# Many-excitation removal of a transmon qubit using a single-junction quantum-
circuit refrigerator and a two-tone microwave drive

Wallace Teixeira  QCD Labs, QTF Centre of Excellence, Department of Applied
Physiscs, Aalto University, P.O. Box 13500, FI-00076 Aalto, Finland
<EMAIL_ADDRESS>Timm Mörstedt  QCD Labs, QTF Centre of
Excellence, Department of Applied Physiscs, Aalto University, P.O. Box 13500,
FI-00076 Aalto, Finland  Arto Viitanen  QCD Labs, QTF Centre of Excellence,
Department of Applied Physiscs, Aalto University, P.O. Box 13500, FI-00076
Aalto, Finland  Heidi Kivijärvi  QCD Labs, QTF Centre of Excellence,
Department of Applied Physiscs, Aalto University, P.O. Box 13500, FI-00076
Aalto, Finland  András Gunyhó  QCD Labs, QTF Centre of Excellence, Department
of Applied Physiscs, Aalto University, P.O. Box 13500, FI-00076 Aalto, Finland
Maaria Tiiri  QCD Labs, QTF Centre of Excellence, Department of Applied
Physiscs, Aalto University, P.O. Box 13500, FI-00076 Aalto, Finland  Suman
Kundu  QCD Labs, QTF Centre of Excellence, Department of Applied Physiscs,
Aalto University, P.O. Box 13500, FI-00076 Aalto, Finland  Aashish Sah  QCD
Labs, QTF Centre of Excellence, Department of Applied Physiscs, Aalto
University, P.O. Box 13500, FI-00076 Aalto, Finland  Vasilii Vadimov  QCD
Labs, QTF Centre of Excellence, Department of Applied Physiscs, Aalto
University, P.O. Box 13500, FI-00076 Aalto, Finland  Mikko Möttönen  QCD Labs,
QTF Centre of Excellence, Department of Applied Physiscs, Aalto University,
P.O. Box 13500, FI-00076 Aalto, Finland  VTT Technical Research Centre of
Finland Ltd., QTF Center of Excellence, P.O. Box 1000, FI-02044 VTT, Finland

###### Abstract

Achieving fast and precise initialization of qubits is a critical requirement
for the successful operation of quantum computers. The combination of
engineered environments with all-microwave techniques has recently emerged as
a promising approach for the reset of superconducting quantum devices. In this
work, we experimentally demonstrate the utilization of a single-junction
quantum-circuit refrigerator (QCR) for an expeditious removal of several
excitations from a transmon qubit. The QCR is indirectly coupled to the
transmon through a resonator in the dispersive regime, constituting a
carefully engineered environmental spectrum for the transmon. Using single-
shot readout, we observe excitation stabilization times down to roughly 500
ns, a 20-fold speedup with QCR and a simultaneous two-tone drive addressing
the $e$–$f$ and $f0$–$g1$ transitions of the system. Our results are obtained
at a $48$-mK fridge temperature and without postselection, fully capturing the
advantage of the protocol for the short-time dynamics and the drive-induced
detrimental asymptotic behavior in the presence of relatively hot other baths
of the transmon. We validate our results with a detailed Liouvillian model
truncated up to the three-excitation subspace, from which we estimate the
performance of the protocol in optimized scenarios, such as cold transmon
baths and fine-tuned driving frequencies. These results pave the way for
optimized reset of quantum-electric devices using engineered environments and
for dissipation-engineered state preparation.

## Introduction

The rapid progress in applications [1, 2, 3] of superconducting quantum
computers brings not only a critical need for improved hardware components,
but also optimized techniques for the control of their basic building blocks,
the qubits. The constantly increasing complexity of the quantum processors and
the emergence of quantum error correction highlight the need for fast and
accurate initialization of the device to a known eigenstate.

Although unconditional and straightforward, the passive relaxation towards
thermal equilibrium is notably inefficient or even insufficient for reset,
especially given the demand for increasing qubit lifetimes in advanced
operations. To evade the initialization errors arising from unwanted
excitation to higher eigenstates and the feedback imposed by measurement-based
protocols, parameter tunability plays an important role in reset strategies.
Recent reset schemes for superconducting circuits employ this tunability
through different ways, such as frequency modulation [4, 5], all-microwave
control [6, 7, 8], and engineered environments [9, 10, 11, 12].

In this work, we experimentally combine the latter two approaches to
demonstrate enhanced removal of excitations from a transmon qubit. On one
hand, microwave driving allows for excitation transfer from the transmon to a
dispersively coupled resonator in a cavity-assisted Raman process [13], and on
the other hand, the so-called quantum-circuit refrigerator (QCR) coupled to
the resonator promotes an overall increase in decay rates through photon-
assisted quasiparticle tunneling [14, 15], with a minimal effect on qubit
lifetimes in its off-state.

Our work is motivated by the recent experimental realization of transmon
initialization using a QCR and two-tone driving [16], showing a
$99.5\%$-reset-fidelity in $180$ ns if no leakage to high-energy states is
considered. Here, we expand the approach by showing the excitation removal
also from the second excited state of the transmon and by studying the
detrimental effects owing to relatively high temperatures. In contrast to Ref.
[16], we employ single-shot qubit readout, allowing us to individually capture
the contributions arising from the different transmon states. Moreover, the
design of our sample is fundamentally different from that measured in Ref.
[16]. In addition to the presence of a dedicated readout resonator, our QCR
structure is based on a single normal-metal–insulator–superconductor (NIS)
junction embedded in an auxilliary resonator, allowing for mitigation of
charge noise and relief in parameter optimization for the readout and reset
compared with the double-junction QCR design [17].

Our experimental results demonstrate not only a moderate speedup in transmon
decay dynamics facilitated by the QCR pulse but also a $20$-fold enhancement
in short-time excitation decay owing to additional two-tone driving, even in
the presence of a relatively hot transmon bath. Importantly, we use a
theoretical model that extends the multilevel structure of the
transmon–auxiliary-resonator system up to the three-excitation subspace,
encompassing the second and third excited states of the transmon. In the case
of an extremely cold transmon bath and fine-tuned driving frequencies, this
model predicts an asymptotic excitation probability $\sim\\!10^{-8}$, even
with conservative QCR pulse amplitudes.

## Results

### General description

To demonstrate the QCR-expedited excitation removal with the two-tone drive,
we carry out experiments with the sample depicted in Fig. 1a. Here, a transmon
qubit is coupled to the flux and drive lines, and to a quarter-wave readout
resonator of frequency $\omega_{\text{RO}}/(2\pi)=7.437$ GHz. To prevent
undesirably high decay rates when not biased, the single-junction QCR is
indirectly coupled to the transmon through an auxiliary quarter-wave resonator
of frequency $\overline{\omega}_{\text{r}}/(2\pi)=4.671$ GHz. Both resonators
are designed to have relatively low quality factors and well-separated
frequencies among each other yet within the strong dispersive regime with
respect to the transmon frequency. The QCR can be carefully biased with both
dc and rf signals to operate in the cooling regime, where the system
temperature may be below the normal-metal temperature. This bias point also
promotes increased decay rates on demand to the low-temperature bath defined
by the QCR.

(a)

(b)

(c)

(d)

Figure 1: Device, pulse protocol, and energy-level diagram of the system. (a)
Micrograph of a sample similar to that utilized in the experiments together
with a simplified driving setup. A transmon is coupled to a readout resonator
and to an auxiliary resonator which is also coupled to the single-junction
quantum-circuit refrigerator (QCR) detailed in the inset. Additional transmon
control is provided through dedicated drive and flux lines. (b) Pulse protocol
describing the state preparation, the QCR-assisted two-tone reset, and the
readout of the system. The transmon is prepared in its first or second excited
state through a sequence of $\pi$ pulses. During the reset stage of length
$t_{\rm{r}}$, the $e$–$f$ and $f0$–$g1$ transitions of the transmon–auxilary-
resonator system are activated with a two-tone drive with adjustable
amplitudes $V_{\rm{ef}}$ and $V_{\rm{f0g1}}$, respectively. Simultaneously,
the QCR is operated with a net-zero sinusoidal pulse with a fixed frequency
and an adjustable amplitude $V_{\rm{QCR}}$. Subsequently, a 2.0-µs pulse is
applied for the dispersive readout, from which the transmon populations are
estimated through a four-component Gaussian mixture model describing the
single-shot data in the IQ plane integrated from $0.4$ to $1.0$ µs (see the
Methods). (c, d) Effective multilevel structure of the transmon–auxiliary-
resonator system during the reset stage truncated to the ten (c) and four (d)
lowest-energy eigenstates of the system. The green and black bidirectional
arrows represent the two-tone-drive-induced coupling between the levels
$|\overline{e,n}\rangle\leftrightarrow|\overline{f,n}\rangle$ and
$|\overline{f,n}\rangle\leftrightarrow|\overline{g,n+1}\rangle$, respectively.
The curly unidirectional arrows indicate the incoherent transitions of the
system, which can be potentially modified by the QCR pulse.

The sample is fabricated on a six-inch prime-grade intrinsic-silicon wafer.
The superconducting circuit consists of a 200-nm Nb film patterned by dry-
etching and the Josephson-junctions are deposited by shadow evaporation using
in-situ oxidation to create the tunnel barrier. The NIS junction is fabricated
in a similar way, replacing Al with Cu on the top electrode. All measurements
are carried out with a dilution refrigerator at the base temperature of $48$
mK and at the flux sweet spot of the transmon. The measured and simulated
parameters are provided in Table 1. Previous characterization of this sample
has also been reported in Ref. [18].

Table 1: Summary of the relevant experimental and simulation parameters. The superscripts ‘off’ and ‘on’ highlight parameters obtained in the QCR-off and QCR-on states, respectively. The asterisk (*) refers to the parameter values estimated from fitting the numerical model to the experimental data or predicted from theory, see the Methods for details. Parameter | Symbol | Value
---|---|---
| transition frequency
---
$g$–$e$
$e$–$f$
$f$–$h$
$f0$–$g1$
aux resonator
RO resonator
| $\overline{\omega}_{\text{ge}}/(2\pi)$
---
$\overline{\omega}_{\text{ef}}/(2\pi)$
$\overline{\omega}_{\text{fh}}/(2\pi)$
$\overline{\omega}^{\text{off(on)}}_{\text{f0g1}}/(2\pi)$
$\overline{\omega}^{\text{off(on)}}_{\text{r}}/(2\pi)$
$\omega_{\text{RO}}/(2\pi)$
|  $4.089$ GHz
---
$3.816$ GHz
$3.486$ GHz
$3.230\ (3.231)$ GHz
$4.671\ (4.670)$ GHz
$7.437$ GHz
| decay time, occupation number
---
$e$–$g$
$f$–$e$*
$h$–$f$*
aux resonator*
|  $1/\gamma_{\text{eg}}^{\text{off(on)}}$, $\bar{n}_{\text{eg}}$
---
$1/\gamma_{\text{fe}}^{\text{off(on)}}$, $\bar{n}_{\text{fe}}$
$1/\gamma_{\text{hf}}^{\text{off(on)}}$, $\bar{n}_{\text{hf}}$
$1/\kappa^{\text{off(on)}}$, $\bar{n}_{\text{r}}$
|  $6.6\ (4.9)$ µs, $0.20$
---
$3.3\ (2.5)$ µs, $0.23$
$2.2\ (1.6)$ µs, $0.28$
$221\ (120)$ ns, $0.15$
| dephasing time
---
$e$–$g$*
$f$–$e$*
$h$–$f$*
| $1/\gamma_{\text{eg}}^{\phi,\text{off(on)}}$
---
$1/\gamma_{\text{fe}}^{\phi,\text{off(on)}}$
$1/\gamma_{\text{hf}}^{\phi,\text{off(on)}}$
|  $1.7\ (1.2)$ µs
---
$0.8\ (0.6)$ µs
$0.6\ (0.4)$ µs
Dynes parameter | $\gamma_{\text{D}}$ | $2.3\times 10^{-3}$
QCR tunneling resistance | $R_{\text{T}}$ |  $13.8$ k$\Omega$

We use four arbitrary waveform generators (AWGs) to implement the pulse
sequence depicted in Fig. 1b. The transmon is prepared with flat-top
$\pi$-pulses, one being addressed to the $g$–$e$ transition,
$\overline{\omega}_{\text{ge}}/(2\pi)=4.0885$ GHz, and another subsequently
addressed to the $e$–$f$ transition,
$\overline{\omega}_{\text{ef}}/(2\pi)=3.8155$ GHz. During the reset stage, two
simultaneous microwave drives, with amplitudes $V_{\text{ef}}$ and
$V_{\text{f0g1}}$ addressing the $e$–$f$ and $f0$–$g1$ transitions of the
transmon–auxiliary-resonator system, respectively, are combined at room
temperature, and the signal is applied to the transmon through its drive line.
In addition, the QCR is simultaneously biased with a $100$-MHz sine wave pulse
of amplitude $V_{\text{QCR}}$ well-below $\Delta/e$, where $e$ is the
elementary charge and $\Delta=0.215$ meV is the superconductor gap parameter
of aluminium. Careful calibration of the drive frequencies of the transmon is
needed due to the ac-Stark shifts caused by the $f0g1$ drive and the Lamb
shift of the auxiliary resonator frequency induced by the QCR.

The mechanism underlying the reset protocol is illustrated in Fig. 1c. In the
dispersive regime, the transmon and the auxiliary resonator form a hybrid
system characterized by the weakly entangled states $|\overline{j,n}\rangle$,
where $j=g,e,f,h,...$, and $n=0,1,2,...$ refer to the energy levels of their
bare states. The transmon excitations are transferred to the resonator with
the help of the two-tone drive, which couples the levels
$|\overline{e,n}\rangle\leftrightarrow|\overline{f,n}\rangle$ and
$|\overline{f,n}\rangle\leftrightarrow|\overline{g,n+1}\rangle$ with Rabi
rates $\Omega_{\rm{ef}}$ and $\Omega_{\rm{f0g1}}\sqrt{n+1}$, respectively. The
frequencies of the two tones are calibrated to be resonant with the
transitions $|\overline{e,0}\rangle\leftrightarrow|\overline{f,0}\rangle$ and
$|\overline{f,0}\rangle\leftrightarrow|\overline{g,1}\rangle$. Owing to the
relatively strong dissipation in the auxiliary resonator, the system rapidly
decays to the ground state $|\overline{g,0}\rangle$ in the absence of heating
effects. This fast decay is potentially enhanced by the application of the QCR
pulse since it may increase the decay rates of the system. However, at
increased temperatures, the effective decoherence channels modified by two-
tone driving tend to exhibit a more involved structure due to the multi-level
nature of the system, leading to more residual excitations than in the absence
of driving as demonstrated below. In this work, we show that the two-tone
protocol assisted by the QCR can, nevertheless, provide a substantial speed up
of the short-time transmon dynamics, even in the presence of heating and small
deviations from the ac-Stark-shifted frequencies. Our results are validated
through a Liouvillian model truncated up to the three-excitation subspace with
ten states as detailed in the Methods.

### Pulse calibration

We begin with the calibration of the pulses for the reset protocol by applying
a QCR pulse with variable amplitude $V_{\text{QCR}}$ and length $t_{\text{r}}$
in the absence of the two-tone drive as shown in Fig. 2. The transmon is
initialized in its first (Fig. 2a) or second excited state (Fig. 2b), and the
amplitude of the voltage signal transmitted through the readout line is
averaged over $20,000$ realizations. For all reset lengths $t_{\text{r}}$, the
delay time between the QCR and readout pulses is fixed to $20$ ns. From
exponential fits to the data (Fig. 2c), and for the first excited state, we
extract the stabilization times $T_{\rm{d}}=6.6$ µs in the QCR-off state,
i.e., at $V_{\text{QCR}}=0.0$ mV and $T_{\rm{d}}=4.9$ µs in the QCR-on state
with amplitude $V_{\text{QCR}}=0.16$ mV. For the second excited state, the
stabilization times at the mentioned amplitudes are $T_{\rm{d}}=10.4$ µs and
$T_{\rm{d}}=5.4$ µs, respectively. Roughly, one can think of $T_{\rm{d}}$ as
the stabilization timescale of the system after being prepared in a given
excited state. Thus, the QCR pulse accelerates the stabilization of the
aforementioned prepared states by a factor of $1.35$ and $1.93$, respectively.
Note that the averaged signal amplitude is a weighted sum of the quantum-sate
probabilities. Consequently, its deviation from the ground-sate value
indicates a finite excitation probability of the transmon without
distinguishing the individual contributions of the excited states which
require single-shot measurements.

(a)

(b)

(c)

Figure 2: QCR pulse calibration. (a, b) $20,000$-averaged-readout signal
amplitude for different QCR pulse amplitudes $V_{\rm{QCR}}$ and reset lengths
$t_{\rm{r}}$ in absence of the two-tone drive, $V_{\rm{ef}}=V_{\rm{f0g1}}=0$.
The readout frequency is set to $f_{\rm{RO}}=7.4372$ GHz. The panels show data
for a state initialized (a) with a single $\pi_{\rm{ge}}$ pulse and (b) with a
sequence of $\pi_{\rm{ge}}$ and $\pi_{\rm{ef}}$ pulses. The dashed lines in
both plots highlight the amplitudes $V_{\rm{QCR}}=0.0$, $0.16$, and $0.22$ mV.
(c) Traces corresponding to the highlighted QCR amplitudes in (a, b) with
matching colors. The solid lines represent exponential fits, from which the
characteristic stabilization times $T_{\rm{d}}$ are extracted accordingly.

Although faster stabilization times are obtained at higher QCR pulse
amplitudes, e.g., $V_{\text{QCR}}=0.22$ mV, the system tends to stabilize to a
different state from that in the QCR-off-state scenario, as suggested by the
saturation of the signal amplitude at high values. This state is expected not
to be the ground state but to follow a distribution close to that of a thermal
state with a nonzero temperature. As shown in previous work [14, 15], for
relatively high normal-metal temperatures, the cooling range of the QCR shifts
to lower amplitudes, where the decay rates are not optimal. Therefore, to
avoid additional heating of the transmon, we conservatively set the QCR-on-
state amplitude to $V_{\text{QCR}}=0.16$ mV in the subsequent measurements. We
highlight that fast on-demand generation of thermal states can be achieved
[19] if the QCR operates with voltages beyond $\Delta/e$, however, this is out
of the scope of the present work.

We proceed with the calibration of the two-tone drive as shown in Fig. 3.
First, we initialize the system with a sequence of $\pi_{\text{ge}}$ and
$\pi_{\text{ef}}$ pulses. Subsequently, we apply a $500$-ns drive of amplitude
$V_{\rm{f0g1}}$ at frequency $f_{\rm{f0g1}}$, and read out the averaged signal
amplitude. The drive frequency is swept around the transition frequency
$\overline{\omega}_{\text{f0g1}}/(2\pi)$ between the levels
$|\overline{f,0}\rangle\leftrightarrow|\overline{g,1}\rangle$. This procedure
is repeated in the QCR-off state (Fig. 3a) and in the QCR-on state (Fig. 3b),
both without the $e$–$f$ drive.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

Figure 3: Two-tone drive and Rabi angular frequency calibration. (a, b)
Averaged-readout signal amplitude after a 500-ns drive of amplitude
$V_{\rm{f0g1}}$ and frequency $f_{\rm{f0g1}}$ in absence of the $e$–$f$ drive,
$V_{\rm{ef}}=0$. The system is initialized with a sequence of $\pi_{\rm{ge}}$
and $\pi_{\rm{ef}}$ pulses. The dips in the data indicate the Stark-shifted
frequency of the $f0$–$g1$ transition, which is fitted with a quadratic
function (solid white curve). The panels show data for QCR amplitudes (a)
$V_{\rm{QCR}}=0.0$ and (b) $0.16$ mV. The dashed lines highlight the
amplitudes $V_{\rm{f0g1}}=0.17$ and $0.25$ mV. (c) Traces corresponding to the
highlighted values of $V_{\rm{f0g1}}$ in (a, b), with the corresponding color
codes and asymmetric Lorentzian fits (solid lines). (d) Temporal dependence of
the signal amplitude in the presence of the $f0$–$g1$ drive with
$V_{\rm{f0g1}}=0.17$ (magenta) and $V_{\rm{f0g1}}=0.25$ mV (blue) for the
transmon initialized in the second excited state. The corresponding drive
frequencies are chosen to be near the Stark-shifted frequencies extracted from
(a, b). (e) Dependence of the Rabi angular frequency $\Omega_{\rm{f0g1}}$ on
the drive amplitude $V_{\rm{f0g1}}$ for the QCR off (magenta) and on (blue).
The dashed lines highlight the values of $V_{\rm{f0g1}}$ used in (d). (f)
Temporal variation of the signal amplitude in the presence of the $e$–$f$
pulse and in the absence of the $f0$–$g1$ and QCR pulses,
$V_{\rm{f0g1}}=V_{\rm{QCR}}=0$. The transmon is initialized in the first
excited state. (g) Dependence of the Rabi angular frequency $\Omega_{\rm{ef}}$
on the drive amplitude $V_{\rm{ef}}$. The dashed lines highlight the values of
$V_{\rm{ef}}$ used in (f). The time-dependent fitting functions of (d, f) are
obtained from the theoretical model described in the Methods.

For relatively high amplitudes, we observe that the $f0$–$g1$ drive induces a
clear ac-Stark shift of the resonance frequency in both QCR-on and QCR-off
states, $\overline{\omega}_{\text{f0g1}}^{\text{on/off}}/(2\pi)$. In both
Figs. 3a and 3b, we select the lowest value of the signal amplitude for each
$V_{\rm{f0g1}}$ and fit the data to a quadratic function, obtaining the base
frequencies $\overline{\omega}_{\text{f0g1}}^{\text{off}}/(2\pi)=3.2302$ GHz,
$\overline{\omega}_{\text{f0g1}}^{\text{on}}/(2\pi)=3.2306$ GHz, and negative
ac-Stark shifts up to $6.4$ MHz.

The simultaneous QCR pulse (Fig. 3b) effectively increases the $f0g1$
resonance frequency up to $0.4$ MHz as a consequence of a negative Lamb shift
of the auxiliary resonator frequency [20]. Additionally, an overall broadening
of the spectral line is observed in the QCR-on state. Since the QCR operates
with a relatively low voltage bias, we attribute this effect to the increase
in the decay rates of the system. Both Lamb shift and line broadening for
selected drive amplitudes are shown in Fig. 3c.

Next, we calibrate the Rabi frequencies of the two-tone drive through time-
domain measurements. To extract their dependence on the corresponding pulse
amplitudes, we use a Liouvillian model truncated to the fourth level of the
system, as detailed in the Methods. First, we set $V_{\text{ef}}=0$ and study
the decay of the second excited state of the transmon in the presence of
$f0$–$g1$ and QCR pulses, as shown in Fig. 3d. In general, we observe damped
oscillations indicating the rapid excitation decay from the auxiliary
resonator. Figure 3e reveals a good linear dependence between
$\Omega_{\text{f0g1}}/(2\pi)$ and $V_{\text{f0g1}}$, with the slope barely
affected by the QCR pulse. Although the $f0$–$g1$ drive frequencies are chosen
according to the quadratic fits in Figs. 3a and 3b, the oscillation pattern in
Fig. 3d is also affected by coarse frequency adjustments resulting from the
low resolution in the $f0g1$ frequency sweep. The fitting from the Liouvillian
model also enables the estimation of average auxiliary resonator decay rates
in the QCR-off and QCR-on states, $1/\kappa^{\text{off}}=221$ ns and
$1/\kappa^{\text{on}}=120$ ns, respectively.

A similar calibration is carried out for the $e$–$f$ drive, as shown in Figs.
3f and 3g. The system is initialized in the first excited state, after which
damped Rabi oscillations are observed. Here, we set
$V_{\text{f0g1}}=V_{\text{QCR}}=0$, such that the oscillations are only
attributed to the weak $e$–$f$ drive (Fig 3f). We observe that also
$\Omega_{\rm{ef}}/(2\pi)$ exhibits a linear dependence with $V_{\rm{ef}}$ (Fig
3g). Due to the single-photon nature of the $e$–$f$ transition, it requires a
much lower driving amplitude for a given Rabi frequency compared to the
$f0$–$g1$ transition which involves two-photon processes. Although the $e$–$f$
drive contributes to negligible ac-Stark shifts, its frequency needs to be
adjusted in the presence of the $f0$–$g1$ drive.

### Excitation dynamics

After the pulse calibration detailed above, we carry out single-shot readout
of the transmon with the pulse configurations described in Table 2. In the
configurations A and B, the QCR is off, and configurations C and D comprise
the QCR-on state. Figures 4a–4f show examples of $8,000$ single-shot data
points each in the in-phase–quadrature-phase (IQ) plane for configuration D.
The labels $|g\rangle$, $|e\rangle$, and $|f\rangle$ indicate the ideal
initial states, which are prepared or not with the corresponding sequence of
$\pi$ pulses.

Table 2: Voltage amplitudes of pulses defined in Fig. 1 for the reset configurations A–D investigated through single-shot measurements. These correspond to the values estimated at the input ports of the sample. Pulse amplitude (µV) | A | B | C | D
---|---|---|---|---
$V_{\text{QCR}}$ | 0.0 | 0.0 | 160.0 | 160.0
$V_{\text{f0g1}}$ | 0.0 | 169.1 | 0.0 | 253.6
$V_{\text{ef}}$ | 0.0 | 10.6 | 0.0 | 16.9

In general, we observe four prominent point clouds associated to the lowest
states of the transmon. While driving errors and measurement-induced
transitions are common sources of leakage [21], we attribute the emergence of
more than two point clouds primarily to the relatively hot transmon
environment. These point clouds persist even in the thermal equilibrium state
(Fig. 4a).

For the reset length $t_{\text{r}}=2.0$ µs (Figs. 4d–4f), the points cluster
predominantly in the leftmost ground-state cloud, indicating the excitation
removal assisted by the QCR and two-tone drive. We highlight that the single-
shot data corresponds to the raw integrated timetraces at an optimized readout
frequency, without the use of weighting functions or postselection. Since the
readout resonator couples directly to the transmon, the auxiliary resonator
states are indistinguishable in the single-shot measurements.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

Figure 4: Single-shot data and excitation dynamics. (a–f) Example
distributions of $8,000$ single-shot measurements acquired after the reset
times (a–c) $t_{\rm{r}}=0$ and (d–f) $t_{\rm{r}}=2$ µs for the different
prepared states as indicated and with the reset pulse amplitudes in
configuration D defined in Table 2. The labels $|g\rangle$, $|e\rangle$, and
$|f\rangle$ refer to the qubit states initialized by starting from the thermal
equilibrium, with an additional $\pi_{\rm{ge}}$ pulse, and with a sequence of
$\pi_{\rm{ge}}$ and $\pi_{\rm{ef}}$ pulses, respectively. The dashed-line
ellipses are obtained from the four-component Gaussian mixture model
calibrated at $t_{\rm{r}}=0$. (g–l) Excitation probability $P_{\rm{exc}}$ as a
function of the reset time $t_{\rm{r}}$ for different prepared states as
indicated. The pulse amplitudes in (g, i, k) are chosen as in configurations A
or B and in (h, j, l) as in C or D of Table 2. Solid curves are obtained from
the master equation simulation of the Liouvillian $\mathcal{L}$ truncated at
the tenth level with corrections for the Stark shifts of the
$|\overline{e,0}\rangle$, $|\overline{g,1}\rangle$, and
$|\overline{f,0}\rangle$ levels. Remaining parameters used in the simulations
are given in Table 1. The grey curves in (j, l) are from the simulation
truncated to the tenth level with drives as in configuration D, but with the
estimated value of $\kappa$ in the QCR-off state. Dotted vertical lines in (h,
j, l) indicate the values of $t_{\textrm{r}}$ in (a–f).

To further investigate the performance of the reset protocol, we study the
dynamics of the excitation probability,

$\displaystyle
P_{\text{exc}}=p_{\text{e}}+p_{\text{f}}+p_{\text{h}}=1-p_{\text{g}},$ (1)

where $p_{j}$ is the probability of the transmon state $|j\rangle$.
Experimentally, $p_{j}$ is quantified from the single-shot measurements by a
normalized number of data points inside the corresponding ellipse, which is
determined using a four-component Gaussian mixture model, see the Methods.
Theoretically, we define the transmon populations as

$\displaystyle
p_{j}^{\text{theory}}=\sum_{n}\langle\overline{j,n}|\hat{\rho}|\overline{j,n}\rangle,$
(2)

where $\hat{\rho}$ is the interaction picture density operator of the
transmon–auxiliary-resonator system, and the interaction picture is defined by
the multilevel Jaynes-Cummings Hamiltonian. The dynamics of $\hat{\rho}$
follows from a Lindblad master equation
$\textrm{d}\hat{\rho}/\textrm{d}t=\mathcal{L}(\hat{\rho})$, where
$\mathcal{L}$ is the Liouvillian superoperator describing the relevant
processes during the reset stage, see the Methods.

Figures 4g–4l show the dynamics of $P_{\text{exc}}$ in the driving
configurations of Table 2 for different initial states. Starting from the
thermal equilibrium state in Figs. 4g and 4h, we observe a weak effect of the
QCR pulse in the absence of two-tone driving in configuration C, as
$P_{\text{exc}}$ exhibits tiny oscillations near the equilibrium value without
the QCR pulse, given in configuration A. From the individual contributions
$p_{j}$, the thermal equilibrium temperature of the transmon is determined as
$T=110$ mK through fitting $p_{j}$ to the closest Boltzmann distribution, and
the average thermal occupation number at the relevant frequencies of the
system given in Table 1. For configurations B and D, the two-tone driving
induces excessive heating indicated by higher $P_{\text{exc}}$ than that
without the drive.

In Figs. 4i–4l where the system is prepared with $\pi$ pulses, we note a
moderate speedup of the decay by the QCR pulse alone, as also suggested in
Fig. 2. Despite the coarse frequency tuning and other potential driving
errors, the benefits of the two-tone pulse for the short-time removal of
$P_{\text{exc}}$ are clearly visible for the considered initial states. In
particular, we obtain in Fig. 4l a rough stabilization time of
$T_{\text{d}}\sim 500$ ns for configuration D if the system is initialized in
the second excited state. This corresponds to an over 20-fold speedup in
comparison to the corresponding intrinsic dynamics.

Figures 4g–4l exhibit a very good agreement between the experimental data and
the master equation simulations. In the absence of two-tone driving, the
simulations very accurately reproduce the expected thermal-equilibrium and
excitation-decay profiles. Furthermore, our model, which takes into account
additional corrections to the energy levels, captures the main dynamics in the
configurations with two-tone driving. With the two-tone Rabi frequencies
$[\Omega_{\text{f0g1}}/(2\pi),\Omega_{\text{ef}}/(2\pi)]=(1.16,1.43)$ MHz in
configuration B and
$[\Omega_{\text{f0g1}}/(2\pi),\Omega_{\text{ef}}/(2\pi)]=(1.73,2.29)$ MHz in
configuration D, we expect the reset to operate in the underdamped regime in
both cases. In addition to the state preparation and measurement (SPAM)
errors, we attribute the minor discrepancies between the simulations and the
experiments to possibly imperfect estimation of the decay rates of the
auxiliary resonator and the higher energy levels of the transmon.

The accelerated decay of $P_{\text{exc}}$ produced by the two-tone drive can
be connected to the eigenspectrum of the Liouvillian $\mathcal{L}$, denoted by
$\\{\lambda_{j}\\}$. The spectrum and the corresponding eigenoperators
trivially yield the dynamics and the steady state. Specifically, the presence
of at least one null eigenvalue, $\lambda_{j}=0$, implies the existence of a
steady state. The negative real parts of the remaining eigenvalues are
associated to the rates of the corresponding incoherent processes [22, 23],
namely, $\Lambda_{j}=|\textrm{Re}(\lambda_{j})|$. In general, the
stabilization timescale is set by the minimal non-zero rate $\Lambda_{j}$. In
practice, however, the influence of this term in the dynamics depends on the
initial state.

Figure 5a compares the smallest rates $\Lambda_{j}$ obtained in configurations
C and D. An increase in these rates for the latter configuration is observed.
Owing to the well-separated timescales set by the different decay rates of the
transmon and the auxiliary resonator, we expect the smallest non-zero rates
$\Lambda_{j}$ to be the most relevant in the reset protocol. However, owing to
the relatively hot bath, two-tone driving tends to increase the asymptotic
excitation probability, $P_{\text{exc}}^{\text{ss}}$, as shown in Fig. 5b.
This compromise can be significantly reduced in the presence of a cold bath
and with fine-tuning of the driving frequencies, as illustrated in Figs. 5c
and 5d. In such cases, two-tone driving can provide substantial speedup in
excitation removal with asymptotic values
$P_{\text{exc}}^{\text{ss}}\sim\\!10^{-8}$.

(a)

(b)

(c)

(d)

Figure 5: Decay rates and asymptotic excitation probability of the
transmon–auxiliary-resonator system. (a) Ten lowest incoherent rates
$\Lambda_{j}$ given by the Liouvillian $\mathcal{L}$ truncated at the tenth
level with drive amplitudes in configurations C and D as indicated. The
auxiliary-resonator decay rate is denoted by $\kappa$. (b) Asymptotic, long-
time excitation probability $P_{\rm{exc}}^{\rm{ss}}$ as a function of the Rabi
angular frequencies $\Omega_{\rm{f0g1}}$ and $\Omega_{\rm{ef}}$ for the model
truncated at the tenth level. This scenario comprises a relatively hot
transmon bath and coarse frequency adjustment as in the modeling of
experimental data in Fig. 4. (c, d) As (a, b) except for an optimized scenario
comprising a cold transmon bath and fine-tuned driving frequencies. In all
plots, decay and dephasing rates are chosen as in Fig. 4.

## Discussion

We have experimentally demonstrated the many-excitation removal from a
transmon qubit assisted by a quantum-circuit refrigerator in combination with
a two-tone microwave drive. Following a careful pulse calibration, we observed
the dynamics of the transmon populations through single-shot measurements of
the different transmon states in various driving configurations and initial
states. To the best of our understanding, this constitutes the first
experimental realization of a two-tone transmon reset protocol using the QCR
with only one junction or single-shot readout.

We observed that the relatively high transmon temperature of $T=110$ mK is
barely affected by the QCR-on state with conservative pulse amplitudes, in
agreement with previous theory on photon-assisted quasiparticle tunneling. We
attribute such a high transmon temperature to the thermalization induced by
the different components in the cryogenic environment, along with potential
quasiparticle generation in the superconducting circuit [24].

Nevertheless, the QCR was still able to roughly halve the stabilization time
of the system when initialized in the second excited state of the transmon.
The stabilization time was further reduced by a factor of $\sim 10$ with
additional two-tone driving, yet some heating, provoking a decrease in the
ground state population was observed and theoretically explained by a simple
Liouvillian model. We attribute this effect to the involved multi-level
structure of the system showcasing the nontrivial interplay between driving
and thermal environment. In principle, a refinement in the reset theory could
be achieved, for example, through more sophisticated models taking into
account multi-photon decay channels or suitably optimized to treat time-
dependent Hamiltonians and dissipative environments [25].

In the near future, the reset protocol can be optimized in different ways.
Colder cryogenic environments achieved by a proper electromagnetic shielding
and reduced electrical noise may decrease the normal-metal temperature of the
QCR, allowing its cooling operation with higher voltages close to $\Delta/e$,
thereby promoting increased decay rates. Furthermore, the decay rates can be
significantly increased upon the fabrication of QCRs targeting smaller
tunneling resistances and smaller Dynes parameters by reducing the cross
section of the NIS junction or modifying oxidation parameters.

In the scenarios above, the optimization of two-tone Rabi frequencies may
allow the slowest eigenmode of the system to theoretically decay at the
maximum rate $\kappa/3$ [8], or in terms of the Liouvillian model,
$\Lambda_{1}=\kappa/6$, with $\kappa$ being orders of magnitude higher than
its value in the QCR-off state. Note that these bounds are derived in the
minimum subspace that encompasses the second-excited state of a non-decaying
transmon. Consequently, the slowest eigenmode is naturally expected to decay
at a higher rate in the presence of additional incoherent processes. More
accurate descriptions of the reset dynamics with a minimal dependence on the
initial state may incorporate the influence of additional eigenmodes.
Therefore, our results pave the way for a detailed understanding of the
dissipative dynamics of superconducting qubits with engineered decay channels,
potentially allowing the exploration of new reset regimes in the future.

The combined effects of the QCR and two-tone driving on multi-qubit systems
are yet to be demonstrated. Nonetheless, the minimal setup for the studied
hybrid reset scheme only requires the integration of QCRs to the
superconducting resonators dispersively coupled to the superconducting qubits,
in addition to their corresponding drive lines. Therefore, this reset strategy
just employs the state-of-art resources available on NISQ devices with a
minimal amount of excess control lines.

## Methods

### Theoretical model

We model the reset of the transmon–auxiliary-resonator system using the
multilevel Jaynes–Cummings (JC) Hamiltonian $\hat{H}_{0}$, the ladder
structure of which is depicted in Figs. 1c and 1d. The spectral decomposition
of $\hat{H}_{0}$ is expressed as
$\hat{H}_{0}=\hbar\sum_{j,n}\overline{\omega}_{{j},n}|\overline{j,n}\rangle\langle\overline{j,n}|$,
where $\hbar\overline{\omega}_{{j},n}$ are the eigenenergies of the system in
the absence of driving and $|\overline{j,n}\rangle$ are the corresponding
dressed eigenstates. Below, we denote the bare states of the system as
$\\{|j\rangle\otimes|n\rangle=|j,n\rangle\\}$, where
$j=\text{g},\text{e},\text{f},\text{h},\dots$, and $n=0,1,2,3\dots$. Owing to
the ladder structure of the multilevel JC Hamiltonian, $|j,n\rangle$ can be
expressed as a linear combination of the dressed states
$\\{|\overline{j^{\prime},n^{\prime}}\rangle\\}$ which involve an equal total
number of excitations.

The reset protocol consists of a $100$-MHz QCR pulse in addition to two
simultaneous microwave pulses addressed to the
$|\overline{e,0}\rangle\leftrightarrow|\overline{f,0}\rangle$ and
$|\overline{g,1}\rangle\leftrightarrow|\overline{f,0}\rangle$ transitions. In
the interaction picture with respect to $\hat{H}_{0}$, we model the open
dynamics of the system density operator, $\hat{\rho}$, through the Lindblad
master equation [26]

$\displaystyle\frac{\text{d}\hat{\rho}}{\text{d}t}=\mathcal{L}_{\rm{c}}(\hat{\rho})+\mathcal{L}_{\downarrow}(\hat{\rho})+\mathcal{L}_{\uparrow}(\hat{\rho})+\mathcal{L}_{\phi}(\hat{\rho}),$
(3)

where
$\mathcal{L}_{\rm{c}}(\hat{\rho})=-\text{i}[\hat{H}^{\prime},\hat{\rho}]/\hbar$
describes the coherent dynamics, with
$\hat{H}^{\prime}=\hat{H}+\hat{H}_{\delta}$ being the effective two-tone-
driven system Hamiltonian, and the last three terms characterize the thermal
emission, absorption, and pure dephasing, respectively.

Next, we discuss the processes described by Eq. (3). First, we assume that the
drive operator is proportional to the annihilation operator of the transmon,

$\displaystyle\hat{b}=(|g\rangle\\!\langle e|+\sqrt{2}|e\rangle\\!\langle
f|+\sqrt{3}|f\rangle\\!\langle h|+\dots)\otimes\sum_{n}|n\rangle\\!\langle
n|.$ (4)

Consequently, the two-tone drive Hamiltonian in the interaction picture can be
made time-independent by neglecting non-resonant terms. By also assuming
$\overline{\omega}_{{\text{f}},0}-\overline{\omega}_{{\text{e}},0}\approx\overline{\omega}_{{\text{f}},1}-\overline{\omega}_{{\text{e}},1}$
and
$\overline{\omega}_{{\text{f}},0}-\overline{\omega}_{{\text{g}},1}\approx\overline{\omega}_{{\text{f}},1}-\overline{\omega}_{{\text{g}},2}$,
up to the three-excitation subspace, it reads

$\displaystyle\hat{H}/\hbar=$ $\displaystyle\
-\text{i}\frac{\Omega_{\rm{ef}}}{2}\left(|\overline{e,0}\rangle\\!\langle\overline{f,0}|+|\overline{e,1}\rangle\\!\langle\overline{f,1}|\right)-\text{i}\frac{\Omega_{\rm{f0g1}}}{2}\left(|\overline{g,1}\rangle\\!\langle\overline{f,0}|+\sqrt{2}|\overline{g,2}\rangle\\!\langle\overline{f,1}|\right)+\
\text{H.c.},$ (5)

where $\Omega_{\rm{ef}}$ and $\Omega_{\rm{f0g1}}$ are the corresponding Rabi
angular frequencies of the drives. To account for shifts of the energy levels
arising from the approximations in Eq. (5) and from different physical
phenomena, such as drive-induced and bath-induced shifts, we add the small
correction term to the energy levels,
$\hat{H}_{\delta}=\hbar\sum_{j,n}\overline{\delta}_{{j},n}|\overline{j,n}\rangle\langle\overline{j,n}|$.
Notably, the incorporation of frequency shifts to the driven system
Hamiltonian is important for accurately estimating the stabilization times and
the steady state of quantum systems [25].

Next, we focus on the thermal emission, absorption, and pure dephasing terms
in Eq. (3). Up to the three-excitation subspace, these processes are,
respectively, expressed as

$\displaystyle\mathcal{L}_{\downarrow}(\hat{\rho})=$ $\displaystyle\
\Gamma_{\rm{eg}}\mathcal{D}\left(|\overline{g,0}\rangle\\!\langle\overline{e,0}|+|\overline{g,1}\rangle\\!\langle\overline{e,1}|+|\overline{g,2}\rangle\\!\langle\overline{e,2}|\right)\hat{\rho}+\Gamma_{\rm{fe}}\mathcal{D}\left(|\overline{e,0}\rangle\\!\langle\overline{f,0}|+|\overline{e,1}\rangle\\!\langle\overline{f,1}|\right)\hat{\rho}+\Gamma_{\rm{hf}}\mathcal{D}\left(|\overline{f,0}\rangle\\!\langle\overline{h,0}|\right)\hat{\rho}$
$\displaystyle+\kappa_{\downarrow}\mathcal{D}\left(|\overline{g,0}\rangle\\!\langle\overline{g,1}|+|\overline{e,0}\rangle\\!\langle\overline{e,1}|+|\overline{f,0}\rangle\\!\langle\overline{f,1}|+\sqrt{2}|\overline{g,1}\rangle\\!\langle\overline{g,2}|+\sqrt{2}|\overline{e,1}\rangle\\!\langle\overline{e,2}|+\sqrt{3}|\overline{g,2}\rangle\\!\langle\overline{g,3}|\right)\hat{\rho},$
$\displaystyle\mathcal{L}_{\uparrow}(\hat{\rho})=$ $\displaystyle\
\Gamma_{\rm{ge}}\mathcal{D}\left(|\overline{e,0}\rangle\\!\langle\overline{g,0}|+|\overline{e,1}\rangle\\!\langle\overline{g,1}|+|\overline{e,2}\rangle\\!\langle\overline{g,2}|\right)\hat{\rho}+\Gamma_{\rm{ef}}\mathcal{D}\left(|\overline{f,0}\rangle\\!\langle\overline{e,0}|+|\overline{f,1}\rangle\\!\langle\overline{e,1}|\right)\hat{\rho}+\Gamma_{\rm{fh}}\mathcal{D}\left(|\overline{h,0}\rangle\\!\langle\overline{f,0}|\right)\hat{\rho}$
$\displaystyle+\kappa_{\uparrow}\mathcal{D}\left(|\overline{g,1}\rangle\\!\langle\overline{g,0}|+|\overline{e,1}\rangle\\!\langle\overline{e,0}|+|\overline{f,1}\rangle\\!\langle\overline{f,0}|+\sqrt{2}|\overline{g,2}\rangle\\!\langle\overline{g,1}|+\sqrt{2}|\overline{e,2}\rangle\\!\langle\overline{e,1}|+\sqrt{3}|\overline{g,3}\rangle\\!\langle\overline{g,2}|\right)\hat{\rho},$
$\displaystyle\mathcal{L}_{\phi}(\hat{\rho})=$ $\displaystyle\
\gamma_{\rm{eg}}^{\phi}\mathcal{D}\left(|\overline{e,0}\rangle\\!\langle\overline{e,0}|+|\overline{e,1}\rangle\\!\langle\overline{e,1}|+|\overline{e,2}\rangle\\!\langle\overline{e,2}|-|\overline{g,0}\rangle\\!\langle\overline{g,0}|-|\overline{g,1}\rangle\\!\langle\overline{g,1}|-|\overline{g,2}\rangle\\!\langle\overline{g,2}|-|\overline{g,3}\rangle\\!\langle\overline{g,3}|\right)\hat{\rho}/2$
$\displaystyle+\gamma_{\rm{fe}}^{\phi}\left(|\overline{f,0}\rangle\\!\langle\overline{f,0}|+|\overline{f,1}\rangle\\!\langle\overline{f,1}|-|\overline{e,0}\rangle\\!\langle\overline{e,0}|-|\overline{e,1}\rangle\\!\langle\overline{e,1}|-|\overline{e,2}\rangle\\!\langle\overline{e,2}|\right)\hat{\rho}/2$
$\displaystyle+\gamma_{\rm{hf}}^{\phi}\left(|\overline{h,0}\rangle\\!\langle\overline{h,0}|-|\overline{f,0}\rangle\\!\langle\overline{f,0}|-|\overline{f,1}\rangle\\!\langle\overline{f,1}|\right)\hat{\rho}/2,$
(6)

where
$\mathcal{D}(\hat{O})\hat{\rho}=\hat{O}\hat{\rho}\hat{O}^{\dagger}-\\{\hat{O}^{\dagger}\hat{O},\hat{\rho}\\}/2$.

In Eqs. (6), we assume that each superoperator $\mathcal{D}$ describes the
exchange of, at most, one excitation between the system and its baths. Multi-
photon processes are known to take place at much lower rates than single-
photon processes, and hence they are neglected. This exchange primarily occurs
between the transmon-like levels
$|\overline{e,n}\rangle\leftrightarrow|\overline{g,n}\rangle$,
$|\overline{f,n}\rangle\leftrightarrow|\overline{e,n}\rangle$,
$|\overline{h,n}\rangle\leftrightarrow|\overline{f,n}\rangle$, and the
auxiliary-resonator-like levels
$|\overline{j,n+1}\rangle\leftrightarrow|\overline{j,n}\rangle$. This
assumption is motivated by the dispersive regime, which yields weakly
entangled eigenstates of $\hat{H}_{0}$. Consequently, we may approximate
$|\overline{j,n}\rangle\approx|j,n\rangle$, with equality holding for the
ground state $|\overline{g,0}\rangle$. Note that this factorization may only
be applicable in the treatment of the open dynamics and does not hold for the
Hamiltonian terms; otherwise, the $f0$–$g1$-like transitions could not be
driven. A comparison between the master equations in the bare and dressed
basis, as well as the impact of the readout resonator on reset, are left for
future work.

The thermal emission and absorption rates in Eqs. (6) can be conveniently
written in terms of the decay rates $\gamma_{jk}$ and $\kappa$ as

$\displaystyle\Gamma_{\rm{eg}}$
$\displaystyle=\gamma_{\rm{eg}}(\bar{n}_{\rm{eg}}+1),\ \
\Gamma_{\rm{ge}}=\gamma_{\rm{eg}}\bar{n}_{\rm{eg}},$
$\displaystyle\Gamma_{\rm{fe}}$
$\displaystyle=\gamma_{\rm{fe}}(\bar{n}_{\rm{fe}}+1),\ \
\Gamma_{\rm{ef}}=\gamma_{\rm{fe}}\bar{n}_{\rm{fe}},$
$\displaystyle\Gamma_{\rm{hf}}$
$\displaystyle=\gamma_{\rm{hf}}(\bar{n}_{\rm{hf}}+1),\ \
\Gamma_{\rm{fh}}=\gamma_{\rm{hf}}\bar{n}_{\rm{hf}},$
$\displaystyle\kappa_{\downarrow}$ $\displaystyle=\kappa(\bar{n}_{\rm{r}}+1),\
\ \kappa_{\uparrow}=\kappa\bar{n}_{\rm{r}},$ (7)

where $\bar{n}_{jk}$ and $\bar{n}_{\rm{r}}$ are the thermal occupation numbers
at the transmon and auxiliary resonator frequencies.

Using the notation of Table 1, we identify the main angular transition
frequencies of the considered system as
$\overline{\omega}_{{jk}}=\overline{\omega}_{{k},0}-\overline{\omega}_{{j},0}$,
$\overline{\omega}_{\text{r}}=\overline{\omega}_{{g},n+1}-\overline{\omega}_{{g},n}$,
and
$\overline{\omega}_{\text{f0g1}}=\overline{\omega}_{\text{f},0}-\overline{\omega}_{\text{g},1}$,
which are obtained through spectroscopic measurements. While the decay rate
$\gamma_{\text{eg}}$ is obtained from a time-domain measurement, we use the
values $\gamma_{\text{fe}}=2\gamma_{\text{eg}}$ and
$\gamma_{\text{hf}}=3\gamma_{\text{eg}}$, which are expected from theory. In
addition, we employ $\gamma_{jk}^{\phi}=4\gamma_{jk}$, assuming that pure
dephasing is a dominant source of decoherence in the system, as verified in
previous independent measurements.

The Liouvillian model described above is utilized for fitting to the time-
dependent readout signals in Figs. 3d and 3f, for the master equation
simulations in Figs. 4g–4l, and for the Liouvillian analysis of Fig. 5. In
these cases, the eigenspectra and the solution of Eq. (3) are obtained through
the vectorization of the Liouvillian using the Melt library in Mathematica
[27].

Specifically in Fig. 3d, we set the initial state to
$\hat{\rho}(0)=|\overline{f,0}\rangle\langle\overline{f,0}|$ with
$\Omega_{\rm{ef}}=0$, and use Eq. (3) truncated at the fourth level. To
extract $\Omega_{\rm{f0g1}}$ for different voltages $V_{\rm{f0g1}}$ from the
readout signal of the transmon state $|f\rangle$, we use the fitting function
$s_{\text{f}}(t)=a_{\text{f}}p_{\text{f}}^{\text{theory}}(t)+b_{\text{f}}$,
where $a_{\text{f}}$ and $b_{\text{f}}$ are real-valued free parameters, and
$p_{\text{f}}^{\text{theory}}(t)$ is defined in Eq. (2). From this fitting, we
also obtain the auxiliary-resonator decay rates $\kappa$, the averaged value
of which is shown in Table 1. We proceed in a similar way to extract the
dependence of $\Omega_{\rm{ef}}$ on $V_{\rm{ef}}$ in Fig. 3f, by setting
$\hat{\rho}(0)=|\overline{e,0}\rangle\langle\overline{e,0}|$,
$\Omega_{\rm{f0g1}}=0$, and using the fitting function
$s_{\text{e}}(t)=a_{\text{e}}p_{\text{e}}^{\text{theory}}(t)+b_{\text{e}}$,
with new real-valued free parameters $a_{{e}}$ and $b_{{e}}$.

For the master equation simulations, we truncate the model at the tenth level
and set the initial states
$\hat{\rho}_{\text{g}}(0)=\hat{\rho}_{\text{thermal}}$ in Figs. 4g and 4h,
$\hat{\rho}_{\text{e}}(0)=\hat{\sigma}_{x}^{\text{ge}}\hat{\rho}_{\text{thermal}}\hat{\sigma}_{x}^{\text{ge}}$
in Figs. 4i and 4j, and
$\hat{\rho}_{\text{f}}(0)=\hat{\sigma}_{x}^{\text{ef}}\hat{\sigma}_{x}^{\text{ge}}\hat{\rho}_{\text{thermal}}\hat{\sigma}_{x}^{\text{ge}}\hat{\sigma}_{x}^{\text{ef}}$
in Figs. 4k and 4l, where $\hat{\sigma}_{x}^{jk}$ is the Pauli
$\hat{\sigma}_{x}$ operator in the corresponding transmon subspace and we have
assumed a thermal natural initial state of the form
$\hat{\rho}_{\text{thermal}}=\text{e}^{-\hat{H}_{0}/(k_{\text{B}}T)}/Z$. This
state exhibits overlap fidelity of $>99.99\%$ with the steady state of Eq. (3)
without two-tone driving. The remaining parameters in these simulations are
given as in Tables 1 and 2. To better match the experimental data, we have
also included additional frequency corrections to the
$|\overline{e,0}\rangle$, $|\overline{g,1}\rangle$, and
$|\overline{f,0}\rangle$ levels in the simulations with two-tone driving,
obtaining the optimized values $\overline{\delta}_{\text{e},0}=-2.0$ MHz and
$\overline{\delta}_{\text{g},1}=0.8$ MHz for configurations B and D,
$\overline{\delta}_{\text{f},0}=-1.2$ MHz for configuration B, and
$\overline{\delta}_{\text{f},0}=-1.6$ MHz for configuration D.

### System readout

The system readout is mostly carried out through standard dispersive
techniques [28]. A $2$-µs pulse is applied to the input port of the readout
line, which is coupled to the readout resonator. The signal transmitted
through the readout line is digitized after an amplification chain composed of
a traveling-wave parametric amplifier (TWPA) at the millikelvin stage, and
low-noise amplifiers at $50$ K and at room temperature. Ensemble-averaged
measurements were first carried out with readout frequency
$f_{\text{RO}}=7.4372$ GHz, which was later optimized to
$f_{\text{RO}}=7.4376$ GHz for single-shot measurements. Such a frequency
provided the best separation in the IQ plane for time traces integrated from
$0.4$ µs to $1.0$ µs of the readout pulse.

In all single-shot measurements, we assume that the point clouds are well-
represented by a weighted sum of four two-dimensional (2D) Gaussian
distributions,

$\displaystyle
G(\mathbf{x})=\sum_{{j}=\text{g},\text{e},\text{f},\text{h}}\frac{w_{{j}}}{2\pi|\Sigma_{{j}}|^{1/2}}\exp\left[-\frac{1}{2}(\mathbf{x}-\mathbf{\mu}_{j})^{\top}\Sigma_{{j}}^{-1}(\mathbf{x}-\mathbf{\mu}_{j})\right],$
(8)

where $\mathbf{x}$ is a 2D vector describing the coordinates of the integrated
points in the IQ plane. Each Gaussian component is characterized by a mean
vector $\mathbf{\mu}_{j}$, covariance matrix $\Sigma_{{j}}$, and weight
$w_{{j}}$, which depend on the physical properties of the device, as well as
the readout probe frequency and integration time.

We use the scikit-learn [29] library in Python to estimate the parameters of
$G(\mathbf{x})$ based on the point distribution at $t_{\text{r}}=0$. The
ellipses corresponding to the four-lowest levels of the transmon are
subsequently obtained, as shown in Figs. 4a–4f. For each set of driving
configurations of Table 2 and reset lengths $t_{\text{r}}$, the populations of
the transmon states are then estimated with the expression

$\displaystyle
p_{j}=\frac{N_{{j}}}{N_{\text{g}}+N_{\text{e}}+N_{\text{f}}+N_{\text{h}}},$
(9)

where $N_{{j}}$ is the number of points inside the $1\sigma$-boundary ellipse
corresponding to the state $|j\rangle$. We observe that the
$|h\rangle$-ellipse is the largest, possibly due to inaccuracies resulting
from the few number of points and the existence of the high-lying transmon
states that were not considered. Nonetheless, the ratio of points inside and
outside the ellipses remains close to $39.35\%$, characteristic from a 2D
Gaussian distribution [30]. Note that this method does not take into account
all state-assignment errors, and hence further single-shot-readout
optimization may be achieved through more advanced machine learning algorithms
[31, 32, 33].

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## Acknowledgements

This work was funded by the Academy of Finland Centre of Excellence program
(project Nos. 352925, and 336810) and grant Nos. 316619 and 349594 (THEPOW).
We also acknowledge funding from the European Research Council under Advanced
Grant No. 101053801 (ConceptQ). We thank Matti Silveri, Joachim Ankerhold,
Gianluigi Catelani, Hao Hsu, Tapio Ala-Nissila, Shuji Nakamura for scientific
discussions, and Leif Grönberg for Nb deposition.

## Author contributions statement

W.T. characterized the sample together with T.M., A.V., H.K., and M.T.,
conducted the QCR-assisted two-tone experiment, and carried out the data
analysis. T.M. designed and carried out the electromagnetic simulations of the
device, fabricated the sample, and built the experimental setup. A.G.
contributed to the measurement code for single-shot readout. S.K. and A.S.
calibrated the TWPA pump. V.V. contributed to the theory and design of the
single-junction QCR. M.M. conceived the experiment and supervised the project.
The manuscript was written by W.T., T.M., and M.M., with comments from all
authors.

## Additional information

Competing interests The authors declare no competing interests.