# Single-Shot Readout of a Superconducting Qubit Using a Thermal Detector

András M. Gunyhó<EMAIL_ADDRESS>QCD Labs, QTF Centre of Excellence,
Department of Applied Physics, Aalto University, P.O. Box 13500, FIN-00076
Aalto, Finland  Suman Kundu  QCD Labs, QTF Centre of Excellence, Department of
Applied Physics, Aalto University, P.O. Box 13500, FIN-00076 Aalto, Finland
Jian Ma  QCD Labs, QTF Centre of Excellence, Department of Applied Physics,
Aalto University, P.O. Box 13500, FIN-00076 Aalto, Finland  Wei Liu  QCD Labs,
QTF Centre of Excellence, Department of Applied Physics, Aalto University,
P.O. Box 13500, FIN-00076 Aalto, Finland  IQM, Keilaranta 19, FI-02150 Espoo,
Finland  Sakari Niemelä  QCD Labs, QTF Centre of Excellence, Department of
Applied Physics, Aalto University, P.O. Box 13500, FIN-00076 Aalto, Finland
Giacomo Catto  QCD Labs, QTF Centre of Excellence, Department of Applied
Physics, Aalto University, P.O. Box 13500, FIN-00076 Aalto, Finland  Vasilii
Vadimov  QCD Labs, QTF Centre of Excellence, Department of Applied Physics,
Aalto University, P.O. Box 13500, FIN-00076 Aalto, Finland  Visa Vesterinen
QTF Centre of Excellence, VTT Technical Research Centre of Finland Ltd., P.O.
Box 1000, 02044 VTT, Finland  Priyank Singh  QCD Labs, QTF Centre of
Excellence, Department of Applied Physics, Aalto University, P.O. Box 13500,
FIN-00076 Aalto, Finland  Qiming Chen  QCD Labs, QTF Centre of Excellence,
Department of Applied Physics, Aalto University, P.O. Box 13500, FIN-00076
Aalto, Finland  Mikko Möttönen  QCD Labs, QTF Centre of Excellence, Department
of Applied Physics, Aalto University, P.O. Box 13500, FIN-00076 Aalto, Finland
QTF Centre of Excellence, VTT Technical Research Centre of Finland Ltd., P.O.
Box 1000, 02044 VTT, Finland

###### Abstract

Measuring the state of qubits is one of the fundamental operations of a
quantum computer. Currently, state-of-the-art high-fidelity single-shot
readout of superconducting qubits relies on parametric amplifiers at the
millikelvin stage. However, parametric amplifiers are challenging to scale
beyond hundreds of qubits owing to practical size and power limitations.
Nanobolometers have a multitude of properties that are advantageous for
scalability and have recently shown sensitivity and speed promising for qubit
readout, but such thermal detectors have not been demonstrated for this
purpose. In this work, we utilize an ultrasensitive bolometer in place of a
parametric amplifier to experimentally demonstrate single-shot qubit readout.
With a readout duration of 13.9 µs, we achieve a single-shot fidelity of 0.618
which is mainly limited by the energy relaxation time of the qubit, $T_{1}=28$
µs. Without the $T_{1}$ errors, we find the fidelity to be 0.927. In the
future, high-fidelity single-shot readout may be achieved by straightforward
improvements to the chip design and experimental setup, and perhaps most
interestingly by the change of the bolometer absorber material to reduce the
readout time to the hundred-nanosecond level and beyond.

Qubit readout is a fundamental operation in quantum computing, both for
determining the result at the end of a computation [1, 2], as well as for
error correction, which is necessary for fault-tolerance [3, 4, 5]. Currently,
one of the most promising platforms for realizing a practically useful quantum
computer is based on superconducting qubits [6, 7, 8]. Importantly, readout is
currently one of the main bottlenecks on the way to quantum-error-corrected
large-scale superconducting quantum processors: most of the error in the error
correction cycles demonstrated in recent state-of-the-art experiments [9, 10]
arose during the readout phase. Thus improving existing readout techniques and
especially discovering completely new advantageous ways to carry out readout
are of great and urgent importance.

The standard method of measuring the state of superconducting qubits in the
framework of circuit quantum electrodynamics is referred to as _dispersive
readout_ [6, 11, 12]. Here, a qubit is dispersively coupled to a readout
resonator, the frequency of which shifts depending on the qubit state. The
current state of the art in dispersive readout is a single-shot readout
fidelity in excess of 99% with an averaging time of less than
$100\text{\,}\mathrm{ns}\text{/}$ for single qubits [13, 14], and 97–98% on
average for the simultaneous multiplexed readout of several qubits [15, 10].

To achieve a signal-to-noise ratio (SNR) sufficient for high-fidelity single-
shot dispersive readout, the output signal from the readout resonator is
typically amplified at the millikelvin stage by a parametric amplifier, such
as a Josephson parametric amplifier (JPA) [16] or a traveling-wave parametric
amplifier (TWPA) [17]. These amplifiers can be quantum limited, i.e., the
noise added by the amplification stems solely from the Heisenberg uncertainty
principle for the in-phase and quadrature components of the amplified signal
[18, 19, 20]. Parametric amplifiers are widely utilized due to their high gain
and low noise, but they suffer from some drawbacks that challenge their
applicability when scaling to large numbers of qubits. Namely, JPAs have a
fairly narrow bandwidth, which renders them less suitable for multiplexed
qubit readout [15]. On the other hand, although TWPAs provide a broad
bandwidth, they have limited dynamic range for the purposes of highly
multiplexed readout and typically incorporate more than $10^{3}$ Josephson
junctions, rendering their on-chip footprint sizeable and high-yield large-
scale fabrication challenging. Importantly, both of these amplifier types
require strong isolation between the amplifier and the qubit-resonator system.
In the case of JPAs, this is because the amplifier works by reflection,
requiring the use of a circulator, and TWPAs require a relatively strong
pumping tone possibly near the frequencies of the qubit and the readout
resonator. Furthermore, TWPAs amplify vacuum and other noise, which may be
reflected by later components in the amplification chain due to imperfect
impedance matching. This noise may leak backwards to the TWPA input and cause
decoherence in the qubit if not isolated. To mitigate these issues, permanent-
magnet microwave isolators are placed between the readout transmission line
and the amplifier input. Such isolators introduce losses in the signal, are
very large in size, costly, and require bulky shielding to protect both the
qubit and the amplifier from the magnetic fields they introduce.

These scalability issues have motivated the development of novel readout
techniques that evade the need for parametric amplifiers. For example, Ref.
[21] introduces a microwave photon counter instead of voltage amplification at
the millikelvin stage. However, this method requires a more involved pulse
sequence, and it suffers from the backaction and possible creation of
quasiparticles due to their tunneling events. In Ref. [22], qubit readout is
achieved without a parametric amplifier by driving the qubit to the second
excited state before readout, at the cost of an increased measurement time.
However, the used non-parametric microwave amplifier introduces high-
temperature noise close to the qubit frequency, which calls for bulky
microwave isolators. Another currently developing approach to scalability is
to deliver signals to and from the cryostat in the optical domain [23, 24].

Interestingly, ultrasensitive nanobolometers [25, 26] have been recently
demonstrated to be fast and sensitive enough for readout of superconducting
qubits, reaching thermal time constants in the hundred nanosecond scale and
energy resolution of a few typical microwave photons [27]. Such bolometers
have a number of characteristic properties that are attractive for scalable
qubit readout. In contrast to a parametric amplifier, the bolometer introduced
in Ref. [25] is driven by a probe tone with a frequency below
$1\text{\,}\mathrm{GHz}\text{/}$, well below the typical frequencies
5–$8\text{\,}\mathrm{GHz}\text{/}$ of the qubit and the resonator. In
addition, the bolometer can be probed with a low power of roughly
$-130\text{\,}\mathrm{dBm}\text{/}$ at the chip, while for example TWPAs
typically require pump powers of $-75\text{\,}\mathrm{dBm}\text{/}$ or higher
[17]. Furthermore, the absorbing port of the bolometer can be conveniently
matched to $50\text{\,}\mathrm{\SIUnitSymbolOhm}\text{/}$ [26]. This presents
a cold bath to the readout transmission line, while allowing for the detection
frequency to vary by orders of magnitude given a fixed probe frequency. These
features may eliminate the need for isolators between the qubit-resonator
system and the bolometer, and together with the small size of the bolometers
render them highly promising from the scalability point of view.

Another advantage of bolometers compared with parametric amplifiers is that
since a bolometer measures power, or photon number, it is not bound to add
quantum noise stemming from the Heisenberg uncertainty principle. The vacuum
noise does not promote detection events in the bolometer since no energy can
be extracted from the vacuum. Thus bolometric readout is fundamentally
different from the usual dispersive readout.

In addition, bolometers are relatively simple to both fabricate and operate
[26]. They do not require engineering of a large number of small Josephson
junctions, and a bolometer requires only a single continuous probe tone with
two parameters, power and frequency, to optimize performance.

Owing to these appealing features, it is of great interest to study whether
the fundamentally different operation principle of bolometers can be harnessed
in the readout of superconducting qubits. However, no thermal detector to date
has been employed for this purpose.

In this work, we integrate an ultrasensitive bolometer at millikelvin
temperature to the readout circuitry of a superconducting qubit (Fig. 1).
After the characterization of the bolometer and the qubit (Fig. 2), we
demonstrate single-shot qubit readout at the fidelity of 0.618 (Fig. 3). Taken
that we have not used the fastest compatible bolometers [27], but those that
are orders of magnitude slower [26], this first demonstration of bolometric
single-shot qubit readout seems a promising milestone for future high-fidelity
scalable readout of superconducting qubits.

### Experimental setup

A schematic diagram of the experimental setup is shown in Fig. 1. A standard
flux-tunable Xmon qubit [28] with a transition frequency
$f_{\mathrm{q}}=$7.655\text{\,}\mathrm{GHz}\text{/}$$ at the flux sweet spot,
anharmonicity $\alpha/(2\pi)=$-273\text{\,}\mathrm{MHz}\text{/}$$, energy
relaxation time $T_{1}=$28\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$$,
and Ramsey dephasing time $T_{2}=$7.6\text{\,}\mathrm{\SIUnitSymbolMicro
s}\text{/}$$ is capacitively coupled to a coplanar readout resonator in a
notch configuration. The resonator has its fundamental resonance frequency at
$f_{\mathrm{r},\mathrm{g}}=$5.473\text{\,}\mathrm{GHz}\text{/}$$ if the qubit
is in its ground state $\ket{\mathrm{g}}$ and a linewidth
$\kappa_{\mathrm{r}}/(2\pi)=$1.0\text{\,}\mathrm{MHz}\text{/}$$. The resonator
is capacitively coupled to the qubit with a coupling strength
$g/(2\pi)=$61\text{\,}\mathrm{MHz}\text{/}$$. These parameters produce a
dispersive shift of $\chi/2\pi=-$0.3\text{\,}\mathrm{MHz}\text{/}$$, i.e., the
resonator frequency shifts to
$f_{\mathrm{r},\mathrm{e}}=f_{\mathrm{r},\mathrm{g}}+2\chi/(2\pi)$ if the
qubit is in its excited state $\ket{\mathrm{e}}$.

To read out the qubit state, a rectangular microwave pulse of length
$t_{\mathrm{RO}}\approx$10\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$$,
frequency $f_{\mathrm{d}}$, and power $P_{\mathrm{d}}$ is applied to the
feedline of the readout resonator. In typical dispersive readout, the drive
frequency is chosen to be in the middle of the dressed resonator frequencies,
$f_{\mathrm{d}}=(f_{\mathrm{r},\mathrm{g}}+f_{\mathrm{r},\mathrm{e}})/2$, and
the photons reflected from the resonator accumulate a phase shift depending on
the qubit state [6]. In contrast, here we operate in a photodetection mode
[29] by driving close to one of the dressed frequencies,
$f_{\mathrm{d}}\approx f_{\mathrm{r},\mathrm{g}}$. In this driving scheme,
information about the qubit state is mostly carried by the power of the
signal, i.e., the number of photons emitted by the resonator into the
feedline. A power difference is necessary for the bolometer, since it is a
thermal detector, sensitive to the power but insensitive to the phase of its
input signal. Note that for the standard dispersive readout, the ratio
$|\chi|/\kappa_{\mathrm{r}}=1/2$ yields the optimal SNR [30], while for
photodetection-based readout, the optimal ratio is higher [29].

The output of the feedline for the readout resonator is connected to the
absorber port of the bolometer. The bolometer, similar to the device of Ref.
[26], resides on a chip different from that of the qubit-resonator system as
shown in Fig. 1. The main component of the bolometer is a resistive
$\textnormal{Au}_{x}\textnormal{Pd}_{1-x}$ ($x\approx 0.6$) nanowire. A
segment of the nanowire works as the resistive absorber and the remaining wire
is interrupted by a series of superconducting Al islands, forming a chain of
SNS junctions. The impedance $Z(T_{\mathrm{e}})$ of this junction chain
depends on the electron temperature of the nanowire $T_{\mathrm{e}}$. The
junction chain is embedded in an effective $LC$ circuit formed by a shunt
capacitor $C_{1}=$134\text{\,}\mathrm{pF}\text{/}$$ in parallel with
$Z(T_{\mathrm{e}})$, which can be modeled as a parallel resistance and
inductance [25]. The nanowire is grounded between the absorber and the
junctions, so that the essentially purely real-valued impedance of the
absorber does not contribute to the $LC$ circuit.

The bolometer is probed by reflecting a continuous tone at power
$P_{\mathrm{p}}$ and frequency $f_{\mathrm{p}}$ from the gate capacitor
$C_{\mathrm{g}}=$0.87\text{\,}\mathrm{pF}\text{/}$$. With a low
$P_{\mathrm{p}}$, the $LC$ circuit resonates at the frequency
$f_{\mathrm{b}}=$585\text{\,}\mathrm{MHz}\text{/}$$ with a linewidth
$7.6\text{\,}\mathrm{MHz}\text{/}$. As radiation is absorbed by the absorber,
$T_{\mathrm{e}}$ increases, which shifts $f_{\mathrm{b}}$ down, and thus by
observing changes in the reflection coefficient $\Gamma$ at the gate
capacitor, it is possible to detect the radiation incident on the bolometer
input. The reflected probe signal is amplified by a low-noise high-electron-
mobility-transistor (HEMT) amplifier at $4\text{\,}\mathrm{K}\text{/}$, and
further amplified, demodulated and digitized at room temperature in a
heterodyne configuration.

Figure 1: Experimental setup. The readout resonator is driven at frequency
$f_{\mathrm{d}}\approx$5.5\text{\,}\mathrm{GHz}\text{/}$$ by a readout tone
which is reflected off of the resonator and absorbed by a metallic nanowire in
the absorber port of the bolometer. The absorbed radiation shifts the electron
temperature $T_{\mathrm{e}}$ and consequently the impedance
$Z(T_{\mathrm{e}})$ of a chain of SNS junctions, marked by a brown cross. A
continuous probe tone with frequency
$f_{\mathrm{p}}\approx$580\text{\,}\mathrm{MHz}\text{/}$$ is reflected off of
the gate capacitor $C_{\mathrm{g}}$ of the bolometer as determined by the
reflection coefficient $\Gamma$. The reflected signal is amplified, digitized,
and used to determine the qubit state.

The shift in $f_{\mathrm{b}}$ due to the readout pulse incoming from the
readout resonator is observed as a change in the digitized voltage [see Fig.
2(a)–(b)]. For low $P_{\mathrm{p}}$, the readout pulse causes $T_{\mathrm{e}}$
and thus the reflected signal to approach a steady state value exponentially
with a thermal time constant $\tau_{\mathrm{b}}$. The time constant depends on
$f_{\mathrm{p}}$, $P_{\mathrm{p}}$, and the power of the readout pulse
$P_{\mathrm{d}}$ in a non-trivial way due to electrothermal feedback [25, 26].
For the relevant parameter regime considered here, $\tau_{\mathrm{b}}$ varies
between $10\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$ and
$1\text{\,}\mathrm{ms}\text{/}$. Notably, we have $\tau_{\mathrm{b}}\gtrsim
T_{1}$, which implies that for qubit readout, we must operate the bolometer in
a calorimetric fashion, i.e., with $t_{\mathrm{RO}}<\tau_{\mathrm{b}}$. This
is highlighted in Fig. 2(a), where the time constant, extracted from a
measurement with a long readout pulse
$t_{\mathrm{RO}}>$1\text{\,}\mathrm{ms}\text{/}$$, is
$\tau_{\mathrm{b}}=$36.2\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$$. With
$t_{\mathrm{RO}}=$10\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$$, which is
more feasible for qubit readout than $t_{\mathrm{RO}}>\tau_{\mathrm{b}}$, the
steady state is far from the reached maximum signal level.

For high enough $P_{\mathrm{p}}$, the electrothermal feedback results in a
bistability for the electron temperature of the SNS junction. This bistability
can be exploited for high-fidelity photodetection by operating the bolometer
in a latching mode using a pulsed probe tone [25]. However, this scheme
introduces significant dead time for the detection, and requires detailed
calibration of the pulse shape. For simplicity, we thus focus on the
continuous detection scheme with sufficiently low $P_{\mathrm{p}}$.

Using time-domain data digitized from the bolometer output, we define the
detector signal as

$\displaystyle S=\bar{V}-\bar{V}_{0},$ (1)

where $\bar{V}_{0}$ is the time-average of the digitized voltage before the
readout pulse and $\bar{V}$ is the time-average of the voltage over some
averaging window $[t_{0},t_{\mathrm{RO}}]$. Since
$t_{\mathrm{RO}}<\tau_{\mathrm{b}}$, increasing the time over which the
average is taken can significantly decrease $\bar{V}$, but may increase the
SNR. In practice, we observe that choosing $t_{0}=0.75\times t_{\mathrm{RO}}$
balances these two effects reasonably well.

Figure 2: Characterization experiments. (a) Example timetrace of the probe
signal reflected from the bolometer with a qubit readout pulse length of
$t_{\mathrm{RO}}=$10\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$$ (dark
gray dots) and $>$1\text{\,}\mathrm{ms}\text{/}$$ (light gray dots). The solid
green line is an exponentially rising fit to the long pulse, and the dashed
horizontal lines indicate the extracted values of $\bar{V}$ and $\bar{V}_{0}$
used to define the detector signal $S$ for the short pulse. The parameter
values in all panels are as shown in Table 1 unless otherwise specified. (b)
As (a) but only for the $10\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$
readout pulse. The blue and red-colored regions indicate where $\bar{V}_{0}$
and $\bar{V}$ are averaged, respectively. These data are taken with 128
ensemble averages. The signal appears noisier than in panel (a), since each
data point in (a) is calculated by averaging 512 adjacent data points. (c)
Detector signal $S$ as a function of the probe frequency $f_{\mathrm{p}}$ and
probe power $P_{\mathrm{p}}$. In the region shaded with red, where
$P_{\mathrm{p}}\gtrsim$-128.5\text{\,}\mathrm{dBm}\text{/}$$, the bolometer
exhibits bistability due to electrothermal feedback. The red cross indicates
the chosen operation point for qubit readout. The colorbar is shared with
panel (e). (d) Detector signal as a function of the nominal power of the
readout pulse $P_{\mathrm{d}}$ for various indicated readout pulse lengths and
for the readout pulse applied off-resonance
($f_{\mathrm{d}}=$5.400\text{\,}\mathrm{GHz}\text{/}$$, filled markers) and
on-resonance
($f_{\mathrm{d}}=$5.473\text{\,}\mathrm{GHz}\text{/}$=f_{\mathrm{r},\mathrm{g}}$,
unfilled markers). The resonator becomes significantly nonlinear at
approximately $-108\text{\,}\mathrm{dBm}\text{/}$, indicated by the black
dotted vertical line. (e) Detector signal as a function of the resonant qubit
drive pulse length and frequency, showing Rabi oscillation. These data are
taken with 512 ensemble averages.

Both the qubit-resonator and bolometer chips are mounted on their individual
sample holders, which are placed in separate magnetic shields and attached to
the mixing-chamber plate of a dilution refrigerator with the base temperature
of $12\text{\,}\mathrm{mK}\text{/}$. Note that in the setup presented in Fig.
1, an additional microwave switch and directional coupler are placed between
the qubit and bolometer. These allow measuring the qubit and driving the
bolometer individually, and are only used for initial separate
characterization of the qubit and the bolometer. The auxiliary readout channel
called for the use of an isolator between the chips, but this can be removed
in future experiments.

## Results

During initial characterization, we apply no driving to the readout resonator
of the qubit and find the bolometer resonance by measuring the reflection
coefficient of the bolometer probe signal as a function of $f_{\mathrm{p}}$
and $P_{\mathrm{p}}$. Next, we apply pulses to the readout resonator with
$f_{\mathrm{d}}$ well detuned from $f_{\mathrm{r},\mathrm{g}}$, and map the
detector signal $S$ as a function of $f_{\mathrm{p}}$ and $P_{\mathrm{p}}$.
These results, shown in Fig. 2(c), provide us with a feasible operation point
$(f_{\mathrm{p}},P_{\mathrm{p}})=($580.5\text{\,}\mathrm{MHz}\text{/}$,$-128.7\text{\,}\mathrm{dBm}\text{/}$)$
where $S$ is maximized with $P_{\mathrm{p}}$ below the region where the
electrothermal feedback induces bistability as discussed above. Table 1
summarizes the parameter values used during the initial characterization.

Table 1: Typical parameter values used during characterization measurements. Note that all powers we report in this manuscript are uncalibrated, nominal values reaching the corresponding chip, based on the estimated attenuation of the lines in our setup. Quantity | Symbol | Value
---|---|---
Bolometer probe frequency | $f_{\mathrm{p}}$ | $580.5\text{\,}\mathrm{MHz}\text{/}$
Bolometer probe power (nominal) | $P_{\mathrm{p}}$ | $-128.7\text{\,}\mathrm{dBm}\text{/}$
Readout resonator drive frequency | $f_{\mathrm{d}}$ |  $5.400\text{\,}\mathrm{GHz}\text{/}$ or $f_{\mathrm{r},\mathrm{g}}$
Readout drive power (nominal) | $P_{\mathrm{d}}$ | $-107.8\text{\,}\mathrm{dBm}\text{/}$
Readout pulse length | $t_{\mathrm{RO}}$ | $10\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$
Number of ensemble-averages |  | 16

With the bolometer operation point fixed, we carry out standard qubit
characterization measurements [31] while monitoring the bolometer signal. For
example, Fig. 2(d) shows $S$ as a function of the qubit readout power
$P_{\mathrm{d}}$, both on resonance $f_{\mathrm{d}}=f_{\mathrm{r},\mathrm{g}}$
and off-resonance at
$f_{\mathrm{d}}=f_{\mathrm{r},\mathrm{g}}-$70\text{\,}\mathrm{MHz}\text{/}$$,
with the qubit drive turned off. Using
$P_{\mathrm{d}}=$-126\text{\,}\mathrm{dBm}\text{/}$$, a maximal contrast of
$32\text{\,}\mathrm{dB}\text{/}$ between the on and off-resonance drives is
achieved with a very long readout pulse. More importantly, Fig. 2(d) further
highlights that the long bolometer time constant is limiting the readout
signal since the contrast is greatly reduced for the microsecond pulses, which
are still roughly an order of magnitude longer than the current state-of-the-
art qubit readout. A reasonable contrast is not achieved with a
$1\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$ readout pulse even for the
highest feasible readout power of $-108\text{\,}\mathrm{dBm}\text{/}$, beyond
which the resonator becomes nonlinear and the quantum non-demolition nature of
the readout breaks down [32, 33].

After finding the resonator and qubit frequencies using the standard single-
tone and two-tone spectroscopies adjusted to our bolometric readout, we carry
out a Rabi oscillation measurement. We initialize the qubit to its ground
state, drive it with a rectangular pulse, and measure the readout resonator
using the bolometer. Here, we employ a $10\text{\,}\mathrm{\SIUnitSymbolMicro
s}\text{/}$ readout pulse and ensemble-average the result 512 times. With the
length and the modulation frequency of the qubit drive pulse are varied, we
obtain a clear chevron pattern as desired [see Fig. 2(e)], from which we
determine a $\pi$ pulse length of $100\text{\,}\mathrm{ns}\text{/}$.

With the qubit frequency and $\pi$ pulse length calibrated, we carry out
single-shot qubit readout by alternating between preparing the qubit in
$\ket{\mathrm{g}}$ and $\ket{\mathrm{e}}$, and recording $S$ with no ensemble-
averaging. For each prepared state, we record $10^{4}$ data points. The data
are binned to produce a histogram, with the bin width chosen using Scott’s
rule [34]. To maximize the readout fidelity
$F=1-P(\mathrm{g}|\mathrm{e})-P(\mathrm{e}|\mathrm{g})$, we optimize the
threshold value for $S$ to decide the measurement outcome, $\ket{\mathrm{g}}$
or $\ket{\mathrm{e}}$. Here, $P(a|b)$ is the probability of measuring the
qubit in the state $\ket{\textrm{a}}$ provided that it was prepared in the
state $\ket{\textrm{b}}$, where the probabilities are obtained from the
measured distributions. The highest fidelity is obtained by increasing the
readout pulse length to
$t_{\mathrm{RO}}=$20\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$$, so that
the above-discussed choice of $t_{0}=0.75\times t_{\mathrm{RO}}$ yields an
averaging time of $5\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$ for
$\bar{V}$, and setting the readout power to
$P_{\mathrm{d}}=$-108\text{\,}\mathrm{dBm}\text{/}$$, which is just below the
point of nonlinearity for the readout resonator. Figure 3(a) shows the
measured probability distributions for the probe signal $S$ with these
parameters, from which we extract the fidelity $F=0.49$.

In an effort to further optimize the single-shot readout fidelity, we carry
out an additional experiment, where we intentionally stretch the readout pulse
to be unreasonably long, $40\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$,
and instead of just storing $S$ derived from the time-averaged quantities
$\bar{V}_{0}$ and $\bar{V}$, we record the full time traces of the bolometer
output signal for 1000 single shots. With these data, we may vary the
effective readout pulse length and averaging time of $\bar{V}$ in post
processing. Figure 3(c) shows the resulting single-shot readout fidelity as a
function of the digitally determined pulse length and averaging time. We
observe a region of relatively high fidelity $F>0.6$ [highlighted by the red
boundary in Fig. 3(c)], with the highest fidelity of 0.618 achieved with
$t_{\mathrm{RO}}=$13.9\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$$ and
$t_{\mathrm{RO}}-t_{0}=$10.6\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$$.
Figure 3(b) shows the probability distributions of the signal with these
parameter values.

The fidelity is overall higher in Fig. 3(b)–(c) than in Fig. 3(a). This is
mostly because in panel (a), $\bar{V}_{0}$ is calculated individually for each
single shot. Since each value of $\bar{V}_{0}$ is averaged over a relatively
short, $1.1\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$ window, it
introduces significant noise to the value of $S$ calculated using Eq. (1). In
contrast, a common value of $\bar{V}_{0}$ is used for the whole data set of
panels (b)–(c), meaning that only the fluctuations of $\bar{V}$ contribute to
the noise of $S$. Rescaling the standard deviation of the data of panels
(b)–(c) by a factor corresponding to the noise from this
$1.1\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$ window reproduces the
increased noise level of panel (a).

For the qubit prepared in $\ket{\mathrm{e}}$, the single-shot probability
distribution is clearly bimodal in Figs. 3(a)–(b). This is expected, since
$t_{\mathrm{RO}}$ is comparable to the qubit energy decay time $T_{1}$, and
thus the qubit experiences significant spontaneous relaxation during the
readout. Even in the case of infinitely narrow distributions and ideal state
preparation, the $T_{1}$ decay produces an error of approximately
$1-\exp\left[-t_{\mathrm{RO}}/(2T_{1})\right]\approx 22\%$ for
$t_{\mathrm{RO}}=$13.9\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$$ and
$T_{1}=$28\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$$ [35]. The long
readout is stems from the relatively long effective thermal time constant of
the bolometer, $\tau_{\mathrm{b}}=$9.4\text{\,}\mathrm{\SIUnitSymbolMicro
s}\text{/}$$, extracted from the data of Fig. 3(c) for the qubit prepared in
$\ket{\mathrm{g}}$. In Fig. 3(c), the trend of relatively high and constant
fidelity obtained for simultaneously increasing readout pulse length and
averaging time indicates that the increase in the fidelity owing to the
increasing bolometer SNR approximately compensates the decrease in the
fidelity owing to increasing qubit decay. With short readout times, the
fidelity is low because the distributions corresponding to the different qubit
states are not well separated.

The data of Fig. 3(a), as well as the $\ket{\mathrm{g}}$ state data of Fig.
3(b) are well modeled by a sum of two Gaussian distributions. For the
$\ket{\mathrm{e}}$ state data of Fig. 3(b), we utilize a model that
incorporates the decay of the qubit during the readout (see Methods). This
advanced model is not needed for the $\ket{\mathrm{e}}$ state data of Fig.
3(a), since the effect of the decay on the distribution is masked by noise.
Using the parameters extracted from these fits, we determine the fidelity with
$t_{\mathrm{RO}}=$13.9\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$$ and the
$T_{1}$ error removed to be $F\approx 0.927$. The $\ket{\mathrm{e}}$ state
data in Fig. 3(b) deviate from the fit above roughly
$150\text{\,}\mathrm{mV}\text{/}$. We attribute this to qubit excitations
outside the computational subspace owing to high readout power.

Figure 3: Single-shot readout. (a) Probability distributions of the single-
shot qubit readout signal $S$ for the qubit prepared in the ground (blue dots)
and excited (red dots) states. The solid lines are fits to a sum of two
Gaussian distributions. The vertical dashed line indicates the threshold which
yields the highest readout fidelity. The bolometer is tuned to its operation
point and the readout power is
$P_{\mathrm{d}}=$-108\text{\,}\mathrm{dBm}\text{/}$$ with a pulse length of
$t_{\mathrm{RO}}=$20\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$$ and an
averaging time of $5\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$ for
$\bar{V}$. (b) As (a) but for
$t_{\mathrm{RO}}=$13.9\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$$ and an
averaging time of $10.6\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$ for
$\bar{V}$ and long averaging for $\bar{V}_{0}$ (see text). Here, the fit for
the excited state is obtained using a model that takes into account the
$T_{1}$ decay of the qubit during the readout (see Methods). (c) Single-shot
readout fidelity as a function of the readout pulse length and averaging time
for $\bar{V}$, both changed in post processing. The red boundary indicates the
region where the fidelity is greater than 0.6. The highest fidelity is
achieved with the parameters indicated by the red cross, where the data of
panel (b) are obtained.

## Conclusions

In summary, we demonstrated that a state-of-the-art thermal detector is
capable of achieving reasonable-fidelity single-shot readout of a
superconducting qubit, i.e., no voltage amplification is required for the
qubit readout signal. We were able to carry out qubit characterization
measurements with significantly less ensemble-averaging than in previous
setups lacking millikelvin amplifiers. Using a
$13.9\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$ readout pulse, we achieve
a single-shot readout fidelity of 0.62, and a fidelity of 0.93 after removing
errors arising from finite qubit lifetime.

Several factors in our experiment can be improved to increase the single-shot
readout fidelity. Most significantly, the SNR in the single-shot readout is
fairly low, which mainly arises from the long effective thermal time constant
of the bolometer, which is on the order of ten microseconds. This motivates to
use a long readout pulse, which degrades the fidelity due to the $T_{1}$ decay
of the qubit. A straightforward way to reduce the time constant is to use a
material with a lower heat capacity in place of AuPd as the absorber. A
promising candidate for this is graphene, which has been observed to lead to
extremely high-sensitivity bolometers with time constants in the hundred-
nanosecond range [36, 27].

Apart from the bolometer chip itself, minor modifications to our experimental
setup may significantly improve the SNR. In the experiments presented here, we
have a number of additional microwave components between the qubit and
bolometer, which were used to independently characterize the qubit and the
bolometer. Removing these components and directly connecting the output of the
qubit readout feedline to the absorber of the bolometer removes the loss of
signal between the two chips. Ultimately, the bolometer could be directly
connected to the readout feedline, either on-chip or in a layered flip-chip
architecture [37]. The losses between the qubit and bolometer are an important
factor in degrading the fidelity especially in our case, since the maximum
power we can apply to the resonator is on the low end of what the bolometer
can detect.

It is also possible to optimize the qubit chip design to be better suited for
photodetection-based readout. In our particular design, the readout resonator
is placed in a notch-type configuration with symmetric coupling to the input
and output feedlines. This is disadvantageous, since effectively half of the
signal reflected from the resonator is lost as it escapes through the input
port. Changing to a transmission configuration with a weak coupling to the
input port is expected to significantly increase the power incident on the
bolometer. The ratio $|\chi|/\kappa_{\mathrm{r}}=0.3$ in our sample is close
to the value of $1/2$ which maximizes the SNR in typical dispersive readout,
but it is sub-optimal for photodetection-based readout where a larger ratio is
desirable. Furthermore, our readout resonator lies below the qubit frequency
which leads to the onset of non-adiabatic and even chaotic resonator dynamics
at a much lower photon number than for an elevated readout frequency [12, 32,
33]. Thus designing the readout circuit to resonate above the qubit frequency
can lead to a greatly increased signal power at the bolometer input.

Finally, the readout fidelity can be improved by advanced pulsing schemes and
improved data analysis methods. Instead of a simple rectangular pulse, the
resonator can be driven by a two-step pulse [13], where the drive amplitude is
initially high for a short amount of time. This may enable the use of a power
level that the bolometer can detect with higher SNR, still maintaining the
quantum-non-demolition nature of the readout. Alternatively, the SNR of the
signal reaching the bolometer may be increased by preparing the qubit in a
higher excited state prior to readout [22], or by using a two-tone drive that
induces an effective longitudinal coupling between the qubit and the bolometer
[38, 39]. Data analysis can be improved by utilizing more sophisticated signal
processing than the simple boxcar time-averaging employed here. For example,
the time-averaging can be weighted by the separation between the average
trajectories corresponding to $\ket{\mathrm{g}}$ and $\ket{\mathrm{e}}$ [13],
or the single-shot trajectories can be classified using a machine learning
algorithm [40].

Note that in our current analysis, we have neglected the falling edge of the
voltage signal that can be taken into use in the advanced analysis methods to
increase the SNR. The averaging over the falling edge does not increase the
$T_{1}$-related infidelity since during this averaging time, the readout pulse
is off and hence possible qubit decay will not lead to a change in the
bolometer signal. In fact, using a graphene bolometer in the calorimetric mode
seems promising to improve the readout fidelity (Methods).

By incorporating the improvements discussed above, we estimate that it may be
possible to carry out single-shot qubit readout at 99.9% level of fidelity
using bolometers (see Methods). Thus, bolometers are a promising component for
scalable high-fidelity readout of superconducting qubits owing to their low
power consumption, large probe tone frequency offset from qubits, resilience
against quantum noise, small footprint, no need for microwave isolators, and
naturally introduced low-temperature bath for qubits that can be impedance
matched at a very broad range of readout frequencies.

## Methods

###  .1 Bolometer fabrication

In the bolometer fabrication, we begin with a four-inch silicon wafer (100) of
high resistivity ($\rho$ $>$ 10 k$\Omega$cm), covered by a 300-nm thermal
oxide. Then, we sputter 200 nm of pure Nb onto the wafer. We use AZ5214E
photoresist in positive mode with hard contact to define the waveguide in a
Karl Suss MA-6 mask aligner. After development, the sample undergoes etching
using a Plasmalab 80Plus Oxford Instruments RIE system. The plasma operates
with a gas flow of $\mathrm{SF_{6}}$/$\mathrm{O_{2}}$ at 40 sccm/20 sccm with
an rf power of 100 W.

We clean the resist residuals in acetone and IPA using an ultrasonic
excitation, and then dry the chip with a nitrogen gun. Next, we use an atomic-
layer deposition (ALD) method to grow a 45-nm dielectric layer of
$\mathrm{Al_{2}O_{3}}$ in a Beneq TFS-500 system. Subsequently, we protect the
dielectric layer at desired capacitor regions using AZ5214E resist, and wet-
etch the rest of the ALD oxide with an ammonium fluoride–hydrofluoric acid
mixture. We then cleave the 4-inch wafer into a 2 $\times$ 2 cm2 chip using
Disco DADdy.

The nanowire is patterned using EPBG5000pES electron beam lithography (EBL)
using a bilayer of MMA/PMMA resist on a single chip. We deposit a 30-nm-thick
AuPd layer in an electron beam evaporator at a rate of 0.5 Å/s. After liftoff
in acetone overnight, we pattern the superconducting leads galvanically
connected to the nanowire by EBL and deposit them with 100 nm of Al at a rate
of 5 Å/s. Finally, we cleave each pixel (5 $\times$ 5 mm2) using a laser
micromachining system and package the chosen chip. We employ Al bonding wires
to connect the chip to the printed circuit board of the sample holder.

###  .2 Qubit fabrication

The qubit samples are fabricated using the following steps: First, a layer of
200 nm of Nb is sputtered onto a high-resistivity silicon substrate. Next, we
define the transmission line, readout resonator, and transmon shunt capacitor
using photolithography, followed by dry reactive ion etching (RIE).

The Al/Al2O3/Al Josephson junctions are subsequently fabricated using standard
electron beam lithography and the Dolan bridge method, where oxidation is used
between the deposition of two Al layers to form the Josephson junctions. To
ensure galvanic contact between Al and Nb, the niobium oxide is removed by
argon milling prior to the deposition of any Al.

The excess metal is lifted off in acetone. The room temperature resistance of
the Al/Al2O3/Al junctions are measured to select a sample that is most likely
to yield the desired qubit frequency. Finally, the chip is diced into
individual samples and the selected sample is wire-bonded to a sample holder.

###  .3 Details of the experimental setup

A diagram of the full experimental setup is shown in Extended Data Fig. 1. A
PXI control computer initiates a measurement by sending a software trigger to
a National Instruments NI-5782 transceiver module connected to a NI PXIe-7962R
analog-to-digital converter (ADC) running custom FPGA code. The transceiver
sends further digital trigger signals to initiate pulses to the readout
resonator and to excite the qubit. The clocks of all devices are synchronized
using a $10\text{\,}\mathrm{MHz}\text{/}$ Rb reference.

The qubit excitation pulse is generated by an Active Technologies AT1212
digital-to-analog converter, upconverted to gigahertz range, and directed into
the cryostat via a series of attenuators and filters. A microwave switch
placed at the mixing-chamber plate is used to select which of the six qubits
on the chip is being driven. The qubit frequency is adjusted by applying a
magnetic field to the qubit-resonator chip using a hand-wound coil outside the
sample holder.

The readout pulse is generated by a microwave source using pulse modulation.
The pulse is reflected off of the resonator on the qubit-resonator chip, after
which it is directed through a circulator, a double pole double throw (DPDT)
switch, a directional coupler and some filters, before reaching the bolometer
absorber port. The components between the qubit-resonator and bolometer chips
are needed only to individually characterize the qubit-resonator chip (by
toggling the DPDT switch) and the bolometer (by applying a pulse through the
directional coupler), and could be removed in future experiments.

To probe the bolometer, a continuous microwave tone is split into two paths, a
reference and a signal. The signal is directed into the cryostat, reflected
off of the probe port of the bolometer, and amplified and demodulated to an
intermediate frequency (IF) using a local oscillator (LO), which is detuned
from the probe tone by a fixed IF frequency of
$70.3125\text{\,}\mathrm{MHz}\text{/}$. The reference is demodulated by the LO
without passing through the cryostat, and is used as an amplitude and phase
reference for the signal. Both IF signals are digitized by the NI-5782 at a
sampling rate of $250\text{\,}\mathrm{M}\mathrm{Sa}\mathrm{/}\mathrm{s}$ and
digitally demodulated into in-phase ($I$) and quadrature ($Q$) components. The
NI-5782 also handles ensemble-averaging the data where applicable, as well as
further downsampling by boxcar averaging a variable number of adjacent points.
From time-domain data in the $IQ$-plane, we calculate the average signal
before the pulse, $\tilde{V}_{0}$, and the average signal during the pulse
$\tilde{V}$, as determined by $t_{0}$ and $t_{\mathrm{RO}}$. The data is then
rotated in the $IQ$-plane such that $\tilde{V}-\tilde{V}_{0}$ lies on the $I$
axis. We define $\bar{V}$ and $\bar{V}_{0}$ discussed in the main text as the
$I$-components of $\tilde{V}$ and $\tilde{V}_{0}$, respectively, after
applying this rotation. For the single-shot experiments, we apply a common
rotation across all shots, with the rotation angle chosen such that the
readout fidelity is maximized. Note that the phase of the signal in the $IQ$
plane is due to the reflection off of the effective $LC$ circuit of the
bolometer, and is thus completely independent of the phase of the photons
emitted by the readout resonator.

The data acquisition and storage is managed by the QCoDeS [41] data
acquisition framework. Data analysis and fitting is carried out using the
NumPy [42], SciPy [43], xarray [44] and lmfit [45] libraries.

Extended Data Fig. 1: Full experimental setup. The labels for the filters,
directional couplers, amplifiers, and the power splitter refer to Mini-
Circuits model numbers. The band-pass filters with two labels denote a low-
pass filter and high-pass filter in series.

###  .4 Sample characterization

Characterization measurements of the bolometer are shown in Extended Data Fig.
2. Extended Data Fig. 2(a) shows the signal coming out of the cryostat
normalized by the reference, as discussed above. The data are averaged for
$1\text{\,}\mathrm{ms}\text{/}$ over 16 repetitions. For each pixel in
Extended Data Fig. 2(b), a $2\text{\,}\mathrm{ms}\text{/}$ pulse is applied to
the bolometer absorber port via the readout resonator with an off-resonant
tone of $f_{\mathrm{d}}=$5.4\text{\,}\mathrm{GHz}\text{/}$$ and
$P_{\mathrm{d}}=$-108\text{\,}\mathrm{dBm}\text{/}$$. From these data, we
extract the time constant $\tau_{\mathrm{b}}$ shown in Extended Data Fig. 2(c)
by fitting the time-domain signal to an exponential model.

Extended Data Fig. 2:  Bolometer characterization. (a) Total transmission
power divided by the reference power, (b) detector signal $S$, and (c)
extracted thermal time constant $\tau_{\mathrm{b}}$ of the bolometer as
functions of the probe frequency $f_{\mathrm{p}}$ and power $P_{\mathrm{p}}$.
The red background approximately indicates the power region where the
bolometer exhibits bistability.

Extended Data Fig. 3 shows standard characterization experiments of the qubit-
resonator system using the bolometer. Panels (a)–(b) show single-tone and two-
tone spectroscopy, respectively, from which we extract an initial estimate for
the resonator and qubit frequencies. In Fig. 3(c), a $\pi$ pulse is applied to
the qubit, and the readout pulse is applied after different delays to extract
the $T_{1}$ decay time. Extended Data Fig. 3(d) shows the result of a Ramsey
experiment, where the qubit is driven by two $\pi/2$ pulses with a varying
idle time between them, for different modulation frequencies of the pulse.
These data are used to extract the $T_{2}$ time, given by the exponentially
decaying envelope of the oscillation. The signal level varies in Extended Data
Fig. 3, since different values of $P_{\mathrm{d}}$ and $t_{\mathrm{RO}}$ were
used during the characterization while optimal values for these parameters
were not yet known. In addition, the cryostat was thermally cycled between the
acquisition of the data of panels (a), (b) and those of (c), (d). This is why
the qubit frequency has decreased in panel (d) compared with panel (b) and
Fig. 2(e).

Extended Data Fig. 3: Qubit characterization using bolometer. (a) Detector
signal $S$ as a function of the bias voltage applied to the flux bias coil and
of the frequency of the resonator drive pulse. The resonator frequency,
indicated by the minimum detector signal at a given flux value, clearly
changes as a function of the flux bias. The data of panel (a) are averaged
over 4 repetitions. (b) Detector signal $S$ as a function of the flux bias
voltage and the frequency of a continuous tone applied to the qubit. The data
is averaged over 32 repetitions. The data of panels (a) and (b) are acquired
using a readout power of
$P_{\mathrm{d}}=$-132.8\text{\,}\mathrm{dBm}\text{/}$$ and a pulse duration
$t_{\mathrm{RO}}=$1\text{\,}\mathrm{ms}\text{/}$$. (c) Detector signal $S$ as
a function of the idling time between a $\pi$ pulse applied to the qubit and
the readout pulse, providing the $T_{1}=$28\text{\,}\mathrm{\SIUnitSymbolMicro
s}\text{/}$$ of the qubit. In these measurements,
$P_{\mathrm{d}}=$-124.8\text{\,}\mathrm{dBm}\text{/}$$ and
$t_{\mathrm{RO}}=$20\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$$, and the
data are averaged over 4096 repetitions. (d) Detector signal as a function of
the idle time between the $\pi/2$ pulses and the qubit drive frequency in a
Ramsey experiment which is used to extract the $T_{2}$ of the qubit. These
data are acquired with $P_{\mathrm{d}}=$-119.8\text{\,}\mathrm{dBm}\text{/}$$
and $t_{\mathrm{RO}}=$25\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$$,
averaged over 1024 repetitions.

###  .5 Detector signal probability distribution with long averaging time

Following Ref. [35], we develop a model for the probability distribution of
the detector signal for the qubit nominally prepared in $\ket{\mathrm{e}}$
(with some preparation error) and for the averaging time being significant
compared to the energy decay time $T_{1}$. Let
$v_{\mathrm{g}/\mathrm{e}}(t,t_{\mathrm{d}})$ be the time domain signal of the
bolometer output when the qubit is in the state $\ket{\mathrm{g}}$ or
$\ket{\mathrm{e}}$ at the beginning of the readout pulse, $t=0$. Here,
$t_{\mathrm{d}}$ denotes the time at which the qubit instantaneously decays to
$\ket{\mathrm{g}}$ if it was initially in $\ket{\mathrm{e}}$ ($v_{\mathrm{g}}$
is independent of $t_{\mathrm{d}}$). We assume that no thermal excitation
events occur, and that the noise in the signal is Gaussian. Formally, the
signal is given by

$\displaystyle
v_{\mathrm{g}/\mathrm{e}}(t,t_{\mathrm{d}})\mathrm{d}t=u_{\mathrm{g}/\mathrm{e}}(t,t_{\mathrm{d}})\mathrm{d}t+\sqrt{P_{\mathrm{N}}}\,\mathrm{d}W(t),$
(2)

where $u_{\mathrm{g}/\mathrm{e}}(t,t_{\mathrm{d}})$ is the evolution of the
output in the absence of noise, $P_{\mathrm{N}}$ is the noise power spectral
density and $\mathrm{d}W(t)$ is the Wiener increment [46].

To incorporate the bolometer dynamics, we model
$u_{\mathrm{g}/\mathrm{e}}(t,t_{\mathrm{d}})$ as exponentially approaching
some steady-state values $c_{\mathrm{g}}$ and $c_{\mathrm{e}}$, for the qubit
in the state $\ket{\mathrm{g}}$ or $\ket{\mathrm{e}}$, respectively [26].
Namely, we assume that the time evolution is of the form

$\displaystyle u_{\mathrm{g}}(t)$
$\displaystyle=c_{\mathrm{g}}(1-e^{-t/\tau_{\mathrm{b}}})$ (3)

and

$\displaystyle\begin{aligned}
u_{\mathrm{e}}(t,t_{\mathrm{d}})&=\hphantom{{}+{}}\theta(t_{\mathrm{d}}-t)c_{\mathrm{e}}\left(1-\mathrm{e}^{-t/\tau_{\mathrm{b}}}\right)\\\
&\hphantom{{}={}}+\theta(t-t_{\mathrm{d}})\left[(c_{\mathrm{g}}-c_{\mathrm{e}}^{\prime})\left(1-\mathrm{e}^{-(t-t_{\mathrm{d}})/\tau_{\mathrm{b}}}\right)+c_{\mathrm{e}}^{\prime}\right],\rule[-15.00002pt]{0.0pt}{0.0pt}\end{aligned}$
(4)

where $\theta(x)$ is the step function and
$c_{\mathrm{e}}^{\prime}=c_{\mathrm{e}}(1-\mathrm{e}^{-t_{\mathrm{d}}/\tau_{\mathrm{b}}})$.

Averaging the bolometer output from $t_{0}$ to $t_{\mathrm{RO}}$ yields the
averaged detector signal

$\displaystyle\bar{V}_{\mathrm{g}/\mathrm{e}}(t_{\mathrm{d}})$
$\displaystyle=\frac{1}{t_{\mathrm{RO}}-t_{0}}\int_{t_{0}}^{t_{\mathrm{RO}}}v_{\mathrm{g}/\mathrm{e}}(t,t_{\mathrm{d}})\mathrm{d}t$
$\displaystyle=\bar{U}_{\mathrm{g}/\mathrm{e}}(t_{\mathrm{d}})+X[0,\sigma^{2}],$
(5)

where
$\bar{U}_{\mathrm{g}/\mathrm{e}}(t_{\mathrm{d}})\equiv(t_{\mathrm{RO}}-t_{0})^{-1}\int_{t_{0}}^{t_{\mathrm{RO}}}u_{\mathrm{g}/\mathrm{e}}(t,t_{\mathrm{d}})\,\mathrm{d}t$
and $X[0,\sigma^{2}]$ is a normally distributed random variable with mean $0$
and standard deviation $\sigma=\sqrt{P_{\mathrm{N}}/(t_{\mathrm{RO}}-t_{0})}$.
Above, we have neglected the constant $\bar{V}_{0}$, which simply shifts
$\bar{V}_{\mathrm{g}/\mathrm{e}}$. With a fixed $t_{\mathrm{d}}$, the
probability of obtaining a given detector voltage $V$ thus obeys the
probability distribution

$\displaystyle
P_{\mathrm{g}/\mathrm{e}}\left(V|t_{\mathrm{d}}\right)=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\left\\{-\frac{\left[V-\bar{U}_{\mathrm{g}/\mathrm{e}}(t_{\mathrm{d}})\right]^{2}}{2\sigma^{2}}\right\\},$
(6)

which is a Gaussian centered around
$\bar{U}_{\mathrm{g}/\mathrm{e}}(t_{\mathrm{d}})$ having a standard deviation
$\sigma$ as defined above.

If the qubit is in the state $\ket{\mathrm{g}}$ at $t=0$, the probability
distribution $P_{\mathrm{g}}(V)$ is simply a Gaussian centered around the
constant value $\bar{U}_{\mathrm{g}}$. When the qubit is initially in
$\ket{\mathrm{e}}$, the probability distribution is obtained by calculating
the average of $P_{\mathrm{e}}(V|t_{\mathrm{d}})$ for all possible
realizations of the qubit decay time $t_{\mathrm{d}}$ weighted by the
probability density $P(t_{\mathrm{d}})$ of the decay occurring at
$t_{\mathrm{d}}$:

$\displaystyle P_{\mathrm{e}}(V)$
$\displaystyle=\int_{0}^{\infty}P(V|t_{\mathrm{d}})P(t_{\mathrm{d}})\,\mathrm{d}t_{\mathrm{d}},$
(7)

where $P(t_{\mathrm{d}})$ is exponentially distributed:
$P(t_{\mathrm{d}})=\mathrm{e}^{-t_{\mathrm{d}}/T_{1}}/T_{1}$. Thus the total
probability distribution for the qubit nominally prepared in
$\ket{\mathrm{e}}$ is the following weighted sum of these distributions:

$\displaystyle
P_{\mathrm{e}}^{\text{tot}}(V)=P_{\mathrm{x}}P_{\mathrm{g}}(V)+(1-P_{\mathrm{x}})P_{\mathrm{e}}(V),$
(8)

where $P_{\mathrm{x}}$ is the probability that the qubit was actually in the
state $\ket{\mathrm{g}}$ at $t=0$. Here, $P_{\mathrm{x}}$ includes state
preparation errors, as well as the $T_{1}$ decay of the qubit during the short
delay between the application of the $\pi$ pulse and the start of the readout
pulse.

In the case where $u_{\mathrm{g}/\mathrm{e}}(t,t_{\mathrm{d}})$ is step-
function-like, Eqs. (7) and (8) have analytical expressions [35]. However, we
do not obtain an analytical expression for $P_{\mathrm{e}}(V)$ given the
temporal evolution arising from Eq. (4), and hence, we calculate the integral
of Eq. (7) numerically. Scaling this by the bin width used in Fig. 3(b)
produces the prediction for the distribution, with $c_{\mathrm{g}}$,
$c_{\mathrm{e}}$, $T_{1}$, $\sigma$, and $P_{\mathrm{x}}$ as the fitting
parameters. For the data of Fig. 3(b), we find
$c_{\mathrm{g}}=$24.7\text{\,}\mathrm{mV}\text{/}$$,
$c_{\mathrm{e}}=$182\text{\,}\mathrm{mV}\text{/}$$,
$T_{1}=$25.8\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$$,
$\sigma=$17.4\text{\,}\mathrm{mV}\text{/}$$ and $P_{\mathrm{x}}=0.20$.

The readout fidelity is given by

$\displaystyle F$
$\displaystyle=1-\int_{-\infty}^{V_{\text{th}}}P_{\mathrm{e}}^{\text{tot}}(V)\,\mathrm{d}V-\int_{V_{\text{th}}}^{\infty}P_{\mathrm{g}}(V)\,\mathrm{d}V,$
(9)

where $V_{\text{th}}$ is the threshold value for assigning the measurement
outcome to $\ket{\mathrm{g}}$ or $\ket{\mathrm{e}}$, and we again assume that
no thermal excitations occur. If we further assume that $P_{\mathrm{x}}=0$ and
that the qubit never relaxes, $P_{\mathrm{e}}^{\text{tot}}(V)$ reduces to a
Gaussian distribution. In this case, the readout fidelity is maximized with
$V_{\text{th}}=(\bar{U}_{\mathrm{e}}+\bar{U}_{\mathrm{g}})/2$, which yields

$\displaystyle F$
$\displaystyle=\operatorname{erf}\left(\frac{\bar{U}_{\mathrm{e}}-\bar{U}_{\mathrm{g}}}{2\sqrt{2\sigma^{2}}}\right),$
(10)

where $\operatorname{erf}(x)$ is the error function.

### Estimate of parameters needed for 99.9% fidelity

Here, we quantitatively discuss how to achieve 99.9% fidelity with bolometric
readout following the improvements to the experiment proposed in the main
text. To be able make a fair comparison between the performance of different
bolometers presented in the literature [26, 27], we carry out this analysis by
assuming that we operate in the calorimetric mode. Namely, we interpret the
distributions of Fig. 3(b) to represent a measurement of the energy of the
absorbed pulses. In such an experiment, the signal is directly proportional to
the energy packet arriving at the bolometer whereas the width of the
distribution is independent of the energy.

Let us first assume that the length of the readout pulse is reduced to 200 ns
without changing the readout power. This means that the bolometer-absorbed
energy, and hence the SNR in Fig. 3(b) is reduced by a factor of 70. By
changing the metallic bolometer to the graphene bolometer, the energy
resolution has been observed to increase by a factor of 13 [27]. Thus to
obtain 99.9% fidelity, one needs to increase the amount of energy absorbed by
the graphene bolometer in $200\text{\,}\mathrm{ns}\text{/}$ by a factor of
$A_{99.9\%}\approx 1.8\times 70/13\approx 10$, where the factor of 1.8 arises
from the fact that in order to reduce the overlap infidelity of Fig. 3(b) from
7% to 0.1%, one needs a factor of 1.8 improvement in the SNR in Eq. (10).

Below, we aim to show that by making different improvements to the measurement
scheme, we can arrive at several factors $A_{\alpha}$ of increment to the
energy absorbed by the bolometer, and taken all these factors into account, it
is possible to exceed the required SNR, i.e.,
$A_{99.9\%}=10<\prod_{\alpha}A_{\alpha}$. Since we have assumed
$200\text{\,}\mathrm{ns}\text{/}$ readout time, having a qubit $T_{1}$ of the
order of $100\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$ is sufficient to
reach 99.9% readout fidelity. Note that the bolometer is a power sensor, and
thus the SNR is directly proportional to the absorbed power for low enough
powers considered here [27] [see also Fig. 2(d)].

Firstly, by switching to a transmission-type setup, where all of the readout
photons are directed into the bolometer instead of half of them escaping
through the input, increases the power incident on the bolometer by a factor
of $A_{\text{t}}=2$. Secondly, removing additional components between the
qubit and bolometer chips reduces losses approximately by
$A_{\text{c}}\approx$1\text{\,}\mathrm{dB}\text{/}$\approx 1.25$. Thirdly, by
optimizing the dispersive shift $\chi$ for photodetection-type readout, we
expect that the photon number occupying the resonator during readout and thus
incident power can be increased by a factor $A_{\chi}=2$ based on Ref. [29].
Fourthly, switching the qubit-resonator system to a configuration where the
resonator resonance frequency lies above the qubit frequency (keeping the
detuning $|f_{\mathrm{r},\mathrm{g}}-f_{\mathrm{q}}|$ constant) allows driving
with a larger number of photons [32], which we estimate to be at least
$A_{\text{a}}=1.5$ times greater than with our current parameters. Fifthly,
increasing the admissible drive power by doubling the resonator drive
frequency to $10\text{\,}\mathrm{GHz}\text{/}$ may further double the photon
energy $hf_{\mathrm{r},\mathrm{g}}$, introducing a factor $A_{2f}=2$. In
total, the improvement to the SNR is given by

$A_{\text{t}}A_{\text{c}}A_{\text{a}}A_{2f}A_{\chi}\approx 15>A_{99.9\%},$
(11)

and thus a high fidelity is feasibly achievable.

## Data availability

The data that support the findings of this study are available at
https://doi.org/10.5281/zenodo.7773981.

## Author contributions

A.M.G. and S.N. conducted the experiments and analysed the data. W.L. designed
and fabricated the bolometer chip. The qubit chip was designed by S.K. and
fabricated by J.M. with help from V. Ve. G.C., P.S. and Q.C. assisted with
characterizing and operating the bolometer. A.M.G. and V. Va. developed the
model for the signal probability distribution. The manuscript was written by
A.M.G. and M.M., with comments from all authors. The work was conceived and
supervised by M.M.

## Acknowledgements

The authors acknowledge funding from the Academy of Finland Centre of
Excellence program (project nos. 352925, 336810, 336817, and 336819), European
Research Council under Advanced Grant no. 101053801 (ConceptQ), the Future
Makers Program of the Jane and Aatos Erkko Foundation, the Technology
Industries of Finland Centennial Foundation and the Finnish Foundation for
Technology Promotion. We thank Arman Alizadeh, Slawomir Simbierowicz, and
Russell Lake for useful discussions.

## Competing interests

M.M. declares that that he is a Co-Founder of the quantum-computer company IQM
Finland Oy. Other authors declare no competing interests.

## References

  * DiVincenzo [2000] D. P. DiVincenzo, The Physical Implementation of Quantum Computation, Fortschritte der Physik 48, 771 (2000).
  * Nielsen and Chuang [2011] M. A. Nielsen and I. L. Chuang,  _Quantum Computation and Quantum Information_ , 10th ed. (Cambridge University Press, Cambridge, New York, 2011).
  * Shor [1995] P. W. Shor, Scheme for reducing decoherence in quantum computer memory, Physical Review A 52, R2493 (1995).
  * Preskill [1997] J. Preskill, Fault-tolerant quantum computation (1997), arxiv:arXiv:quant-ph/9712048 .
  * Terhal [2015] B. M. Terhal, Quantum error correction for quantum memories, Reviews of Modern Physics 87, 307 (2015).
  * Blais  _et al._ [2004] A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation, Physical Review A 69, 062320 (2004).
  * Barends  _et al._ [2014] R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O‘Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, Logic gates at the surface code threshold: Superconducting qubits poised for fault-tolerant quantum computing, Nature 508, 500 (2014).
  * Kjaergaard  _et al._ [2020] M. Kjaergaard, M. E. Schwartz, J. Braumüller, P. Krantz, J. I.-J. Wang, S. Gustavsson, and W. D. Oliver, Superconducting Qubits: Current State of Play, Annual Review of Condensed Matter Physics 11, 369 (2020).
  * Zhao  _et al._ [2022] Y. Zhao, Y. Ye, H.-L. Huang, Y. Zhang, D. Wu, H. Guan, Q. Zhu, Z. Wei, T. He, S. Cao, F. Chen, T.-H. Chung, H. Deng, D. Fan, M. Gong, C. Guo, S. Guo, L. Han, N. Li, S. Li, Y. Li, F. Liang, J. Lin, H. Qian, H. Rong, H. Su, L. Sun, S. Wang, Y. Wu, Y. Xu, C. Ying, J. Yu, C. Zha, K. Zhang, Y.-H. Huo, C.-Y. Lu, C.-Z. Peng, X. Zhu, and J.-W. Pan, Realization of an Error-Correcting Surface Code with Superconducting Qubits, Physical Review Letters 129, 030501 (2022).
  * Acharya  _et al._ [2023] R. Acharya, I. Aleiner, R. Allen, T. I. Andersen, M. Ansmann, F. Arute, K. Arya, A. Asfaw, J. Atalaya, R. Babbush, D. Bacon, J. C. Bardin, J. Basso, A. Bengtsson, S. Boixo, G. Bortoli, A. Bourassa, J. Bovaird, L. Brill, M. Broughton, B. B. Buckley, D. A. Buell, T. Burger, B. Burkett, N. Bushnell, Y. Chen, Z. Chen, B. Chiaro, J. Cogan, R. Collins, P. Conner, W. Courtney, A. L. Crook, B. Curtin, D. M. Debroy, A. Del Toro Barba, S. Demura, A. Dunsworth, D. Eppens, C. Erickson, L. Faoro, E. Farhi, R. Fatemi, L. Flores Burgos, E. Forati, A. G. Fowler, B. Foxen, W. Giang, C. Gidney, D. Gilboa, M. Giustina, A. Grajales Dau, J. A. Gross, S. Habegger, M. C. Hamilton, M. P. Harrigan, S. D. Harrington, O. Higgott, J. Hilton, M. Hoffmann, S. Hong, T. Huang, A. Huff, W. J. Huggins, L. B. Ioffe, S. V. Isakov, J. Iveland, E. Jeffrey, Z. Jiang, C. Jones, P. Juhas, D. Kafri, K. Kechedzhi, J. Kelly, T. Khattar, M. Khezri, M. Kieferová, S. Kim, A. Kitaev, P. V. Klimov, A. R. Klots, A. N. Korotkov, F. Kostritsa, J. M. Kreikebaum, D. Landhuis, P. Laptev, K.-M. Lau, L. Laws, J. Lee, K. Lee, B. J. Lester, A. Lill, W. Liu, A. Locharla, E. Lucero, F. D. Malone, J. Marshall, O. Martin, J. R. McClean, T. McCourt, M. McEwen, A. Megrant, B. Meurer Costa, X. Mi, K. C. Miao, M. Mohseni, S. Montazeri, A. Morvan, E. Mount, W. Mruczkiewicz, O. Naaman, M. Neeley, C. Neill, A. Nersisyan, H. Neven, M. Newman, J. H. Ng, A. Nguyen, M. Nguyen, M. Y. Niu, T. E. O’Brien, A. Opremcak, J. Platt, A. Petukhov, R. Potter, L. P. Pryadko, C. Quintana, P. Roushan, N. C. Rubin, N. Saei, D. Sank, K. Sankaragomathi, K. J. Satzinger, H. F. Schurkus, C. Schuster, M. J. Shearn, A. Shorter, V. Shvarts, J. Skruzny, V. Smelyanskiy, W. C. Smith, G. Sterling, D. Strain, M. Szalay, A. Torres, G. Vidal, B. Villalonga, C. Vollgraff Heidweiller, T. White, C. Xing, Z. J. Yao, P. Yeh, J. Yoo, G. Young, A. Zalcman, Y. Zhang, N. Zhu, and Google Quantum AI, Suppressing quantum errors by scaling a surface code logical qubit, Nature 614, 676 (2023).
  * Koch  _et al._ [2007] J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Charge-insensitive qubit design derived from the Cooper pair box, Physical Review A 76, 042319 (2007).
  * Blais  _et al._ [2021] A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, Circuit Quantum Electrodynamics, Reviews of Modern Physics 93, 025005 (2021).
  * Walter  _et al._ [2017] T. Walter, P. Kurpiers, S. Gasparinetti, P. Magnard, A. Potočnik, Y. Salathé, M. Pechal, M. Mondal, M. Oppliger, C. Eichler, and A. Wallraff, Rapid High-Fidelity Single-Shot Dispersive Readout of Superconducting Qubits, Physical Review Applied 7, 054020 (2017).
  * Sunada  _et al._ [2022] Y. Sunada, S. Kono, J. Ilves, S. Tamate, T. Sugiyama, Y. Tabuchi, and Y. Nakamura, Fast readout and reset of a superconducting qubit coupled to a resonator with an intrinsic Purcell filter, Physical Review Applied 17, 044016 (2022).
  * Heinsoo  _et al._ [2018] J. Heinsoo, C. K. Andersen, A. Remm, S. Krinner, T. Walter, Y. Salathé, S. Gasparinetti, J.-C. Besse, A. Potočnik, A. Wallraff, and C. Eichler, Rapid High-fidelity Multiplexed Readout of Superconducting Qubits, Physical Review Applied 10, 034040 (2018).
  * Aumentado [2020] J. Aumentado, Superconducting Parametric Amplifiers: The State of the Art in Josephson Parametric Amplifiers, IEEE Microwave Magazine 21, 45 (2020).
  * Esposito  _et al._ [2021] M. Esposito, A. Ranadive, L. Planat, and N. Roch, Perspective on traveling wave microwave parametric amplifiers, Applied Physics Letters 119, 120501 (2021).
  * Clerk  _et al._ [2010] A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, Introduction to quantum noise, measurement, and amplification, Reviews of Modern Physics 82, 1155 (2010).
  * Bergeal  _et al._ [2010] N. Bergeal, F. Schackert, M. Metcalfe, R. Vijay, V. E. Manucharyan, L. Frunzio, D. E. Prober, R. J. Schoelkopf, S. M. Girvin, and M. H. Devoret, Phase-preserving amplification near the quantum limit with a Josephson ring modulator, Nature 465, 64 (2010).
  * Macklin  _et al._ [2015] C. Macklin, K. O’Brien, D. Hover, M. E. Schwartz, V. Bolkhovsky, X. Zhang, W. D. Oliver, and I. Siddiqi, A near–quantum-limited Josephson traveling-wave parametric amplifier, Science 350, 307 (2015).
  * Opremcak  _et al._ [2021] A. Opremcak, C. H. Liu, C. Wilen, K. Okubo, B. G. Christensen, D. Sank, T. C. White, A. Vainsencher, M. Giustina, A. Megrant, B. Burkett, B. L. T. Plourde, and R. McDermott, High-Fidelity Measurement of a Superconducting Qubit Using an On-Chip Microwave Photon Counter, Physical Review X 11, 011027 (2021).
  * Chen  _et al._ [2022] L. Chen, H.-X. Li, Y. Lu, C. W. Warren, C. J. Križan, S. Kosen, M. Rommel, S. Ahmed, A. Osman, J. Biznárová, A. F. Roudsari, B. Lienhard, M. Caputo, K. Grigoras, L. Grönberg, J. Govenius, A. F. Kockum, P. Delsing, J. Bylander, and G. Tancredi, Transmon qubit readout fidelity at the threshold for quantum error correction without a quantum-limited amplifier (2022), arxiv:arXiv:2208.05879 .
  * Lecocq  _et al._ [2021] F. Lecocq, F. Quinlan, K. Cicak, J. Aumentado, S. A. Diddams, and J. D. Teufel, Control and readout of a superconducting qubit using a photonic link, Nature 591, 575 (2021).
  * Delaney  _et al._ [2022] R. D. Delaney, M. D. Urmey, S. Mittal, B. M. Brubaker, J. M. Kindem, P. S. Burns, C. A. Regal, and K. W. Lehnert, Superconducting-qubit readout via low-backaction electro-optic transduction, Nature 606, 489 (2022).
  * Govenius  _et al._ [2016] J. Govenius, R. E. Lake, K. Y. Tan, and M. Möttönen, Detection of Zeptojoule Microwave Pulses Using Electrothermal Feedback in Proximity-Induced Josephson Junctions, Physical Review Letters 117, 030802 (2016).
  * Kokkoniemi  _et al._ [2019] R. Kokkoniemi, J. Govenius, V. Vesterinen, R. E. Lake, A. M. Gunyhó, K. Y. Tan, S. Simbierowicz, L. Grönberg, J. Lehtinen, M. Prunnila, J. Hassel, A. Lamminen, O.-P. Saira, and M. Möttönen, Nanobolometer with ultralow noise equivalent power, Communications Physics 2, 1 (2019).
  * Kokkoniemi  _et al._ [2020] R. Kokkoniemi, J.-P. Girard, D. Hazra, A. Laitinen, J. Govenius, R. E. Lake, I. Sallinen, V. Vesterinen, M. Partanen, J. Y. Tan, K. W. Chan, K. Y. Tan, P. Hakonen, and M. Möttönen, Bolometer operating at the threshold for circuit quantum electrodynamics, Nature 586, 47 (2020).
  * Barends  _et al._ [2013] R. Barends, J. Kelly, A. Megrant, D. Sank, E. Jeffrey, Y. Chen, Y. Yin, B. Chiaro, J. Mutus, C. Neill, P. O’Malley, P. Roushan, J. Wenner, T. C. White, A. N. Cleland, and J. M. Martinis, Coherent Josephson qubit suitable for scalable quantum integrated circuits, Physical Review Letters 111, 10.1103/PhysRevLett.111.080502 (2013).
  * Nesterov  _et al._ [2020] K. N. Nesterov, I. V. Pechenezhskiy, and M. G. Vavilov, Counting statistics of microwave photons in circuit QED, Physical Review A 101, 052321 (2020).
  * Gambetta  _et al._ [2008] J. Gambetta, A. Blais, M. Boissonneault, A. A. Houck, D. I. Schuster, and S. M. Girvin, Quantum trajectory approach to circuit QED: Quantum jumps and the Zeno effect, Physical Review A 77, 012112 (2008).
  * Chen [2018] Z. Chen,  _Metrology of Quantum Control and Measurement in Superconducting Qubits_ , Ph.D. thesis, University of California Santa Barbara (2018).
  * Cohen  _et al._ [2022] J. Cohen, A. Petrescu, R. Shillito, and A. Blais, Reminiscence of classical chaos in driven transmons (2022), arxiv:arXiv:2207.09361 .
  * Khezri  _et al._ [2022] M. Khezri, A. Opremcak, Z. Chen, A. Bengtsson, T. White, O. Naaman, R. Acharya, K. Anderson, M. Ansmann, F. Arute, K. Arya, A. Asfaw, J. C. Bardin, A. Bourassa, J. Bovaird, L. Brill, B. B. Buckley, D. A. Buell, T. Burger, B. Burkett, N. Bushnell, J. Campero, B. Chiaro, R. Collins, A. L. Crook, B. Curtin, S. Demura, A. Dunsworth, C. Erickson, R. Fatemi, V. S. Ferreira, L. F. Burgos, E. Forati, B. Foxen, G. Garcia, W. Giang, M. Giustina, R. Gosula, A. G. Dau, M. C. Hamilton, S. D. Harrington, P. Heu, J. Hilton, M. R. Hoffmann, S. Hong, T. Huang, A. Huff, J. Iveland, E. Jeffrey, J. Kelly, S. Kim, P. V. Klimov, F. Kostritsa, J. M. Kreikebaum, D. Landhuis, P. Laptev, L. Laws, K. Lee, B. J. Lester, A. T. Lill, W. Liu, A. Locharla, E. Lucero, S. Martin, M. McEwen, A. Megrant, X. Mi, K. C. Miao, S. Montazeri, A. Morvan, M. Neeley, C. Neill, A. Nersisyan, J. H. Ng, A. Nguyen, M. Nguyen, R. Potter, C. Quintana, C. Rocque, P. Roushan, K. Sankaragomathi, K. J. Satzinger, C. Schuster, M. J. Shearn, A. Shorter, V. Shvarts, J. Skruzny, W. C. Smith, G. Sterling, M. Szalay, D. Thor, A. Torres, B. W. K. Woo, Z. J. Yao, P. Yeh, J. Yoo, G. Young, N. Zhu, N. Zobrist, D. Sank, A. Korotkov, Y. Chen, and V. Smelyanskiy, Measurement-Induced State Transitions in a Superconducting Qubit: Within the Rotating Wave Approximation (2022), arxiv:arXiv:2212.05097 .
  * SCOTT [1979] D. W. SCOTT, On optimal and data-based histograms, Biometrika 66, 605 (1979).
  * Gambetta  _et al._ [2007] J. Gambetta, W. A. Braff, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, Protocols for optimal readout of qubits using a continuous quantum nondemolition measurement, Physical Review A 76, 012325 (2007).
  * McKitterick  _et al._ [2013] C. B. McKitterick, D. E. Prober, and B. S. Karasik, Performance of graphene thermal photon detectors, Journal of Applied Physics 113, 044512 (2013).
  * Kosen  _et al._ [2022] S. Kosen, H.-X. Li, M. Rommel, D. Shiri, C. Warren, L. Grönberg, J. Salonen, T. Abad, J. Biznárová, M. Caputo, L. Chen, K. Grigoras, G. Johansson, A. F. Kockum, C. Križan, D. P. Lozano, G. J. Norris, A. Osman, J. Fernández-Pendás, A. Ronzani, A. F. Roudsari, S. Simbierowicz, G. Tancredi, A. Wallraff, C. Eichler, J. Govenius, and J. Bylander, Building blocks of a flip-chip integrated superconducting quantum processor, Quantum Science and Technology 7, 035018 (2022).
  * Touzard  _et al._ [2019] S. Touzard, A. Kou, N. E. Frattini, V. V. Sivak, S. Puri, A. Grimm, L. Frunzio, S. Shankar, and M. H. Devoret, Gated Conditional Displacement Readout of Superconducting Qubits, Physical Review Letters 122, 080502 (2019).
  * Ikonen  _et al._ [2019] J. Ikonen, J. Goetz, J. Ilves, A. Keränen, A. M. Gunyho, M. Partanen, K. Y. Tan, D. Hazra, L. Grönberg, V. Vesterinen, S. Simbierowicz, J. Hassel, and M. Möttönen, Qubit Measurement by Multichannel Driving, Physical Review Letters 122, 10.1103/PhysRevLett.122.080503 (2019).
  * Lienhard  _et al._ [2022] B. Lienhard, A. Vepsäläinen, L. C. Govia, C. R. Hoffer, J. Y. Qiu, D. Ristè, M. Ware, D. Kim, R. Winik, A. Melville, B. Niedzielski, J. Yoder, G. J. Ribeill, T. A. Ohki, H. K. Krovi, T. P. Orlando, S. Gustavsson, and W. D. Oliver, Deep-Neural-Network Discrimination of Multiplexed Superconducting-Qubit States, Physical Review Applied 17, 014024 (2022).
  * Nielsen  _et al._ [2022] J. H. Nielsen, M. Astafev, W. H. P. Nielsen, D. Vogel, lakhotiaharshit, A. Johnson, A. U. C. Hardal, Akshita, s. chatoor, Farzad, Liang, G. Ungaretti, S. Pauka, T. Morgan, Adriaan, G. Prawiroatmodjo, B. Nijholt, P. Eendebak, qSaevar, P. Eendebak, S. Droege, J. Darulova, R. van Gulik, N. Pearson, ThorvaldLarsen, A. Corna, and damazter, QCoDeS/Qcodes: 0.34.0 - June 2022 (2022-06-13), Zenodo (2022).
  * Harris  _et al._ [2020] C. R. Harris, K. J. Millman, S. J. van der Walt, R. Gommers, P. Virtanen, D. Cournapeau, E. Wieser, J. Taylor, S. Berg, N. J. Smith, R. Kern, M. Picus, S. Hoyer, M. H. van Kerkwijk, M. Brett, A. Haldane, J. F. del Río, M. Wiebe, P. Peterson, P. Gérard-Marchant, K. Sheppard, T. Reddy, W. Weckesser, H. Abbasi, C. Gohlke, and T. E. Oliphant, Array programming with NumPy, Nature 585, 357 (2020).
  * Virtanen  _et al._ [2020] P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, İ. Polat, Y. Feng, E. W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Henriksen, E. A. Quintero, C. R. Harris, A. M. Archibald, A. H. Ribeiro, F. Pedregosa, and P. van Mulbregt, SciPy 1.0: Fundamental algorithms for scientific computing in Python, Nature Methods , 1 (2020).
  * Hoyer and Joseph [2017] S. Hoyer and H. Joseph, Xarray: N-D labeled Arrays and Datasets in Python, Journal of Open Research Software 5, 10 (2017).
  * Newville  _et al._ [2022] M. Newville, R. Otten, A. Nelson, T. Stensitzki, A. Ingargiola, D. Allan, A. Fox, F. Carter, Michał, R. Osborn, D. Pustakhod, lneuhaus, S. Weigand, A. Aristov, Glenn, C. Deil, Mark, A. L. R. Hansen, G. Pasquevich, L. Foks, N. Zobrist, O. Frost, Stuermer, azelcer, A. Polloreno, A. Persaud, J. H. Nielsen, M. Pompili, S. Caldwell, and A. Hahn, Lmfit/lmfit-py: 1.1.0, Zenodo (2022).
  * Jacobs [2010] K. Jacobs,  _Stochastic Processes for Physicists: Understanding Noisy Systems_ (Cambridge University Press, Cambridge, 2010).