applied Physics Letters_ 95, 233508 (2009).  URL /aip/apl/article/95/23/233508/120944/Improving-the-coherence-time-of-superconducting.
  * [47] Megrant, A. _et al._ Planar superconducting resonators with internal quality factors above one million.  _Applied Physics Letters_ 100, 113510 (2012).  URL /aip/apl/article/100/11/113510/126200/Planar-superconducting-resonators-with-internal.
  * [48] Chang, J. B. _et al._ Improved superconducting qubit coherence using titanium nitride.  _Applied Physics Letters_ 103, 12602 (2013).  URL /aip/apl/article/103/1/012602/235870/Improved-superconducting-qubit-coherence-using.
  * [49] Stern, M. _et al._ Flux qubits with long coherence times for hybrid quantum circuits.  _Physical Review Letters_ 113, 123601 (2014).  URL https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.113.123601.
  * [50] Bruno, A. _et al._ Reducing intrinsic loss in superconducting resonators by surface treatment and deep etching of silicon substrates.  _Applied Physics Letters_ 106, 182601 (2015).  URL /aip/apl/article/106/18/182601/27784/Reducing-intrinsic-loss-in-superconducting.
  * [51] Deng, H. _et al._ Titanium Nitride Film on Sapphire Substrate with Low Dielectric Loss for Superconducting Qubits.  _Physical Review Applied_ 19, 024013 (2023).  URL https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.19.024013.
  * [52] He, H., Wang, W., Liu, F., Yuan, B. & Shan, Z.  Suppressing the Dielectric Loss in Superconducting Qubits through Useful Geometry Design.  _Entropy_ 24, 952 (2022).  URL https://www.mdpi.com/1099-4300/24/7/952/htmhttps://www.mdpi.com/1099-4300/24/7/952.
  * [53] Martinis, J. M.  Surface loss calculations and design of a superconducting transmon qubit with tapered wiring.  _npj Quantum Information_ 8, 1–12 (2022).  URL https://www.nature.com/articles/s41534-022-00530-6.
  * [54] Krinner, S. _et al._ Engineering cryogenic setups for 100-qubit scale superconducting circuit systems.  _EPJ Quantum Technology_ 6 (2019).  URL https://doi.org/10.1140/epjqt/s40507-019-0072-0.
  * [55] Kono, S. _et al._ Breaking the trade-off between fast control and long lifetime of a superconducting qubit.  _Nature Communications_ 11, 1–6 (2020).  URL https://www.nature.com/articles/s41467-020-17511-y.
  * [56] Xia, M. _et al._ Fast superconducting qubit control with sub-harmonic drives (2023).  URL http://arxiv.org/abs/2306.10162.
  * [57] Barends, R. _et al._ Coherent josephson qubit suitable for scalable quantum integrated circuits.  _Physical Review Letters_ 111, 080502 (2013).  URL https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.111.080502.
  * [58] Jeffrey, E. _et al._ Fast accurate state measurement with superconducting qubits.  _Physical Review Letters_ 112, 190504 (2014).  URL https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.112.190504.
  * [59] Esteve, D., Devoret, M. H. & Martinis, J. M.  Effect of an arbitrary dissipative circuit on the quantum energy levels and tunneling of a Josephson junction.  _Physical Review B_ 34, 158–163 (1986).  URL https://journals.aps.org/prb/abstract/10.1103/PhysRevB.34.158.
  * [60] Houck, A. A. _et al._ Controlling the spontaneous emission of a superconducting transmon qubit.  _Physical Review Letters_ 101, 080502 (2008).  URL https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.101.080502.
  * [61] Blais, A., Grimsmo, A. L., Girvin, S. M. & Wallraff, A.  Circuit quantum electrodynamics.  _Reviews of Modern Physics_ 93, 025005 (2021).
  * [62] Fedorov, G. P. & Ustinov, A. V.  Automated analysis of single-tone spectroscopic data for cQED systems.  _Quantum Science and Technology_ 4, 045009 (2019).  URL https://iopscience.iop.org/article/10.1088/2058-9565/ab478bhttps://iopscience.iop.org/article/10.1088/2058-9565/ab478b/meta.
  * [63] Zhang, H. _et al._ Universal Fast-Flux Control of a Coherent, Low-Frequency Qubit.  _Physical Review X_ 11, 011010 (2021).  URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.11.011010.
  * [64] Hyyppä, E. _et al._ Unimon qubit.  _Nature Communications_ 13, 1–14 (2022).  URL https://www.nature.com/articles/s41467-022-34614-w.

## Acknowledgments

We express our gratitude to the KAUTE Foundation for their support through the
Ph.D. grant awarded to the author. We acknowledge the funding organizations
European Research Council under the Advanced Grant no. 101053801 (ConceptQ),
Quantum Technology Finland, and the Finnish Quantum Flagship. Special thanks
are extended to Sergei Lemziakov, Dmitrii Lvov, and Joonas Peltonen for their
invaluable assistance with the Otanano cryogenic facility. The author
acknowledges Giacomo Catto, Timm Mörstedt, and Priyank Singh for useful
discussions related to the fabrication and we thank Jukka-Pekka Kaikkonen from
VTT Technical Research for providing us with the unpatterned Niobium-on-
Silicon wafer. Furthermore, we are thankful for the enriching general
discussions on the subject matter with Michael Hatridge, Florian Blanchet,
Arman Alizadeh, Wallace Santos Teixeira, and Vasilii Vadimov.

## Authors contributions

A.S. and M.M. conceived the experiment. A.S. and S.K. designed and carried out
the electromagnetic simulations of the device. H.S. and A.S. wrote the
measurement codes for the experiment. Q.C. helped with the theoretical
derivations of subharmonic drive. A.S. fabricated the device, conducted the
experiments, and analyzed the results with feedback from S.K. A.S., S.K., and
M.M. wrote the manuscript with comments from all the authors.

## Competing interests

M.M. declares that he is a Co-Founder and Shareholder of IQM Finland Oy.

## Supplementary Information for ”Decay-protected superconducting qubit with
fast control enabled by integrated on-chip filters”

## Supplementary Note S1: Design and simulation of the on-chip filters

To facilitate the design and the implementation of the on-chip filters, we
establish a simulation procedure to obtain the Rabi frequency and the $T_{1}$
of the qubit resulting from the coupling to the on-chip filters. To this end,
we estimate the quantum properties of superconducting circuits from
classicalelectromagnetic simulations of lumped and distributed-element
circuits [59, 60, 61]. By simulating the input admittance ($Y_{\mathrm{in}}$)
in parallel with the qubit from qubit–environment coupling, one obtains the
external coupling rate using
$\gamma_{\mathrm{q}}(\omega)=\mathrm{Re}[Y_{\mathrm{in}}(\omega)]/C_{\mathrm{q}}$,
where $C_{\mathrm{q}}$ is the shunt capacitance of the qubit to the ground.
Here, the environment refers to any element that couples to the qubit such as
standard drive lines, on-chip filters, readout resonators, and flux lines.
However, care needs to be taken when addressing the contribution from the
individual coupling elements, for example, by tuning out the other coupled
channels by either grounding them or removing them completely from the
simulation geometry.

Supplementary Fig. S1: Comparison different models for the Rabi frequency and
the energy decay time of a weakly coupled qubit to a standard drive. a
Simplified model of a transmon qubit coupled to a coplanar-waveguide (CPW)
transmission line rendered in Sonnet. A box-wall port 1 is used at the drive
port, and a co-calibrated lumped port 2 is placed on the qubit island to
investigate scattering, $S_{\mathrm{21}}$, and input admittance,
$Y_{\mathrm{in}}$, parameters of the qubit. Note that $Y_{\mathrm{in}}$ is
calculated when the other ports are terminated using the impedances set by the
port terminations, in this case, to 50 $\Omega$. An estimate of the qubit-
drive coupling capacitance $C_{\mathrm{d}}$ is obtained from the finite-
element method (FEM) simulation carried out in Sonnet. b Equivalent lumped
model of the qubit-drive system built in Microwave Office AWR design
environment. A transmission line element is used to model a CPW drive line
with a characteristic impedance $Z_{\mathrm{0}}$ of 50 $\Omega$ and relative
effective dielectric constant $E_{\mathrm{eff}}$ of 6.45 obtained using
$E_{\mathrm{eff}}=(\epsilon_{\mathrm{Si}}+\epsilon_{\mathrm{air}})/2$, where
$\epsilon_{\mathrm{Si}}=11.9$ and $\epsilon_{\mathrm{air}}=1$ are the relative
dielectric permittivity of a silicon substrate and air, respectively. The
coupling capacitance $C_{\mathrm{d}}$ is taken from the Sonnet simulation. c
Combination of a FEM simulation and a lumped model is set up inside AWR,
allowing us to simulate complex and large geometries while avoiding the time
and memory constraints usually incurred in FEM simulations of such geometries.
From Sonnet simulation, we obtain an S-parameter file in touchstone format and
export it to AWR, where FEM-lumped-model simulation yields $Y_{\mathrm{in}}$
in parallel with the qubit due to the coupling to the drive. d, e Comparison
of the estimated (d) Rabi frequency and (e) $T_{1}$ of the qubit from lumped,
FEM-lumped, and analytical approaches.

Once the external coupling rate is known, the Rabi frequency and the $T_{1}$
can be estimated using
$\Omega_{\mathrm{R}}(\omega)=2\sqrt{\gamma_{\mathrm{q}}(\omega)}\beta$ and
$T_{1}(\omega)=1/\gamma_{\mathrm{q}}(\omega)$, respectively, as discussed in
the Methods of the main text. Using this approach, we compare the Rabi
frequency and the $T_{1}$ estimates obtained from FEM and lumped-element
simulations to the theoretical model for a qubit weakly coupled to a standard
transmission line, as shown in Supplementary Fig. S1. We observe a very good
correspondence between the simulated and the theoretical results, confirming
the validity of the established approach.

In Supplementary Fig. S2, we describe the approach for full-circuit simulation
for the qubit Q3 coupled to the $\lambda/2$ filter, the readout resonator, and
the flux line. Supplementary Figure S2a shows a simplified model of a large
simulation geometry by trimming out the meanders of the $\lambda/2$ filter and
adding co-calibrated lumped ports that allow for building the full-circuit
scheme with a lumped-element network simulator such as Microwave office AWR.
After building the full-circuit scheme in AWR, we simulate the input
admittance in parallel with the qubit from the port of interest. To estimate
the Rabi frequency and $T_{1}$ just from the on-chip filter, we ground all the
other co-calibrated ports except the qubit and the filter ports.

Supplementary Fig. S2: Full-circuit simulation scheme. a, b (a) Simplified
model of the $\lambda/2$ filter of (b) is generated in Sonnet by removing the
meanders and adding two co-calibrated ports 6 and 7 at both ends. Similarly,
co-calibrated ports 2, 4, and 5 are added to the qubit island, readout
resonator, and flux line, respectively in the Sonnet FEM simulation. Again,
box-wall port 1 is used for the qubit drive through the $\lambda/2$ filter.
The FEM simulation for this geometry is carried out in the frequency range of
1–10 GHz, and the resulting 7-port S-parameter file is exported. c Full-
circuit simulation scheme built in AWR utilizing the S-parameter file obtained
from the FEM simulations, with the rest of the circuitry implemented with
lumped elements. The meanders of the $\lambda/2$ filter corresponding to a
length of about 11 mm are added as a transmission line element between ports 6
and 7 of the S-parameter network. A Purcell filter together with three readout
resonators R1, R2, and R3 is constructed with just the lumped components. The
parameters of readout resonators and Purcell filters such as coupling
capacitances, mutual inductances, and filter and resonator lengths are
adjusted to match the designed values. Readout resonator R3 is coupled to the
qubit Q3 at port 4 of the S-parameter network, while the flux line at port 5
is grounded. To estimate the Rabi frequency and the $T_{1}$ from just the
$\lambda/2$ filter, we also ground the resonator port, illustrated in (a) with
a switch symbol.

To investigate the frequency splitting of the stopband observed in the
characterization of the $\lambda/2$ filter in the experiments, we simulate a
lumped model of a $\lambda/2$ filter that couples capacitively to qubit port
at two separate locations, as depicted in Supplementary Fig. S3. We place a
transmission line element between the coupling points, as shown in the inset
of Supplementary Fig. S3b. We introduce a top and a bottom capacitor with
capacitances $C^{\mathrm{top}}_{\mathrm{d}}$ and
$C^{\mathrm{bot}}_{\mathrm{d}}$, respectively, at the coupling points. By
adding an asymmetry offset $\Delta C^{\mathrm{top}}_{\mathrm{d}}$ to the top
capacitor such that
$C^{\mathrm{top}}_{\mathrm{d}}=C^{\mathrm{bot}}_{\mathrm{d}}-\Delta
C^{\mathrm{top}}_{\mathrm{d}}$, we observe a frequency splitting of the
stopband as a function of the offset relative to the bottom capacitor. The
observed splitting of about 1 GHz in the experiment corresponds to a relative
asymmetry offset of $4.9$% as shown in Supplementary Fig. S3a.

Supplementary Fig. S3: Stopband frequency splitting due to an asymmetric two-
point coupling of the $\lambda/2$ filter. a Transmission amplitude from the
qubit port to the drive line port as a function of frequency for the indicated
asymmetry offset capacitances $\Delta C^{\mathrm{top}}_{\mathrm{d}}$ which are
introduced to the top capacitor with capacitance
$C^{\mathrm{top}}_{\mathrm{d}}=C^{\mathrm{bot}}_{\mathrm{d}}-\Delta
C^{\mathrm{top}}_{\mathrm{d}}$. b Frequency splitting of the stopband as a
function of the asymmetry offset obtained as the separation of the dips in the
transmission amplitude in calcualtions identical to those exemplified in (a).
For a relative asymmetry offset of approximately $4.9$% (dashed purple
vertical line) with respect to the bottom capacitance
$C^{\mathrm{bot}}_{\mathrm{d}}$, we obtain the frequency splitting of roughly
1 GHz, equal to that observed in the experiment. The inset shows the lumped
model of the $\lambda/2$ filter used in (a).

## Supplementary Note S2: Device fabrication

A 200-nm-thick niobium film deposited on a six-inch high-resistivity silicon
wafer is obtained from VTT Technical Research Centre of Finland. We spin-coat
the wafer with a few-micron-thick optical resist AZ-5214E, and then dice the
wafer into 33 mm $\times$ 33 mm squares using a Disco DAD3220 dicing saw.
Following dicing, the dies are left overnight in acetone, then subjected to 10
minutes of sonication in acetone and a quick dip in isopropanol.

For patterning niobium capacitors, on-chip filters, resonators, and qubit
islands, we utilize optical lithography with a Maskless Aligner MLA150 from
Heidelberg. The process begins with HMDS priming of the die in the YES-3 prime
oven. Next, we spin-coat the die with AZ-5214E resist using LabSpin6 at 4000
RPM for 40 seconds, achieving a resist thickness of roughly 1 $\mu\mathrm{m}$.
This is immediately followed by a soft bake at 90 ${}^{\circ}\mathrm{C}$ for
1.5 minutes.

A CAD file, featuring a 3-by-3 grid of 10 mm $\times$ 10 mm device geometries,
is converted into a native job file for optical exposure of the die with
MLA150. The mask aligner is set to operate with a 375-nm-wavelength laser
diode source at a dose of 120 $\mathrm{mJ/cm^{2}}$. We then develop the resist
in a 1:5 mixture of AZ-351B:DI $\mathrm{H_{2}O}$ for 50 seconds. Subsequently,
the niobium layer is dry-etched using Plasmalab 80Plus from Oxford Instruments
to reveal micron-size features. For reactive ion etching, we use SF6 (40 SCCM)
and $\mathrm{O_{2}}$ (20 SCCM) for six minutes. The die is left in acetone
overnight for lift-off, followed by $\mathrm{O_{2}}$ descum in the Plasmalab
to remove any remaining resist residue.

For the junctions in the qubit, we fabricate Manhattan-style Josephson
junctions in a SQUID form using electron beam lithography (EBL). We first
spin-coat the die with MMA 8.5 EL13 polymer-based resist at 1550 RPM for a
minute, followed by a post-bake at 160 ${}^{\circ}\mathrm{C}$ for another
minute. A layer of PMMA A4 resist is then added following similar steps. The
resist is exposed to an electron beam using EBPG5200 from Raith, followed by
development in a series of solvents: MIBK 1:3, methyl glycol 1:2, and
isopropanol, each for 20 seconds. Aluminum is evaporated in a Plassys
MEB700S2-III UHV at a rate of 0.2 nm/s to a thickness of 30 nm, then oxidized
at a pressure of 3 mbar for seven minutes to create a tunnel barrier. This is
followed by a second aluminum evaporation to a thickness of 40 nm. After
overnight acetone soaking for lift-off, a second round of EBL and evaporation
with similar parameters is carried out to establish contact between aluminum
and niobium patterns. We dice a 33 mm $\times$ 33 mm die, yielding a total of
nine devices from a single fabrication round.

## Supplementary Note S3: Experimental setup

Following the fabrication of the device, we wire-bonded the device to an SMA-8
sample holder and placed the device inside a magnetic shield, which is
installed at the base plate of a Bluefors XLD500 dilution refrigerator.
Throughout the experiment, the refrigerator maintained a temperature range of
18–24 mK. An overview of the experimental setup is presented in Supplementary
Fig. S4.

We utilize three separate microwave lines to control the qubits Q1, Q2, and Q3
and a single line for readout (RO) input. Each control line has 62 dB and the
RO input has 72 dB of attenuation just from the attenuators spread over
multiple stages of the cryostat. For the readout output, we have employed two
isolators and a high-electron-mobility transistor (HEMT) amplifier at the 4-K
stage. In addition, low-pass and infrared (IR) filters are installed in all of
the lines to block higher-frequency noise. Owing to the limited resource for
direct-current (DC) lines, we use a magnetic coil that is installed on the
sample holder to simultaneously bias all three qubits.

For the room temperature setup, we use the Presto microwave platform from
Intermodulation Products for time-domain measurements and a vector network
analyzer from Rohde & Schwarz for continuous measurements, which we use in the
initial characterizations of the Purcell filter and of the readout resonators.
For the DC voltage source, we employ both Presto and the SIM900 from Stanford
Research Systems, depending on the availability of SIM900 since the instrument
generates stable DC voltage to bias the qubits.

Supplementary Fig. S4: Schematic room temperature and cryogenic setup for the
experiments. The SMA-8 sample holder is not shown for clarity. The arrows
heads on the lines indicate the main intended direction of propagation for the
control and readout signals. The nominal temperatures of the different plates
of the cryostat are shown on the right and the definitions of the circuit
elements are shown on the left.

## Supplementary Note S4: Device characterization and measurement sequence

Immediately after the cool down, we implement continuous spectroscopic
measurements to locate the readout resonator frequencies. In Supplementary
Fig. S5a, b, we observe a broadband feature inherent to a Purcell filter
centered at a frequency of 6.8 GHz. We find three sharp features corresponding
to individual readout resonators within the Purcell filter band. In
Supplementary Fig. S5c–e, we conduct resonator spectroscopy measurements for
each qubit with varying flux bias voltage. The observed periodic pattern is
typical of readout resonators with frequencies below the qubit frequency at
the qubit sweet spot, leading to avoided crossings between the resonators and
the qubits at a finite flux.

Supplementary Fig. S5: Characterization of Purcell filter and readout
resonators using vector network analyzer. a,b (a) Broadband transmission
amplitude of a Purcell filter centered at 6.8 GHz and (b) a magnification on
the sharp features corresponding to three readout resonators within the
Purcell filter band as functions of probe frequency. c,d,e, Spectroscopy data
similar to those in (b) of the readout resonators (c) Q1, (d) Q2, and (e) Q3
as functions of the flux bias voltage.

In Supplementary Table S1, we summarize the parameters obtained from standard
single-tone and two-tone time-domain measurements and from the initial
characterization of our device.

Supplementary Fig. S6: Measurement sequence for on-chip-filter characterization for qubit Q2. a,b Normalized readout signal for the spectroscopy measurements for (a) the readout resonator and (b) the qubit Q2 as a function of the flux bias voltage and the probe frequency. c Measured resonator frequency (blue marker) and qubit frequency (turquoise marker) as a function of the bias voltage, obtained from experiments similar to those of (a) and (b), respectively. Analytical fit of the resonator frequency (red solid) and qubit frequency (purple solid) to the measured data. c,d,e For a fixed pulse amplitude, we implement (c) qubit spectroscopy by sweeping the probe frequency $f$ around qubit frequency $f_{\mathrm{q}}$, (d) Rabi oscillation by varying the qubit pulse length, and (e) $T_{1}$ measurement by changing an idle delay between the $\pi$ pulse of the qubit and the readout pulse. The measured (markers) data and corresponding fit (solid line) at two different flux bias voltages in (d)–(f) correspond to the sweet spot of the qubit at around 7.85 GHz (blue markers, purple line) and to the filter frequency around 5 GHz (turquoise markers, orange line). The fitting parameters are shown on top of the corresponding figures.  Supplementary Table S1: Summary of measured device parameters. The asterisk (*) denotes a case where, owing to a very low $T_{1}$ of qubit Q3 at the sweet spot, we estimate the dispersive shift from measured vacuum Rabi coupling strength $g$, detuning between the qubit and the readout resonator frequencies $\Delta=\omega_{\mathrm{q}}-\omega_{\mathrm{r}}$, and the anharmonicity of the qubit $\alpha$ using $\chi=\alpha\frac{g^{2}}{\Delta(\Delta+\alpha)}$. Parameters |  Symbols |  Q1 (Standard) |  Q2 ($\mathrm{\lambda/4}$ filter) |  Q3 ($\mathrm{\lambda/2}$ filter)
---|---|---|---|---
Resonator frequency |  $\mathrm{\omega}_{\mathrm{r}}/2\pi$ |  6.8913 GHz |  6.9886 GHz |  6.7650 GHz
Resonator linewidth |  $\mathrm{\kappa}/2\pi$ |  2.076 MHz |  2.805 MHz |  10.756 MHz
Qubit frequency |  $\omega_{\mathrm{q}}/2\pi$ |  7.773 GHz |  7.852 GHz |  7.640 GHz
Coupling strength |  $g/2\pi$ |  72.455 MHz |  72.000 MHz |  68.750 MHz
Anharmonicity |  $\alpha/2\pi$ |  $-232.916$ MHz |  $-229.847$ MHz |  $-234.520$ MHz
Dispersive shift |  $\chi/2\pi$ |  $-2.300$ MHz |  $-2.025$ MHz |  $-1.978$ MHz (*)
Relaxation time |  $T_{1}$ |  7.446 $\mu$s |  0.481 $\mu$s |  0.06 $\mu$s
Dephasing time |  $T^{\mathrm{*}}_{\mathrm{2}}$ |  8.289 $\mu$s |  0.929 $\mu$s |  —
Hahn echo time |  $T^{\mathrm{E}}_{\mathrm{2}}$ |  8.475 $\mu$s |  0.934 $\mu$s |  —

We show a typical measurement sequence for the characterizion of the on-chip
filters in Fig. S6 for qubit Q2 coupled to the $\lambda/4$ filter. We begin by
determining the flux-bias-voltage-dependent frequencies of the resonator and
of the qubit through the single-tone and two-tone spectroscopy measurements,
respectively, as depicted in Fig. S6a, b. We fit the resonator and the qubit
spectroscopic data to the analytical model of Ref. [62], as shown in Fig. S6c.
With this fit, we can automate the search of the resonator and the qubit
frequencies in large frequency ranges for a given bias voltage.

To measure the Rabi frequency and the qubit $T_{1}$, we first determine the DC
voltage for the desired qubit flux bias according to Fig. S6c. Subsequently,
we set the correct readout frequency according to the fit in Fig. S6c. Next,
we again carry out qubit spectroscopy at very low power, fitting the data to a
Lorentzian curve to accurately determine the qubit frequency.

Using the carefully extracted qubit frequency, we implemented Rabi
oscillations and fit the data to a decaying cosine function to determine the
$\pi$-pulse length. The Rabi oscillation measurements are conducted at fixed
pulse voltages of 0.05 V and 0.025 V for the $\lambda/4$ and $\lambda/2$
filters, respectively. We then measure the $T_{1}$ time by applying a $\pi$
pulse, as obtained from the fit, and by varying the delay before the final
readout pulse. We fit the measured data to a decaying exponential function to
obtain the $T_{1}$ time and its uncertainty. The entire measurement sequence
is automated, requiring only the bias voltage as an input parameter.

## Supplementary Note S5: Observations and discussion on $T_{1}$

We observe a maximum measured $T_{1}$ of around 10 $\mu\mathrm{s}$ across all
qubits. To investigate the reason behind this bound, we measure the $T_{1}$ of
qubit Q1 in the vicinity of its resonator frequency. We fit the measured data
to the Purcell decay and the dielectric-loss models, as shown in Supplementary
Fig. S7a. For the dielectric-loss model, we follow a similar approach used for
the Fluxonium and the Unimon qubit [63, 64]. However for the transmon qubit,
we assume the capacitive loss to be dominant and utilize the relation

$\displaystyle 1/T_{1,\mathrm{diel}}$
$\displaystyle=\frac{16E_{\mathrm{C}}}{\hbar
Q_{\mathrm{cap}}}\coth\bigg{(}{\frac{\hbar\omega_{\mathrm{q}}}{2k_{\mathrm{B}}T}}\bigg{)}\big{|}\bra{0}\hat{n}\ket{1}\big{|}^{2}$
(8) $\displaystyle=\frac{\sqrt{8E_{\mathrm{J}}E_{\mathrm{C}}}}{\hbar
Q_{\mathrm{cap}}}\coth\bigg{(}{\frac{\hbar\omega_{\mathrm{q}}}{2k_{\mathrm{B}}T}}\bigg{)},$
(9)

where $\hbar$ is the reduced Planck constant, $k_{\mathrm{B}}$ is the
Boltzmann constant, $\omega_{\mathrm{q}}$ is the angular frequency of the
qubit, $E_{\mathrm{J}}$ is the Josephson energy, $E_{\mathrm{C}}$ is the
charging energy, $T$ is the temperature of the dissipative environment,
$\hat{n}=\textrm{i}\big{(}\frac{E_{\mathrm{J}}}{8E_{\mathrm{C}}}\big{)}^{\frac{1}{4}}(\hat{b}^{\dagger}-\hat{b})$
is the number operator of the qubit with $\hat{b}$ and $\hat{b}^{\dagger}$ as
the annihilation and creation operators. The ket vectors $\ket{0}$ and
$\ket{1}$ represents the ground and the excited state of the qubit,
respectively.

We fit the equation (9) to the measured data to obtain a realistic estimate
for the loss tangent $\tan\delta=1/Q_{\mathrm{cap}}$. From the fit, we obtain
a value of $\tan\delta=3\times 10^{-6}$, suggesting that dielectric loss is
the primary contributor to the observed $T_{1}$ bound. For the Purcell decay
model, we employ the relation $1/T_{\mathrm{1,Purcell}}=\kappa
g^{2}/|\omega_{\mathrm{q}}-\omega_{\mathrm{r}}|^{2}$, where $g$ is the
coupling strength between the qubit and the resonator, $\kappa$ is the
linewidth of the resonator, and $\omega_{\mathrm{r}}$ is the angular frequency
of the resonator.

Supplementary Fig. S7: Energy decay time $T_{1}$ of each qubit at different
frequencies. a Measured $T_{1}$ of qubit Q1 (markers) and corresponding
results of fitted models (dashed lines), Purcell decay (red) and dielectric-
loss (yellow), as functions of qubit frequency. From the fit to the
dielectric-loss model, we obtain a loss tangent of $3\times 10^{-6}$,
manifesting as the dominant loss mechanism in our devices far from the
resonance of the qubit and its readout resonator. b,c Measured $T_{1}$ of
qubit (b) Q2 and (c) Q3 with an additional off-chip low-pass filter with a
cut-off frequency at around 3 GHz installed in qubit control line.

Furthermore, following a similar approach to the one used in Ref. [56], we
show that the use of off-chip low-pass filter in the qubit control line,
facilitating a passband at sub-harmonic frequency while introducing a stopband
at qubit frequency, creates an unwanted resonance in the qubit drive line that
modulates the measured $T_{1}$ in extreme ranges, as depicted in Supplementary
Fig. S7b, c. By employing the integrated on-chip filters, we are not faced
with such an issue and the resulting engineering and scaling challenge.