Present address: ]Institute of Physics, Czech Academy of Sciences, 162 00
Prague, Czech Republic

# Flip-chip-based fast inductive parity readout of a planar superconducting
island

M. Hinderling  IBM Research Europe - Zurich, Säumerstrasse 4, 8803 Rüschlikon,
Switzerland  S. C. ten Kate  IBM Research Europe - Zurich, Säumerstrasse 4,
8803 Rüschlikon, Switzerland  D. Z. Haxell  IBM Research Europe - Zurich,
Säumerstrasse 4, 8803 Rüschlikon, Switzerland  M. Coraiola  IBM Research
Europe - Zurich, Säumerstrasse 4, 8803 Rüschlikon, Switzerland  S. Paredes
IBM Research Europe - Zurich, Säumerstrasse 4, 8803 Rüschlikon, Switzerland
E. Cheah  Solid State Physics Laboratory, ETH Zurich, Otto-Stern-Weg 1, 8093
Zürich, Switzerland  F. Krizek  [  IBM Research Europe - Zurich, Säumerstrasse
4, 8803 Rüschlikon, Switzerland  Solid State Physics Laboratory, ETH Zurich,
Otto-Stern-Weg 1, 8093 Zürich, Switzerland  R. Schott  Solid State Physics
Laboratory, ETH Zurich, Otto-Stern-Weg 1, 8093 Zürich, Switzerland  W.
Wegscheider  Solid State Physics Laboratory, ETH Zurich, Otto-Stern-Weg 1,
8093 Zürich, Switzerland  D. Sabonis  IBM Research Europe - Zurich,
Säumerstrasse 4, 8803 Rüschlikon, Switzerland  F. Nichele  Author to whom
correspondence should be addressed<EMAIL_ADDRESS>IBM Research Europe -
Zurich, Säumerstrasse 4, 8803 Rüschlikon, Switzerland

###### Abstract

Properties of superconducting devices depend sensitively on the parity (even
or odd) of the quasiparticles they contain. Encoding quantum information in
the parity degree of freedom is central in several emerging solid-state qubit
architectures. Yet, accurate, non-destructive, and time-resolved parity
measurement is a challenging and long-standing issue. Here we report on
control and real-time parity measurement in a superconducting island embedded
in a superconducting loop and realized in a hybrid two-dimensional
heterostructure using a microwave resonator. Device and readout resonator are
located on separate chips, connected via flip-chip bonding, and couple
inductively through vacuum. The superconducting resonator detects the parity-
dependent circuit inductance, allowing for fast and non-destructive parity
readout. We resolved even and odd parity states with signal-to-noise ratio SNR
$\approx 3$ with an integration time of 20 $\mu$s and detection fidelity
exceeding 98 $\%$. Real-time parity measurement showed state lifetime
extending into millisecond range. Our approach will lead to better
understanding of coherence-limiting mechanisms in superconducting quantum
hardware and provide novel readout schemes for hybrid qubits.

parity, flip-chip, hybrid, superconductivity, inductive, readout

## Introduction

Control and measurement of isolated quantum systems is a crucial task for the
realization of practical quantum devices [1, 2]. Within the last decades,
quantum dots emerged as an ideal platform for the realization of critical
components for charge [3, 4] and spin [5, 6] qubits due to their ability to
trap, control and sense individual charge carriers using electrostatic means.
Superconducting quantum dots posses an additional degree of freedom, given by
the parity (even or odd) of unpaired excitations [7, 8, 9, 10, 11, 12]. Parity
is an attractive alternative to charge and spin for encoding long-lived
quantum states, for example in $0-\pi$ [13], topological [14], and Kerr cat
qubits [15]. Performing real-time, non-destructive charge parity readout is
therefore a fundamental requirement in proposed topological quantum computing
schemes [16, 17, 18, 19, 20, 21] and Andreev spin qubits [22, 23, 24]. More
generally, continuous monitoring of the parity of superconducting systems
could shine new light on parity dynamics and on fundamental mechanisms
limiting coherence in superconducting hardware.

Investigating effects related to parity in superconductor-semiconductor hybrid
systems is a subject of intense research [25, 26, 27, 28]. However, direct
measurements of the parity in superconducting systems is often challenging.
Successful demonstrations made use of parity-dependent switching supercurrents
[29, 12, 30, 31, 32, 25], detected with dc transport techniques. Such
approaches are typically slow, require macroscopic leads for current
injection, and measurements lead to a destruction of the parity state. High-
frequency parity and charge readout studies revolved around reflectometry
techniques, where the device-under-test was embedded in an impedance-matching
tank circuit [33, 34] and addressed in the radio frequency [35] or microwave
domain [36, 37, 38], allowing the detection of parity transitions. Finally,
experiments in the high-frequency domain have been, so far, almost exclusively
performed in semiconductor nanowires. This is due to the challenging, and
sometimes competing, fabrication requirements for integrating microwave
hardware on planar hybrid systems [39, 40, 41, 42, 43], where fast parity-
readout and QP dynamics studies are yet to be demonstrated.

In this Article, we present a novel approach to parity readout in a confined
superconducting system. A planar superconducting island is embedded in a
superconducting loop, which is inductively coupled to a high $Q$ factor
microwave resonator using a flip-chip approach. Parity in the superconducting
island directly maps into the direction of the circulating supercurrent in the
loop, allowing for fast, absolute and non-destructive parity readout. The
parity effect in the island is observed at zero magnetic field concomitantly
with the presence of a low-energy bound state in the island. We demonstrate
parity detection with signal-to noise ratio $\mathrm{SNR}\approx 3$ for an
integration time of $20~{}\mu$s with detection fidelity for both parity states
above 98 $\%$. By continuously monitoring the state of the island, parity
switching events are observed, enabling the estimation of parity lifetime
approaching millisecond timescales, with no measurable parity transitions
within the parity-blockade regime. Combination of inductive readout
technology, flip-chip approach and scalable planar heterostructure will pave a
path towards coherent addressing and manipulation of large scale hybrid
devices.

Figure 1: Device and parity detection scheme. a, False-colored scanning
electron micrograph of the superconducting island. The Al (blue) was
selectively removed to form a 250 nm long and 400 nm wide superconducting
island coupled to wide superconducting T-shaped leads. The gate (yellow) with
voltage $V_{\mathrm{P}}$ is used to suppress the parallel transport paths on
both sides of the island and for tuning charge occupation. Gates with voltages
$V_{\mathrm{L}}$ and $V_{\mathrm{R}}$ (yellow) tune the coupling to the leads.
b, Schematic representation of the device under study, showing a
superconducting island with charging energy $E_{\mathrm{C}}$ embedded in a
superconducting loop tuned by the flux $\it{\Phi}$. Coupling between island
and leads is tuned via electrostatically controlled Josephson junctions with
Josephson energies $E_{\mathrm{J1}}$ and $E_{\mathrm{J2}}$. c, Micrograph of
the superconducting loop incorporating the gate-tunable island. d, Schematic
of the equivalent parity detection circuit. The readout hang resonator (green)
is inductively coupled to the superconducting loop (blue). The readout is
performed by probing the complex transmission parameter
$S_{\mathrm{21}}=10\mathrm{log}(P_{\mathrm{out}}/P_{\mathrm{in}})$ through a
coplanar transmission line (purple) fabricated on the same chip as the
resonator and capacitively coupled to the resonator. An additional drive line
(red) present on the resonator chip enables two-tone microwave spectroscopy
measurements. e, Magnitude of the transmission response $\lvert S_{21}\rvert$
in the Coulomb blockade regime ($V_{\mathrm{L}}=-1.755$ V and
$V_{\mathrm{R}}=-1.610$ V) as a function of plunger voltage $V_{\mathrm{P}}$
and offset readout frequency $f_{\mathrm{r}}-f_{\mathrm{0}}$, with
$f_{\mathrm{0}}=6.38198$ GHz. The shift of the resonator frequency indicates a
transition between the two parity states. f, Linecuts from resonator response
in e at positions indicated by the green and pink markers. An 80 kHz shift of
the resonator frequency is present between even (e) and odd (o) parity states.
g, A linecut from e at the turquoise marker position as a function of
$V_{\mathrm{P}}$ emphasizes the change in the resonator magnitude response
between even and odd parity state, demonstrating parity detection.

## Device and readout scheme

Figure 1a displays a false-colored scanning electron micrograph of the active
part of the device. The device consisted of a floating superconducting island
(blue) with dimensions of 250 nm $\times$ 400 nm that was coupled to the loop
by two semiconducting junctions each of length $\approx 50$ nm. The junctions
were controlled with gate voltages $V_{\mathrm{L}}$ and $V_{\mathrm{R}}$
(yellow), realizing a tunable coupling to the superconducting leads (blue).
The middle gate, energized by voltage $V_{\mathrm{P}}$, was used both for the
depletion of parallel transport paths around the island and for tuning the
offset charge on the island. Figures 1b,c show a schematic and a scanning
electron micrograph of the full device, respectively. Relevant device
parameters are the charging energy $E_{\mathrm{C}}$ of the island and the
gate-controlled Josephson energies $E_{\mathrm{J1}}$ and $E_{\mathrm{J2}}$,
describing the energy required to add (remove) a single electron to (from) the
island and the coupling between the island and the leads, respectively. To
enable flux-biasing, the island was embedded into a grounded superconducting
loop (blue). The loop was patterned into the same InAs/Al heterostructure, and
was controlled via a magnetic flux $\it{\Phi}$, generated by a home-made
superconducting coil mounted on top of the printed circuit board that hosted
the device.

Microwave measurements relied on the flip-chip approach that was recently
pioneered in hybrid materials using nanowire-based [44] and planar devices
[43]. The planar device and microwave components were fabricated on separate
chips to achieve high-quality devices and readout resonators ($Q_{\mathrm{i}}$
= 35000, $f_{\mathrm{0}}=6.36198$ GHz). The chips were connected with indium
bump bonding. While the microwave resonators were inductively coupled to the
device via a vacuum gap ($d=5~{}\mu$m) to sense the supercurrent in the device
loop, galvanic connections between both chips were used to tune the
electrostatic gates and provide a grounding contact to the loop.

The simplified microwave measurement scheme is shown in Fig. 1d. The
transmission line (purple) was capacitively coupled to the readout resonator
(green), which changed its frequency and loaded quality factor
$Q_{\mathrm{L}}$ due to current circulating in the loop (blue). Parity readout
was performed by applying a probe signal $P_{\mathrm{in}}$ with frequency
$f_{\mathrm{r}}$ into the left port of the transmission line and detecting the
response $P_{\mathrm{out}}$ at the right output port at the same frequency
$f_{\mathrm{r}}$. For two-tone spectroscopy measurements, the additional drive
line shown in Fig. 1d (red) was used to apply a continuous drive tone with
power $P_{\mathrm{d}}$ and frequency $f_{\mathrm{d}}$ while monitoring the
resonator transmission $P_{\mathrm{out}}$ at frequency $f_{\mathrm{r}}$. More
details about the fabrication, packaging and measurement setup are reported in
the Methods section. Data presented here were collected on one device, whereas
data from a second device are presented in the Supplementary Section S1.

## Parity effect in a superconducting island

We first discuss the response of the resonator as a function of gate voltage
$V_{\mathrm{P}}$, when $V_{\mathrm{L}}$ and $V_{\mathrm{R}}$ were set negative
enough to result in a finite charging energy, but still allowed for a
supercurrent to flow in the loop. Furthermore, the system was tuned to a
regime with strong parity effect. Figure 1e displays the magnitude of
transmission response $\lvert S_{21}\rvert$ as a function of $V_{\mathrm{P}}$
and offset readout frequency $f_{\mathrm{r}}-f_{\mathrm{0}}$, with
$f_{\mathrm{0}}=6.38198$ GHz. As the gate voltage $V_{\mathrm{P}}$ was
increased, a sharp change of the resonator frequency was observed in a narrow
gate voltage range around $V_{\mathrm{P}}=-1.67$ V. In Fig. 1f, we show
linecuts of the resonator magnitude response $\lvert S_{21}\rvert$ as a
function of readout frequency $f_{\mathrm{r}}$ at two voltage $V_{\mathrm{P}}$
values (dark green and pink markers) indicated in panel e. At these values of
$V_{\mathrm{P}}$ both a shift of the resonator frequency by 80 kHz and an
accompanying 12$\%$ change in its linewidth were visible. Remarkably, the
frequency shift can be associated to a parity-dependent change in the
direction of the circulating current, i.e. in the formation of a
$\pi$-junction for an odd-parity ground state. The circulating current results
in a sign change of the effective inductance
$L=[(2\pi/\it{\Phi}_{\mathrm{0}})\partial
I_{\mathrm{s}}(\varphi)/\partial\varphi]^{\mathrm{-1}}$, producing a shift in
the resonator frequency. Damping is associated to the formation of additional
loss channels during the transition through a charge degeneracy point,
consistent with the variable load impedance model [45]. Monitoring the
resonator response at fixed $f_{\mathrm{r}}$ (turquoise marker position in
Fig. 1e) as a function of $V_{\mathrm{P}}$, as shown in Fig. 1g, allowed us to
track the parity ground state of the superconducting island.

Figure 2: Charge states of a superconducting island a, Electrostatic energy
diagram of a Coulomb-blockaded superconducting island with charging energy
$E_{\mathrm{C}}$ smaller than a subgap energy state $\delta$ (2$e$ periodic
regime). b, Experimentally measured transmission response magnitude $\lvert
S_{21}\rvert$ at $\it{\Phi}=\it{\Phi}_{\mathrm{0}}/\mathrm{2}$ as a function
of plunger gate voltage $V_{\mathrm{P}}$ and right barrier gate voltage
$V_{\mathrm{R}}$. The 2$e$ periodic Coulomb blockade oscillations observed as
a function of both voltages are consistent with the situation depicted in a.
c, Same as a but for $E_{\mathrm{C}}>\delta$, showing parity effect with
alternating even and odd parity valleys. d, Same as b but in a gate regime
where the parity effect is present, leading to periodically alternating
spacings for even (e) and odd (o) parity sectors. e, Overview of resonator
response magnitude $\lvert S_{21}\rvert$ at
$\it{\Phi}=\it{\Phi}_{\mathrm{0}}/\mathrm{2}$ for broad ranges of
$V_{\mathrm{P}}$ and $V_{\mathrm{R}}$. Tuning of parity effect is observed as
a modulation of periodicity between Coulomb blockade peaks as the voltage
$V_{\mathrm{R}}$ is varied. f, Two-tone spectroscopy measurement at
$\it{\Phi}=\it{\Phi}_{\mathrm{0}}/\mathrm{2}$ showing the resonator response
magnitude $\lvert S_{21}\rvert$ as a function of drive frequency
$f_{\mathrm{d}}$ and right barrier voltage $V_{\mathrm{R}}$ in a gate regime
where the parity effect is present.

The energy diagram of a confined superconducting island in the absence of low
energy excitations is shown in Fig. 2a. The energy of the island is
characterized by periodic parabolas describing the even parity states (blue)
with charging energy $E_{\mathrm{C}}$, the induced superconducting gap
$\Delta$ (red) and a generic subgap state with energy $\delta$ (green), which
is in this case above $E_{\mathrm{C}}$. In this situation the parity of the
island remains even, and sweeping gate voltages changes the charge occupation
by two electrons at a time. Experimentally, this was observed in Fig. 2b as a
series of periodic features in transmission response magnitude $\lvert
S_{21}\rvert$ as a function of $V_{\mathrm{P}}$ and $V_{\mathrm{R}}$, measured
at $\it{\Phi}=\it{\Phi}_{\mathrm{0}}/\mathrm{2}$. The regions where the
$\lvert S_{21}\rvert$ signal was enhanced are associated with crossing points
of electrostatic energy parabolas in Fig. 2a. At these points, corresponding
to resonant tunnelling of Cooper pairs through the island, a sizeable
supercurrent circulated in the loop, whereas between resonances the charge
configuration was stable and the supercurrent was blocked. Therefore, $\lvert
S_{21}\rvert$ monitored at a fixed $f_{\mathrm{r}}$ allowed to distinguish
between the current flowing at charge-state degeneracy and its suppression in
the blockade regime.

A significantly different situation is obtained when an excitation with energy
$\delta<E_{\mathrm{C}}$ is present in the spectrum (see Fig. 2c). In this
case, crossing points between the lowest energy parabolas (blue and green)
become unevenly spaced, and the superconducting island accepts single
electrons. This was measured in Fig. 2(d). The abrupt and periodic change in
color of the $\lvert S_{21}\rvert$ response separates the two parity sectors.
The odd parity (o) state is associated to narrow and dark color regions
separated by larger bright and wider color regions where parity is even (e).
The additional features in the center of the even parity sector (h), running
parallel in gate space to the main parity transitions, are associated to
transport processes involving excited states of the island. The crossings of
the associated higher energy parabolas [higher energy than shown in Fig. 2c]
enable new transport channels and lead to a measurable effect on the current
circulating in the loop visible in the even parity valley [46].

Figure 2e displays $\lvert S_{21}\rvert$ at
$\it{\Phi}=\it{\Phi}_{\mathrm{0}}/\mathrm{2}$ in a broader parameter space,
where transitions between the two regimes discussed above were visible.
Results in Fig. 2e strongly suggest that the charging energy of the
superconducting island was generally smaller than the superconducting gap.
However, discrete Andreev states could be occasionally tuned to energies
$\delta<E_{\mathrm{C}}$, giving rise to parity transitions. In the case of
Fig. 2(e), such a discrete state was predominantly controlled by
$V_{\mathrm{R}}$.

More quantitative insights are gained by performing two-tone spectroscopy
measurements, as shown in Fig. 2f. We observe a driven transition (parabolas
indicated by blue arrows), which is continuous and periodic as a function of
$V_{\mathrm{R}}$, with a periodicity consistent with 2$e$ charging. We
interpret the driven transition as excitations from the even parity ground
state to its excited state (blue curves in Fig. 2a,c) [47]. Here, the minimum
transition frequency of each oscillation is consistent with the total
Josephson energy of the system $\approx 11$ GHz. As a discrete state is tuned
below the charging energy, the ground state parity changes to odd, resulting
in sharp color changes over finite regions of $V_{\mathrm{R}}$. From the
transition frequency where ground state transitions first occurred, a charging
energy $E_{\mathrm{C}}\approx 6.7$ GHz was estimated (see Supplementary
Section S2). Horizontal resonances are attributed to unintentional standing
waves in the measurement circuit. Additional two-tone spectroscopy
measurements are presented in Extended Data Fig. 7.

Figure 3: Signal-to-noise ratio and detection fidelity. a, Transmission
magnitude response $R$ as a function of plunger gate voltage $V_{\mathrm{P}}$
measured at $V_{\mathrm{L}}=-1.755$ V and $V_{\mathrm{R}}=-1.609$ V, using a
readout power $P_{\mathrm{in}}=-44$ dBm (before attenuation) and an
integration time $\tau_{\mathrm{int}}=50~{}\mu$s. Even (e) and odd (o) parity
states are identified as different discrete levels in $R$ response. b,
Histogram of the resonator magnitude response $R$ decomposed into in-phase
($I$) and quadrature ($Q$) components encoding the two parity states. Signal
($S$) is characterized by the separation between the maxima of two centers
while noise ($N$) is responsible for the broadening of the distribution. The
data range used for the histogram is between the two red markers in a. c,
Parity readout infidelity for the even (1$-F_{\mathrm{e}}$, green) and odd
(1$-F_{\mathrm{o}}$, pink) sectors, together with readout invisibility $1-V$
(black dashed) as a function of threshold voltage $V_{\mathrm{T}}$ separating
the two parity sectors for $\tau_{\mathrm{int}}=50~{}\mu$s. d, Signal-to-noise
ratio (SNR) together with a fit to an uncorrelated noise model (solid black
line) and extracted minimum invisibility $1-V_{\mathrm{max}}$ as a function of
integration time $\tau_{\mathrm{int}}$. e, Dependence of SNR on readout power
$P_{\mathrm{in}}$ (before attenuation) for $\tau_{\mathrm{int}}=50~{}\mu$s and
$f_{r}=6.36206$ GHz, showing a turnover behavior.

## Estimation of signal-to-noise ratio and readout fidelity

Fast and continuous monitoring of charge parity provides access to the fast
dynamics of the system. The performance of our approach is quantified by the
signal-to-noise ratio (SNR) and the measurement fidelity $F$. First, we
acquired in-phase ($I$) and quadrature ($Q$) components of the resonator
response at fixed readout frequency $f_{\mathrm{r}}=6.36206$ GHz and
calculated the transmission response magnitude ${R=\sqrt{I^{2}+Q^{2}}}$ as a
function of $V_{\mathrm{P}}$ for variable data integration times
$\tau_{\mathrm{int}}$. An exemplary $R$ trace measured with
$\tau_{\mathrm{int}}=50~{}\mu$s is depicted in Fig. 3a, where two well-
distinguishable levels of response are associated with even and odd parity
states. Higher order transport processes mentioned above are also visible as a
dip in $R$ centered at $V_{\mathrm{P}}=-1.6955$ V. The $I$ and $Q$ quadrature
map of Fig. 3b provides access to amplitude and phase of the signal, which we
next use to estimate the levels of signal ($S$) and noise ($N$). For this, we
fit the resulting histogram to a two-dimensional bi-modal Gaussian
distribution. The signal level is defined as the distance between centers of
the fitted Gaussians, whereas the noise is defined as the standard deviation
of the Gaussians responsible for state broadening (see Extended Data Fig. 9
for details).

To calculate the fidelity $F$ of the parity-state assignment, we first project
the histogram data along the axis connecting the centers of the two Gaussians
to recover a 1D histogram of state distribution. Following this, the
fidelities for the even and odd states are defined as
${F_{\mathrm{e}}=1-\int_{V_{\mathrm{T}}}^{\infty}n_{\mathrm{e}}(V)dV}$ and
${F_{\mathrm{o}}=1-\int_{-\infty}^{V_{\mathrm{T}}}n_{\mathrm{o}}(V)dV}$,
respectively [48, 35], where $V_{\mathrm{T}}$ is the threshold voltage, namely
the value of the output voltage that defines the separation between the
assignment of data points to even or odd parity sectors, and $n_{e}$ ($n_{o}$)
is the probability density of the even (odd) state. The results for the fixed
integration time $\tau_{\mathrm{int}}=50~{}\mu$s are shown in Fig. 3c where
both state infidelities 1$-F_{\mathrm{e}}$ (green) and 1$-F_{\mathrm{o}}$
(pink) are plotted as a function of $V_{\mathrm{T}}$. To define the optimum
value of $V_{\mathrm{T}}$ for the two parity states, the visibility is
calculated as $V=F_{\mathrm{e}}+F_{\mathrm{o}}-1$. Maximizing $V$ allows us to
find the optimal regime for $F_{\mathrm{e}}$ and $F_{\mathrm{o}}$. In this
case the detection is optimized when 1$-V$ reaches a minimum at a threshold
voltage $V_{\mathrm{T}}=306~{}\mu$V with maximum obtained visibility
$V_{\mathrm{max}}=0.9996$ (three nines).

The extraction procedure for SNR and $V_{\mathrm{max}}$ is repeated for
multiple $\tau_{\mathrm{int}}$ values and summarized in Fig. 3d. Both SNR and
$V_{\mathrm{max}}$ increase with increasing $\tau_{\mathrm{int}}$. The SNR
trace in Fig. 3d was fitted to an uncorrelated noise model [34],
${\mathrm{SNR(\tau_{\mathrm{int}})}=(S_{\mathrm{\tau_{\mathrm{int}}=1\mu
s}}/N_{\mathrm{\tau_{\mathrm{int}}=1\mu s}})\sqrt{\tau_{\mathrm{int}}/(1~{}\mu
s)}}$, where $S_{\mathrm{\tau_{\mathrm{int}}=1\mu s}}=87~{}\mu$V is the signal
at $\tau_{\mathrm{int}}=1~{}\mu$s (used as a fit parameter) and
$N_{\mathrm{\tau_{\mathrm{int}}=1\mu s}}=135~{}\mu$V is the noise at
$\tau_{\mathrm{int}}=1~{}\mu$s, determined by fitting
${N(\tau_{\mathrm{int}})\sim 1/\sqrt{\tau_{\mathrm{int}}}}$. This allowed the
estimation of integration time $\tau_{\mathrm{min}}=2.4~{}\mu$s for which
$\mathrm{SNR}=1$ and parity sensitivity
$S=e\sqrt{\tau_{\mathrm{min}}}=1.5\times 10^{-3}~{}e/\sqrt{\mathrm{Hz}}$. The
measurement fidelities were found to vary from 85$\%$ at
$\tau_{\mathrm{int}}=10~{}\mu$s reaching 99.99999$\%$ (seven nines) at
$\tau_{\mathrm{int}}=100~{}\mu$s. The values obtained here for SNR, fidelity
and visibility are comparable to previously reported charge and parity
detection studies in hybrid materials [35, 27].

Finally, Fig. 3e displays the SNR dependence on readout power
$P_{\mathrm{in}}$ (before cryogenic attenuation) measured with
$\tau_{\mathrm{int}}=50~{}\mu$s at a readout frequency
${f_{\mathrm{r}}=6.36202}$ GHz. The initial increase of $P_{\mathrm{in}}$
leads to a correlated increase in SNR reaching a maximum of
$\mathrm{SNR}\approx 6.7$ at $P_{\mathrm{in}}=-38$ dBm. Beyond this optimal
value, the SNR shows a strong downward trend. This turnover behavior is mainly
caused by signal $S$, while noise $N$ is only slightly affected in the
$P_{\mathrm{in}}$ range under study. Here, the signal is set by the frequency
shift and damping of the readout resonator, which both decrease for larger
$P_{\mathrm{in}}$ (see Extended Data Fig. 6). For low powers the decreased
damping improves the signal, but past the threshold the reduction in frequency
shift starts to dominate, which results in smaller signals. We additionally
note that the SNR values presented here are likely affected by the $1/f$ noise
due to the low sweep rate of the gate voltage $V_{\mathrm{P}}$, and therefore
act as a lower bound for SNR in time-resolved measurements. The SNR dependence
on $f_{\mathrm{r}}$ is shown in Extended Data Fig. 8.

## Time-resolved parity monitoring

Figure 4: Continuous parity monitoring. a, Resonator response magnitude $R$ in
the regime where the parity effect is visible as a function of measurement
time $t$ and voltage $V_{\mathrm{L}}$, for $\tau_{\mathrm{int}}=30~{}\mu$s.
Around the regions where parities are degenerate, the switching rates are
enhanced compared to the rates away from degeneracy. b, Blue; time trace of
resonator response $R$ close to parity degeneracy, showing parity switching
effects detected in real time. Green; similar trace but further away from
degeneracy where the switching rate is reduced. Purple; time trace in the
blockade regime, where the parity is fixed. c, Welch power spectral density
(PSD) as a function of frequency $f$ for the blue and green traces in b,
fitted to an asymmetric random telegraph noise model including a $1/f$ noise
term. d, Parity switching rates $\Gamma_{\mathrm{e\rightarrow o}}$ and
$\Gamma_{\mathrm{o\rightarrow e}}$ as a function of voltage $V_{\mathrm{L}}$
in a. In the blockade regime the lower and upper bound on
$\Gamma_{\mathrm{e\rightarrow o}}$ and $\Gamma_{\mathrm{o\rightarrow e}}$,
shown as red and black lines, are limited by the length of the total recorded
time trace (900 ms) and the integration time
($\tau_{\mathrm{int}}=30~{}\mu$s).

After assessing good parity sensitivity and detection fidelity, we now study
the time-resolved dynamics in the planar island. To detect parity transition
events, the resonator response magnitude $R$ was monitored for a time window
of 900 ms with $\tau_{\mathrm{int}}=30~{}\mu$s for multiple values of
$V_{\mathrm{L}}$. The results are summarized in Fig. 4a, where we show a
subset of that time window spanning 250 ms. Deep in the even parity sector,
where the charge configuration is fixed, the resonator response level is
stable in time (purple trace in Fig. 4b). In this case the charging energy of
the superconducting island energetically penalizes changes in charge
configuration, leading to a stable parity occupation. The step-like
transitions observed between the blue and green traces in Fig. 4b are
associated to parity switching events following a random telegraph noise
behavior. The frequency of switching events as well as the preferred parity
state are tuned by $V_{\mathrm{L}}$. Additional measurements on gate-dependent
parity switching are presented in Extended Data Fig. 10.

The average transition rates in such a two-level system is quantified in the
spectral domain using power spectral density (PSD) analysis [49, 50]. For this
the time traces were converted to a Welch PSD [51], as shown in Fig. 4c for
the green and blue time traces from Fig. 4b. The PSD was then fitted with an
asymmetric random telegraph noise model including a $1/f$ noise term [52].
This allowed us to infer the two parity switching rates
$\Gamma_{\mathrm{e\rightarrow o}}$ and $\Gamma_{\mathrm{o\rightarrow e}}$
associated to transitions between even-odd and odd-even parity sectors. For
the green trace we obtain ${\Gamma_{\mathrm{e\rightarrow o}}=53\pm 1}~{}$Hz
and ${\Gamma_{\mathrm{o\rightarrow e}}=1123\pm 17}~{}$Hz. For the blue trace
we obtained ${\Gamma_{\mathrm{e\rightarrow o}}=163\pm 4}$ Hz and
${\Gamma_{\mathrm{o\rightarrow e}}=1624\pm 24}$ Hz, consistent with a reduced
asymmetric switching configuration.

The parity switching rate extraction procedure was repeated at several
$V_{\mathrm{L}}$ values from Fig. 4a and the results are summarized in Fig.
4d. In the regions close to parity degeneracy, both rates develop a strong
dependence on $V_{\mathrm{L}}$ and exchange their dominating behavior as a
function of detuning on both sides of the degeneracy. In the blockaded regime,
the lowest detectable parity switching rate is limited by the length of the
total recorded time trace of 900 ms with
${\mathrm{min}(\Gamma_{\mathrm{e\rightarrow
o}})=\mathrm{min}(\Gamma_{\mathrm{o\rightarrow e}})=1/(900~{}\mathrm{ms})}$,
whereas the highest detectable rate is limited by the integration time
$\tau_{\mathrm{int}}=30~{}\mu$s, ${\mathrm{max}(\Gamma_{\mathrm{e\rightarrow
o}})=\mathrm{max}(\Gamma_{\mathrm{e\rightarrow o}})=1/(30~{}\mu\mathrm{s})}$.
The two limits are schematically indicated as red and black regions in Fig.
4d. The parity switching rates $\Gamma_{\mathrm{e\rightarrow o}}$,
$\Gamma_{\mathrm{o\rightarrow e}}$ and the corresponding state lifetimes
$T_{\mathrm{e}}$=1/$\Gamma_{\mathrm{e\rightarrow o}}$,
$T_{\mathrm{o}}$=1/$\Gamma_{\mathrm{o\rightarrow e}}$ reported here are
comparable to values measured for superconducting-semiconducting nanowire
qubits and similar hybrid devices [27, 53, 26]. For the time response shown in
blue in Fig. 4b, we found $T_{\mathrm{e}}=6.1\pm 0.2$ ms and
$T_{\mathrm{o}}=616\pm 9~{}\mu$s, producing a first reported estimate on
parity lifetime in a superconducting planar heterostructure.

## Conclusion

We have reported on a flip-chip-based inductive detection scheme of the charge
parity in a planar superconducting island. Our method combined the advantages
of high-bandwidth readout technology with planar gate-tunable superconductor-
semiconductor materials to achieve a high signal-to-noise ratio for non-
destructive readout of parity. While the parity effect in our devices was
enabled by the low energy excitation in the island, the readout technique is
general and does not depend on the presence of such states. The combination of
flip-chip approach and inductive readout offers significant improvements
whenever the experimental implementation involves circulating currents. Our
results open interesting avenues for the realization of theoretical proposals
in the field of Andreev spin and topological qubits. Furthermore, it will
facilitate the long-range coupling of different qubit subsystems, especially
when large-scale devices become available using planar device technology. The
long measured parity lifetimes constitute an important advantage in the
context of hybrid qubits, where large numbers of gate operations could be
performed before the parity information is lost. Additionally, the
availability of parity lifetime measurements will allow to constrain
theoretical proposals for hybrid and protected qubit designs based on hybrid
planar materials.

## Methods

### Fabrication

Superconducting islands were fabricated from an InAs/Al heterostructure grown
on an InP substrate using electron beam lithography techniques. The
semiconductor was etched in a solution of H2O : C6H8O7 : H3PO4 : H2O2 with
composition 220 : 55 : 3 : 3. The epitaxial Al was etched with a short dip
into 50 ∘C Transene D to define the island and the superconducting loop.
Under-bump metallization was evaporated on the device contacts [43], and the
entire chip was covered by a dielectric layer consisting of 3 nm AlO2 and 15
nm HfO2, grown by thermal and plasma ALD, respectively. Next, the fine
features of the gate electrodes were deposited by electron evaporation of 5 nm
Ti and 20 nm Au, followed by the evaporation of thicker gate features
(Ti/Al/Ti/Au, 5/250/5/100 nm). For both the inner and outer gates, lift-off
was done in DMSO at 120 ∘C. A 45 nm layer of Al2O3 was then deposited with
plasma ALD. With wet etching in BHF 7:1, RIE etching and wet etching in the
semiconductor etchant, the semiconductor stack was deep etched in the region
between the devices and the bottom edge of the chip, to reduce resonator
losses. Finally, the chip was diced into a 3 mm by 3 mm piece.

The readout resonators were fabricated on a high-resistivity Si wafer. First,
200 nm of Nb was sputtered over the entire chip, followed by deposition of a
57 nm AlO2 hard mask by plasma ALD. Resonator features were defined by RIE in
the hard mask. After resist removal in hot DMSO, ICP etching with Cl2 and Ar
was used to pattern the Nb layer. Subsequently, the hard mask was removed with
BHF 7:1. Next, the In bumps for flip-chip bonding were patterned by optical
lithography and In evaporation. Finally, the wafer was diced into 6 mm by 9 mm
chips. The resonator chip is identical to that shown in Fig. 1 of Ref. [43],
and includes three $\lambda/4$ CPW resonators capacitively coupled to a common
transmission line, drive lines, a ground plane and DC control lines.

### Flip-chip technique

After fabrication of the devices and resonators on separate chips, the two
chips were coupled via flip-chip bonding using indium bumps. These bumps
provided connections from wire bonding pads on the Si chip to the DC control
lines of the gate electrodes and ground of the devices on the InP chip.
Furthermore, several more In bumps were included in the final chip to ensure
mechanical stability of the structure. The devices were carefully aligned with
the shorted ends of the $\lambda/4$ resonators and with the drive lines in
order to ensure good coupling. The separation between the two chips was
$5~{}\mathrm{\mu m}$, and was controlled by the size of the In bumps as well
as the temperature and force used during the flip-chip bonding. To minimize
dielectric losses, the device chip was considerably smaller than the resonator
chip and was positioned to minimize its overlap with the resonators.

### Measurement setup

Measurements were performed in a BlueFors cryogen-free dilution refrigerator
with a mixing chamber base temperature of 9 mK. The wiring of the dilution
refrigerator is illustrated in Extended Data Fig. 1. The measurements depicted
in Figs. 1 and 2 of the Main Text were taken with a Keysight VNA B2911A vector
network analyser, while those in Figs. 3 and 4 were acquired with a Zurich
Instruments SHFQA 8.5 GHZ Quantum Analyzer. The readout tone with frequency
$f_{\mathrm{r}}$ was applied at port $P_{\mathrm{in}}$. For two-tone
spectroscopy measurements (see Fig. 2f), a continuous drive tone with
frequency $f_{\mathrm{d}}$ was applied at port $P_{\mathrm{out}}$ in Extended
Data Fig. 1. Drive signals were attenuated by 66 dB and 39 dB, respectively,
at different temperature stages of the dilution refrigerator. The signal
transmitted through the readout transmission line passed through a DC block
followed by a circulator, a dual isolator and an rf low-pass filter. The
signal was then amplified with a cryogenic high electron mobility transistor
amplifier at 4 K and a room-temperature amplifier before being detected at
port labelled as $P_{\mathrm{out}}$. Gates were controlled with QDevil
digital-to-analog converter DC voltage sources with 19 $\mu$V resolution. The
DC lines were filtered using a home-made low-pass filter at room temperature,
QDevil rf and RC filters at base temperature, and an RC filter on the printed
circuit board onto which the chip was mounted. The magnetic flux in the loop
was applied via a home-made superconducting coil mounted on top of the printed
circuit board.

### Device tuneup

For each device, the superconducting island was electrically defined by tuning
the left and right island barrier gates with voltages $V_{\mathrm{L}}$ and
$V_{\mathrm{R}}$, respectively as well as the plunger gate with voltage
$V_{\mathrm{P}}$. Extended Data Fig. 2a shows flux dependence measured with
all gate voltages grounded, resulting in a large circulating current. Extended
Data Fig. 2b shows flux dependence measured with all gates set to large
negative voltages, resulting in suppression of the circulating current and
absence of a resonator response. Dependences of $\lvert S_{21}\rvert$ on
various gate voltages and in different gate configurations are shown in
Extended Data Figs. 3a-e. The operating regime was found first by setting
$V_{\mathrm{P}}=-1.5$ V and $\it{\Phi}=\it{\Phi}_{\mathrm{0}}/\mathrm{2}$ and
by mapping $\lvert S_{21}\rvert$ as a function of $V_{\mathrm{L}}$ and
$V_{\mathrm{R}}$. A corner diagram as shown in Extended Data Fig. 3f was
recorded, indicative of quantum dot formation. Here, fine tuning of each gate
led to the electrostatic regime presented in the Main Text. Additional
electrostatic configurations, characterized by different coupling to the
leads, are presented in Extended Data Figs. 4 and 5. Power dependent
measurements of Coulomb blockade transitions and resonator response are shown
in Extended Data Fig. 6.

### Compensated resonator measurements

Measurements in Figs. 2b,d,e were taken with resonator compensation. Each time
the slow axis variable was swept to the next value, $\lvert S_{21}\rvert$ as a
function of readout frequency $f_{\mathrm{r}}$ was recorded, similar to Fig.
1f, and the operating frequency was offset by 90 kHz with respect to the
frequency of the minimum. During post-processing, the median of the resonator
response along the fast axis was subtracted from all data points, resulting in
maps of compensated resonator response as a function of two gate voltages. The
maps were taken with $P_{\mathrm{in}}=-44$ dBm, integration time of
$420~{}\mu$s and 15 averages.

### Signal-to-noise ratio, Fidelity and Visibility calculation

For the signal-to-noise (SNR) calculation, the in-phase ($I$) and quadrature
($Q$) components of the readout resonator was recorded as a function of the
plunger gate voltage $V_{\mathrm{P}}$ with different integration times
$\tau_{\mathrm{int}}$ per data point, ranging from 10 to 100 $\mu$s. The
readout frequency was fixed at $f_{\mathrm{r}}=6.36206$ GHz and the power at
$P_{\mathrm{in}}=-44$ dBm. At each value of the plunger gate voltage, 5000
measurements of $I$ and $Q$ were acquired (10,000 for
$\tau_{\mathrm{int}}=10~{}\mu$s). Plotting the response magnitude
$R=\sqrt{I^{2}+Q^{2}}$ (see Fig. 3a) as a function of $V_{\mathrm{P}}$ reveals
the two parity states with the even state influenced by higher order transport
processes. In addition, the amount of recorded data as a function of
$V_{\mathrm{P}}$ is much larger in the even parity state than in the odd
state. Therefore, we selected a data set with 120 $V_{\mathrm{P}}$ values
around a parity transition to have similar amount of data in the even and odd
parity state. The $I$ and $Q$ values were then plotted as a histogram with bin
size (Fig. 3b). The histogram was fitted with a sum of two 2D Gaussians, Eq. 1

$A_{1}\cdot e^{-(((b_{I1}-x)/c_{I1})^{2}+((b_{Q1}-y)/c_{Q1})^{2})/2}\\\
+A_{2}\cdot e^{-(((b_{I2}-x)/c_{I2})^{2}+((b_{Q2}-y)/c_{Q2})^{2})/2}$ (1)

where $A_{i}$ is the height of the Gaussian, $b_{Ii}$ ($b_{Qi}$) the center
point and $c_{Ii}$ ($c_{Qi}$) the width of the Gaussian on the $I$-axis
($Q$-axis), for $i=1,2$. The distance between the center points of the two
Gaussians defined the strength of the signal $S$, whereas the quadratic sum of
the average widths of the Gaussians, projected along $S$ and perpendicular to
$S$ using the polar form of ellipse equation, defined the noise $N$. SNR was
then defined as $S/N$ and is given by Eq. 2

$\begin{split}SNR&=S/N\\\ S&=\sqrt{(b_{I1}-b_{I2})^{2}+(b_{Q1}-b_{Q2})^{2}}\\\
N&=\sqrt{((|c_{\parallel S1}|+|c_{\perp S1}|)/2)^{2}+((|c_{\parallel
S2}|+|c_{\perp S2}|)/2)^{2}}\end{split}$ (2)

where $c_{\parallel Si}$ and $c_{\perp Si}$ are the widths of the Gaussian
along and perpendicular to $S$ for $i=1,2$. The SNR for all integration times
is plotted in Fig. 3d and it was fitted to
$\mathrm{SNR(\tau_{\mathrm{int}})}=(S_{\mathrm{\tau_{\mathrm{int}}=1\mu
s}}/N_{\mathrm{\tau_{\mathrm{int}}=1\mu s}})\sqrt{\tau_{\mathrm{int}}/(1~{}\mu
s)}$, where $S_{\mathrm{\tau_{\mathrm{int}}=1\mu s}}$ represents the signal at
$\tau_{\mathrm{int}}=1~{}\mu$s and was used as a fit parameter and
$N_{\mathrm{\tau_{\mathrm{int}}=1\mu s}}=135~{}\mu$V is the noise at
$\tau_{\mathrm{int}}=1~{}\mu$s and determined by fitting
$N(\tau_{\mathrm{int}})\sim 1/\sqrt{\tau_{\mathrm{int}}}$ assuming $S$ to be
independent of $\tau_{\mathrm{int}}$ [34].

Readout visibility is defined as $V=F_{\mathrm{e}}+F_{\mathrm{o}}-1$, where
$F_{\mathrm{e}}$ ($F_{\mathrm{o}}$) is the fidelity of the even (odd) state,
respectively. These fidelities are defined as
$F_{\mathrm{e}}=1-\int_{V_{\mathrm{T}}}^{\infty}n_{\mathrm{e}}(V)dV$ and
$F_{\mathrm{o}}=1-\int_{-\infty}^{V_{\mathrm{T}}}n_{\mathrm{o}}(V)dV$, where
$n_{\mathrm{e}}$ ($n_{\mathrm{o}}$) is the probability density of the even
(odd) state [48]. The fidelity of a state describes the probability of an
assignment of resonator response to the correct parity sector, where
$V_{\mathrm{T}}$ is the voltage threshold, describing the value of
transmission output voltage dividing the assignment to two parity sectors.
Optimal assignment of $V_{\mathrm{T}}$ for detection is the value for which
the Visibility $V$ is maximized. Maximising $V$ simultaneously also maximizes
both detection fidelities $F_{\mathrm{e}}$ and $F_{\mathrm{o}}$. To calculate
the $F_{\mathrm{e}}$, $F_{\mathrm{o}}$ and $V$ a linecut of the data in the IQ
plane (see Extended Data Fig. 9a,d) was taken along signal $S$ axis (100
points). The resulting 1D histogram as a function of the voltage
$V_{\parallel}$ was normalized by dividing the data by the sum of all counts
(see Extended Data Fig. 9b,e). The normalized histogram was fitted with two 1D
Gaussians given by Eq. 3

$A_{1}\cdot e^{-((b_{1}-x)/c_{1})^{2}/2}+A_{2}\cdot
e^{-((b_{2}-x)/c_{2})^{2}/2}$ (3)

where $A_{i}$ is the height of the Gaussian, $b_{i}$ the center point and
$c_{i}$ the width of the Gaussian for $i=1,2$. The integral in
$F_{\mathrm{e}}$ ($F_{\mathrm{o}}$) was found through numerical integration of
the corresponding 1D Gaussian with the composite trapezoidal rule using the
python function numpy.trapz. We started (stopped) the integral by the
threshold voltage $V_{\mathrm{T}}$. Finally, the integral was normalized
because the histogram data was normalized to the sum of both 1D Gaussians,
resulting in Eq. 4

$\begin{split}F_{\mathrm{e}}=1-\int_{V_{\mathrm{T}}}^{\infty}A_{2}\cdot
e^{-((b_{2}-V_{\parallel})/c_{2})^{2}/2}dV_{\parallel}\\\
/\int_{-\infty}^{\infty}A_{2}\cdot
e^{-((b_{2}-V_{\parallel})/c_{2})^{2}/2}dV_{\parallel}\\\
F_{\mathrm{o}}=1-\int_{-\infty}^{V_{\mathrm{T}}}A_{1}\cdot
e^{-((b_{1}-V_{\parallel})/c_{1})^{2}/2}dV_{\parallel}\\\
/\int_{-\infty}^{\infty}A_{1}\cdot
e^{-((b_{1}-V_{\parallel})/c_{1})^{2}/2}dV_{\parallel}.\end{split}$ (4)

Fidelities and visibility as a function of $V_{\mathrm{T}}$ are plotted in
Fig. 3c and Extended Data Fig. 9c,f.

### Parity switching rate estimation

To estimate the parity switching rate we acquired the time trace of the
transmission response magnitude $R$ for different left barrier gate voltages
$V_{\mathrm{L}}$, while the other gate voltages were fixed at
$V_{\mathrm{R}}=-1.59800$ V and $V_{\mathrm{P}}=-1.69219$ V. The readout
frequency was $f_{\mathrm{r}}=6.36206$ GHz and the power was
$P_{\mathrm{in}}=-40.5$ dBm. At each value of $V_{\mathrm{L}}$, 30001 data
points were recorded. The integration time was set to 30 $\mu$s, resulting in
a 900 ms long time trace (neglecting the sampling rate of 0.5 ns). The data of
the time trace at $V_{\mathrm{L}}=-1.7277$ V (blue time trace in Fig. 4b),
which showed parity switching between even and odd parity state due to random
telegraph noise, was converted into a histogram distributed into 20 bins along
$R$. That histogram was fitted with two 1D Gaussians as described in Eq. 3.
All time traces were normalized based on this fit parameters so that the
transmission response magnitude $R$ became 1 and 0 for the even and odd parity
state, respectively.

Subsequently, the time traces were used to calculate the Welch power spectral
density (PSD). The calculation was done using the python function
scipy.signal.welch with sampling frequency $f_{s}=1/(30~{}\mu\mathrm{s})$ and
with segment length 5001. The resulting PSD was fitted to an asymmetric random
telegraph noise model including $1/f$ noise in the system using Eq. 5 to
estimate the transition rates from the even to odd $\Gamma_{e\rightarrow o}$
and from the odd to even $\Gamma_{o\rightarrow e}$ parity state [49, 50].

$\mathrm{PSD}(f)=\frac{8\Gamma_{e\rightarrow o}\Gamma_{o\rightarrow
e}}{(\Gamma_{e\rightarrow o}+\Gamma_{o\rightarrow e})((\Gamma_{e\rightarrow
o}+\Gamma_{o\rightarrow e})^{2}+(2\pi f)^{2})}\\\ +B/f+C$ (5)

where $B$ represents the influence of $1/f$ noise in the system and $C$ is a
constant describing the detection limit. The detection limit is given by the
SNR, and we start to approach it using 30 $\mu$s integration time, which can
be seen by the flattening PSD curve at high frequencies. Therefore, $C$ was
determined as the average of the PSD for the highest 100 frequency points.
Since $1/f$ noise influences the slope of the PSD curve, the parameter $B$ was
estimated empirically for every time trace with $2\cdot 10^{-6}$ times its PSD
range. The transition rates $\Gamma_{e\rightarrow o}$ and
$\Gamma_{o\rightarrow e}$ were used as free fit parameters. Since Eq. 5 is
symmetric with respect to $\Gamma_{e\rightarrow o}$ and $\Gamma_{o\rightarrow
e}$, the initial guess determined which of the two quantities was larger.

The maximum and minimum detectable transition rates were set by the total time
of the time trace and the integration time, respectively. Therefore, the lower
bound of the rates is $1/(900~{}\mathrm{ms})$ (total measurement time) and the
upper bound is $1/(30~{}\mu\mathrm{s})$ (integration time). For resonator
response levels that were stable within the measured time, as the purple time
trace in Fig. 4b, its $\Gamma_{e\rightarrow o}$ and $\Gamma_{o\rightarrow e}$
were assigned to the boundary rates.

## Data availability

Data presented in this work will be available on Zenodo. The data that support
the findings of this study are available upon reasonable request from the
corresponding author.

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## Acknowledgments

We thank the Cleanroom Operations Team of the Binnig and Rohrer Nanotechnology
Center (BRNC) for their help and support. We are greatful to R. Žitko and L.
Pavešić for helpful discussions. W.W. acknowledges support from the Swiss
National Science Foundation (Grant No. 200020 207538). F. N. acknowledges
support from the European Research Council (Grant No. 804273) and the Swiss
National Science Foundation (Grant No. 200021 201082).

## Author contributions

F.N. and D.S. conceived the experiments. E.C., F.K., R.S. and W.W. developed
and provided the heterostructure material. M.H. and S.C.t.K. designed the
devices. M.H. fabricated the sample and S.P. performed the indium evaporation.
M.H. performed the electrical measurements with contributions from D.S. and
S.C.t.K.. M.H., S.C.t.K. and D.S. analyzed and interpreted the data with
inputs from F.N., D.Z.H. and M.C. M.H., D.S. and S.C.t.K. prepared the
manuscript with feedback from all the authors.

## Supplementary Information

###  .1 Extended Data Figures

Extended Data Fig. 1: Schematics of the measurement setup. The resonator
readout and device drive tones with frequencies $f_{\mathrm{r}}$ and
$f_{\mathrm{d}}$ were applied using a vector network analyzer (VNA) or Quantum
Analyser, corresponding to $P_{\mathrm{in}}$ and $P_{\mathrm{d}}$,
respectively. After 66 dB and 39 dB attenuation of $P_{\mathrm{in}}$ and
$P_{\mathrm{d}}$ at different temperature stages of the dilution refrigerator,
the signals reached the resonator chip. After the transmission, the readout
signal passed through a DC block followed by a circulator, a dual isolator and
an rf low-pass filter. The signal was amplified with a cryogenic high electron
mobility transistor amplifier at 4 K and a room-temperature amplifier, before
being detected ($P_{\mathrm{out}}$) by the VNA or Quantum Analyser. An extra
rf line, which could be used for reflectometry measurements, stayed grounded
during the experiments. The gates were controlled with digital-to-analog
converter DC voltage sources. The DC lines were filtered using a home-made
low-pass filter at room temperature, QDevil rf and RC filters at base
temperature and an RC filter on the printed circuit board. The applied DC
signal reached the device via the resonator chip through the indium bumps. The
external flux was applied by sourcing a DC current through a coil, mounted on
top of the sample space. There was no additional shielding of the sample space
because the refrigerator was equipped with a vector magnet which, despite not
being utilised for the experiments. Extended Data Fig. 2: Flux dependence in
Device 1. a, Normalized resonator magnitude response $\lvert S_{21}\rvert$ as
a function of flux $\it{\Phi}$ and offset readout frequency
$f_{\mathrm{r}}-f_{\mathrm{0}}$ at $V_{\mathrm{G}}\equiv
V_{\mathrm{P}}=V_{\mathrm{L}}=V_{\mathrm{R}}=0$. The distinct modulation of
$\lvert S_{21}\rvert$ with respect to $\it{\Phi}$ reveals the coupling between
resonator and device and indicates a large supercurrent in the device loop
carried by many Andreev bound states located between the two superconducting
leads [43]. b, Same as a but at $V_{\mathrm{G}}\equiv
V_{\mathrm{P}}=V_{\mathrm{L}}=V_{\mathrm{R}}=-1.8$ V. Depletion of the InAs
quantum well between the leads suppresses the supercurrent. Extended Data Fig.
3: Tuneup of Device 1. a, Normalized resonator magnitude response $\lvert
S_{21}\rvert$ as a function of offset readout frequency
$f_{\mathrm{r}}-f_{\mathrm{0}}$ and barrier voltage $V_{\mathrm{R}}$ at
$\it{\Phi}=\it{\Phi}_{\mathrm{0}}/\mathrm{2}$,
$V_{\mathrm{P}}=V_{\mathrm{L}}=0$. b, Same as a but sweeping $V_{\mathrm{L}}$
and keeping $V_{\mathrm{P}}=V_{\mathrm{R}}=0$. Transport channels parallel to
the island limited the effect of the barrier gates $V_{\mathrm{R}}$ and
$V_{\mathrm{L}}$, respectively. c, Same as a but at $V_{\mathrm{P}}=-1.5$ V.
d, Same as b but at $V_{\mathrm{P}}=-1.5$ V. Suppression of parallel
conductance channels increased the influence of the barrier gates on $\lvert
S_{21}\rvert$. The coupling between island and lead decreases by reducing the
density of Andreev bound states between them. This is observed for
$V_{\mathrm{R}},V_{\mathrm{L}}<-1$ V. Appearing resonant transport damped the
resonator and caused a shift in its resonance frequency which both decrease
for more negative barrier gate voltages. e, Resonator magnitude response
$\lvert S_{21}\rvert$ as a function of offset readout frequency
$f_{\mathrm{r}}-f_{\mathrm{0}}$ and plunger voltage $V_{\mathrm{P}}$ at
$\it{\Phi}_{\mathrm{0}}/\mathrm{2}$, $V_{\mathrm{L}}=V_{\mathrm{R}}=0$. f,
Resonator magnitude response $\lvert S_{21}\rvert$ as a function of barrier
gate voltages $V_{\mathrm{L}}$ and $V_{\mathrm{R}}$ at $V_{\mathrm{P}}=-1.5$ V
and $f_{\mathrm{r}}=6.362$ GHz. The corner at $V_{\mathrm{L}}\approx-1.7$ V
and $V_{\mathrm{R}}\approx-1.5$ V defines the gate range where Coulomb blockad
was observed. Extended Data Fig. 4: Flux dependence in Device 1 in different
gate settings. a, Normalized resonator magnitude response $\lvert
S_{21}\rvert$ as a function of offset readout frequency
$f_{\mathrm{r}}-f_{\mathrm{0}}$ and plunger gate voltage $V_{\mathrm{P}}$ at
$V_{\mathrm{R}}=-1.47$ V, $V_{\mathrm{L}}=-1.71$ V and
$\it{\Phi}=\it{\Phi}_{\mathrm{0}}/\mathrm{2}$. The barrier gates put the
system in a state where there are isolated Andreev bound states (ABSs), but
the island is still strongly coupled to the leads. The energy of these bound
states is tuned by $V_{\mathrm{P}}$, leading to anti-crossings when the energy
of the ABSs decreases below the resonator frequency consistent with a
transmission close to one. b, Same as a but at $V_{\mathrm{L}}=-1.56$ V and
$V_{\mathrm{L}}=-1.72$ V. The weaker coupling of the island to the leads gives
sharp transitions in the regime where ABS energies are below the resonator
frequency. We assign these sharp lines to Coulomb transitions between even and
odd parity states. Extended Data Fig. 5: Additional measurement data on Device
1. a, Normalized resonator magnitude response $\lvert S_{21}\rvert$ as a
function of offset readout frequency $f_{\mathrm{r}}-f_{\mathrm{0}}$ and
plunger gate voltage $V_{\mathrm{P}}$ at $V_{\mathrm{R}}=-1.610$ V,
$V_{\mathrm{L}}=-1.755$ V and $\it{\Phi}=\it{\Phi}_{\mathrm{0}}/\mathrm{2}$.
The 2$e$ periodic Coulomb blockade oscillations are consistent with the
charging energy of the island being lower than the bound state energy
($E_{\mathrm{C}}<\delta$). At Coulomb resonances, a shift in the resonator
frequency is observed due to enhanced circulating current in the device loop,
which changes the inductance of the device. b, Same as a, but at
$V_{\mathrm{R}}=-1.620$ V. Here, $\delta<E_{\mathrm{C}}$ that allows the
number of quasiparticles on the superconducting island to be even or odd. The
inductance of the even (e) and odd (o) parity states are different resulting
in distinct resonator responses. A slight modification of the even state is
associated to higher order transport processes (h) [46]. In the future, such
quasiparticle poisoning from strong coupling to the leads could be suppressed
with larger $E_{\mathrm{C}}$. When the charging energy of the superconducting
island is larger, a larger energy of the bound state is sufficient to enter
the even-odd parity regime, resulting in less coupling between the island and
the leads. In that case, stronger coupling between resonator and device,
resonators with higher $Q$-factors, or an improvement in the readout setup is
required to achieve the same readout performance. c, Normalized resonator
magnitude response $\lvert S_{21}\rvert$ as a function of barrier gate
voltages $V_{\mathrm{L}}$ and $V_{\mathrm{R}}$ at $V_{\mathrm{P}}=-1.7$ V. The
gate-voltage map shows Coulomb blockade oscillations because both barrier
gates are able to tune the chemical potential of the superconducting island,
but with less efficiency compared to $V_{\mathrm{P}}$. The bound state energy
$\delta$ is mainly modulated by $V_{\mathrm{R}}$. If $\delta$ becomes lower
than $E_{\mathrm{C}}$, a transition from a 2$e$ periodic to an even-odd regime
happens. The odd parity ground state emerge via splitting of the Coulomb
resonance peak. Extended Data Fig. 6: Power dependence measurements of Device
1. a, Normalized resonator magnitude response $\lvert S_{21}\rvert$ as a
function of input readout power $P_{\mathrm{in}}$ and plunger gate voltage
$V_{\mathrm{P}}$ in the regime with 2$e$ periodic Coulomb blockade
oscillations ($V_{\mathrm{L}}=-1.760$ V, $V_{\mathrm{R}}=-1.623$ V) at
$\it{\Phi}=\it{\Phi}_{\mathrm{0}}/\mathrm{2}$. b, Same as a, but in the even-
odd regime ($V_{\mathrm{L}}=-1.795$ V, $V_{\mathrm{R}}=-1.623$ V). c,
Normalized resonator magnitude response $\lvert S_{21}\rvert$ as a function of
input readout power $P_{\mathrm{in}}$ and barrier gate voltage
$V_{\mathrm{L}}$ at $\it{\Phi}=\it{\Phi}_{\mathrm{0}}/\mathrm{2}$. d,
Normalized resonator magnitude response $\lvert S_{21}\rvert$ as a function of
offset readout frequency $f_{\mathrm{r}}-f_{\mathrm{0}}$ and input readout
power $P_{\mathrm{in}}$ on a Coulomb resonance peak indicated by the green
marker in a. At low input powers the resonator frequency is independent of
$P_{\mathrm{in}}$, but the noise increases for decreasing $P_{\mathrm{in}}$.
Above $P_{\mathrm{in}}=-44$ dBm the resonator frequency shift decreases for
increasing $P_{\mathrm{in}}$. e, Same as d, but on a transition between even
and odd parity state indicated by the blue marker in b. We observe a frequency
shift and damping of the readout resonator as a function of $P_{\mathrm{in}}$.
The damping increases significantly for $P_{\mathrm{in}}<-38$ dBm. f, Same as
e, but in a Coulomb blockade regime indicated by the turquoise marker in c.
Except for large input powers the resonator frequency stays constant while the
noise increases for decreasing $P_{\mathrm{in}}$. Extended Data Fig. 7:
Additional two-tone spectroscopy measurements in Device 1. a, Resonator
response magnitude $\lvert S_{21}\rvert$ as a function of drive frequency
$f_{\mathrm{d}}$ and right barrier gate voltage $V_{\mathrm{R}}$ at
$V_{\mathrm{L}}=-1.755$ V, $V_{\mathrm{P}}=-1.6955$ V and
$\it{\Phi}=\it{\Phi}_{\mathrm{0}}/\mathrm{2}$. While the horizontal lines are
associated to unintentional resonances in the measurement circuit, the
vertical stripes with opposite magnitude shift (black and white inverted)
indicate the odd ground state. The continuous parabola-curved lines, which are
periodic with $V_{\mathrm{R}}$, are associated to driven transitions from the
even parity ground state to its excited state. Deep in the 2$e$ periodic
regime ($V_{\mathrm{R}}<-1.62$ V where $E_{\mathrm{C}}\ll\delta$) the
transition parabolas have constant minimum transition frequency
$f_{\mathrm{d}}\approx 11$ GHz corresponding to the total Josephson energy of
the device. The transition parabolas get modulated in the even-odd regime and
close to it ($V_{\mathrm{R}}>-1.62$ V) because the barrier gate voltage tunes
the gate induced charge of the island, the bound state energy $\delta$ and the
coupling to the right lead. The explanation of the modulated transitions goes
beyond the simple picture in Fig. 2 of the Main Text. b, Resonator response
magnitude $\lvert S_{21}\rvert$ as a function of drive frequency
$f_{\mathrm{d}}$ and plunger gate voltage $V_{\mathrm{P}}$ at
$V_{\mathrm{R}}=-1.627$ V (green marker position in a). The spectroscopy shows
2$e$ periodic transition parabolas because $V_{\mathrm{P}}$ tunes only the
gate induced charge of the island. Weaker transitions shifted by 1$e$ are
assigned to higher order transport processes [46]. c, Same as b but in the
even-odd regime at the pink marker position in a. Extended Data Fig. 8:
Frequency-dependent signal-to-noise ratio. Signal-to-noise ratio (SNR) as a
function of readout frequency $f_{\mathrm{r}}$ at a readout power
$P_{\mathrm{in}}=-44$ dBm measured with an integration time
$\tau_{\mathrm{int}}=50~{}\mu$s and plotted together with the resonator
response magnitude $R$ in the even parity state. The SNR dependence shows the
pronounced maximum at a readout frequency $f_{\mathrm{r}}=6.36206$ GHz, which
is due to the signal term (the noise term is independent of $f_{\mathrm{r}}$).
The signal term is determined by the change of the resonator response
magnitude between the even and odd parity state
($\absolutevalue{R_{\mathrm{e}}-R_{\mathrm{o}}}$) resulting in the resonance-
like shape, with a pronounced peak at $f_{\mathrm{r}}$ values, where the rate
of change in $R$ is strong. Extended Data Fig. 9: Readout fidelity
determination. a, Quadrature map of the two parity states for the integration
time $\tau_{\mathrm{int}}=20~{}\mu$s plotted together with the line
$V_{\mathrm{\parallel}}$ along the signal axis. b, Linecut from a along
$V_{\mathrm{\parallel}}$ normalized (black data points) and plotted together
with the fit to the sum of two Gaussians (orange). c, Even and odd state
fidelities $F_{\mathrm{e}},F_{\mathrm{o}}$ as well as visibility $V$ as a
function of threshold voltage $V_{\mathrm{T}}$. The readout fidelity
$F_{\mathrm{e}}$ ($F_{\mathrm{o}}$) is calculated by numerical integration of
the fit in b starting (stopping) at $V_{\mathrm{T}}$. The readout visibility
is calculated as $V=F_{\mathrm{e}}+F_{\mathrm{o}}-1$. d, Same as a but for
$\tau_{\mathrm{int}}=60~{}\mu$s. e, Same as b, but with the linecut along the
signal axis from d. The sum of two Gaussians cannot accurately fit the data
due to a systematic error in the measurment. The slow sweep rate of the
plunger gate voltage $V_{\mathrm{P}}$ during data acquisition resulted in a
finite step width between even and odd parity state as seen in Fig. 3 in the
Main Text. The finite step width can be mitigated by sweeping $V_{\mathrm{P}}$
faster, however, it does not affect the readout fidelity calculation presented
here. f, Same as c using the fit curve from e. Extended Data Fig. 10: Gate-
dependent parity switching measurements in Device 1. a, Normalized resonator
magnitude response $R$ as a function of time $t$ and left barrier gate voltage
$V_{\mathrm{L}}$ at $V_{\mathrm{R}}=-1.599$ V, $V_{\mathrm{P}}=-1.6925$ V and
$\it{\Phi}=\it{\Phi}_{\mathrm{0}}/\mathrm{2}$. At each $V_{\mathrm{L}}$ value,
the parity was monitored for a time window of 900 ms
($\tau_{\mathrm{int}}=30~{}\mu$s, 30000 points). Parity switches are observed
only close to transitions between even and odd ground state and only in a
small gate voltage range around $V_{\mathrm{L}}=-1.73$ V. Deep in a fixed
parity sector, the charging energy of the superconducting island suppresses
the parity switches. b, Same as a, but as a function of $V_{\mathrm{R}}$ at
$V_{\mathrm{L}}=-1.729$ V. Here, the parity switches are observed at all
transitions between the even and odd state. c, Overview of normalized
resonator magnitude response $R$ as a function of barrier gate voltages
$V_{\mathrm{R}}$ and $V_{\mathrm{L}}$. For $V_{\mathrm{L}}<-1.73$ V a
transition from a 2$e$ periodic to an even-odd regime as a function of
$V_{\mathrm{R}}$ is observed, as described in Extended Data Fig. 5. For more
positive $V_{\mathrm{L}}$, in the even-odd regime the resonator magnitude
response $R$ of the even parity state changes while $R$ of the odd parity
state stays almost unaffected. The reason for this is unkown but parity
switches detected in a and b appear only close to that gate voltage regime
(turquoise and blue dashed lines show gate voltages used in a and b,
respectively).

### S1: Flux dependence and coupling factor $g$

Figures S.1a,b shows the resonator response as a function of plunger gate
voltage and magnetic flux in the $2e$-periodic regime. The even-odd regime,
shown in Figs. S.1c,d, is observed only when the energy of a subgap state is
lower than the charging energy of the superconducting island
($\delta<E_{\mathrm{C}}$). In our system the subgap state is an Andreev bound
state (ABS) with energy

$\delta(\varphi,\tau)=\pm\Delta\sqrt{1-\tau\sin^{2}(\varphi/2)},$

where $\Delta$ is the induced superconducting gap of the Al, $\tau$ is the
transmission of the ABS and $\varphi\approx
2\mathrm{\pi}\it{\Phi}/\it{\Phi}_{\mathrm{0}}$ is the phase difference across
the two superconducting leads [54, 55]. Since $\delta(\varphi,\tau)$ is
minimal at $\varphi=\pi$ and $E_{\mathrm{C}}$ is independent of $\varphi$, the
even-odd regime emerges at $\pi-$phase. Therefore, a flux dependence
measurement allows to estimate $E_{\mathrm{C}}$ of the island if
$\delta(\varphi,\tau)$ is known. We observe an anti-crossing between readout
resonator and ABS in the resonator magnitude response $\lvert S_{21}\rvert$
which was measured as a function of $\varphi$ at similar gate voltages
($V_{\mathrm{L}}=-1.755$ V, $V_{\mathrm{R}}=-1.609$ V). From that measurement,
we estimated $\tau\approx 0.98$ while $\Delta\approx 184~{}\mu$eV is known
from two-tone spectroscopy measurements of similar devices [43]. With these
values, we estimated the charging energy of the superconducting island as
$E_{\mathrm{C}}\approx 28~{}\mu$eV.

Extended Data Fig. S.1: Flux dependence measurements of Device 1. a, Resonator
magnitude response $\lvert S_{21}\rvert$ as a function of total phase
$\varphi$ across the superconducting leads and plunger gate voltage
$V_{\mathrm{P}}$ in the regime with 2$e$ periodic Coulomb blockade
oscillations ($V_{\mathrm{L}}=-1.755$ V, $V_{\mathrm{R}}=-1.617$ V). The
Coulomb resonances are resolvable only around $\pi-$phase. b, Zoomed-in
measurement of a around $\pi-$phase. c, Same as a, but in the even-odd regime
($V_{\mathrm{L}}=-1.755$ V, $V_{\mathrm{R}}=-1.609$ V). The odd parity state
(o) as well as higher order transport effects (h) are observed only around
$\pi-$phase. d, Zoomed-in measurement of b around $\pi-$phase.

The Coulomb resonance and blockade features in our measurements were observed
only around $\pi-$phase, as shown in Fig. S.1. This is due to the inductive
coupling between the microwave resonator and the device loop, where
circulating currents were mainly carried by a discrete Andreev bound state
with transmission $\tau$. Thus, the coupling depended on $\varphi$ and $\tau$.
Following Ref. [56], we calculate the coupling factor

$g(\varphi,\tau)=\frac{\sqrt{z}}{h}\frac{\Delta}{2}\frac{\delta(\pi,\tau)}{\delta(\varphi,\tau)}\tau\sin^{2}(\varphi/2)$

with $h$ Planck’s constant and $z=\pi
M^{2}\omega^{2}_{\mathrm{R}}/(Z_{\mathrm{R}}R_{\mathrm{Q}})$ a constant
coupling parameter, where $M\approx 14.7$ pH is the calculated mutual
inductance between resonator and device loop,
$\omega_{\mathrm{R}}/(2\pi)=6.362$ GHz is the measured resonator frequency,
$Z_{\mathrm{R}}\approx 56~{}\Omega$ is the characteristic resonator impedance
and $R_{\mathrm{Q}}=h/(4e^{2})$. The coupling $g$ peaks around $\pi-$phase and
broadens with lower $\tau$ (see Fig. S.2) consistent with the observations in
Fig. S.1.

Extended Data Fig. S.2: Coupling between readout resonator in Device 1.
Coupling factor $g$ between readout resonator and device loop as a function of
total phase $\varphi$ across the superconducting leads for two transmission
$\tau$ values of the bound state.

### S2: Characterization of Device 2

A second device, called Device 2, was investigated for this study. Device 2
was identical to Device 1 except for the size of the superconducting island,
which was approximately 400 nm by 400 nm. Device 2 was fabricated on the same
chip as Device 1, but was coupled to a different resonator with resonance
frequency 5.920 GHz. A scanning electron micrograph of Device 2 is shown in
Fig. S.3a. Basic parity readout is demonstrated in Figs. S.3b-d. An extended
gate-dependent study of Device 2 is shown in Figs. S.4 and S.5, with results
similar to those obtained on Device 1.

Extended Data Fig. S.3: Parity detection measurements in Device 2. a, False-
colored scanning electron micrograph of Device 2. The gates (yellow) with
voltages $V_{\mathrm{P}}$, $V_{\mathrm{L}}$ and $V_{\mathrm{R}}$ were used for
tuning the chemical potential of the island and the coupling strength to the
leads, respectively. b, Resonator response magnitude $\lvert S_{21}\rvert$ as
a function of offset readout frequency $f_{\mathrm{r}}-f_{\mathrm{0}}$ and
plunger gate voltage $V_{\mathrm{P}}$ at $V_{\mathrm{R}}=-1.0750$ V,
$V_{\mathrm{L}}=-1.5031$ V and $\it{\Phi}=\it{\Phi}_{\mathrm{0}}/\mathrm{2}$.
The modification of the resonator frequency indicates a transition between the
even and odd parity states. c, Linecuts of resonator response at two marker
positions indicated in b showing the shift of the resonator between the even
(e, green) and odd (o, pink) parity states. d, A linecut from b at the
turquoise marker position as a function of $V_{\mathrm{P}}$ emphasizes the
change in the resonator magnitude response between even and odd parity state.
Extended Data Fig. S.4: Charge state detection in Device 2. a, Normalized
resonator magnitude response $\lvert S_{21}\rvert$ as a function of plunger
voltage $V_{\mathrm{P}}$ and left barrier voltage $V_{\mathrm{L}}$ at
$\it{\Phi}=\it{\Phi}_{\mathrm{0}}/\mathrm{2}$ showing 2$e$ periodic Coulomb
blockade oscillations as a function of both voltages. a, Same as a, but
showing parity effect with alternating even (e) and odd (o) parity sectors.
Vertical stripes appeared due to flux drifts during the measurement. The
coupling between resonator and device decreases rapidly as the flux drifts
away from $\it{\Phi}=\it{\Phi}_{\mathrm{0}}/\mathrm{2}$ as discussed in Fig.
S.2. Higher order transport features (h) [46] are more pronounced compared to
Device 1, because the coupling to the leads was larger in Device 2. For an
island with small charging energy $E_{\mathrm{C}}$ the bound state energy
$\delta$ must be small to enter the even-odd regime leading to a strong
coupling to the leads. c, Same as a and b, but sweeping $V_{\mathrm{L}}$ in a
wide range. We observe (i) pure 2$e$ periodic Coulomb oscillations
($V_{\mathrm{L}}<-1.515$ V), (ii) 2$e$ periodic Coulomb oscillations with
higher order transport processes (for $V_{\mathrm{L}}$ between $-1.515$ V and
$-1.503$ V) and (iii) even-odd regime ($V_{\mathrm{L}}>-1.503$ V). Extended
Data Fig. S.5: Additional measurement on Device 2. a, Normalized resonator
magnitude response $\lvert S_{21}\rvert$ as a function of offset readout
frequency $f_{\mathrm{r}}-f_{\mathrm{0}}$ and plunger gate voltage
$V_{\mathrm{P}}$ at $V_{\mathrm{R}}=-1.075$ V, $V_{\mathrm{L}}=-1.514$ V and
$\it{\Phi}=\it{\Phi}_{\mathrm{0}}/\mathrm{2}$. The 2$e$ periodic Coulomb
blockade oscillations are consistent with the charging energy of the island
being below the bound state energy ($E_{\mathrm{C}}<\delta$). At Coulomb
resonances, a shift in the resonator frequency is observed due to enhanced
circulating current in the device loop, which changes the inductance of the
device. b, Same as a, but at $V_{\mathrm{L}}=-1.506$ V. Here,
$\delta<E_{\mathrm{C}}$ that allows the number of quasiparticles on the
superconducting island to be even or odd. The inductance of the even (e) and
odd (o) parity states are different resulting in distinct resonator responses.
The modification of the even state is more pronounced than in Fig. 5 because a
smaller coupling to the leads suppresses the higher order transport processes.
c, Normalized resonator magnitude response $\lvert S_{21}\rvert$ as a function
of barrier gate voltages $V_{\mathrm{L}}$ and $V_{\mathrm{R}}$ at
$V_{\mathrm{P}}=-1.855$ V. The gate-voltage map shows Coulomb blockade
oscillations because both barrier gates are able to tune the chemical
potential of the superconducting island. We observe two isolated bound states,
one on the left side and one in the upper right area of the gate-voltage map,
which were relevant and tunable with $V_{\mathrm{L}}$ and $V_{\mathrm{R}}$,
respectively. When at the same gate voltages the energy of both bound states
is lower than $E_{\mathrm{C}}$, i.e. $V_{\mathrm{L}}=-1.495$ V and
$V_{\mathrm{R}}=-1.095$ V, the even-odd regime becomes 1$e$ periodic.


Accepted as Demo paper at ICPR 2020

# From Tinkering to Engineering:
Measurements in Tensorflow Playground

Axel Harstad  University of California, Berkeley
and NTNU
<EMAIL_ADDRESS>Henrik Hoeiness  University of California, Berkeley
and NTNU
<EMAIL_ADDRESS>Gerald Friedland  University of California, Berkeley
<EMAIL_ADDRESS>
###### Abstract

In this article, we present an extension of the Tensorflow Playground [7],
called Tensorflow Meter (short TFMeter). TFMeter is an interactive neural
network architecting tool that allows the visual creation of different
architecture of neural networks. In addition to its ancestor, the playground,
our tool shows information-theoretic measurements while constructing,
training, and testing the network. As a result, each change results in a
change in at least one of the measurements, providing for a better engineering
intuition of what different architectures are able to learn. The measurements
are derived from various places in the literature, including recent work by
[4]. In this paper, we describe our web application that is available
online[8] and argue that in the same way that the original Playground is meant
to build an intuition about neural networks, our extension educates users on
available measurements, which we hope will ultimately improve experimental
design and reproducibility in the field.

##  I Introduction

When performing neural network experiments many details need to be addressed,
such as the number of neurons, network topology, activation function, and
secondary parameters like learning rate or regularization. Today, the most
common way to find these parameters seems to be through a “guess and check”
approach. While this can work fine anecdotally, it is certainly not ideal, as
parameter search can take exponential time, burns billions of computing
cycles, may lead to oversized networks where energy is wasted during both
training and testing, and, if the experiments do not work, it is back to the
drawing board. As a result, there is currently a movement to improve the
reproducibility of machine learning experiments through engineering
measurements.

In accordance with this idea, we present our extension to the Tensorflow
Playground [7, 6], which we call TFMeter (see Figure 1. In our extension, all
functionality of the original Tensorflow Playground is preserved but we
introduce information-theoretic measurements, such as Memory Equivalent
Capacity and bitwise generalization. These measures were presented in various
places in the literature, such as [5, 2, 4, 3]. In addition, TFMeter features
the ability to manually remove network links, and add direct connections (to
simulate a Residual Network). Furthermore, it is possible to run the
architected network on one’s own dataset. We also indicate bias by measuring
class balance and comparing it to the accuracy for each class. This gives the
user a flexible platform where one can obtain an intuition for neural networks
and deep learning.

Figure 1: TFMeter [8] in action. This network is classifying data from a
dataset uploaded by the user. Neurons biases and weights are visualized with
heatmaps and boldness of connections.The top-right corner shows a graph
plotting the loss over time along with the network’s generalization ratio. The
network’s Memory Equivalent Capacity is found in the top-middle section, and
the dataset’s expected capacity demand in the top-left corner, along with a
bias indicator. Classification accuracies for each class are shown bottom-
right.

##  II Measurements

Figure 2: Two networks illustrating Memory Equivalent Capacity: In both
networks, the second layer is dependent on the first hidden layer. The greater
number of neurons in a) makes no difference to the overall capacity of the
network, so both networks end up with equal capacity.

The capacity of a Neural Network is defined as the number of functions a
network can implement. TFMeter [8], just like its ancestor, only supports
binary classification. This allows us to adopt the definition of capacity
presented in [2, 5], where individual neurons are treated as electrical
elements and thus a network’s memorization capacity (number of points it can
overfit) can be calculated using circuit rules [4]. For a given neural network
architecture, we use these engineering rules to compute the memorization
capacity of a network or Memory Equivalent Capacity (MEC). In TFMeter, the MEC
is shown just above the corresponding network, see Figure 1.

Figure 3: Classifications of two networks. The Memory Equivalent Capacity of
network a) is set to equal the dataset’s expected capacity demand and b) is
above it. Network a) yields a generalization ratio of $G=41.67$, while b)
yields $G=17.24$. Both networks reach $100\,\%$ accuracy.

MEC allows us to compare different network architectures independent of a task
(dataset and labels). For example, consider the two networks in Figure 2. The
first network has more neurons in the subsequent layers than the second
network, but their MEC is the same. This indicates that the second network is
more efficient than the first, as using the first network would result in
wasting memory and computation resources. Furthermore, as the second network
has less parameters, it should have a better chance to avoid overfitting [4].

We have seen that MEC allows the comparison of different architectures
independent of the task. However, sizing a network properly to a task requires
an estimate of the required capacity. [3] proposes a heuristic method to
estimate the required neural network capacity for any given dataset and binary
labeling. Experimental results for various datasets show that it works well
for estimating the expected required capacity. Therefore, we use this
heuristic method to estimate the expected capacity demand for a given dataset.
If features are selected, TFMeter takes that into account. The expected
capacity demand is shown in the top-left corner, see Figure 1.

The expected capacity requirement of a dataset is useful because it provides
an estimate for how large the network needs to be. For example, in Figure 3,
we used two different networks for the same binary classification task. The
expected required capacity for this dataset was 12 bits. The first network had
a MEC equal to the expected required capacity, while the second network had a
MEC of 35 bits. Both networks reached $100\,\%$ test accuracy. That is, the
larger network is oversized. This relation between network capacity and
performance is quantifiable and is called generalization.

TFMeter [8] uses the measurement of generalization defined in [4], which is
$G=\frac{\\#\ correctly\ predicted\ instances}{Memory\ Equivalent\ Capacity}$.
A machine learner with generalization $G\leq 1$ implies no generalization,
while $G>1$ implies successful generalization.

The machine learner with the lowest capacity and highest accuracy is the one
that uses the representation function(s) most effectively. Therefore, it has
the lowest chance of failing when applied to unseen data. This is consistent
with Occam’s razor [1], which dictates that for two competing correct
hypotheses one should follow the one with the least assumptions.

## References

  * [1] A. Blumer, A. Ehrenfeucht, D. Haussler, and M. K.Warmuth, _Occam’s razor_ , Information processing letters 24: 377–380, 1987
  * [2] T. M. Cover, _Geometrical and statistical properties of systems of linear inequalities with applications in pattern recognition_ , 1965.
  * [3] G. Friedland, _Reproducibility and Experimental Design for Machine Learning on Audio and Multimedia Data_ , MM ’19: Proceedings of the 27th ACM International Conference on Multimedia, pp. 2709–2710, October 2019.
  * [4] G. Friedland, and M. Krell, _A Capacity Scaling Law for Artificial Neural Networks_ , 2018
  * [5] D. J. C. MacKay, _Information Theory, Inference, and Learning Algorithms_ , 2003.
  * [6] D. Smilkov and S. Carter, _TensorFlow Playground, http://playground.tensorflow.org/_ , 2017
  * [7] D. Smilkov, S. Carter, D. Sculley, F. Viégas, M. Wattenberg, _Direct-Manipulation Visualization of Deep Networks_ , 2017.
  * [8] Tensorflow Meter, _http://tfmeter.icsi.berkeley.edu_ , March 16th, 2020.