††thanks: These authors contributed equally to this work.
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>These authors contributed equally to this
work.
<EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
#  Supplementary Information
“Quantum Control of an Oscillator with a Kerr-cat Qubit”

Andy Z. Ding  Benjamin L. Brock  Alec Eickbusch  Akshay Koottandavida
Nicholas E. Frattini
Rodrigo G. Cortiñas  Vidul R. Joshi  Stijn J. de Graaf  Benjamin J. Chapman
Suhas Ganjam  Luigi Frunzio  Robert J. Schoelkopf  Michel H. Devoret
Departments of Applied Physics and Physics, Yale University, New Haven, CT
06520, USA

###### Contents

  1. I System Hamiltonian
    1. I.1 Encoding a qubit in a Kerr-cat
    2. I.2 Coupling a KCQ to a cavity
    3. I.3 KCQ induced cavity dephasing
  2. II System setup
    1. II.1 Device fabrication
    2. II.2 Package processing
    3. II.3 Wiring diagram
  3. III System design and characterization
    1. III.1 Design and simulation
    2. III.2 System characterization with two-tone spectroscopy
  4. IV KCQ calibration and characterization
    1. IV.1 Calibrating CQR
    2. IV.2 Calibrating KCQ gates
    3. IV.3 Calibrating the size of the cat $\alpha$
    4. IV.4 KCQ spectroscopy
  5. V Coherence characterization
    1. V.1 SNAIL coherence characterization
    2. V.2 KCQ coherence characterization
  6. VI Conditional displacement and cavity displacement calibrations
    1. VI.1 CD calibration
    2. VI.2 Displacement calibration
    3. VI.3 CD rate as a function of cat size
  7. VII Fock state preparation in the storage cavity
    1. VII.1 Swapping a photon between a SNAIL and a storage cavity
    2. VII.2 Storage Fock qubit cardinal state tomography
    3. VII.3 Storage Fock qubit logical Pauli measurement
  8. VIII SNAIL-cavity cross-Kerr characterization
  9. IX SNAIL thermal population measurement
  10. X System stability measurement
  11. XI Frequency-selective dissipation on KCQ
    1. XI.1 Basic calibration of the FSD
    2. XI.2 FSD on a large Kerr-cat

##  I System Hamiltonian

###  I.1 Encoding a qubit in a Kerr-cat

In this section, we define the Bloch sphere of the KCQ [1], which is used
throughout this work. The degenerate ground state manifold of the KCQ
Hamiltonian $H_{\mathrm{KCQ}}=-Ka^{\dagger 2}a^{2}+\epsilon_{2}a^{\dagger
2}+\epsilon_{2}^{*}a^{2}$ is spanned by the orthornormal cat states
$|\mathcal{C}^{\pm}_{\alpha}\rangle=\mathcal{N}_{\alpha}^{\pm}(|+\alpha\rangle\pm|-\alpha\rangle)$,
where $a$ is the lowering operator of the SNAIL mode, $\alpha=\epsilon_{2}/K$
is the cat size and $\mathcal{N}_{\alpha}^{\pm}=1/\sqrt{2\left(1\pm
e^{-2\alpha^{2}}\right)}$ is the normalization coefficient. Without loss of
generality, we take alpha to be real, $\alpha\in\mathbb{R}$. In the limit
where $|\alpha|^{2}\gg 1$, $\mathcal{N}_{\alpha}^{\pm}\rightarrow 1/\sqrt{2}$.
We define the logical Pauli operators as

$\begin{split}\sigma_{x}&=|\mathcal{C}^{+}_{\alpha}\rangle\langle\mathcal{C}^{+}_{\alpha}|-|\mathcal{C}^{-}_{\alpha}\rangle\langle\mathcal{C}^{-}_{\alpha}|\\\
\sigma_{y}&=i|\mathcal{C}^{+}_{\alpha}\rangle\langle\mathcal{C}^{-}_{\alpha}|-i|\mathcal{C}^{-}_{\alpha}\rangle\langle\mathcal{C}^{+}_{\alpha}|\\\
\sigma_{z}&=|\mathcal{C}^{+}_{\alpha}\rangle\langle\mathcal{C}^{-}_{\alpha}|+|\mathcal{C}^{-}_{\alpha}\rangle\langle\mathcal{C}^{+}_{\alpha}|\end{split}$
(1)

We can then define the projector
$\mathcal{P}_{\mathcal{C}}=|\mathcal{C}^{+}_{\alpha}\rangle\langle\mathcal{C}^{+}_{\alpha}|+|\mathcal{C}^{-}_{\alpha}\rangle\langle\mathcal{C}^{-}_{\alpha}|$
to project the bosonic operators of the SNAIL $a$, $a^{\dagger}$ onto the KCQ
subspace.

###  I.2 Coupling a KCQ to a cavity

In this section we derive the static effective Hamiltonian of our setup
consisting of a KCQ coupled to a storage cavity. Our calculation follows the
method in [2, 3] where we use Feynman-like diagrams to systematically compute
the static terms describing different cascaded nonlinear processes in our
setup.

The Hamiltonian of the system consisting of a SNAIL coupled to a cavity is
given by

$H_{\mathrm{full}}=H_{\mathrm{SNAIL}}+H_{\mathrm{cavity}}+H_{c}+H_{d}$ (2)

where
$H_{\mathrm{SNAIL}}=\omega_{a}^{\prime}a^{\prime\dagger}a^{\prime}+\sum_{n>2}g_{n}(a^{\prime}+a^{\prime\dagger})^{n}$
is the SNAIL Hamiltonian,
$H_{\mathrm{cavity}}=\omega_{b}^{\prime}b^{\prime\dagger}b^{\prime}$ is the
Hamiltonian of the microwave cavity,
$H_{c}=g(a^{\prime}+a^{\prime\dagger})(b^{\prime}+b^{\prime\dagger})$ is the
capacitive coupling between the two modes,
$H_{d}=2\real(e^{i\omega_{s}t})(\epsilon_{s}a^{\prime\dagger}+\epsilon^{\ast}_{s}a^{\prime})+2\real(e^{i\omega_{d}t})(\epsilon_{d}a^{\prime\dagger}+\epsilon^{\ast}_{d}a^{\prime})$
is the drive Hamiltonian, $\epsilon_{s}$ is the strength of the squeezing
drive on the SNAIL, and $\epsilon_{d}$ is the resonant drive on the SNAIL. In
these expressions, $a^{\prime}$ and $b^{\prime}$ are the bare modes of the
SNAIL and the cavity, with bare frequencies $\omega_{a}^{\prime}$ and
$\omega_{b}^{\prime}$, respectively. The capacitive coupling hybridizes the
modes and shifts their frequencies. Therefore, we reserve the notations $a$
and $b$ for the hybridized modes that we can measure experimentally.

We truncate the SNAIL nonlinearity to the fourth order to compute the static
effective Hamiltonian of our system,

$H=\Delta_{s}a^{\dagger}a-Ka^{\dagger 2}a^{2}+\epsilon_{2}a^{\dagger
2}+\epsilon_{2}a^{2}-K_{b}b^{\dagger
2}b^{2}-\chi_{ab}a^{\dagger}ab^{\dagger}b$ (3)

where we have, to the leading order of $(g/\Delta)^{2}$ or the leading non-
negligible terms,

$\begin{split}\Delta_{s}&=\left(\frac{24g_{4}}{9g_{3}^{2}}-\frac{9}{\omega_{a}}\right)\epsilon_{2}^{2}\\\
K&=-6g_{4}+\frac{30g_{3}^{2}}{\omega_{a}}\\\
K_{b}&=-6g_{4}\left(\frac{g}{\Delta}\right)^{4}+9g_{3}^{2}\left(\frac{g}{\Delta}\right)^{2}\left(\frac{1}{2\omega_{a}-\omega_{b}}-\frac{1}{2\omega_{a}+\omega_{b}}-\frac{4}{\omega_{b}}\right)\\\
\chi_{ab}&=-24g_{4}\left(\frac{g}{\Delta}\right)^{2}+36g_{3}^{2}\left(\frac{g}{\Delta}\right)^{2}\left(\frac{1}{2\omega_{a}-\omega_{b}}+\frac{1}{2\omega_{a}+\omega_{b}}+\frac{2}{\omega_{a}}\right).\end{split}$
(4)

Plugging in the parameters from Table. 1, we calculate $(g/\Delta)^{2}=0.5\%$,
so we can safely ignore higher order terms in the expansion of $\Delta_{s}$,
$K$, and $\chi_{ab}$. In the case of $K_{b}$ the leading order of the
expansion is already of order $(g/\Delta)^{4}$. Given that $K_{a}/2\pi=0.93$
MHz, the inherited self-Kerr of the storage from the SNAIL is negligible,
which makes this setup suitable for encodings such as GKP states [4] that are
susceptible to Kerr effects.

To gain intuition for the dispersive coupling between a KCQ and a storage, it
is useful to consider the projection of the SNAIL number operator
$a^{\dagger}a$ onto the KCQ subspace

$\begin{split}\mathcal{P}_{\mathcal{C}}a^{\dagger}a\mathcal{P}_{\mathcal{C}}&=\alpha^{2}r^{2}|\mathcal{C}^{+}_{\alpha}\rangle\langle\mathcal{C}^{+}_{\alpha}|+\alpha^{2}r^{-2}|\mathcal{C}^{-}_{\alpha}\rangle\langle\mathcal{C}^{-}_{\alpha}|\\\
&=\alpha^{2}\left(\mathbb{I}-2e^{-2\alpha^{2}}\sigma_{x}\right).\\\
\end{split}$ (5)

After projection, the number operator of the SNAIL mode gets turned into an
identity and a $\sigma_{x}$ rotation. Of particular interest to us is the
scaling factor in front of the $\sigma_{x}$ rotation term. As we increase the
squeezing amplitude, this rotation gets exponentially suppressed.

###  I.3 KCQ induced cavity dephasing

FIG. S 1: Spectrum of a KCQ coupled to a cavity as a function of cat size
$\alpha$, obtained from numerical diagonalization of the system Hamiltonian in
Eq. 3.

To better understand the interaction between a KCQ and a storage cavity, we
diagonalize the Hamiltonian in Eq. 3 using our experimental parameters in
Table 1. The spectrum we obtain from the diagonalization as a function of
$|\alpha|^{2}$ is shown in Fig. S 1. At an energy scale comparable with $K$ in
Fig. S 1(a), the spectrum agrees with the result from [5], where the excited
states of the KCQ become degenerate as we increase $|\alpha|^{2}$. The gaps
between different lines, such as $E_{\mathrm{gap}}$ and $E_{\mathrm{gap,P}}$
are determined by the drive strength and $K$ of the KCQ, where the subscript
$P$ indicates that the states have definite photon number parity. When we zoom
in on the lines at different cat sizes in Fig.1(b) and (c), we see the
spectrum of the KCQ split into different harmonic sectors due to coupling with
Fock states in the storage cavity. The frequency of each harmonic sector is a
function of the average photon number of the KCQ state.

As an example, in the case of Fig. S 1(b) where $|\alpha|^{2}=4$, only the
degenerate ground states of the KCQ, spanned by
$|\mathcal{C}_{\alpha}^{\pm}\rangle$ are inside the double well potential. The
storage photons split the ground states evenly by
$\chi_{ab}\bar{n}_{\mathrm{cat}}$, where
$\bar{n}_{\mathrm{cat}}=|\alpha|^{2}$. This indicates that the storage photons
are not able to distinguish between the different cat states in the KCQ ground
state manifold, the storage is simply Stark-shifted by
$\chi_{ab}\bar{n}_{\mathrm{cat}}$. Therefore, the storage cavity and the KCQ
will not entangle with each other during idling time when the KCQ stays in its
ground state. The first excited states are parity states right above the
wells. Though the storage photons still split each level evenly, different
levels in the KCQ Stark-shift the storage cavity by different amounts due to
their different average photon numbers
$\bar{n}_{P^{-}}\neq\bar{n}_{P^{+}}\neq\bar{n}_{\mathrm{cat}}$. As a result,
the storage cavity is capable of distinguishing between different levels in
the KCQ, since it rotates at slightly different frequencies depending on
whether the KCQ is in its ground or excited states.

This understanding can be extended to the case of larger KCQs with more than
one level inside the wells, as in Fig. S 1(c) with $|\alpha|^{2}=8.2$. In this
case, the spectral lines of the first excited states are merging while the
second excited states are still outside of the wells. In this case, the Stark
shift of the storage cavity when the KCQ is in its first excited state is
still distinct from that of the ground state of the KCQ, as we have
$\bar{n}_{\mathrm{cat}}^{(1)}\approx\bar{n}_{\mathrm{cat}}+1$. In the limit of
large $\alpha$, we have two distinct regions: states inside the well and
states outside the well. Fig. S 2 shows the energy level diagram in this
asymptotic limit.

FIG. S 2: Energy level diagram of a system with a KCQ coupled to a storage
cavity.

In the absence of KCQ heating, the cavity coherence is protected from
dephasing at large $\alpha$ because the cross-Kerr interaction simplifies into
a Stark-shift of the storage cavity, which can be tracked in software. When
the KCQ is in an excited state, however, the storage cavity is Stark-shifted
by a different amount. Thus, when the KCQ heats to an excited state for an
unknown amount of time, the storage accumulates an unknown amount of phase,
leading to dephasing. This dephasing is what we have measured in Fig. 3(c) of
the main text.

To model the cavity coherence in the absence of KCQ heating, as well as the
suppression of cavity dephasing at small $\epsilon_{2}$, we perform a master
equation simulation using Eq. 3 in the previous section as our model of the
system and photon loss as the only dissipation. We manage to achieve good
agreement between data and simulation for the regime $|\alpha|^{2}<1$ only by
setting $T_{1,a}=100$ $\mu s$ for the SNAIL, which is significantly different
from our characterized SNAIL coherence in Table 1. This is due to a lack of
understanding of the heating mechanism of the KCQ [5, 6, 7]. As a strongly
driven nonlinear quantum system, the environment seen by the KCQ is not
Markovian, rendering our master equation simulation only a preliminary attempt
at understanding the dynamics of the system. As a result, we did not push the
limit of the simulation to match our data by adding in dephasing or SNAIL
heating. This is also because we are using the simulation result as a “guide-
to-the-eye” for our data on cavity coherence under active cooling on the KCQ,
showcasing that in the absence of KCQ heating, the cavity dephasing will be
suppressed when we increase the squeezing amplitude.

##  II System setup

###  II.1 Device fabrication

Following the recipe in [8], the on-chip superconducting circuits used in our
experiment were fabricated on a c-plane sapphire substrate grown using the
heat-exchange method (HEM). The substrate wafer was cleaned using a piranha
solution (2:1 $\mathrm{H_{2}SO_{4}:H_{2}O_{2}}$) for 20-25 minutes and rinsed
under running deionized (DI) water before being annealed in a FirstNano
EasyTube 6000 furnace. The annealing step has been shown to improve substrate
surface and bulk loss [8]. During the annealing process, the furnace was
preheated to 200°C with constant oxygen flow before loading the sample. It was
then heated up to 1200°C at a ramp rate of 400°C per hour. Afterwards, the
furnace was held at 1200°C for 1 hour before cooled down gradually to room
temperature over 6 hours with constantly flowing gas mixture (4:1
$\mathrm{N_{2}:O_{2}}$).

The tantalum layer was uniformly sputtered on the substrate using the Kurt J.
Lesker CMS 18 DC magnetron sputtering system. The substrate was loaded into
the sputtering chamber with a niobium thermalization disc in the back for
better heating uniformity. Afterward, the substrate was heated to 400°C and
was held for 15 minutes for dehydration bakeout. The substrate was then heated
to the deposition temperature at 800°C and idled for 10 minutes for
temperature uniformity. The tantalum was sputtered at 300 W with 6 mTorr of Ar
gas at a rate of 2.5Å/s for 10 minutes. The system then idled again for 10
minutes at 800°C for the post-deposition bake. Subsequently, the system was
cooled down at a controlled rate of 10°C per minute down to 500°C. This was to
release the stress in the tantalum film into the substrate adiabatically to
avoid material damage at the metal-substrate interface. Afterward, the
substrate was cooled down to room temperature at a faster rate. It took 3-5
hours to reach room temperature before the substrate was unloaded.

Before optical lithography, the substrate surface was primed with HMDS vapor
for better resist adhesion and uniformity. During the process, the substrate
was loaded into a TA Series Yield Engineering System (YES) oven right after
the annealing process. The oven chamber was then purged with HMDS vapor three
times before baking out the wafer at 150°C for 25 minutes. At the end of the
process, the chamber was re-purged with HMDS vapor three times before the
substrate was removed.

The sapphire substrate was then coated with SC1805 photoresist using Laurell
spin coater WS-400 at 500 rpm for 10 seconds before ramping up to 4000 rpm for
1 minute and 30 seconds for film uniformity. The resist was baked at 115°C for
1 minute. The edge beads were removed using acetone before the wafer was
loaded into the Heidelberg Maskless Aligner 150 direct laser writer for
exposure with a calibrated dose of 92 mJ/mm2. Then the resist was developed
with Microposit MF319 developer for 1 minute and then treated with oxygen
plasma using the AutoGlow 200 plasma system for 1 minute and 30 seconds at 150
W and 300 mTorr of oxygen.

The tantalum layer was etched using the dry etch recipe from [8] with an
Oxford 80 reactive ion etch system. Before the sample was loaded, the chamber
was prepped using the standard cleaning recipe, which consists of two main
steps. We ran the recipe with an empty chamber before the actual etching. The
first step is 10 minutes of cleaning with a mixture of O2 and SF6 at flow
rates 50 sccm and 10 sccm, respectively. The power was at 100 W and the
pressure was maintained at 50 mTorr. The second step is another 10 minutes of
cleaning with a mixture of O2 and Ar at 50 sccm and 10 sccm, respectively. The
power was kept at 50 W and the pressure was 50 mTorr. These two steps
conditioned the chamber to have more reproducible results. The tantalum etch
recipe required us to flow SF6 at a rate of 20 sccm at a pressure of 10 mTorr
for 3 minutes to etch through 150 nm of tantalum. After etching, the substrate
was cleaned with 3 minutes of sonication in N-Methylpyrrolidone (NMP), acetone
and isopropyl alcohol (IPA) sequentially to get rid of the photoresist and was
blown dry with nitrogen.

We thoroughly cleaned the wafer before electron beam (E-beam) lithography to
reduce surface loss. The substrate was cleaned with a piranha solution (2:1
$\mathrm{H_{2}SO_{4}:H_{2}O_{2}}$) for 20-25 minutes to get rid of organic
contaminants. Then it was rinsed under DI water for 5-10 minutes before being
transferred in Transene 10:1 buffered oxide etch (BOE) for 20 minutes to etch
away most of the tantalum oxides [9]. The DI water rinse prior to BOE is
crucial because a mixture of piranha and BOE solutions can attack tantalum, a
phenomenon we learned through trial and error. The BOE step was followed by a
20-25 minute rinse in DI water.

The SNAIL loops and the Josephson junctions were patterned using electron-beam
lithography. After a dehydration bake at 180°C for 5 minutes, the substrate
was spin-coated with 800 nm of MMA (8.5) MAA EL13 and 200 nm of 950K PMMA A4,
with a 5-minute bake at 180°C after each layer was spun. Then, a 15 nm
aluminum anticharging layer was deposited on top of the bilayer to avoid
charging effects. The bridge-free junctions were then written using a Raith
EBPG 5200+ at a base dose of 290 $\mu$C/cm2. Afterwards, the anticharging
layer was removed using MF312 developer for 2 minutes. The Ebeam patterns were
developed in 3:1 IPA:H2O for 2 minutes at 6°C and the wafer was blown dry with
nitrogen.

The Josephson junctions were deposited using a Plassys UMS300 electron-beam
evaporator. After being loaded in the load lock and pumped to base pressure,
the sample was etched with an Ar ion beam at 400 V and 22 mA at $\pm
45\degree$ angles for 34 seconds each. This step removed the oxide on the
tantalum sidewalls and cleaned the substrate regions where aluminum was to be
deposited. Subsequently, the aluminum was evaporated at angles of $\pm
20\degree$ in the evaporation chamber with an oxidation step in the middle
using an 85:15 Ar:O2 mixture at 15 Torr for 15 minutes. After the depositions,
the surface of the devices was capped with a layer of aluminum oxide grown at
50 Torr for 5 minutes with the same gas mixture in the oxidation chamber
before the substrate was unloaded.

The liftoff was performed by submerging the substrate in NMP at 75°C for 4
hours. Then it was cleaned by being sonicated in RT NMP, acetone, and IPA for
1 minute each. Then the wafer was blown dry with nitrogen. A layer of SC1827
photoresist was coated on the wafer before it was diced with an ADT ProVectus
7100 dicer. The chips were cleaned by being sonicated in NMP, acetone, and IPA
for 2 minutes each and were tested by probing at room temperature before being
loaded into our experimental package.

###  II.2 Package processing

The main 3D aluminum package was mechanically machined out of high-purity
(5N5) aluminum. It was then cleaned by sonicating in NMP, acetone and IPA for
5 minutes each. Afterward, it went through a chemical etching treatment using
Transene Aluminum Etchant Type A to get rid of contaminants from the machining
process [10]. During the etching process, the package was mounted on a custom-
designed PTFE cage and submerged in 1L of the etchant solution that was heated
to 50°C for about 2 hours in total. During the etching process, the etchant
bath needs to be periodically changed to avoid secondary reaction product
redepositing on the surface of the package. Also, the bath was stirred at 600
RPM. Afterwards, the package was rinsed thoroughly with DI water for 10
minutes, before blown dry with nitrogen gas. The auxiliary parts, including
the chip clamps, the copper brackets and the screws were cleaned by sonication
in NMP, acetone and IPA for 3 minutes each.

###  II.3 Wiring diagram

FIG. S 3: The wiring diagram of our setup.

Our experimental device is clamped to the mixing chamber of a dilution
refrigerator, as depicted in Fig. S3, and surrounded with two layers of
shielding. The outermost shield is a cryoperm $\mathrm{\mu}$-metal magnetic
shield, and the innermost shield is made of copper and coated with Berkeley
black [11] to absorb stray infrared radiation. To characterize and control our
experimental device inside the dilution refrigerator, we built a set of
microwave lines shown in the wiring diagram in Fig. S. 3. There are 10 lines
in total, and each line is numbered next to its respective label on the
diagram.

Lines 1, 3, and 6 provide the microwave signals for resonantly driving the
storage cavity, SNAIL mode, and readout cavity, respectively. On these lines a
signal generator supplies a local oscillator (LO) tone to an IQ mixer, which
up- or down-converts the LO tone via the inputs to its in-phase (I) and
quadrature (Q) ports. These IQ pairs are sourced from an FPGA-based digital-
to-analog converter (DAC) with custom software [12], providing us full control
of the amplitude and phase of our drives with $2$ ns sampling resolution.

Lines 2, 4, and 5 provide the microwave signals for parametrically driving the
conditional displacement (CD), squeezing, and cat-qubit readout (CQR)
processes, respectively. On line 2 we mix together the SNAIL LO and storage LO
(using), filter the output to isolate the desired difference frequency, and
amplify this tone (with a Mini-Circuits ZVA-183-S+ amplifier) so it can
provide sufficient power to the LO input of an IQ mixer. This tone is then
mixed with an IQ DAC pair from our FPGA to give us full control of the
amplitude and phase of our CD drive, and then amplified to achieve sufficient
power to drive this parametric process. Line 4 is designed in the same way,
except we are mixing the SNAIL LO with itself (using a Marki MLD-0416LSM
frequency doubler) to generate a tone at twice the frequency of this signal.
Line 5 is designed in the same way, except we are mixing together the SNAIL LO
with the readout LO (using a Marki M2H-0220HP mixer) to generate a tone at the
difference frequency between these signals.

After conditioning all of these drive signals, they are combined in different
ways and sent into the dilution refrigerator. The signal on line 1 is
delivered to the storage pin (see Fig. S4(a) for our nomenclature regarding
the pins). After entering the fridge it is attenuated (nominally to thermalize
noise in the transmission line to the temperature of the mixing chamber) and
filtered (with a K&L 6L250-12000/T26000-O/O tubular low-pass filter and an
eccosorb filter) before arriving at the storage pin. The eccosorb filter is
used to attenuate infrared Cooper-pair-breaking radiation. The signals on
lines 2 and 6 are combined and delivered to the readout pin. After entering
the fridge this combined signal is similarly attenuated and filtered before
arriving at the readout pin, but in this case it also passes through a
cryogenic circulator (Quinstar OXE89). The signals on lines 3, 4, and 5 are
combined and delivered to the SNAIL pin. After entering the fridge this
combined signal is similarly attenuated and filtered before arriving at the
SNAIL pin, but in this case we use a $20$ dB directional coupler instead of an
attenuator at the mixing chamber. This syphons off $1\%$ of the signal and
sends the remaining $99\%$ back up the fridge to the $4$K plate to be
dissipated as heat. In this way we are able to decouple the process of
attenuation from heat generation; if we were to use a 20 dB attenuator
instead, the fridge would heat up due to the high power needed for the
parametric drives being delivered to the SNAIL pin.

Line 7 is our output line. Photons in the readout cavity that leak out through
the readout pin are first amplified by our quantum-limited SNAIL parametric
amplifier (SPA) [13], then a cryogenic HEMT amplifier, and finally a room-
temperature amplifier (MITEQ AMF-5F-04001200). The resulting signal is then
mixed together with the readout LO tone (using a Marki IRW-0618LXW-2 image
reject mixer), resulting in $50$ MHz signal that is digitially acquired using
the analog-to-digital (ADC) of our FPGA.

Lines 8 and 9 are used for operating our SPA. Line 8 simply provides a pump
tone to the SPA, while line 9 delivers a DC current that threads flux through
the SPA to tune its operating point. This DC current is carried by a twisted
pair of superconducting NbTi wires from the $4$K plate down to the mixing
chamber of our dilution refrigerator. To tune the SPA we only need to drive a
current of a few hundred $\mu$A, so we are able to modularize this DC line
using normal-metal connectors at both the $4$K plate and the mixing chamber.

Line 10 delivers a DC current that threads flux through our on-chip SNAIL,
enabling us to tune its operating point. The DC current is again carried by a
twisted pair of superconducting NbTi wires from the $4$K plate down to the
mixing chamber, but in this case we need to drive about $100$ mA of current to
tune the SNAIL to its operating flux bias point of $0.32\Phi_{0}$. If we were
to use normal-metal connectors anywhere below the $4$K plate, the heat
generated would overwhelm the cooling power of the fridge. To get around this,
we use a continuous NbTi twisted pair running from $4$K to the mixing chamber,
following the method introduced in [14].

##  III System design and characterization

FIG. S 4: The setup and basic spectroscopy result. (a) A cartoon of our
experimental setup modeled from above. The sapphire chip (colored in blue)
with the superconducting circuits (colored in silver) sits in between two 3D
aluminum post cavities (colored in silver) with coupling pins (colored in
brown) for delivering microwave control lines. The SNAIL has its drive pin
perpendicular to the chip at the bottom end of the chip tunnel. Above the
chip, there is a copper coil (colored brown) wound with Nb-Ti wires for
threading magnetic flux into our SNAIL loops. On the right side of (a), we
have the optical micrograph of our SNAIL circuit, where the SNAIL loops with
the Josephson junctions are made of aluminum, and the rest of the structures
are made out of tantalum. (b) The two-tone spectroscopy result, where we drive
the coupling pin to the SNAIL with a CW spectroscopy tone while measuring the
reflection coefficient of the readout cavity. Whenever photons are absorbed by
a mode the readout is Stark-shifted, which manifests as a change in the phase
of the reflection coefficient S11. The green curve is our best fit for the
SNAIL frequency as a function of flux. The light green curve is the difference
between the CQR buffer mode frequency and the fitted SNAIL frequency,
corresponding to the CQR parametric process. The light blue curve is twice the
fitted SNAIL frequency, corresponding to the squeezing parametric processes.

###  III.1 Design and simulation

Our experimental package consists of a sapphire chip (50 mm $\times$ 5.6 mm)
sandwiched between two 3D aluminum post cavities, as is depicted in Fig. S
4(a). Superconducting circuits are fabricated on the chip out of tantalum and
aluminum, as described above. The package was designed using electromagnetic
finite element solver HFSS. The SNAIL circuit is at the center of the chip,
highlighted in the orange boxes. The SNAIL consists of two superconducting
loops to lower its anharmonicity [13, 15]. As the SNAIL circuit is a flux-
tunable device, we need to deliver external magnetic flux to our device inside
our experimental setup. To do this, we follow the experimental setup developed
in [16, 14], using a copper coil nested inside the aluminum package as well as
an on-chip flux transformer. When we run DC current in the twisted pair, the
current will turn the copper coil into an electromagnet, which then threads
flux into the flux transformer loop on the sapphire chip underneath. The flux
generates current in the loop that travels down the chip close to the SNAIL
loops, where its electromagnetic field threads flux into the SNAIL circuit,
providing it with the external DC flux bias. In our design process, we
optimized the geometry of the pick-up loop of the flux transformer so that its
eigenmodes do not crowd into frequencies of interest in the package. The
common mode of the flux transformer is at $\omega_{c}/2\pi=9.26$ GHz, and its
differential mode is at 9.76 GHz. The common mode has a cross-Kerr
$\chi_{ac}/2\pi=2.31\pm 0.1$ KHz to the SNAIL mode with a total linewidth of
$\kappa_{c}/2\pi=131.4\pm 0.4$ kHz. Compared with the readout mode, it enjoys
a much larger $\chi/\kappa$ ratio and is therefore more suitable for
dispersive readout. As a result, in our basic system characterization
experiments, we use the common mode of the flux transformer as our readout
mode to perform two-tone spectroscopy, which is discussed in more detail in
the next subsection.

The SNAIL circuit hosts the KCQ, and it is driven with microwave signals
delivered via the SNAIL pin shown at the bottom end of the chip tunnel in Fig.
S 4. The SNAIL pin is coupled to the SNAIL at a rate of $(150$ $\mu$s$)^{-1}$,
which corresponds to the Purcell limit of this mode. Inspired by the design in
[14], we design two on-chip meandering resonators serving as buffer modes for
the squeezing drive (on the left at 7.760 GHz with a linewidth of $1.35\pm
0.02$ MHz) and the CQR drive (on the right at 5.308 GHz with a linewidth of
$1.24\pm 0.01$ MHz). The squeezing drive at 7.996 GHz is detuned from the
squeezing buffer by 240 MHz, and the CQR drive at 5.362 GHz is detuned from
its buffer mode by 56 MHz. Both parametric drives are very far away from their
designed buffer modes in frequency. Nevertheless, we can still efficiently
drive these parametric processes without using excessively noisy amplifiers
(see the wiring diagram in Fig. S 3): we use Minicircuits ZVAs instead of
ZVEs, as were used in [17, 5, 2]. The capacitor pads of the SNAIL circuit have
a slight asymmetry to avoid coupling the two post cavities directly via the
common mode of the circuit. The post cavity on the right (7.025 GHz) is our
high-Q storage cavity. The post cavity on the left serves two purposes. Its
fundamental mode (3.133 GHz, with a linewidth of $0.28\pm 0.01$ MHz in total)
is the buffer mode for the CD parametric drive, and its first harmonic (9.362
GHz) is the readout cavity for the SNAIL/KCQ. The CD drive at 3.027 GHz is
detuned from its buffer by 106 MHz, but we can still drive $g_{CD}/2\pi=3.1$
MHz (see CD calibration below).

###  III.2 System characterization with two-tone spectroscopy

To experimentally validate the microwave design of our package, as well as
extract the SNAIL parameters, we perform a two-tone spectroscopy experiment as
a function of the SNAIL’s flux bias. In this experiment, we drive the SNAIL
pin with a CW spectroscopy tone swept from 2-10 GHz at -50 dBm while measuring
the reflection coefficient of the cavity via the readout + CD pin. Whenever
photons are absorbed by a mode, the readout is Stark-shifted, which manifests
as a change in the phase of the reflection coefficient S11. In the results
shown in Fig. S 4(b), we identify the resonance feature tuning with flux from
3.5 - 4.5 GHz as our SNAIL mode, the feature tuning from 4.9-5.9 GHz as our
CQR parametric process, and the feature tuning from 6.9-8.9 GHz as our
squeezing parametric process. We extract the SNAIL parameters listed in the
top section of Table 1 by fitting the measured SNAIL frequencies using the
model
$H_{\mathrm{SNAIL}}=\omega_{a}^{\prime}a^{\prime\dagger}a^{\prime}+\sum_{n>2}g_{n}(a^{\prime}+a^{\prime\dagger})^{n}$
up to $n=4$. We also identify other prominent features in the two-tone
spectroscopy result, such as the CQR parametric process and the squeezing
process. We subtract the fitted SNAIL frequencies from the CQR buffer
frequency and the squeezing buffer frequency to obtain the lines in light
green and light blue. They serve as guide-to-the-eye for the measured CQR and
squeezing processes. The readout mode and the CD buffer mode are barely
visible as they are not as strongly coupled to the SNAIL drive pin by design.

Parameter | Value | Measurement or estimate method
---|---|---
SNAIL charging energy $E_{\mathrm{C}}/h$ | 62 MHz | Two-tone spectroscopy and SNAIL fit
Number of SNAILs | 2 | Design
SNAIL asymmetry $\alpha_{\mathrm{SNAIL}}$ | 0.11 | Two-tone spectroscopy and SNAIL fit
SNAIL 1 large junction inductance | 0.6 nH | Two-tone spectroscopy and SNAIL fit
SNAIL 1 small junction inductance | 5.5 nH | Two-tone spectroscopy and SNAIL fit
SNAIL linear stray inductance $L_{\mathrm{lin}}$ | 1.5 nH | Two-tone spectroscopy and SNAIL fit
SNAIL frequency at $\Phi/\Phi_{0}=0$ | 4.46 GHz | Two-tone spectroscopy
SNAIL frequency at $\Phi/\Phi_{0}=0.5$ | 3.46 GHz | Two-tone spectroscopy
SNAIL operating bias point $\Phi_{\mathrm{op}}/\Phi_{0}$ | 0.32 | Design
SNAIL operating frequency $\omega_{a}/2\pi$ | 3.998 GHz | Two-tone spectroscopy
SNAIL cubic nonlinearity $g_{3}/2\pi$ | 6 MHz | Two-tone spectroscopy and SNAIL fit
SNAIL self-Kerr nonlinearity $K/2\pi$ | 0.93 $\pm$ 0.03 MHz | Kerr refocusing msmt.
SNAIL single-photon decay time $T_{1,a}$ | 16.0 $\pm$ 0.4 $\mu$s | Coherence msmt.
SNAIL Ramsey decay time $T_{2,a}$ | 7.2 $\pm$ 0.1 $\mu$s | Coherence msmt.
SNAIL Hahn echo decay time $T_{2\mathrm{E},a}$ | 9.5 $\pm$ 0.1 $\mu$s | Coherence msmt.
Storage cavity frequency $\omega_{b}/2\pi$ | 7.025 GHz | Two-tone spectroscopy
Storage cavity frequency tuning range with external flux | 1.2 MHz | Two-tone spectroscopy
Storage cavity single-photon decay time $T_{1}$ | 204 $\pm$ 9 $\mu$s | Coherence msmt.
Storage cavity Ramsey decay time $T_{2}$ | 381 $\pm$ 8 $\mu$s | Coherence msmt.
Storage cavity to SNAIL cross-Kerr $\chi_{ab}/2\pi$ | 2.91 $\pm$ 0.03 kHz | Coherence msmt.
Storage cavity self-Kerr nonlinearity $K_{b}/2\pi$ |  $<1$ Hz | Simulation
Readout cavity frequency $\omega_{r}/2\pi$ | 9.362 GHz | Direct RF reflection msmt.
Readout cavity total linewidth $\kappa_{r}/2\pi$ | 0.396 MHz | Direct RF reflection msmt.
Readout cavity internal linewidth | 0.04 MHz | Direct RF reflection msmt.
Readout cavity to SNAIL cross-Kerr $\chi_{ar}/2\pi$ | 1.51 $\pm$ 0.01 kHz | Direct RF reflection msmt.
CQR buffer mode frequency | 5.308 GHz | Two-tone spectroscopy
CQR buffer mode linewidth | 1.24 $\pm$ 0.01 MHz | Two-tone spectroscopy
Squeezing buffer mode frequency | 7.760 GHz | Two-tone spectroscopy
Squeezing buffer mode linewidth | 1.35 $\pm$ 0.02 MHz | Two-tone spectroscopy
CD buffer mode frequency | 3.133 GHz | Two-tone spectroscopy
CD buffer mode linewidth | 0.28 $\pm$ 0.01 MHz | Two-tone spectroscopy
Supplementary Table 1: Summary of device parameters. The design simulations
are carried out with Ansys HFSS and the black-box quantization method [15, 18,
19].

##  IV KCQ calibration and characterization

###  IV.1 Calibrating CQR

FIG. S 5: Calibrating the CQR. (a). The pulse sequence for measure the QNDness
of the CQR. (b). A highly-averaged CQR signal obtained from performing the
experiment in (a). (c). The corresponding histogram of the CQR signal in (b),
where we perform two consecutive readouts and then postselect the result of
the second one based on that of the first one $|\pm\alpha\rangle$. The yellow
(green) circles are measured data postselected on measuring
$|\pm\alpha\rangle$, where the solid lines are the Gaussian fits.

We perform logical readout on the KCQ following the same pulse sequence that
was first experimentally demonstrated in [17]. The basic principle of the
readout process is that we perform a conditional displacement on the readout
cavity conditioned on whether the KCQ is in the state $|+\alpha\rangle$ or
$|-\alpha\rangle$. When we drive at $\omega_{r}-\omega_{a}$, we obtain the
interaction Hamiltonian

$H_{\mathrm{CQR}}=g_{\mathrm{CQR}}\left(r^{\dagger}+r\right)\sigma_{z}$ (6)

where $r$ is the readout mode and $g_{\mathrm{CQR}}$ is linearly proportional
to the size of the Kerr-cat $\alpha$ [17, 5]. The intuition of the physical
process is that we are driving a beamsplitter between the readout cavity and
the stabilized KCQ. This process converts photons in the KCQ to photons in the
readout cavity, which exit through the readout pin and then travel up the
amplification chain to the ADC of the readout FPGA card. The phase of the
signal carries ”which-well” information about the KCQ. More detailed
treatments of the CQR process can be found in [17, 5]. For all the CQR
measurements included in both the main text and the supplement, we drive the
beamsplitter for 600 ns and simultaneously integrate the signal coming from
the readout cavity for $2.5\mu$s. Fig. S 5 shows the pulse sequence and the
result of a readout QNDness measurement, where we perform two consecutive CQRs
on the KCQ and post-select the result of the second measurement on the
outcomes $|\pm\alpha\rangle$ of the first measurement [17]. We measure the
QNDness to be $\mathcal{Q}=(p(+\alpha|+\alpha)+p(-\alpha|-\alpha))/2=91\%$,
where $p(\pm\alpha|\pm\alpha)$ is the probability that the second measurement
agrees with the first [17, 20].

###  IV.2 Calibrating KCQ gates

As discussed in the main text, we need two gates on the KCQ to realize
universal control: the $\sigma_{z}(\theta)$ rotation and the
$\sigma_{x}(\pi/2)$ rotation (i.e., the Kerr gate). Both gates were first
experimentally demonstrated in [17]. In this section, we describe our process
for tuning up these gates.

FIG. S 6: KCQ gate calibration (a). Pulse sequence for characterizing the
Kerr-refocusing in our SNAIL. We prepare a coherent state $|\pm\alpha\rangle$
in the SNAIL through a CQR measurement, and then switch off the squeezing
drive to let the state evolve under Kerr of the SNAIL. We take “snapshots” of
the evolution by switching on the squeezing drive after time $t$ and reading
out the state with CQR. (b). The readout signal for the Kerr refocusing
measurement. The blue circles are measured data and the red solid line is
obtained from a master equation simulation. (c). Longer Kerr refocusing data,
obtained by increasing the wait time $t$. The solid line is found using the
same master equation simulation as in (b). (d). The control sequence for
calibrating a $\sigma_{z}(\theta)$ rotation on the KCQ. The gaps in the
squeezing drive pulse train are the Kerr gates. (e). Rabi oscillations between
different cats with different parities $|\mathcal{C}^{\pm}_{\alpha}\rangle$ as
a function of single photon drive phase and duration. (f). Calibrating the
amplitude $\epsilon_{z,\pi}$ for a $\sigma_{z}(\pi)$ rotation, which enables
arbitrary $\sigma_{z}(\theta)$ rotations. (g). KCQ Rabi rate as a function of
the square root of the amplitude of our digital control on the squeezing
drive. (h). $\bar{n}_{\mathrm{cat}}$ in the KCQ as a function of the amplitude
of our digital control on the squeezing drive.

The Kerr-refocusing phenomenon, first demonstrated in [21], is the foundation
of the discrete $\sigma_{x}(\pi/2)$ rotation. Under the Kerr-nonlinear
Hamiltonian of the SNAIL mode in its rotating frame
$H=-Ka^{\dagger}a^{\dagger}aa$, the coherent state $|+\alpha\rangle$ in the
SNAIL evolves into the parity-less cat state
$|\mathcal{C}^{-i}_{\alpha}\rangle$ at $t=\pi/K$, and then refocuses into
$|-\alpha\rangle$ at $t=2\pi/K$. We can calibrate this refocusing effect using
the measurement sequence shown in Fig. S 6(a). We first prepare a coherent
state $|\pm\alpha\rangle$ in the SNAIL via adiabatic squeezing of the SNAIL at
$\omega_{S}=2\omega_{a}$ and a projective CQR measurement. Then, we abruptly
switch off the squeezing drive and let the system evolve under Kerr for
different duration $t$, before abruptly switching on the squeezing drive to
“catch” the cat state with the right phase. Lastly, we perform a CQR
measurement to determine whether the KCQ has refocused to its initial state.
As shown in Fig. S 6(b), we observe this refocusing peak at $544$ ns,
corresponding to a Kerr gate duration of $272$ ns. From this experiment, we
extract the Kerr nonlinearity of the SNAIL to be $K/2\pi=0.93$ MHz. A detailed
discussion of the Kerr gate is included in the supplement material of [5].

Our Kerr gate quality is limited by $T_{1}$ errors on the SNAIL. We perform a
master equation simulation where the only dissipation on the system is the
single photon loss on the SNAIL. We plot the result of the simulation on top
of the experimental data in Fig. S 6(b), and we find excellent agreement
between simulation and data, indicating that our Kerr gate fidelity is limited
by $T_{1,a}$ of the SNAIL. To further validate our understanding, we increase
the duration of the Kerr evolution to include five coherent state refocusing
events in Fig. S 6(c) and we still obtain excellent agreement between our
simulation and data.

With this Kerr gate, we can now validate the presence of a KCQ in our SNAIL.
We replicate the experiment first performed in [17], where we prepare the KCQ
in state $|\mathcal{C}^{\pm i}_{\alpha}\rangle$ and then apply a single-photon
drive on the KCQ at $\omega_{S}/2$. When the single photon drive is in phase
with the squeezing drive, it breaks the symmetry of the system and thus lifts
the degeneracy between the states $|+\alpha\rangle$ and $|-\alpha\rangle$,
leading to a Rabi rotation of $\sigma_{z}(\theta)$ around the KCQ Bloch sphere
at a rate

$\Omega_{z}=\mathrm{Re}(4\epsilon_{z}\alpha)$ (7)

to the first order in perturbation theory [5]. Note that the ”Rabi”
terminology comes from the different convention used in [17] for the Bloch
sphere orientation. When the single-photon drive is out of phase with the
squeezing drive, the degeneracy of the potential is restored and the rotation
of the system disappears. We demonstrate this phase-dependent rotation rate
with a pulse sequence shown in Fig. S 6(d). We prepare a parity-less cat state
$|\mathcal{C}^{\pm i}_{\alpha}\rangle$ in the SNAIL via a projective CQR
measurement and a Kerr gate. Then we drive the SNAIL at $\omega_{S}/2$ at
different phases relative to the squeezing drive for time $t$. Afterward, we
read out the KCQ along the $\sigma_{y}$ quadrature by performing a Kerr gate
and a CQR. In the experimental result in Fig. S 6(e), we see the dependence of
the Rabi rate $\Omega_{z}$ on the phase of the single photon drive
$\epsilon_{z}$, proving that we are generating a bona fide KCQ.

Following this demonstration, we use the same pulse sequence to calibrate the
$\sigma_{z}(\pi)$ and $\sigma_{z}(\pi/2)$ rotation pulses. We use Gaussian
pulses ($\sigma$=80 ns, total duration 320 ns) generated by our digital
control for this Rabi drive. In the experiment, we calibrate the amplitude of
the Gaussian pulses by keeping the single photon drive in phase with the
squeezing drive while varying its amplitude $\epsilon_{z}$, before reading out
the KCQ along the $\sigma_{y}$ axis. An example of such a calibration result
is shown in Fig. S 6(f). We fit the result to a cosine function to calibrate
the pulse amplitude $\epsilon_{z,\pi}$ corresponding to a $\sigma_{z}(\pi)$
rotation. We realize arbitrary $\sigma_{z}(\theta)$ rotations by varying this
amplitude. Combined with the $\sigma_{x}(\pi/2)$ rotation, this gives us
universal control on the KCQ.

###  IV.3 Calibrating the size of the cat $\alpha$

As indicated in Eq. 7, the Rabi rate of the KCQ is also a function of the size
of the cat $\alpha$, which we utilize to calibrate the strength of the
squeezing drive [17, 5]. $\alpha$ scales linearly with the digital control
amplitude of $\epsilon_{2}$. If we increase $\epsilon_{2}$ while holding
$\epsilon_{z}$ constant, we can calibrate the scaling factor by fitting for
the slope of the measured KCQ Rabi rates as a function of digital control
amplitude. However, this calibration requires an independent measurement of
$\epsilon_{z}$ beforehand, which is done by measuring the Rabi rate of the
SNAIL under the drive $\epsilon_{z}a^{\dagger}+\epsilon_{z}^{*}a$ with
$\epsilon_{2}=0$. This gives us $\epsilon_{z}/2\pi=0.4$ MHz. With this, we can
calibrate $\epsilon_{2}$ as described. The results are shown in Fig. S 6(g)
where there is a clear linear dependence of the Rabi rate $\Omega_{z}$ on the
square root of $\epsilon_{2}$ at large drive amplitudes. The slope of this
line enables us to calibrate $\bar{n}_{\mathrm{cat}}=|\alpha|^{2}$ as a
function of $\epsilon_{2}$ as is shown in Fig. S 6.

###  IV.4 KCQ spectroscopy

FIG. S 7: KCQ spectroscopy (a). The pulse sequence for KCQ spectroscopy. After
preparing a $|\pm\alpha\rangle$ state in the KCQ via CQR, we probe the
spectrum of the KCQ with a tone that is detuned from $\omega_{a}$ for time
$t$. When the probe is on resonance with a transition, it will transfer some
population to the other well state, which can be resolved using CQR at the end
of the measurement sequence. (b). The result of our KCQ spectroscopy
experiment. The purple lines are the measured spectroscopy data and the green
dashed lines are from diagonalizing the Hamiltonian with parameters
independently characterized in other experiments.

In this section, we perform KCQ spectroscopy following the method introduced
in [5]. This result is useful for performing frequency-selective dissipation
(FSD) on the KCQ in later sections. To perform the spectroscopy, we use the
pulse sequence shown in Fig. S 7(a). We first prepare $|\pm\alpha\rangle$ in
the KCQ by ramping up the squeezing drive and performing a projective CQR
measurement. Then, we send in an $800$ ns spectroscopy probe tone with
variable frequency $\omega_{\mathrm{spec}}$, near half of the squeezing
frequency. If the tone is on resonance with one of the transitions to the
excited states, some of the population will be transferred to the other well
state $|\mp\alpha\rangle$ via the excited state, which leads to a reduced CQR
signal.

The spectrum of our KCQ is in Fig. S 7(b). The spectral lines for the first
two excited states are performed at a smaller probe tone power than the one
used for resolving the second pair of excited states. We stitch the datasets
from these separate measurements together as part of the data post-processing.
This is the same method used by the authors of [5] for their KCQ spectroscopy
measurements. It agrees well with our theoretical expectation (in dashed
lines). As we increase the squeezing drive amplitude, the spectral lines merge
as the states fall inside the wells and become degenerate [5]. Fig. S 7(b)
tells us that there is more than one level inside the wells when
$|\alpha|^{2}\gtrsim 5$.

##  V Coherence characterization

###  V.1 SNAIL coherence characterization

Owing to the small cross-Kerr $\chi_{ab}/2\pi=2.91$ kHz between the SNAIL and
the readout cavity compared with the total linewidth of the readout cavity
$\kappa_{r}/2\pi=0.396$ MHz (Table 1), our dispersive readout of the SNAIL
does not have enough SNR for most of our experiments [22]. Therefore, we are
not able to use the standard dispersive readout methods [23, 24] for
characterizing the SNAIL coherence. Instead, we adopt a method where we
prepare states in the SNAIL, adiabatically ramp up the squeezing drive to map
these states to the KCQ Bloch sphere [17], and perform logical readout via
CQR. Between the state preparation and the logical readout, the state inside
the SNAIL evolves under the dissipation seen by the bare SNAIL, which we use
to exact its coherence.

To measure the single-photon decay time $T_{1}$ of the SNAIL, we prepare the
excited state $|e\rangle$ in the SNAIL using a square pulse of duration
$1.08\mu$s, such that the frequency distribution of the pulse is a sinc
function with a notch at the anharmonicity of the SNAIL. This pulse shaping
minimizes leakage to the $|f\rangle$ state of the SNAIL. Then, we idle the
SNAIL for time $t$, before adiabatically ramping up the squeezing drive for 2
$\mu$s. We pulse-shape the ramp with a tanh function to minimize leakage. This
process maps the YZ plane of the SNAIL qubit Bloch sphere onto the XY plane of
the KCQ Bloch sphere so that energy relaxation of the SNAIL from $|e\rangle$
to $|g\rangle$ is mapped to a parity flip of the KCQ (e.g.,
$|\mathcal{C}^{+}_{\alpha}\rangle$ to $|\mathcal{C}^{-}_{\alpha}\rangle$). To
read out the parity of the KCQ, we perform a $\sigma_{z}(\pi/2)$ rotation and
a Kerr gate, followed up with CQR. Then we fit the parity as a function of $t$
to extract the energy relaxation rate of the SNAIL $T_{1,a}=16.0\pm 0.4$
$\mu$s (see Fig. S 8(a)).

To measure the Ramsey decay time of the SNAIL, we prepare a coherent state in
the KCQ via a projective CQR measurement. Then we adiabatically map the
coherent state $|\pm\alpha\rangle$ of the KCQ to the $|\pm\rangle$ state of
the SNAIL. We then idle the SNAIL for time $t$ before mapping it back onto the
KCQ for logical readout. To measure the Hahn echo decay time, we perform the
same measurement, but we apply a $\pi$ pulse halfway through the idling time
to echo out low-frequency noise. We obtain $T_{2R}=7.2\pm 0.1\mu$s and
$T_{2E}=9.5\pm 0.1\mu$s, from the measurements shown in Fig. S 8(b) and (c),
respectively. In these examples, we use a software detuning to artificially
rotate our measurement axis at $1$ MHz for better fits.

FIG. S 8: Coherence characterizations of the SNAIL and the KCQ at
$\Phi/\Phi_{0}=0.32$. (a), (b) and (c): Measurements of the SNAIL $T_{1,a}$,
$T_{2,a}$ and $T_{2\mathrm{E},a}$, realized by adiabatically mapping the SNAIL
states to the KCQ states and performing CQR. (d) and (e): Measurements of the
KCQ lifetimes $T_{Z}$ and $T_{X,Y}$, corresponding to the coherent states
$|\pm\alpha\rangle$ and cat states $|\mathcal{C}^{\pm}_{\alpha}\rangle$
respectively, using the method in [5]. (f). $T_{Z}$ and $T_{X,Y}$ as functions
of $|\alpha|^{2}$, reproducing the results from [5].

###  V.2 KCQ coherence characterization

To characterize the KCQ coherence at our operating point $\Phi/\Phi_{0}=0.32$,
we use the same method in [5]. We measure the lifetime $T_{Z}$ of the coherent
states $|\pm\alpha\rangle$ by first preparing a coherent state in the KCQ via
a projective CQR measurement. Then we idle the KCQ with a constant squeezing
drive for duration $t$ and perform CQR. We measure the lifetime of the cat
states $T_{X,Y}$ by performing a Kerr gate on the $|\pm\alpha\rangle$ state
after the first CQR measurement to prepare $|\mathcal{C}^{\pm
i}_{\alpha}\rangle$ states in the KCQ. After idling for time $t$ with a
constant squeezing drive, we perform another Kerr gate to rotate the KCQ so
that we are reading it out along the Y axis of the KCQ Bloch sphere. By
fitting the population decay shown in Fig. S 8 (d) and (e), we obtain
$T_{Z}=147.4\pm 3.8\mu$s and $T_{X,Y}=2.32\pm 0.04\mu$s, respectively.

We also investigate $T_{\mathrm{X,Y,Z}}$ as a function of the squeezing drive
strength to replicate a central result of [5]. Essentially, we repeat the
measurements described in the previous paragraph on the KCQ at different
squeezing drive amplitudes. However, there are two complications with
implementing this directly: the Kerr gate fidelity degrades as we increase
$|\alpha|^{2}$ [5], and the CQR needs to be recalibrated for each value of
$|\alpha|^{2}$. To avoid such complications, we perform all the KCQ state
preparation and CQR processes at $|\alpha|^{2}=4$. We then change the
squeezing drive amplitude for the various idling times required for coherence
characterization. Our result in Fig. S 8(f) qualitatively agrees with the one
in Fig.3(c) of [5].

##  VI Conditional displacement and cavity displacement calibrations

###  VI.1 CD calibration

FIG. S 9: Conditional displacement and cavity displacement calibrations (a) CD
calibration result, where we take a cut of the vacuum CF tomography along
$\mathrm{Re}[\beta]$. The blue circles are measured data and the red solid
line is the Gaussian fit. (b) Displacement calibration. After calibrating the
CD, we displace the storage cavity and take cuts along $\mathrm{Re}[\beta]$
and $\mathrm{Im}[\beta]$ of the real and imaginary parts of the CF tomography.
(c) and (d) Theory predictions and measured data for displacements of 1, 1.5,
and 3 (left to right) inside the storage cavity. For each figure, the upper
panel is the real part of the CF, while the lower panel is the imaginary part.

As a first step towards understanding the coupling between a KCQ and a storage
cavity, we need to calibrate the strength of the CD operation. Compared with
other cQED implementations where a transmon is dispersively coupled to a
high-Q cavity, as in [25, 20, 26], our CD is fundamentally different. As
derived earlier, the Hamiltonian of our system, when we are parametrically
driving the CD interaction with a time-dependent pulse while stabilizing the
KCQ, is given by

$H=-Ka^{\dagger 2}a^{2}+\epsilon_{2}a^{\dagger
2}+\epsilon_{2}^{*}a^{2}-\chi_{ab}a^{\dagger}ab^{\dagger}b+g_{\mathrm{BS}}(t)a^{\dagger}b+g^{*}_{\mathrm{BS}}(t)ab^{\dagger},$
(8)

in the regime where $g_{\mathrm{BS}}(t)\ll 4\epsilon_{2}$. We project this
Hamiltonian onto the KCQ subspace by using the projector
$\mathcal{P}_{\mathcal{C}}=|\mathcal{C}^{+}_{\alpha}\rangle\langle\mathcal{C}^{+}_{\alpha}|+|\mathcal{C}^{-}_{\alpha}\rangle\langle\mathcal{C}^{-}_{\alpha}|$
and obtain

$H=\left(g_{\mathrm{CD}}(t)b^{\dagger}+g_{\mathrm{CD}}^{*}(t)b\right)\sigma_{z}-\chi_{ab}\alpha^{2}b^{\dagger}b,$
(9)

where $g_{\mathrm{CD}}=\alpha g_{\mathrm{BS}}$ in the large cat limit where
$\alpha\gg 1$, as discussed in the main text. The second part of Eq. 9 is a
Stark shift of the storage cavity due to photons in the KCQ. This Stark shift
can be used to characterize $\chi_{ab}$ between the two modes, but for now we
focus our attention on the first part of Eq. 9,

$H_{\mathrm{CD}}=\left(g_{\mathrm{CD}}(t)b^{\dagger}+g_{\mathrm{CD}}^{*}(t)b\right)\sigma_{z}.$
(10)

We see that the CD operation in our setup is parametrically activated by
driving a beamsplitter between the SNAIL and the storage, which is
fundamentally different than other cQED implementations [26, 25]. The solution
to the time-dependent Schrödinger equation
$i\hbar\partial_{t}U=H_{\mathrm{CD}}U$ is given by

$U(t)=\exp\left(-\frac{i}{\hbar}\left(\int_{0}^{t}d\tau
g_{\mathrm{CD}}(\tau)a^{\dagger}+\int_{0}^{t}d\tau
g_{\mathrm{CD}}^{*}(\tau)a\right)\sigma_{z}\right),$ (11)

which is the CD unitary
$U_{\mathrm{CD}}=D(-\beta/2)|-\alpha\rangle\langle-\alpha|+D(+\beta/2)|+\alpha\rangle\langle+\alpha|$
where $\beta=2\int_{0}^{t}d\tau g_{\mathrm{CD}}(\tau)$.

We calibrate the strength of our CD operation by measuring the characteristic
function (CF) [27] of the vacuum state
$\mathcal{C}(\beta)=\exp(-1/2|\beta|^{2})$ [25]. Experimentally, we measure
the CF of the ground state of the storage cavity along $\mathrm{Re}(\beta)$
with the KCQ and fit the results to a Gaussian distribution to extract the
calibration scaling factor (Fig. S 9(a)). With $|\alpha|^{2}=4$ and a 348 ns
pulse (including a 24 ns tanh ramp at both the beginning and end), we obtain
$g_{\mathrm{CD}}/2\pi=6.2$ MHz. This method relies on the assumption that the
storage cavity has a negligible thermal population. Excessive storage thermal
population would decrease the width of the Gaussian distribution, leading to a
larger scaling factor and thus introducing systematic errors for subsequent
calibrations.

###  VI.2 Displacement calibration

Having calibrated the amplitude of our CD drive, we can now calibrate the
amplitude of our unconditional displacement drive on the storage cavity. To do
so, we prepare a coherent state of unknown amplitude in the storage cavity by
displacing it from vacuum, perform CF tomography using the KCQ, and fit the
result to its theoretical form given by
$\mathcal{C}(\beta)=\exp(-|\beta|^{2}/2)\exp(\beta\eta^{*}-\eta\beta^{*})$.
The best fit for $\eta$ tells us the unknown amplitude of the displacement. In
our experiment, rather than performing a full tomography on the displaced
state, we sample two cuts along the $\mathrm{Re}[\beta]$ and
$\mathrm{Im}[\beta]$ axis, which suffices to determine $\eta$ robustly.

After calibrating both the CD and displacement drives, we validate our
calibration by preparing coherent states $|\eta=1\rangle,|\eta=1.5\rangle$,
and $|\eta=3\rangle$, and performing full CF tomography. We find excellent
agreement between theory and experiment, shown in Fig. S 9(c) and (d).

###  VI.3 CD rate as a function of cat size

FIG. S 10: The rate of CD as a function of cat size $\mathbf{\alpha}$.

As described earlier, when we drive a beamsplitter between the KCQ and the
storage cavity with strength $g_{\mathrm{BS}}$, we obtain the conditional
displacement Hamiltonian

$H_{\mathrm{CD}}=\left(g_{\mathrm{CD}}(t)b^{\dagger}+g_{\mathrm{CD}}^{*}(t)b\right)\sigma_{z},$
(12)

where $g_{\mathrm{CD}}=\alpha g_{\mathrm{BS}}$. To validate that
$g_{\mathrm{CD}}$ is proportional to cat size $\alpha$, we characterize the CD
strength using the method described above at different cat sizes $\alpha$. The
measurement result in Fig. S 10 displays the predicted linear relationship
between $g_{\mathrm{CD}}$ and $\alpha$, with a slope of $g_{\mathrm{BS}}=1.8$
MHz.

There are two important things to note about this measurement. First, we fix
$T_{\mathrm{CD}}=348$ ns so that one unit of CD can be performed with a
reasonably small fraction of the digital control amplitude range. This is
necessary for our tomography experiments, as we need to use large CDs to
perform full CF tomography. If we were to use $T_{\mathrm{CD}}=100$ ns, we
would expect to obtain $g_{\mathrm{CD}}/2\pi\approx 20$ MHz for
$|\alpha|^{2}=4$. Second, there is a trade-off when going to large $\alpha$s
to achieve a faster CD. As can be seen from the measurement results in Fig. S
8(f), when we increase $\alpha$, $T_{X,Y}$ decreases. This leads to a
reduction in the SNR of CF tomography since the coherence of the KCQ becomes
comparable to the duration of the measurement ($\sim 1\mu$s).

FIG. S 11: Calibrating photon swap between a SNAIL and a storage cavity (a).
The control sequence for calibrating the swap. We first prepare the SNAIL in
its excited state and then drive a beamsplitter interaction between the SNAIL
and the storage at frequency $\omega_{b}-\omega_{a}+\Delta$ for time $t$. We
then map the SNAIL state to the KCQ and perform a logical readout along the
$X$ axis of the KCQ Bloch sphere. (b). The population in the SNAIL as a
function of swap frequency and duration.

##  VII Fock state preparation in the storage cavity

###  VII.1 Swapping a photon between a SNAIL and a storage cavity

As discussed in the main text, to characterize the storage coherence we must
prepare cardinal states of the storage Fock qubit spanned by the $|0\rangle$
and $|1\rangle$ states. To do so, we prepare the corresponding state in the
SNAIL, and then drive a beamsplitter (BS) interaction between the SNAIL and
the storage to swap the single-photon state into the storage cavity. To
calibrate this swap, we use the measurement pulse sequence in Fig. S 11(a).
First, we excite the SNAIL from $|g\rangle$ to $|e\rangle$ with the square
pulse for $t=1088$ ns. Then we drive the beamsplitter interaction with varying
frequency and duration. Next, we adiabatically ramp up the squeezing drive to
map the YZ plane of the SNAIL qubit Bloch sphere onto the XY plane of the KCQ
Bloch sphere. Finally, we read out the KCQ along the X axis via a
$\sigma_{z}(\pi/2)$ rotation and a Kerr gate. The dataset in Fig. S 12(b)
displays the standard chevron-like pattern. We fit the cut at the center of
the pattern to a damped sine wave and extract the duration
$t_{\mathrm{SWAP}}=1.1\mu$s and detuning $\Delta=0.17$ MHz.

FIG. S 12: Storage Fock qubit tomography. (a) and (b) Theoretical prediction
and experimental data of the CF of the cardinal states on the Bloch sphere of
the storage Fock qubit. For each figure, the upper panels are the real part of
the CF and the lower panels are the imaginary part. From left to right, the
states are $|0\rangle$, $|0\rangle+|1\rangle$, $|0\rangle+i|1\rangle$,
$|0\rangle-|1\rangle$, $|0\rangle-i|1\rangle$ and $|1\rangle$. Note that the
contrast of (b) is much lower than that in (a). This is mostly due to the
faulty Kerr gate on the KCQ.

###  VII.2 Storage Fock qubit cardinal state tomography

With a full swap pulse calibrated, we can now prepare all the cardinal states
of the storage Fock qubit and validate this preparation via CF tomography.
This is the first characterization of a non-Gaussian state with a KCQ. Fig. S
12(a) shows the theoretical CFs for all the cardinal states and Fig. S 12(b)
shows our experimental results. These tomography experiments are performed
using the same pulse sequence included in Fig. 2 of the main text. Though the
contrast is limited by our Kerr gate fidelity, we find excellent agreement
between theory and experiment.

###  VII.3 Storage Fock qubit logical Pauli measurement

To measure the coherence of the storage cavity, we measure the decay of the
expectation values of the Pauli operators of the storage Fock qubit $\langle
X(t)\rangle$, $\langle Y(t)\rangle$ and $\langle Z(t)\rangle$. We use a method
adapted from [28] where we project the displacement operator onto the
$|0\rangle$, $|1\rangle$ subspace in order to express these expectation values
in terms of the characteristic function.

The single-mode CF for a bosonic mode is defined as

$C(\beta)=\mathrm{Tr}\left(\rho D(\beta)\right)$ (13)

where $D(\beta)=\exp(\beta a^{\dagger}-\beta^{*}a)$ is the displacement
operator for the storage cavity and $\rho$ is the density matrix of the
storage cavity. The CF is measured using our CD operation [25, 26, 27] as
described in the main text. The logical codewords of the storage Fock qubit
are given by $|+Z\rangle=|0\rangle$ and $|-Z\rangle=|1\rangle$. We define the
Fock qubit subspace projector as $P=|0\rangle\langle 0|+|1\rangle\langle 1|$,
and we project $D(\beta)$ as

$PD(\beta)P=\begin{bmatrix}\langle 0|D(\beta)|0\rangle&\langle
0|D(\beta)|1\rangle\\\ \langle 1|D(\beta)|0\rangle&\langle
1|D(\beta)|1\rangle\end{bmatrix}=e^{-\frac{|\beta|^{2}}{2}}\begin{bmatrix}1&\beta\\\
\beta^{*}&1-|\beta|^{2}\end{bmatrix}$ (14)

where we use

$|\beta\rangle=e^{-\frac{|\beta|^{2}}{2}}\sum^{\infty}_{n=0}\frac{\beta^{n}}{\sqrt{n!}}|n\rangle$
(15)

and

$\langle
1|D(\beta)|1\rangle=e^{-\frac{|\beta|^{2}}{2}}\left(1-|\beta|^{2}\right).$
(16)

We then equate Eq. 14 to the four Pauli matrices, $X$, $Y$, $Z$ and $I$ and
solve for the different $\beta$s. We make the ansatz that $X$ and $Z$ take the
form

$X=aPD(\beta_{1})P+bPD(\beta_{2})P=ae^{-\frac{|\beta_{1}|^{2}}{2}}\begin{bmatrix}a&a\beta_{1}\\\
a\beta_{1}^{*}&a-a|\beta_{1}|^{2}\end{bmatrix}+e^{-\frac{|\beta_{2}|^{2}}{2}}\begin{bmatrix}b&b\beta_{2}\\\
b\beta_{2}^{*}&b-b|\beta_{2}|^{2}\end{bmatrix}=\begin{bmatrix}0&1\\\
1&0\end{bmatrix},$ (17)
$Z=cPD(\beta_{1})P+dPD(\beta_{2})P=e^{-\frac{|\beta_{1}|^{2}}{2}}\begin{bmatrix}c&c\beta_{1}\\\
c\beta_{1}^{*}&c-c|\beta_{1}|^{2}\end{bmatrix}+e^{-\frac{|\beta_{2}|^{2}}{2}}\begin{bmatrix}d&d\beta_{2}\\\
d\beta_{2}^{*}&d-d|\beta_{2}|^{2}\end{bmatrix}=\begin{bmatrix}1&0\\\
0&-1\end{bmatrix}.$ (18)

To solve this, we make an additional ansatz that $\alpha=-\beta$, $a=-b$, and
$c=d$, from which we obtain the solutions

$\begin{split}&\beta_{1}=\sqrt{2}i,\\\ &\beta_{2}=-\sqrt{2}i,\\\
&a=-b=e/2\sqrt{2}i,\\\ &c=d=e/2,\\\ \end{split}$ (19)

where $e$ is Euler’s number. To measure $I$, we only need to measure the CF at
$\beta_{0}=0$, as can be observed from Eq. 14. For $Y$, we need to make a
slightly more complicated ansatz involving three displacements

$\begin{split}Y&=fPD(\beta_{1})P+hPD(\beta_{2})P+jPD(\beta_{3})P\\\
&=e^{-\frac{|\beta_{1}|^{2}}{2}}\begin{bmatrix}f&f\beta_{1}\\\
f\beta_{1}^{*}&f-f|\beta_{1}|^{2}\end{bmatrix}+e^{-\frac{|\beta_{2}|^{2}}{2}}\begin{bmatrix}h&h\beta_{2}\\\
h\beta_{2}^{*}&h-h|\beta_{2}|^{2}\end{bmatrix}+e^{-\frac{|\beta_{3}|^{2}}{2}}\begin{bmatrix}j&j\beta_{3}\\\
j\beta_{3}^{*}&j-j|\beta_{3}|^{2}\end{bmatrix}=\begin{bmatrix}0&-i\\\
i&0\end{bmatrix}\end{split}$ (20)

to cancel out the real part of the off-diagonal terms while preserving the
imaginary part. Here we obtain the solutions

$\begin{split}&\beta_{3}=\sqrt{2},\\\ &f=h=-ei/2\sqrt{2},\\\
&j=ei/\sqrt{2},\\\ \end{split}$ (21)

where $e$ is again Euler’s constant.

All together, we have

$\begin{split}&I=D(0),\\\
&X=\frac{e}{2\sqrt{2}i}\left[D(\sqrt{2}i)-D(-\sqrt{2}i)\right],\\\
&Y=\frac{ie}{\sqrt{2}}\left[D(\sqrt{2})-D(\sqrt{2}i)-D(-\sqrt{2}i)\right],\\\
&Z=\frac{e}{2}\left[D(\sqrt{2}i)+D(-\sqrt{2}i)\right].\\\ \end{split}$ (22)

These equations tell us that we should measure the CF at these four points,
$\beta={0,\sqrt{2},\sqrt{2}i,-\sqrt{2}i}$, in order to reconstruct the density
matrix of the Fock qubit in the storage cavity. This method can also be used
for characterizing cavity coherences for transmon-cavity systems in the weak
dispersive regime.

Data analysis for Fig. 4(b) in the main text Monitoring the expectation value
of the identity $I$ in Eq. 22 is crucial in our experiments, since it gives us
information about changes in the system calibrations and characterization. In
the experiment described in Fig. 4 in the main text, we notice that $\langle
I(t)\rangle$ increases as a function of time when we are performing the FSD.
Through further investigation, we found that the quality of the KCQ
$\sigma_{x}(\pi/2)$ rotation improves as we increase the idling time in the
presence of FSD. For good measure, we performed separate measurements to
confirm that the storage cavity thermal population stays the same while
applying FSD on the KCQ. Critically, this affects the contrast of the CF
tomography, and thereby skews the coherence of the storage cavity extracted
from this tomography data. To get rid of this effect, we divide the measured
Pauli operator expectation values by the measured identity operator
expectation values in our data analysis.

##  VIII SNAIL-cavity cross-Kerr characterization

FIG. S 13: Cavity Stark shift as a function of $|\alpha|^{2}$. (a) and (b) The
Stark shifts of the storage and readout cavities as functions of
$|\alpha|^{2}$, respectively. From the linear fits, we can extract
$\chi_{ab}/2\pi=2.91\pm 0.03$ kHz and $\chi_{ar}/2\pi=1.51\pm 0.01$ kHz.

We calibrate the cross-Kerr between the SNAIL and a cavity mode by measuring
the Stark shift of the cavity at a function of the cat size $\alpha$ in the
KCQ. The Stark shift is given by the second term in Eq. 9, where the frequency
shift of the cavity is linearly proportional to $\alpha$, with the cross-Kerr
as the proportionality constant.

Specifically, for calibrating $\chi_{ab}$ between the storage cavity and the
SNAIL, we measure the rate at which the storage cavity rotates at different
values of $\alpha$. To do this, we repeat the experiment in Fig.3 of the main
text, where we prepare a single photon state in the storage at a certain
$\alpha$ and monitor how fast it is rotating by reconstructing its density
matrix as a function of time. We plot the rotation frequency of the storage
cavity as a function of $|\alpha|^{2}$ in Fig. S 13(a), where the slope gives
us $\chi_{ab}/2\pi=2.91\pm 0.03$ kHz. After obtaining $\chi_{ab}$, we can add
in a software detuning at $\chi_{ab}|\alpha|^{2}$ at different $|\alpha|^{2}$
for the main experiment in Fig.3 of the main text to cancel out the effects of
the deterministic Stark shift for better fit results and better consistency of
experimental settings.

We also use this method to measure $\chi_{ar}$, the cross-Kerr between the
SNAIL and the readout. In this case, we measure the oscillation frequency of
the ringdown of the readout cavity at different values of $|\alpha|^{2}$. For
better measurement results, we detune the readout local oscillator by $2$MHz
for the measurements. The results in Fig. S 13(b) also display a linear
relationship between the readout Stark shift and $|\alpha|^{2}$ with a slope
of $\chi_{ar}/2\pi=1.51\pm 0.01$ kHz.

##  IX SNAIL thermal population measurement

FIG. S 14: SNAIL thermal population measurement via a swap test. (a) The pulse
sequence for measuring the thermal population in the SNAIL. We swap the states
between the SNAIL and the storage cavity after the system reaches thermal
equilibrium and we then perform a CF tomography of the storage along the
$\mathrm{Re}[\beta]$ axis. (b). The measurement results of the CF tomography
without the swap (upper) and with the swap (lower). The relative change in the
width of the Gaussian gives us the thermal population in the SNAIL.

As is discussed in Fig.2 of the main text, we measure the pure dephasing rate
of the storage cavity to be $\Gamma_{\phi}=(5.69\pm 2.84\mathrm{ms})^{-1}$ in
the presence of the SNAIL (at $\alpha=0$). To confirm the hypothesis that this
dephasing is due to SNAIL heating, we design an experiment to measure the
thermal population inside the SNAIL where we take advantage of the existing
calibrations and methods. This measurement consists of swapping the states in
the SNAIL and the storage after reaching thermal equilibrium, and then
performing CF tomography with the KCQ. The width of the Gaussian distribution
of the thermal state tells us the thermal population of the SNAIL. The CF of a
thermal state is given by

$\mathcal{C}(\beta)=\exp(-(n_{\mathrm{th}}+\frac{1}{2})|\beta|^{2}),$ (23)

where the width of the Gaussian is given by
$\sigma_{\mathrm{th}}=1/\sqrt{2n_{\mathrm{th}}+1}$. More thermal population
results in a narrower Gaussian distribution.

The pulse sequence of this measurement is shown in Fig. S 14. It is almost
identical to the experiment shown in Fig.2(a) of the main text, with the
distinction that we do not excite the SNAIL before the swap pulse. The results
in Fig. S 14(b) show the cut of the storage CF along the $\mathrm{Re}[\beta]$
axis without the swap (upper) and with the swap (lower). From these Gaussian
fits, we find that the CF after the swap has a standard deviation that is
$0.027\pm 0.004$ lower than that of the CF without the swap. Assuming
negligible thermal population in the storage cavity, we obtain
$n_{th}=0.028\pm 0.005$ in the SNAIL, giving an effective temperature of $54$
mK. Using the equation from [29]

$\Gamma^{\mathrm{th}}_{\phi}=\frac{n_{\mathrm{th}}\kappa_{1a}\chi_{ab}^{2}}{\kappa_{1a}^{2}+\chi_{ab}^{2}}$
(24)

derived in the limit of $n_{\mathrm{th}}\ll 1$ from [30], where
$\kappa_{1a}=1/T_{1a}$. Plugging in our value for $n_{\mathrm{th}}$ we obtain
$\Gamma_{\phi}=(7.96\pm 1.59\mathrm{ms})^{-1}$, within the errorbar of our
independent measurement of the pure dephasing rate in the main text. The
errorbar of this result is a lot smaller than the one from the previous
method, because this data is much better averaged. This shows that the
dephasing of our storage cavity at $|\alpha|^{2}=0$ is dominated by SNAIL
heating.

##  X System stability measurement

FIG. S 15: System stability measurement. We monitor the coherence of the
system over 5 days to test the stability of our coherences. Most of the
coherences are fluctuating within $10\%$, while the coherence of the storage
cavity is fluctuating within $16\%$.

The main dataset in Fig.3 of the main text took more than three weeks to
complete. We routinely recalibrated the system once a week during this time.
This necessitated us to test the long-term stability of our setup. To this
end, we ran a one-week-long stability measurement of the system where we
monitored the coherences of the bare SNAIL, the KCQ, and the storage cavity
over 110 hours. Each set of coherence measurements takes $\sim$2 hours to
complete and we did not recalibrate the system in the middle of the test. The
measurement result is in Fig. S 15. Each data point for the storage $T_{1}$
and $T_{2}$ is averaged over the same number of shots as the data in Fig.3(b)
of the main text. The coherences of the bare SNAIL ($T_{1a}$ and $T_{2a}$) and
the KCQ ($T_{X}$ and $T_{Z}$) were fluctuating within $10\%$ of their mean
values, whereas the storage $T_{1}$ and $T_{2}$ were fluctuating within $12\%$
and $16\%$, respectively.

##  XI Frequency-selective dissipation on KCQ

###  XI.1 Basic calibration of the FSD

FIG. S 16: Calibrating FSD on the KCQ. (a). The control sequence of the
calibration experiment. After preparing a $|\mathcal{C}^{+}_{\alpha}\rangle$
state in the KCQ with $\alpha=2$, we perform two successive Kerr gates to
induce heating in the KCQ. Then we drive the FSD process with different
frequencies and amplitudes before we measure the excited state population via
CQR. (b). An example of a readout signal of a heated KCQ. The readout blob in
the middle, marked by the dashed circle, is the population in the excited
states above the wells. (c). KCQ excited state population as a function of
cooling amplitude and frequency. The parameters of the point with the lowest
population are adopted for our calibrated FSD pulse.

In this section, we discuss the calibration of the frequency-selective
dissipation (FSD) on the KCQ introduced in the main text. Here, we calibrate
the amplitude and frequency of the pulse used to implement the FSD operation.
To perform this calibration, we artificially introduce heating to the KCQ,
apply the FSD with variable amplitude and detuning, and then perform CQR. We
perform this measurement with $|\alpha|^{2}=4$ for which the first excited
state manifold is outside of the well (see Fig. 3 of the main text), such that
our CQR has a third blob in the center of the I/Q plane correlated with the
excited state population. The goal is to find the FSD parameters that minimize
the excited state population.

The experimental pulse sequence is shown in Fig. S 16(a). At first, we prepare
an even parity cat $|\mathcal{C}^{+}_{\alpha}\rangle$ in the KCQ by
adiabatically ramping up the KCQ from the $|g\rangle$ state of the SNAIL [17].
Then, we artificially heat the SNAIL by performing two consecutive Kerr gates
on the KCQ. As our Kerr gate is mostly limited by single-photon loss on the
SNAIL, there is a significant excited state population after the Kerr gates.
Subsequently, we perform an FSD by turning on the CQR drive for 5 $\mu$s with
variable detuning $\Delta$ and digital control amplitude. We then wait for
$400$ ns for the readout cavity to empty itself before our CQR measurement. An
example histogram of the measurement is shown in Fig. S 16(b), where we
indicated the region corresponding to the excited state population. Our
measured excited state population as a function of detuning and amplitude is
shown in Fig. S 16(c). We find it is minimized at $\Delta=12.5$ MHz, which
agrees with the gap energy independently measured in KCQ spectroscopy in Fig.
S 7, confirming that we are indeed dissipating population from the first pair
of excited states of the KCQ.

###  XI.2 FSD on a large Kerr-cat

FIG. S 17: FSD on a larger cat. (a). Coherence of the storage in the presence
of a $|\alpha|^{2}=7$ KCQ with $T_{2}=226\pm 8\mu$s. (b). Coherence of the
storage cavity when dissipating on the first excited state in the KCQ with
$T_{2}=433\pm 12\mu$s. (c). Coherence of the storage cavity when dissipating
on the second excited state in the KCQ with $T_{2}=334\pm 9$ $\mu$s. The
frequencies of the oscillation are different due to the different Stark shifts
on the storage cavity from the FSD pulse.

We next use the FSD to investigate whether KCQ-induced storage dephasing is
due to heating outside of the well or due to heating to the first excited
state. To this end, we use a larger Kerr-cat with $|\alpha|^{2}=7$ where the
first pair of excited states is inside the wells and the second pair is
outside. We then apply the FSD separately on these levels to see the effects
on storage dephasing. To do so, we make use of the KCQ spectroscopy
experiment, which tells us that for $|\alpha|^{2}=7$ the first pair of excited
states is detuned from the ground state by $20$ MHz while the second pair of
excited states is detuned by $38$ MHz. By setting our FSD pulses at these
detunings, we are able to target these levels separately. Finally, we use
these FSD pulses during the idling time of our storage dephasing measurement
(described in Fig.3 of the main text ).

The results of these measurements are shown in Fig. S 17. In the top panel
(a), we apply no FSD on the KCQ, and we obtain $T_{2}=226\pm 8$ $\mu$s. In the
middle panel (b), we apply FSD to the first pair of excited states inside the
wells, and we obtain $T_{2}=433\pm 12$ $\mu$s. This is in excellent agreement
with $2T_{1}=432\pm 17$ $\mu$s, showing that we have eliminated dephasing to
within the precision of our measurement. In the bottom panel (c), we apply FSD
on the second pair of excited states just outside of the wells, and we obtain
$T_{2}=334\pm 9$ $\mu$s. This indicates that even heating inside the wells to
the first excited state manifold causes storage dephasing, even though these
states are degenerate.

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