# Efficient decoupling of a non-linear qubit mode from its environment

F. Pfeiffer<EMAIL_ADDRESS>Technical University of Munich, TUM
School of Natural Sciences, Department of Physics, Garching 85748, Germany
Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, Garching
85748, Germany  M. Werninghaus  Technical University of Munich, TUM School of
Natural Sciences, Department of Physics, Garching 85748, Germany  Walther-
Meißner-Institut, Bayerische Akademie der Wissenschaften, Garching 85748,
Germany  C. Schweizer  Fakultät für Physik, Ludwig-Maximilians-Universität
München, Schellingstraße 4, D-80799 München, Germany  N. Bruckmoser  Technical
University of Munich, TUM School of Natural Sciences, Department of Physics,
Garching 85748, Germany  Walther-Meißner-Institut, Bayerische Akademie der
Wissenschaften, Garching 85748, Germany  L. Koch  Technical University of
Munich, TUM School of Natural Sciences, Department of Physics, Garching 85748,
Germany  Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften,
Garching 85748, Germany  N. J. Glaser  Technical University of Munich, TUM
School of Natural Sciences, Department of Physics, Garching 85748, Germany
Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, Garching
85748, Germany  G. Huber  Technical University of Munich, TUM School of
Natural Sciences, Department of Physics, Garching 85748, Germany  Walther-
Meißner-Institut, Bayerische Akademie der Wissenschaften, Garching 85748,
Germany  D. Bunch  Technical University of Munich, TUM School of Natural
Sciences, Department of Physics, Garching 85748, Germany  Walther-Meißner-
Institut, Bayerische Akademie der Wissenschaften, Garching 85748, Germany  F.
X. Haslbeck  Technical University of Munich, TUM School of Natural Sciences,
Department of Physics, Garching 85748, Germany  Walther-Meißner-Institut,
Bayerische Akademie der Wissenschaften, Garching 85748, Germany  M. Knudsen
Technical University of Munich,TUM School of Computation, Information and
Technology, Department of Computer Science, Garching 85748, Germany  Walther-
Meißner-Institut, Bayerische Akademie der Wissenschaften, Garching 85748,
Germany  G. Krylov  Technical University of Munich, TUM School of Natural
Sciences, Department of Physics, Garching 85748, Germany  Walther-Meißner-
Institut, Bayerische Akademie der Wissenschaften, Garching 85748, Germany  K.
Liegener  Technical University of Munich, TUM School of Natural Sciences,
Department of Physics, Garching 85748, Germany  Walther-Meißner-Institut,
Bayerische Akademie der Wissenschaften, Garching 85748, Germany  A. Marx
Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, Garching
85748, Germany  L. Richard  Technical University of Munich, TUM School of
Natural Sciences, Department of Physics, Garching 85748, Germany  Walther-
Meißner-Institut, Bayerische Akademie der Wissenschaften, Garching 85748,
Germany  J. Romeiro  Technical University of Munich, TUM School of Natural
Sciences, Department of Physics, Garching 85748, Germany  Walther-Meißner-
Institut, Bayerische Akademie der Wissenschaften, Garching 85748, Germany  F.
Roy  Technical University of Munich, TUM School of Natural Sciences,
Department of Physics, Garching 85748, Germany  Walther-Meißner-Institut,
Bayerische Akademie der Wissenschaften, Garching 85748, Germany  J. Schirk
Technical University of Munich, TUM School of Natural Sciences, Department of
Physics, Garching 85748, Germany  Walther-Meißner-Institut, Bayerische
Akademie der Wissenschaften, Garching 85748, Germany  C. Schneider  Technical
University of Munich, TUM School of Natural Sciences, Department of Physics,
Garching 85748, Germany  Walther-Meißner-Institut, Bayerische Akademie der
Wissenschaften, Garching 85748, Germany  M. Singh  Technical University of
Munich, TUM School of Natural Sciences, Department of Physics, Garching 85748,
Germany  Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften,
Garching 85748, Germany  L. Södergren  Technical University of Munich, TUM
School of Natural Sciences, Department of Physics, Garching 85748, Germany
Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, Garching
85748, Germany  I. Tsitsilin  Technical University of Munich, TUM School of
Natural Sciences, Department of Physics, Garching 85748, Germany  Walther-
Meißner-Institut, Bayerische Akademie der Wissenschaften, Garching 85748,
Germany  F. Wallner  Technical University of Munich, TUM School of Natural
Sciences, Department of Physics, Garching 85748, Germany  Walther-Meißner-
Institut, Bayerische Akademie der Wissenschaften, Garching 85748, Germany  C.
A. Riofrío  BMW Group, Munich, Germany  S. Filipp<EMAIL_ADDRESS>Technical University of Munich, TUM School of Natural Sciences, Department of
Physics, Garching 85748, Germany  Walther-Meißner-Institut, Bayerische
Akademie der Wissenschaften, Garching 85748, Germany

###### Abstract

To control and measure the state of a quantum system it must necessarily be
coupled to external degrees of freedom. This inevitably leads to spontaneous
emission via the Purcell effect, photon-induced dephasing from measurement
back-action, and errors caused by unwanted interactions with nearby quantum
systems. To tackle this fundamental challenge, we make use of the design
flexibility of superconducting quantum circuits to form a multi-mode element –
an artificial molecule – with symmetry-protected modes. The proposed circuit
consists of three superconducting islands coupled to a central island via
Josephson junctions. It exhibits two essential non-linear modes, one of which
is flux-insensitive and used as the protected qubit mode. The second mode is
flux-tunable and serves via a cross-Kerr type coupling as a mediator to
control the dispersive coupling of the qubit mode to the readout resonator. We
demonstrate the Purcell protection of the qubit mode by measuring relaxation
times that are independent of the mediated dispersive coupling. We show that
the coherence of the qubit is not limited by photon-induced dephasing when
detuning the mediator mode from the readout resonator and thereby reducing the
dispersive coupling. The resulting highly protected qubit with tunable
interactions may serve as a basic building block of a scalable quantum
processor architecture, in which qubit decoherence is strongly suppressed.

††preprint: APS/123-QED

##  I  Introduction

The decoupling of quantum systems from their environment is of fundamental
importance for realizing quantum states with long coherence times. However,
for controlling and measuring the system a controlled interaction with the
environment, however weakly, is inevitably required. Atoms in free space, for
example, are decoupled intrinsically well from the radiation field due to
their typically small dipole moment. To manipulate their internal electronic
state, strong mono- or bi-chromatic laser pulses [1, 2] or high-lying Rydberg
states with large dipole moments [3, 4, 5] have to be employed. In contrast,
the interaction of superconducting quantum circuits acting as artificial atoms
can be designed at will [6]. Because of their relatively large coupling
strengths the complete and controlled decoupling from external noise sources
can then become challenging and in large systems for quantum computing, cross-
talk between superconducting qubits is a major hurdle for further scaling [7,
8, 9]. Understanding and circumventing the limits imposed by engineered
couplings to control and measure the systems, in particular as the coherence
times of superconducting qubits continue to improve through advances in
fabrication and qubit design [10, 11, 12, 13] has, therefore, become
imperative.

For the control of superconducting qubits one typically relies on the weak
coupling to control lines and the high signal-to-noise of microwave signal
generators. State readout, however, requires a strong coupling between a qubit
and a meter system, typically a cavity resonator operated in the dispersive
regime, to achieve quantum-limited measurements. Here, an effective non-linear
dispersive interaction
$\propto\hat{a}_{q}^{\dagger}\hat{a}_{q}\hat{a}_{r}^{\dagger}\hat{a}_{r}$
between qubit mode $\hat{a}_{q}$ and resonator mode $\hat{a}_{r}$ originates
from a linear coupling
$\propto\hat{a}_{q}^{\dagger}\hat{a}_{r}+\hat{a}_{q}\hat{a}_{r}^{\dagger}$
when the detuning from the resonator is large compared to the coupling. Since
for an efficient and fast measurement the resonator itself needs to be coupled
to external degrees of freedom, the linear coupling leads to Purcell-induced
decay [14]. Additionally, residual excitations of the readout mode, e.g.,
caused by thermal noise or by un-filtered noise from the control electronics,
lead to dephasing by randomly shifting the qubit frequency through the
dispersive interaction [15, 16, 17, 18, 19, 20].

Protection against photon-induced dephasing and Purcell decay can be achieved
by controlling the dispersive [21] or directly the linear coupling [22, 23].
However, this requires active control of the qubit frequency to change the
detuning of the modes, or active control of the magnitude of the linear
coupling between qubit and resonator. Both methods introduce additional
decoherence channels and, moreover, Purcell-decay cannot be suppressed during
readout, when the effective dispersive coupling is maximized. A further
strategy is to shape the density of environmental states of the qubit via
additional circuit elements acting as filters between the readout resonator
and the transmission line used for probing the system, known as a Purcell
filter [24, 25, 26, 27, 28, 29]. Alternatively, one can exploit symmetry
properties of quantum states by forming a decoherence free subspace based on
the destructive interference of decoherence processes [30]. For example,
collective subradiant states [31], realized using trapped ions [32, 33], cold
atoms [34] or superconducting qubits [35, 36, 37, 38], or particular states of
multi-level atoms [39] can exhibit a suppression of spontaneous emission.
Similarly, superconducting quantum circuits can be used to synthesize
artificial multi-level systems, hosting qubit modes protected from intrinsic
[40] or environmentally induced decoherence [41]. Here, we focus on the
latter, implemented using weakly-anharmonic multi-mode circuits, which have
also been used for efficient entanglement operations [42, 43, 44, 45],
alternative readout schemes [46, 47, 48], two-qubit gates with suppressed
cross-talk [49], as weakly tunable qubit [50] or as sensors for charge
fluctuations [51]. In particular, a two-mode circuit [41, 52, 53, 54], has
been proposed to protect either against Purcell decay or photon-induced
dephasing by controlling the linear or the dispersive coupling to its readout
resonator. However, this two-mode qubit design is not able to simultaneously
protect against both effects. Moreover, by introducing dynamic tunability of
the couplings via flux control, the mode used to encode the qubit becomes
sensitive to flux noise.

To mitigate this effect and allow for simultaneous Purcell and photon-induced
dephasing protection of a flux-insensitive qubit mode, we design and
characterize a three-mode circuit which includes a fixed-frequency qubit mode
that is decoupled from the readout resonator. A second, independently tunable
mode is used to mediate an indirect dispersive coupling through a non-linear
cross-Kerr coupling between the modes. This indirect dispersive shift can be
dynamically controlled by detuning the secondary mode with respect to the
readout resonator. Consequently, the qubit mode can simultaneously exhibit
negligible linear and dispersive coupling with the readout resonator,
protecting it from Purcell decay and photon-induced dephasing while remaining
unaffected by flux noise.

##  II Protected Three-Mode Qubit

The underlying idea for the protected three-mode qubit circuit, shown in Fig.
1(a), is to generate a flux-insensitive qubit mode and a tunable mediator
mode. The qubit mode $\mathcal{A}$ is realized by coupling two superconducting
islands, $1$ and $2$, to a middle island $0$ via Josephson junctions, such
that charges oscillate between the outer islands $1$ and $2$, with the middle
island remaining charge neutral [Fig. 1(c)]. Connecting a third island to the
middle island via a superconducting interference device (SQUID), introduces
two additional flux-tunable modes $\mathcal{B}$ and $\mathcal{C}$. Mode
$\mathcal{B}$ exhibits charge oscillations from islands $1$ and $2$ to island
$3$ and will be used as a tunable mediator. The remaining mode $\mathcal{C}$,
in which charges oscillate between the three outer islands $1$, $2$, $3$ and
the center island, can be shifted to high frequencies with a suitable choice
of parameters.

Figure 1: (a) Circuit diagram with outer islands $1,2,3$ and middle island
$0$. The most relevant capacitances are $C_{12},\;C_{13},\;C_{23}$;
capacitances to the center island $C_{0i}$ and to the ground $C_{gi}$ are not
explicitly depicted. The Josephson energies between outer and center island
are labeled by $E_{Ji}$, with $E_{J3}(\phi_{\rm ext})$ denoting the flux-
dependent Josephson energy of the SQUID loop. (b) Energy level diagram
restricted to the two-excitation subspace of mode $\mathcal{A}$ and
$\mathcal{B}$ relative to the frequency of the readout resonator $\omega_{r}$.
(c) Normal modes under symmetry constraints $C_{13}=C_{23}\neq C_{12}$ and
${E_{J1}=E_{J2}=E_{J3}}$. Fixed frequency mode $\mathcal{A}$ (red) and
frequency tunable modes $\mathcal{B}$ (blue) and $\mathcal{C}$ (green). The
oscillating charge distributions are indicate by positive (’+’) and negative
(’-’) charge, ’$\pm$’ indicates charge neutrality. (d) False-color microscope
image of the qubit sample including the coupler to the readout resonator
(left) and the flux line (top). Islands are labeled as in the circuit diagram
in (a), the close up shows the Josephson junctions and SQUID connected to
island $0$.

Note, that the qubit mode $\mathcal{A}$ remains confined to islands $1$ and
$2$ and since no current passes through the SQUID, this mode is flux-
insensitive. It remains linearly decoupled from the flux-tunable modes when
the islands $1$ and $2$ couple symmetrically to all other circuit elements,
i.e., when both the Josephson energies of the junctions to the middle island
are equal, $E_{J1}$=$E_{J2}$, and the capacitances to island $0$ and $3$ are
identical, $C_{01}=C_{02}$ and $C_{13}=C_{23}$. To lift the degeneracy between
mode $\mathcal{A}$ and $\mathcal{B}$, an asymmetry $C_{13}=C_{23}\neq C_{12}$
is necessary.

If the effective Josephson energy $E_{J3}(\phi_{\rm ext})$ of the SQUID loop
is equal to $E_{J1},E_{J2}$ the modes $\mathcal{B}$ and $\mathcal{C}$ are
decoupled as well, making the modes $\mathcal{A},\mathcal{B},\mathcal{C}$ the
normal modes of the circuit, as depicted in Fig. 1(c). Here, $\phi_{\rm ext}$
denotes the magnetic flux threaded through the loop. The non-linearity of the
Josephson junctions eventually leads to non-linear cross-Kerr couplings
between the modes, which are crucial for mediating the interactions between
the qubit mode $\mathcal{A}$ and its environment via the mediator mode
$\mathcal{B}$.

For a quantitative analysis of the protected three-mode qubit we write its
circuit Hamiltonian

$\displaystyle\hat{H}_{\rm c}=$
$\displaystyle\frac{1}{2}\bm{\hat{q}}^{T}\mathbb{C}^{-1}\bm{\hat{q}}-\sum_{i}E_{Ji}\cos\hat{\phi}_{i}$
(1)

in terms of node fluxes $\hat{\phi}_{i}$ and charge operators $\hat{q}_{i}$.
The energy scales are determined by the Josephson energies
${E_{Ji},\;(i=1,2,3)}$ as well as by the charging energies of the islands and
their cross-couplings included in the capacitance matrix $\bm{C}$. To
guarantee protection from charge noise, we choose to operate each relevant
mode in the transmon regime [55, 41, 51] by choosing Josephson energies which
are large compared to the respective charging energies for each mode (see
Appendix A). Approximating the potential to fourth order and expanding the
Hamiltonian with respect to modes $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$
yields the effective Hamiltonian

$\displaystyle\hat{H}_{\rm q}$
$\displaystyle=\sum_{m}\omega_{m}\hat{a}_{m}^{\dagger}\hat{a}_{m}+\dfrac{\alpha_{m}}{2}\hat{a}_{m}^{\dagger}\hat{a}_{m}^{\dagger}\hat{a}_{m}\hat{a}_{m}$
$\displaystyle+\sum_{m\neq
n}g_{mn}(\hat{a}_{m}^{\dagger}\hat{a}_{n}+\hat{a}_{m}\hat{a}_{n}^{\dagger})+\alpha_{mn}\hat{a}_{m}^{\dagger}\hat{a}_{m}\hat{a}_{n}^{\dagger}\hat{a}_{n},$
(2)

with bosonic operators $\hat{a}_{m}$, mode frequencies $\omega_{m}$, mode-mode
couplings $g_{mn}$, as well as self- and cross-Kerr terms
$\alpha_{m},\alpha_{mn}$ with
$m,n\in\\{\mathcal{A},\mathcal{B},\mathcal{C}\\}$ (see Appendix A).

For small capacitive couplings of the outer islands to the center island $0$,
${C_{0i}\ll C_{ij}},i,j=1,2,3$, mode $\mathcal{C}$ acquires a large charging
energy $\propto{C_{0i}}^{-1}$ and consequently
$\omega_{\mathcal{C}}\gg\omega_{\mathcal{A}},\omega_{\mathcal{B}}$. The two
relevant low-energy modes $\mathcal{A}$ and $\mathcal{B}$ form a V-shaped
energy-level diagram shown in Fig. 1(b) for the two-excitation subspace with
the qubit encoded in the transition
$\ket{0_{\mathcal{A}},0_{\mathcal{B}}}\leftrightarrow\ket{1_{\mathcal{A}},0_{\mathcal{B}}}$.
Transitions between states of different occupation number of the individual
modes are modified by the self-Kerr couplings $\alpha_{m}$, while the cross-
Kerr interaction $\alpha_{\mathcal{A}\mathcal{B}}$ shifts the transitions
$\ket{q_{\mathcal{A}},0_{\mathcal{B}}}\leftrightarrow\ket{q_{\mathcal{A}},1_{\mathcal{B}}}$
dependent on the qubit-state $q\in\\{0,1\\}$.

The exchange-type linear couplings $g_{mn}$ between the modes occur only due
to deviations from the considered circuit symmetries. Introducing an asymmetry
in the capacitances $C_{13}$ and $C_{23}$ leads to a linear coupling
$g_{\mathcal{A}\mathcal{B}}\propto C_{\Delta,3}=C_{13}-C_{23}$ between mode
$\mathcal{A}$ and mode $\mathcal{B}$, which can be used for fast exchange-type
operations between these modes with potential applications for the controlled
coupling of multiple three-mode qubits. In addition, asymmetries in the
Josephson energies, e.g., due to unwanted fabrication or design uncertainties,
result in inductive coupling contributions
$g_{\mathcal{A}\mathcal{B}},g_{\mathcal{A}\mathcal{C}}\propto E_{\delta
J,12}=E_{J1}-E_{J2}$. Inevitably, mode $\mathcal{B}$ and $\mathcal{C}$
interact via a flux-tunable coupling $g_{BC}\propto E_{\Delta
J,123}=2E_{J3}(\phi_{\rm ext})-E_{J1}-E_{J2}$. Note that this has no effect on
the qubit mode $\mathcal{A}$ if it remains decoupled from the modes
$\mathcal{B}$ and $\mathcal{C}$.

##  III Flux-noise in-sensitivity

We realize the protected three-mode qubit in a circular design as shown in
Fig. 1(d). The sample is fabricated on a high-ohmic silicon substrate with
niobium ground planes and Al/AlOx/Al Josephson junctions. The qubit
capacitances are designed to be in the regime $C_{0i}\ll
C_{ij},\;(i,j=1,2,3)$, such that the modes $\mathcal{A}$ and $\mathcal{B}$ lie
at low frequencies around $5~{}\rm{GHz}$ and mode $\mathcal{C}$ is shifted to
a frequency above $20~{}\rm{GHz}$. Because of fabrication and design
asymmetries in the Josephson energies and capacitances, the uncoupled modes
$\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$ hybridize and we refer to theses
dressed modes by $\tilde{\mathcal{A}}$, $\tilde{\mathcal{B}}$,
$\tilde{\mathcal{C}}$. An on-chip flux line controls the magnetic flux
$\phi_{\rm ext}$ threading the SQUID loop and is used to control the mode
$\tilde{\mathcal{B}}$ frequency. For readout and control, the protected three-
mode qubit is capacitively coupled to a readout resonator positioned at
$\omega_{r}/2\pi=6.990~{}\rm{GHz}$ with a coupling
$\kappa/2\pi=1.32(5)~{}\rm{MHz}$ to a transmission line. The measured mode
frequencies, self- and cross-Kerr terms, dispersive shifts, readout fidelities
and single-qubit gate fidelities are summarized in Appendix C.

To characterize the flux dependence of the modes, the spectrum is measured
using two-tone spectroscopy [15] for different flux-bias values $\phi_{\rm
ext}$, as shown in Fig. 2(a).

Figure 2: (a) Two tone spectroscopy ($|S_{21}|$) fitted with eigenenergies
calculated via exact diagonalization (E.D.) of the circuit model (dashed).
Comparison to the effective model with coupled modes (dotted) and bare
($g_{\mathcal{A}\mathcal{B}}=g_{\mathcal{A}\mathcal{C}}=0$) modes (dash-
dotted). The inset shows the pulse scheme for control and readout of the
qubit. After every pulse sequence (S) on the microwave (MW) line, a flux pulse
to $\phi_{\rm ext}=0$ is performed on the flux line prior to readout, to
maximize the readout fidelity. (b) and (c) Flux dependence of $T_{2}$ and
$T_{2}^{*}$ times of mode $\tilde{\mathcal{A}}$ (red) and
$\tilde{\mathcal{B}}$ (blue) at different bias points, compared to the flux
noise limit (dashed). The dotted lines between (a), (b) and (c) indicate the
part of the spectrum at which the corresponding $T_{2},T_{2}^{*}$ times are
measured.

At each point, a flux pulse to $\phi_{\rm ext}=0$ before the readout maximizes
the readout fidelity [Fig. 2(a); inset], as further discussed below in Sec.
IV. Mode $\tilde{\mathcal{A}}$ remains constant with a frequency of
$5.017~{}\rm{GHz}$ at $\phi_{\rm ext}=0$. Mode $\tilde{\mathcal{B}}$ is
tunable in the range between $3.811$ and $6.408~{}\rm{GHz}$ and shows an
avoided crossing with a coupling strength
$g_{\mathcal{A}\mathcal{B}}/2\pi=248~{}\rm{MHz}$ with mode
$\tilde{\mathcal{A}}$. From the measured spectrum, we extract the device
parameters by fitting the experimental data to spectra obtained via exact
diagonalization of the circuit Hamiltonian in Eq. (1) [Fig. 2(a); dashed
lines]. For comparison with the numerical solution, the energy spectrum is
derived from the effective Hamiltonian in Eq. (2) [Fig. 2(a); dotted line]
based on the obtained circuit parameters. While the numeric solution is in
good agreement with the experimental data, the corresponding effective model
slightly deviates. This is expected as the approximation is only exact in the
transmon regime. However, the effective model can explain the weak flux
dependence of mode $\tilde{\mathcal{A}}$ resulting from the mode-mode coupling
$g_{\mathcal{A}\mathcal{B}}$, which leads to hybridization with the flux-
tunable mode $\mathcal{B}$. The spectrum of the effective Hamiltonian with the
couplings $g_{\mathcal{A}\mathcal{B}}$ and $g_{\mathcal{A}\mathcal{C}}$ set to
zero, shows the flux-independent nature of the bare mode $\mathcal{A}$
[dotted-dashed lines in Fig. 2 (a)].

To demonstrate the flux-noise insensitivity of the dressed mode
$\tilde{\mathcal{A}}$, we compare mode $\tilde{\mathcal{A}}$ and
$\tilde{\mathcal{B}}$ by measuring both the $T_{2}$ and the $T_{2}^{*}$
coherence time against the applied external flux $\phi_{\rm ext}$ using
standard spin-echo and Ramsey pulse sequences. Here,
$T_{2}^{(*)}=(\Gamma_{1}/2+\Gamma_{\phi})^{-1}$ has contributions from
spontaneous decay $\Gamma_{1}=1/T_{1}$ and pure dephasing $\Gamma_{\phi}$. The
flux-noise contribution to $\Gamma_{\phi}$ is proportional to
$|\frac{\partial\omega_{m}}{\partial\phi_{\rm ext}}|$ [55] and thus minimal
flux noise-sensitivity is expected at flux bias points at which the derivative
vanishes, i.e. $\phi_{\rm ext}=0$ and $\phi_{\rm ext}=\phi_{0}/2$, and we
therefore consider only adjacent regions. As shown in Fig. 2(b), mode
$\tilde{\mathcal{B}}$ has an inherent strong flux dependence, so its coherence
degrades rapidly between $\phi_{\rm ext}=0$ and $\phi_{\rm
ext}=0.15~{}\phi_{0}$ due to flux noise. For mode $\tilde{\mathcal{A}}$, we
expect a constant $T_{2}^{*}$ time when mode $\tilde{\mathcal{B}}$ is
sufficiently detuned from mode $\tilde{\mathcal{A}}$. The observed slow
decrease in $T_{2}^{*}$ between $\phi_{\rm ext}=0$ and $\phi_{\rm
ext}=0.2\phi_{0}$ is predominantly caused by an increased hybridization with
the tunable mode $\tilde{\mathcal{B}}$. This effect is enhanced around
$\phi_{\rm ext}=\phi_{0}/2$ as mode $\tilde{\mathcal{B}}$ exhibits stronger
flux tuning. Additionally, at $\phi_{\rm ext}=\phi_{0}/2$, the $T_{2}^{*}$
time of mode $\tilde{\mathcal{A}}$ is affected by charge noise. This is
because mode $\tilde{\mathcal{B}}$, which hybridizes with mode
$\tilde{\mathcal{A}}$, is not completely in the transmon regime.

The $T_{2}^{*}$-times are compared to the flux noise limit, which is
approximated using the dephasing rate
$\Gamma_{\phi}=\sqrt{A_{\phi}}\cdot|\frac{\partial\omega_{m}}{\partial\phi_{\rm
ext}}|$ [55]. Here, a $1/f$-like power spectral density $S_{\phi}(\omega)=2\pi
A_{\phi}/|\omega|$ is assumed with a fitted flux noise amplitude
$A_{\phi}=1.69(8)\;\mu\phi_{0}$ for a designed SQUID loop size of
$210~{}\rm\mu m^{2}$. The obtained flux noise amplitude $A_{\phi}$ is
consistent with observed values for superconducting qubits [56, 57, 58].

Both charge and flux-noise typically exhibit a $1/f$-like power spectral
density and can therefore be mitigated using a spin-echo pulse sequence. This
becomes evident in the measured $T_{2}$ times [Fig. 2(b)], which are much
longer than the respective $T_{2}^{*}$ times at all flux points where charge-
or flux-noise is dominant.

Counter-intuitively, the measured $T_{2}$-time of mode $\tilde{\mathcal{A}}$
increases when moving away from the flux-insensitive point $\phi_{\rm ext}=0$,
opposed to the corresponding $T_{2}^{*}$ times. This is caused by the
dispersive interaction with the readout resonator, which is controlled by the
detuning of mode $\tilde{\mathcal{B}}$ from the resonator. At the point
$\phi_{\rm ext}=0$, the interaction strength is maximal and thus photon-
induced dephasing limits the $T_{2}$-time. Reducing the dispersive
interaction, by moving mode $\tilde{\mathcal{B}}$ away from the resonator,
mitigates this effect and consequently leads to an increase in the
$T_{2}$-time. At $\phi_{\rm ext}=\phi_{0}/2$, where the longest $T_{2}$-times
are measured, the dispersive interaction is further suppressed, leading to
protection from photon-induced dephasing, as discussed in the following.

##  IV Protection against Purcell-decay and photon-induced dephasing

Aside from supporting a flux-insensitive qubit mode, a further advantage of
the protected three-mode qubit becomes evident when considering the coupling
to other elements. Here we focus on the cross-Kerr mediated interaction
between the qubit mode and a readout resonator. A direct linear exchange-type
coupling $\propto\hat{a}_{q}^{\dagger}\hat{a}_{r}+h.c.$ would lead to Purcell-
enhanced spontaneous emission. In contrast, a cross-Kerr type interaction with
an intermediate mode which itself is coupled linearly to the readout resonator
can protect the qubit mode from this effect [41]. The mediator mode is able to
control a dispersive interaction between the resonator and the qubit mode,
which remains linearly decoupled and, therefore, protected against Purcell-
decay. In addition, by reducing the dispersive interaction when detuning the
mediator mode from the resonator frequency, photon-induced dephasing can be
suppressed at the same time.

In general, the capacitive coupling of the protected three-mode qubit to a
readout resonator results in the Hamiltonian

$\displaystyle\hat{H}_{\rm qr}=\hat{H}_{\rm
q}+\omega_{r}\hat{a}_{r}^{\dagger}\hat{a}_{r}+\sum_{m}g_{m}(\hat{a}^{\dagger}_{r}\hat{a}_{m}+\hat{a}_{r}\hat{a}_{m}^{\dagger}).$
(3)

Here,

$\displaystyle
g_{m}=\sqrt{\frac{\hbar}{2Z_{r}}}\bra{0}\sum_{i}(\bm{C}^{-1})_{ir}\hat{q}_{i}\ket{1_{m}}$
(4)

is the linear coupling between the circuit modes
($m\in\\{\mathcal{A},\mathcal{B},\mathcal{C}\\}$) and the resonator with an
impedance $Z_{r}$. The coupling $g_{m}$ depends not only on the capacitive
coupling matrix $\bm{C}$, but also by the matrix element
$\bra{0}\hat{q}_{i}\ket{1_{m}}$ on the charge distribution of the modes on the
islands $i$. The characteristic charge distribution of the individual modes
indicated in Fig. 1(c) can be exploited to engineer mode-selective coupling to
the readout resonator. Coupling capacitively only to circuit island $3$
results in a decoupling of the readout resonator from mode $\mathcal{A}$. In
this case the island $3$ remains charge neutral for mode $\mathcal{A}$, and
the expectation value
$g_{\mathcal{A}}\propto\bra{0}\hat{q}_{3}\ket{1_{\mathcal{A}}}$ is zero, while
$g_{\mathcal{B}}$ remains finite. Note that the same effect can be achieved by
symmetrically coupling to islands $1$ and $2$.

In the dispersive regime $|\Delta_{m}|=|\omega_{m}-\omega_{r}|\gg g_{m}$ and
for single circuit-mode excitations the interaction of the circuit modes with
the resonator is described by the dispersive Hamiltonian [43]

$\displaystyle\hat{H}_{\rm disp.}=\hat{H}_{\rm
q}+\omega_{r}\hat{a}^{\dagger}_{r}\hat{a}_{r}+\sum_{m}2\chi_{m}\hat{a}^{\dagger}_{r}\hat{a}_{r}\hat{a}^{\dagger}_{m}\hat{a}_{m},$
(5)

with the dispersive shifts $\chi_{m}$ (see Appendix B). For the qubit mode
$\mathcal{A}$ the dispersive shift is given by

$\displaystyle\chi_{\mathcal{A}}=$ $\displaystyle
g_{\mathcal{A}}^{2}\left(\frac{1}{\Delta_{\mathcal{A}}}-\frac{1}{\Delta_{\mathcal{A}}+\alpha_{\mathcal{A}}}\right)+\frac{g_{\mathcal{B}}^{2}}{2}\left(\frac{1}{\Delta_{\mathcal{B}}}-\frac{1}{\Delta_{\mathcal{B}}+\alpha_{\mathcal{A}\mathcal{B}}}\right).$
(6)

The first term in Eq. (6) is the typical _direct_ dispersive shift due to the
linear coupling $g_{\mathcal{A}}$. The second term in Eq. (6) describes the
_indirect_ dispersive shift mediated by coupling mode $\mathcal{B}$ and its
cross-Kerr type interaction $\alpha_{\mathcal{A}\mathcal{B}}$ with the qubit
mode. It is caused by virtual transitions between states with one excitation
either in the resonator or the coupling mode,
$\ket{1_{A},0_{B},n=1}\leftrightarrow\ket{1_{A},1_{B},n=0}$ in second-order
perturbation theory [59]. Here, $n$ denotes the resonator photon number. Since
the high-frequency $\mathcal{C}$-mode is far detuned its contribution can be
neglected. In the case of a mode-selective coupling with $g_{\mathcal{A}}=0$
and $g_{\mathcal{B}}\neq 0$, the dispersive interaction $\chi_{\mathcal{A}}$
can be non-zero, mediated exclusively through mode $\mathcal{B}$ and the non-
linear cross-Kerr term $\alpha_{\mathcal{A}\mathcal{B}}$ between the modes.

The advantage of our protected three-mode qubit design is that
$\chi_{\mathcal{A}}$ can be controlled via the independently tunable mode
$\mathcal{B}$, while the linear coupling $g_{\mathcal{A}}$ remains suppressed.
This is in contrast to the tunable coupling qubit in Refs. [41, 52, 53, 54],
in which the dispersive interaction cannot be modified without changing the
linear coupling. The protected three-mode qubit can therefore be operated in
two different regimes of $\chi_{\mathcal{A}}$: the readout regime, where
$\chi_{\mathcal{A}}$ is maximal, and the control regime with minimal
$\chi_{\mathcal{A}}$ and protection against photon-induced dephasing. In both
regimes the qubit mode $\mathcal{A}$ remains Purcell protected and insensitive
to flux noise.

We realize the above regimes by capacitively coupling the qubit to the readout
resonator mainly via circuit island $3$ to minimize the linear coupling
$g_{\mathcal{A}}$ and to suppress the indirect component of
$\chi_{\mathcal{A}}$. While the coupling $g_{\mathcal{A}}$ can be completely
eliminated in the design, we set a weak coupling to allow for qubit control
through the readout resonator [60]. The dispersive interaction strength is
maximal at the flux-bias value $\phi_{\rm ext}=0$ where the detuning of the
mode $\tilde{\mathcal{B}}$ from the resonator is minimal. In contrast, for
$\phi_{\rm ext}=\phi_{0}/2$ mode $\tilde{\mathcal{B}}$ is maximally detuned
from the resonator and the dispersive interaction is minimized. In the
following, we refer to these operation points as readout point $\phi_{\rm r}$
($\phi_{\rm ext}=0$) and control point $\phi_{\rm c}$ ($\phi_{\rm
ext}=\phi_{0}/2$), respectively. Randomized benchmarking of single-qubit gates
consisting of $100~{}{\rm ns}$ DRAG pulses [61, 62] yields fidelities of
$\mathcal{F}_{\rm RB}>99.9\%$ at both operating points for the qubit mode
$\tilde{\mathcal{A}}$. Moreover, the readout fidelity remains consistently
high for a $2~{}\mu{\rm s}$ readout pulse when pulsing to $\phi_{\text{r}}$
immediately before the readout operation, reducing from $96.3\%$ at
$\phi_{\text{r}}$ to $94.1\%$ at $\phi_{\text{c}}$, cf. Appendix C.

To confirm the predicted tunability of the couplings, the dispersive shifts of
the two modes are measured as a function of applied flux, as shown in Fig.
3(a). At each flux point, the dispersive shift is determined by measuring the
shift of the resonator frequency with the qubit prepared in its ground state
or excited state. The error bars, estimated from the fit of the resonator
frequency shift, are comparable to the size of the data points.

Figure 3: (a) Dispersive interaction
$\chi_{\tilde{\mathcal{A}}},\chi_{\tilde{\mathcal{B}}}$ of modes
$\tilde{\mathcal{A}}$ and $\tilde{\mathcal{B}}$ as a function of reduced
external flux $\phi_{\rm ext}/\phi_{0}$. Comparison with sum (solid) of direct
(dotted) and indirect (dashed) contributions to the dispersive shifts
$\chi_{m}$ ($m=\tilde{\mathcal{A}},\tilde{\mathcal{B}}$) calculated via Eq.
(25). The arrows indicate the readout point $\phi_{\rm r}$ and control point
$\phi_{\rm c}$. (b) $T_{2}$ of mode $\tilde{\mathcal{A}}$ vs. noise photon
number at different flux points and comparison to the dephasing limit
$T_{2,\rm limit}$ (dash-dotted line). (c) $T_{1}$ of both modes vs. resonator
detuning $\Delta_{\tilde{\mathcal{B}}}$ of mode $\tilde{\mathcal{B}}$ and
comparison with the corresponding Purcell limit (dashed line). The large dots
represent the median of the individual $T_{1}$ measurements (small dots).
Arrows indicate the flux-bias points that correspond to the realized detunings
$\Delta_{\tilde{\mathcal{B}}}$.

Since the transmon approximation is not fully valid over the whole tuning
range of mode $\tilde{\mathcal{B}}$, Eq. (6) is adapted to model the data, see
Eq. (25) in Appendix B. The dispersive shifts calculated via equation (25) and
the experimental data show good agreement. Separating the equation into its
direct and indirect part, reveals that the dispersive shift of mode
$\tilde{\mathcal{B}}$ is dominated by the direct contribution. In contrast,
the small linear coupling $g_{\tilde{\mathcal{A}}}$ and the large detuning of
mode $\tilde{\mathcal{A}}$ to the readout resonator suppress the direct
dispersive shift contribution. The indirect dispersive shift remains as the
dominant contribution. At the readout point $\phi_{\rm r}$,
$\chi_{\tilde{\mathcal{A}}}(\phi_{\rm r})/2\pi=-0.32(1)~{}\rm{MHz}$ is maximal
providing the highest readout fidelity (see Appendix C). At the control point
$\phi_{\rm c}$, $\chi_{\tilde{\mathcal{A}}}(\phi_{\rm
c})/2\pi=-19(4)~{}\rm{kHz}$ is strongly suppressed, thus protecting mode
$\tilde{\mathcal{A}}$ from photon-induced dephasing through residual
population of the readout resonator.

To demonstrate the protection against photon-induced dephasing at the control
point $\phi_{\rm c}$, we use a broadband noise source, see Appendix E, to
artificially inject photons into the resonator and track $T_{2}$ at the flux
bias points $\phi_{\rm r}$ and $\phi_{\rm c}$, as shown in Fig. 3(b). The
average noise photon number is calculated via AC Stark shift measurements at
the readout point $\phi_{\rm r}$, $\langle
n\rangle=(\omega_{\tilde{\mathcal{A}}}-\omega_{\tilde{\mathcal{A}},\rm
noise})/(2\chi_{\tilde{\mathcal{A}}})$ with $\omega_{\tilde{\mathcal{A}},\rm
noise}$ denoting the shifted frequency in presence of the additional noise
photons. At this point, an increase in noise photons leads to a decrease in
$T_{2}$ according to the photon-induced dephasing rate [17, 16]

$\displaystyle\Gamma_{\phi}=\frac{\kappa}{2}\mathrm{Re}\left[\sqrt{\left(1+\frac{2i\chi_{\tilde{\mathcal{A}}}}{\kappa}\right)^{2}+\frac{8i\chi_{\tilde{\mathcal{A}}}n_{\rm
th}}{\kappa}}-1\right],$ (7)

which limits the coherence to $T_{2,\rm
limit}=(\Gamma_{1}/2+\Gamma_{\phi})^{-1}$. For the evaluation of
$\Gamma_{\phi}$, we assume an initial thermal population $n_{\rm
initial}=0.005$, i.e. the total photon number is given by $n_{th}=n_{\rm
initial}+\langle n\rangle$. The resulting estimated $T_{2,\rm limit}$ agrees
well with the measured values. In contrast, at the control point $\phi_{\rm
c}$ the dispersive shift $\chi_{\tilde{\mathcal{A}}}/2\pi=-19(4)$ kHz is
strongly suppressed. Consequently, $T_{2}$ remains constant over all measured
noise photon numbers, thus showcasing the protection against shot noise with
an estimated pure dephasing limit of
$T_{2,\phi}=1/\Gamma_{\phi}=29(12)~{}\rm{ms}$.

To demonstrate the Purcell protection of mode $\tilde{\mathcal{A}}$, we
measure its $T_{1}$-time and compare it to mode $\tilde{\mathcal{B}}$ around
the readout point $\phi_{\rm r}$ for different values of the detuning
$\Delta_{\tilde{\mathcal{B}}}$ of $\tilde{\mathcal{B}}$. Fifty $T_{1}$
measurements are performed at each value of $\Delta_{\tilde{\mathcal{B}}}$ to
capture fluctuations in $T_{1}$ that are typically observed in superconducting
qubits [63, 64, 65]. It is assumed that the $T_{1}$ fluctuations originate
from fluctuating two-level system defects, which we also expected to be the
limiting factor for the measured $T_{1}$-times. A comparison of both modes
along with a Purcell-limit estimate of mode $\tilde{\mathcal{B}}$ given by
$T_{1,\rm
Purcell}=1/\kappa(\Delta_{\tilde{\mathcal{B}}}/g_{\tilde{\mathcal{B}}})^{2}$
is shown in Fig. 3(c). Mode $\tilde{\mathcal{B}}$ is strongly Purcell-limited
at the readout point, with increasing detuning the $T_{1}$-time increases
according to the Purcell limit. Deviations from the Purcell limit for larger
detunings indicate the presence of another dominating loss channel. In
contrast, mode $\tilde{\mathcal{A}}$ maintains a constant $T_{1}$-time,
indicating that Purcell decay into the readout mode is strongly suppressed.

To estimate the Purcell limit on mode $\tilde{\mathcal{A}}$ imposed by the
linear couplings $g_{\tilde{\mathcal{A}}}$, we use
$g_{\tilde{\mathcal{A}}}/2\pi=23.4(2)$ MHz at $\phi_{\rm r}$ and
$g_{\tilde{\mathcal{A}}}/2\pi=6.5(1)$ MHz at $\phi_{\rm c}$, estimated from
circuit parameters extracted from the measured energy spectrum [Fig. 2(a)] and
dispersive shifts [Fig. 3(a)], see Appendix B. We find corresponding Purcell
limits $T_{1,\rm
Purcell}=1/\kappa(\Delta_{\tilde{\mathcal{A}}}/g_{\tilde{\mathcal{A}}})^{2}$
of $855(12)~{}\mu$s and $10.1(3)~{}$ms, respectively, well beyond the measured
$T_{1}$-times. While ideally $g_{\tilde{\mathcal{A}}}$ is constant for varying
flux bias values, the coupling between the modes in our device leads to a
residual flux dependence. In fact, the large coupling
$g_{\mathcal{B}\mathcal{C}}\propto E_{\Delta J,123}$ between modes
$\mathcal{B}$ and $\mathcal{C}$, which is estimated to vary between
$g_{\mathcal{B}\mathcal{C}}/2\pi=1.144\;\rm{GHz}$ at $\phi_{\rm r}$ and
$g_{\mathcal{B}\mathcal{C}}/2\pi=-2.567\;\rm{GHz}$ at $\phi_{\rm c}$,
indirectly influences mode $\tilde{\mathcal{A}}$ because of its hybridization
with mode $\mathcal{B}$.

##  V Conclusion and Outlook

We have introduced the protected three-mode qubit, a multi-mode circuit which
accommodates a qubit mode $\mathcal{A}$ that is insensitive to flux noise and
protected from resonator-induced Purcell decay and dephasing caused by
residual noise photons. A key element in the realized multi-mode circuit is
the formation of a qubit mode that is by design independent of external
magnetic fields. A second ingredient is the hierarchy of couplings between the
modes. The qubit mode couples ideally only via a cross-Kerr type interaction
to a second circuit mode $\mathcal{B}$, acting as a mediator mode to other
circuit elements via a linear exchange-type interaction. When coupling to a
readout-resonator, the dispersive shift of the qubit mode and, therefore, the
strength of the measurement-induced back-action can be adjusted via the
independently tunable mediator mode without affecting the $T_{1}$ time of the
qubit.

Both Purcell protection and the flux noise-insensitivity are fundamentally
limited by circuit asymmetries and the resulting hybridization of mode
$\mathcal{A}$ and $\mathcal{B}$. In the present device a designed capacitance
asymmetry $C_{13}\neq C_{23}$ and fabrication uncertainty-caused Josephson
junction asymmetries of $d=(E_{J1}-E_{J2})/(E_{J1}+E_{J2})=7\%$ lead to an
enhanced mode-mode coupling $g_{\mathcal{A}\mathcal{B}}$. Assuming that a
symmetric capacitance $C_{13}=C_{23}$ can be fabricated with high precision
and that junction asymmetries can be controlled in future devices via an
improved junction fabrication process [66] or by post-fabrication adjustments
of the junction parameters [67], the mode-mode coupling can be reduced to
$g_{\mathcal{A}\mathcal{B}}/2\pi=15~{}\textrm{MHz}$, assuming a spread of
$d=1\%$. Furthermore, the observed sensitivity to charge noise at the control
point $\phi_{\rm c}$ can be eliminated in future designs by ensuring the
transmon regime over the entire tuning range of mode $\tilde{\mathcal{B}}$.

The readout fidelity can be further improved by increasing the coupling
$\kappa$ between the resonator and the transmission line. This requires a
larger dispersive interaction, which can be achieved by a smaller detuning of
the mediator mode with respect to the readout resonator. Alternatively, the
cross-Kerr coupling between qubit and mediator mode can be used directly by
resonantly coupling the mediator mode to the readout mode [46, 47, 48]. This
generates large qubit state-dependent shifts while maintaining the quantum
non-demolition character of the measurement.

Figure 4: Circuit diagram sketch of two protected three-mode qubits coupled to
each other and to their individual readout resonator, which is coupled to a
transmission line. The qubit modes (red) are decoupled from each other and
their respective readout resonator. The mediator modes (blue) are used to
mediate couplings to the corresponding readout resonator and between the two
protected-three mode qubits via a mode-selective capacitive coupling.

Similar to the demonstrated tunable indirect dispersive coupling to the
readout resonator, the mediator mode $\mathcal{B}$ has the potential to be
used for coupling multiple protected three-mode qubits. When coupled to one or
more neighbouring protected three-mode qubits, residual interactions between
the respective qubit modes can be largely suppressed without the need of
further cancellation strategies or passive ZZ cancellation via additional
coupling paths [68, 69, 70]. Mode-selective linear coupling between mediator
modes of adjacent three-mode qubits, as illustrated in Fig. 4, can be utilized
for indirect coupling between protected qubit modes either via linear
exchange-type or non-linear cross-Kerr-type interactions with the mediator
modes. Using non-linear couplings only, has the advantage of suppressing
spurious cross-talk, originating from stray linear couplings [71, 72]. This
has been demonstrated in Ref. [49] for two-mode qubits coupled to a common bus
resonator. However, the employed RIP gate [73] is based on the weak indirect
dispersive interaction with the bus resonator, which leads to slow gate times.
Instead, coupled mediator modes $\mathcal{B}$ of adjacent protected three-mode
qubits could act as a common element to which the corresponding protected
qubit modes $\mathcal{A}$ non-linearly couple via the cross-Kerr interaction
$\alpha_{\mathcal{A}\mathcal{B}}$. Since the cross-Kerr interaction is much
larger than the dispersive couplings, this may offer the potential for fast
high-fidelity two-qubit gates with suppressed cross-talk.

In summary, with the robustness to coupling-induced incoherent and coherent
errors, the protected three-mode qubit is an ideal building block for a
quantum processor architecture that retains the performance of a single qubit
at large scale. The investigation of architectures with multiple protected
qubits and the demonstration of two-qubit gates between the protected modes is
a promising future research direction.

##  VI Acknowledgements

We thank Peter Rabl for critically reading the manuscript. This research was
funded by the BMW Group. We acknowledge financial support from the German
Federal Ministry of Education and Research via the funding program ’Quantum
technologies - from basic research to the market’ under contract number
13N15680 (GeQCoS) and under contract number 13N16188 (MUNIQC-SC), by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via project
number FI2549/1-1 and via the Germany’s Excellence Strategy EXC-2111-390814868
‘MCQST’ as well as by the European Union with the project grants No. 765267
(QuSCo), 955479 (MOQS) and 847471 (QUSTEC). The research is part of the Munich
Quantum Valley, which is supported by the Bavarian state government with funds
from the Hightech Agenda Bayern Plus. C.S. has received funding from the
European Union’s Framework Programme for Research and Innovation Horizon 2020
(2014–2020) under the Marie Sklodowska-Curie Grant Agreement No. 754388
(LMUResearchFellows) and from LMUexcellent.

##  Appendix A Effective model

The Lagrangian of the circuit depicted in Fig. 1(a) is given by

$\displaystyle\hat{\mathcal{L}}=$
$\displaystyle\frac{1}{2}\dot{\bm{\phi}}^{T}\mathbb{C}\dot{\bm{\phi}}-\sum_{i}E_{Ji}\cos\hat{\phi}_{i}$
(8) $\displaystyle=$
$\displaystyle\frac{1}{2}\dot{\bm{\phi}}^{T}\mathbb{C}\dot{\bm{\phi}}+\frac{1}{2}\bm{\phi}^{T}\mathbb{J}\bm{\phi}-\sum_{i}\frac{E_{J,i}}{4!}\hat{\phi}_{i}^{4}+\mathcal{O}(\hat{\phi}_{i}^{6}),$
(9)

with $i\in\\{1,2,3\\}$ and the capacitance matrix $\mathbb{C}$ given by

$\displaystyle\begin{pmatrix}C_{12}+C_{13}+C_{01}&-C_{12}&-C_{13}\\\
-C_{12}&C_{12}+C_{23}+C_{02}&-C_{23}\\\
-C_{13}&-C_{23}&C_{13}+C_{23}+C_{03}\end{pmatrix}.$ (10)

Here $C_{ij}$ are renormalized capacitances when taking the ground capacitance
of each island into account and eliminating the static mode. For simplicity,
we set $C_{0i}=C_{0}$ for all $i\in\\{1,2,3\\}$ in the following discussion.
Under the symmetry constraints $C_{13}=C_{23}\neq C_{12}$ and
$E_{J1}=E_{J2}=E_{J3}$ the normal modes of the linear Lagrangian, as
illustrated in Fig. 1(c), are given in terms of the node variables by

$\displaystyle\bm{\theta}=\mathbb{T}\cdot\bm{\phi}=\begin{pmatrix}-1/\sqrt{2}&1/\sqrt{2}&0\\\
-1/\sqrt{6}&-1/\sqrt{6}&2/\sqrt{6}\\\
1/\sqrt{3}&1/\sqrt{3}&1/\sqrt{3}\end{pmatrix}\cdot\bm{\phi},$ (11)

with the orthonormal transformation matrix $\mathbb{T}$. The transformed
Lagrangian reads

$\displaystyle\hat{\mathcal{L}}=\frac{1}{2}\dot{\bm{\theta}}^{T}\bm{\tilde{C}}\dot{\bm{\theta}}+\frac{1}{2}\bm{\theta}^{T}\bm{\tilde{J}}\bm{\theta}-\sum_{i}E_{J,i}(\sum_{j}T_{ji}\hat{\theta}_{j})^{4}+\mathcal{O}(\hat{\phi}_{i}^{6}),$
(12)

with the transformed capacitance

$\displaystyle\bm{\tilde{C}}$ $\displaystyle=\bm{T}\cdot\bm{C}\cdot\bm{T}^{T}$
$\displaystyle=\begin{pmatrix}\frac{1}{2}(2C_{0}+4C_{12}+C_{\Sigma,3})&\frac{\sqrt{3}}{2}C_{\Delta,3}&0\\\
\frac{\sqrt{3}}{2}C_{\Delta,3}&C_{0}+\frac{3}{2}C_{\Sigma,3}&0\\\
0&0&C_{0}\end{pmatrix}$ (13)

using $C_{\Delta,3}=C_{13}-C_{23}$ and $C_{\Sigma,3}=C_{13}+C_{23}$ and the
Josephson matrix

$\displaystyle\bm{\tilde{J}}$ $\displaystyle=\bm{T}\cdot\bm{J}\cdot\bm{T}^{T}$
$\displaystyle=\\!\begin{pmatrix}\frac{1}{2}E_{J,\Sigma}&\frac{1}{2\sqrt{3}}E_{J,\Delta}&-\frac{1}{\sqrt{6}}E_{J,\Delta}\\\
\frac{1}{2\sqrt{3}}E_{J,\Delta}&\\!\\!\frac{1}{6}(E_{J,\Sigma}+4E_{J3})&\\!\\!\\!\frac{1}{3\sqrt{2}}(2E_{J3}-E_{J,\Sigma})\\\
-\frac{1}{\sqrt{6}}E_{J,\Delta}&\\!\\!\frac{1}{3\sqrt{2}}(2E_{J3}-E_{J,\Sigma})&\\!\\!\frac{1}{3}(E_{J,\Sigma}+E_{J3})\end{pmatrix}$
(14)

with $E_{J,\Delta}=E_{J1}-E_{J2}$ and $E_{J,\Sigma}=E_{J1}+E_{J2}$.

A Legendre transformation using
$(2e)\mathbb{n}=\frac{\partial\mathcal{L}}{\partial\dot{\bm{\theta}}}=\tilde{C}\dot{\bm{\theta}}$
is applied to obtain the Hamiltonian. Introducing annihilation and creation
operators of the harmonic modes

$\displaystyle\hat{\theta}_{m}$
$\displaystyle=\left(\dfrac{2E_{C,m}}{E_{J,m}^{\prime}}\right)^{1/4}(\hat{a}_{m}^{\dagger}+\hat{a}_{m}),$
$\displaystyle\hat{n}_{m}$
$\displaystyle=\dfrac{i}{2}\left(\dfrac{E_{J,m}^{\prime}}{2E_{C,m}}\right)^{1/4}(\hat{a}_{m}^{\dagger}-\hat{a}_{m}),$
$\displaystyle E_{C,m}$
$\displaystyle=\dfrac{e^{2}}{2}(\tilde{C}^{-1})_{mm},\;E_{J,m}^{\prime}=\tilde{J}_{mm},$
(15)

and approximating the potential up to fourth order results in the effective
Hamiltonian

$\displaystyle\hat{H}_{\rm q}$
$\displaystyle=\sum_{m}\omega_{m}\hat{a}_{m}^{\dagger}\hat{a}_{m}+\sum_{m}\dfrac{\alpha_{m}}{2}\hat{a}_{m}^{\dagger}\hat{a}_{m}^{\dagger}\hat{a}_{m}\hat{a}_{m}$
$\displaystyle+\sum_{m\neq
n}g_{mn}(\hat{a}_{m}^{\dagger}\hat{a}_{n}+\hat{a}_{m}\hat{a}_{n}^{\dagger})+\alpha_{mn}\hat{a}_{m}^{\dagger}\hat{a}_{m}\hat{a}_{n}^{\dagger}\hat{a}_{n}$
(16)

when applying the rotating wave approximation. The mode frequencies,
couplings, self- and cross-Kerr terms are, in this order, given by

$\displaystyle\omega_{m}$ $\displaystyle=\omega_{m,0}+\alpha_{m}+\sum_{m\neq
n}\dfrac{\alpha_{mn}}{2},\;\omega_{m,0}=\sqrt{8E_{C,m}E_{J,m}^{\prime}},$
$\displaystyle g_{mn}$
$\displaystyle=e^{2}(\tilde{C}^{-1})_{mn}\left(\frac{E_{J,m}^{\prime}E_{J,n}^{\prime}}{4E_{C,m}E_{C,n}}\right)^{1/4}$
$\displaystyle+\tilde{J}_{mn}\left(\frac{4E_{C,m}E_{C,n}}{E_{J,m}^{\prime}E_{J,n}^{\prime}}\right)^{1/4},$
$\displaystyle\alpha_{m}$
$\displaystyle=-\frac{\beta_{m}}{E_{J,m}^{\prime}}E_{C,m},\;\beta_{m}=E_{J,m}(T_{m1}^{4}+T_{m2}^{4}+T_{m3}^{4}),$
$\displaystyle\alpha_{mn}$
$\displaystyle=-2\frac{\gamma_{mn}}{\sqrt{E_{J,m}^{\prime}E_{J,n}^{\prime}}}\sqrt{E_{C,m}E_{C,n}},$
$\displaystyle\gamma_{mn}$
$\displaystyle=\sum_{i}E_{J,i}T_{mi}^{2}T_{ni}^{2}.$ (17)

Here the indices $m,n\in\\{1,2,3\\}$ can be identified with the modes
$\mathcal{A},\mathcal{B},\mathcal{C}.$ The mode-mode couplings $g_{mn}$
originate from the off-diagonal terms in the transformed capacitance
$\bm{\tilde{C}}$ and Josephson matrix $\bm{\tilde{J}}$. Therefore, their
dependence on the circuit asymmetries can be read off directly from these off-
diagonal matrix entries. In the symmetric case, where $E_{J1}=E_{J2}=E_{J3}$,
the self- and cross-Kerr terms simplify to

$\displaystyle\alpha_{\mathcal{A}}$
$\displaystyle=-\frac{E_{C,\mathcal{A}}}{2},\;\alpha_{\mathcal{B}}=-\frac{E_{C,\mathcal{B}}}{2},\;\alpha_{\mathcal{C}}=-\frac{E_{C,\mathcal{C}}}{3},$
$\displaystyle\alpha_{\mathcal{A},\mathcal{B}}$
$\displaystyle=-\frac{1}{3}\sqrt{E_{C,\mathcal{A}}E_{C,\mathcal{B}}},\;\alpha_{\mathcal{A},\mathcal{C}}=-\frac{2}{3}\sqrt{E_{C,\mathcal{A}}E_{C,\mathcal{C}}},$
$\displaystyle\alpha_{\mathcal{B},\mathcal{C}}$
$\displaystyle=-\frac{2}{3}\sqrt{E_{C,\mathcal{B}}E_{C,\mathcal{C}}},$ (18)

showing that the self-Kerr terms of mode $\mathcal{A}$ and $\mathcal{B}$ are
diminished by a factor of 2, compared to the conventional transmon self-Kerr
term.

##  Appendix B Coupling to the resonator

When capacitively coupling the multi-mode circuit described by the circuit
Hamiltonian $\hat{H}_{c}$ in Eq. (1) to a harmonic readout resonator, we
obtain the Hamiltonian

$\displaystyle\hat{H}_{\rm qr}=\hat{H}_{\rm
c}+\omega_{r}\hat{a}_{r}^{\dagger}\hat{a}+\sum_{k=1}^{3}(\bm{C}^{-1})_{kr}\hat{q}_{k}\hat{q}_{r}$
(19)

with resonator frequency $\omega_{r}$, capacitance matrix $\bm{C}$, and
resonator charge operator
$\hat{q}_{r}=i\sqrt{\hbar/2Z_{r}}(\hat{a}_{r}^{\dagger}-\hat{a}_{r})$, where
$Z_{r}$ is the resonator impedance. Expressing the Hamiltonian $\hat{H}_{\rm
qr}$ in terms of the eigenstates and eigenvalues $\\{\ket{i},\omega_{i}\\}$ of
$\hat{H}_{\rm c}$ results in:

$\displaystyle\hat{H}_{\rm qr}=$
$\displaystyle\sum_{i}\omega_{i}\ket{i}\bra{i}+\omega_{r}\hat{a}_{r}^{\dagger}\hat{a}_{r}+\sum_{ij}g_{i,j}\ket{i}\bra{j}(\hat{a}_{r}^{\dagger}-\hat{a}_{r}),$
$\displaystyle g_{i,j}=$ $\displaystyle
i\sqrt{\hbar/2Z_{r}}\sum_{k=1}^{3}(\bm{C}^{-1})_{kr}\bra{i}\hat{q}_{k}\ket{j}.$
(20)

This has the general form of a multi-level system coupled to a harmonic mode.
In the dispersive regime $|\omega_{i}-\omega_{j}-\omega_{r}|\gg|g_{i,j}|$ the
Hamiltonian reduces to [59]

$\displaystyle\hat{H}_{\rm disp}$
$\displaystyle=\omega_{r}\hat{a_{r}}^{\dagger}\hat{a}_{r}+\sum_{j}(\omega_{j}+\Lambda_{j})\ket{j}\bra{j}+\sum_{j}\chi_{j}\hat{a}^{\dagger}\hat{a}\ket{j}\bra{j},$
$\displaystyle\Lambda_{j}$
$\displaystyle=\sum_{i}\chi_{ij},\;\chi_{j}=\sum_{i}(\chi_{ij}-\chi_{ji}),\;\chi_{ij}=\dfrac{|g_{i,j}|^{2}}{\omega_{j}-\omega_{i}-\omega_{r}}.$
(21)

From the eigenstates and eigenvalues $\\{\ket{i},\omega_{i}\\}$, as obtained
from exact diagonalization of the circuit Hamiltonian $\hat{H}_{\rm c}$ in the
charge basis, the couplings $g_{i,j}$ and corresponding dispersive shifts
$\chi_{i}$ can be evaluated numerically. When instead approximating the
circuit Hamiltonian $\hat{H}_{\rm c}$ using the effective Hamiltonian
$\hat{H}_{q}$ in Eq. (16), and expressing $\hat{q}_{k}$ in terms of the mode
operators $\hat{a}_{m}$ in Eq. (19), the resonator-qubit Hamiltonian
$\hat{H}_{qr}$ can be expressed in the form

$\displaystyle\hat{H}_{\rm qr}=\hat{H}_{\rm
q}+\omega_{r}\hat{a}_{r}^{\dagger}\hat{a}_{r}+\sum_{m}g_{m}(\hat{a}^{\dagger}_{r}\hat{a}_{m}+\hat{a}_{r}\hat{a}_{m}^{\dagger}),$
(22)

with $g_{m}=g_{0,m}$ and $m=\mathcal{A},\mathcal{B},\mathcal{C}$, equivalent
to the Hamiltonian in Eq. (3). In the dispersive regime
$|\omega_{m}-\omega_{r}|\gg g_{m}$ the Hamiltonian reduces to [43]

$\displaystyle\hat{H}_{\rm disp.}=\hat{H}_{\rm
q}+\omega_{r}\hat{a}^{\dagger}_{r}\hat{a}_{r}+\sum_{m}2\chi_{m}\hat{a}^{\dagger}_{r}\hat{a}_{r}\hat{a}^{\dagger}_{m}\hat{a}_{m},$
(23)

in the case of a single circuit mode excitation, with the dispersive shifts
$\chi_{m}$ given by

$\displaystyle\chi_{m}=$ $\displaystyle
g_{m}^{2}\left(\frac{1}{\Delta_{m}}-\frac{1}{\Delta_{m}+\alpha_{m}}\right)+$
$\displaystyle\sum_{n\neq
m}\frac{g_{n}^{2}}{2}\left(\frac{1}{\Delta_{n}}-\frac{1}{\Delta_{n}+\alpha_{mn}}\right),$
(24)

with $n,m\in\\{\mathcal{A},\mathcal{B},\mathcal{C}\\}$. Note, the above
dispersive shifts can be derived from the general expression in Eq. (21), by
applying selection rules to the couplings $g_{i,j}$, which are present in the
transmon regime [55] with additional contributions in case of multi-mode
circuits [41, 43]. When not operating in the transmon regime, the relations
$g_{m,2m}=\sqrt{2}g_{0,m}$ and $g_{n,mn}=g_{0,m}$ are no longer valid and the
dispersive shifts are more precisely given by

$\displaystyle\chi_{m}=$ $\displaystyle
g_{m}^{2}\frac{1}{\Delta_{m}}-\frac{g_{m,2m}^{2}}{2}\frac{1}{\Delta_{m}+\alpha_{m}}+$
$\displaystyle\sum_{n\neq
m}\frac{g_{n}^{2}}{2}\frac{1}{\Delta_{n}}-\frac{g_{n,mn}^{2}}{2}\frac{1}{\Delta_{n}+\alpha_{mn}}.$
(25)

This expression we use for modeling the experimentally measured dispersive
shifts shown in Fig. 3(a). The self- and cross-Kerr terms are calculated via
exact diagonalization of the circuit Hamiltonian $\hat{H}_{\rm c}$ Eq. (1),
for which we include the full capacitive coupling matrix of the qubit and the
resonator. The obtained eigenstates and eigenvalues, are then used to
numerically evaluate the couplings using Eq. (20). Here, the capacitance
matrix and the Josephson energies are estimated by fitting the circuit model
to the experimentally measured spectrum [see Fig. 2(a)] and the dispersive
shifts at $\phi_{\rm ext}=0$ using a least-mean-squared cost-function and an
numeric optimizer provided by SciPy [74]. To estimate the uncertainties of the
fitted model parameters, we sample 100 artificial data sets within the initial
fit residuals and then repeat the optimization for each data set. This yields
a distribution of each model parameter with a corresponding standard deviation
and is know as bootstrapping [75].

##  Appendix C Device parameters

The device parameters at the readout and control point are summarized in Table
1.

Table 1: Parameters of mode $\tilde{\mathcal{A}}$ and $\tilde{\mathcal{B}}$ at the readout point $\phi_{\rm r}$ ($\phi_{\rm ext}=0$) and control point $\phi_{\rm c}$ ($\phi_{\rm ext}=\phi_{0}/2$). The mode frequency $\omega_{m}$ is measured via a Ramsey sequence, the self-Kerr terms $\alpha_{m}$, cross-Kerr couplings $\alpha_{\tilde{\mathcal{A}}\tilde{\mathcal{B}}}$ and dispersive shift $\chi_{m}$ are measured in spectroscopy. The single qubit gate fidelity $\mathcal{F}_{\rm RB}$ is obtained via randomized benchmarking (RB). The single-shot measurement fidelity is given by $\mathcal{F}_{\rm meas.}$.  | $\tilde{\mathcal{A}}(\phi_{\rm r})$ | $\tilde{\mathcal{B}}(\phi_{\rm r})$ | $\tilde{\mathcal{A}}(\phi_{\rm c})$ | $\tilde{\mathcal{B}}(\phi_{\rm c})$
---|---|---|---|---
$\omega_{m}/2\pi$ (GHz) | 5.017 | 6.408 | 5.101 | 3.811
$\alpha_{m}/2\pi$ (MHz) | -117 | -111 | -98 | -403
$\alpha_{\tilde{\mathcal{A}}\tilde{\mathcal{B}}}/2\pi$ (MHz) | -72 | -72 | -119 | -119
$\chi_{m}/2\pi$ (MHz) | -0.32(1) | -0.70(1) | -0.019(4) | -0.05(1)
$\mathcal{F}_{\rm RB}$ (%) | 99.933(2) | 99.844(7) | 99.948(3) |
$\mathcal{F}_{\rm meas.}$ (%) | 96.3 | 94.2 | 94.1 |

Note, that the self-Kerr terms are reduced by a factor of two compared to a
conventional transmon qubit [55], as expected from the model described in
Appendix A. The use of optimal control [61, 62] still allows high-fidelity
single-qubit gates as determined by a randomized benchmarking sequence with
$100~{}{\rm ns}$ DRAG pulses resulting in fidelities of $\mathcal{F}_{\rm
RB}>99.9$% at both operating points for the qubit mode $\tilde{\mathcal{A}}$
[see Fig. 5].

Figure 5: Comparison of single-qubit gate randomized benchmarking (RB) of mode
$\tilde{\mathcal{A}}$ at the readout point $\phi_{\rm r}$ and control point
$\phi_{\rm c}$ and corresponding error per gate (EPG).

The readout fidelities shown in Tab. 1 are determined for a $2~{}\mu{\rm s}$
readout pulse. The single-qubit gate fidelity and the readout fidelity of mode
$\tilde{\mathcal{B}}$ at the control point has not been determined, since it
is charge noise limited at this operating point.

Figure 6: Wiring diagram of the experimental setup.

##  Appendix D Experimental setup

The wiring diagram of the experimental setup is shown in Fig. 6. Measurements
are performed in a BlueFors XLD1000 dilution refrigerator with a base
temperature of 8 mK. A common input microwave line is used for readout and
qubit control. Microwave pulses are generated using a Zurich Instruments SHFQC
qubit controller and a Zurich Instruments HDAWG arbitrary waveform generator
in combination with a Rhode & Schwarz SGS100A local oscillator (LO) with
integrated mixer, which is able to provide more input power than the SHFQC and
is also used as the noise source. The on-chip flux line is controlled via a
Zurich Instruments HDAWG arbitrary waveform generator. Microwave attenuators
and filters provide isolation from thermal noise and noise from the room
temperature signal sources. The readout signal is amplified using a LNF-
LNC4_8F high electron mobility transistor (HEMT) and BZ-04000800-081045-152020
low noise amplifier (LNA). Single shot readout is performed by the Zurich
Instruments SHFQC waveform generator and analyzer.

##  Appendix E Noise photon calibration

We use the power-dependent background noise of the Rhode & Schwarz SGS100A LO
to artificially populate the readout resonator with noise photons. The noise
power spectral density of the output signal of the LO is measured at room
temperature as a function of its power setting, as shown in Fig. 7(a), using a
spectrum analyzer (Rhode & Schwarz FSW26). It exhibits a broadband background
noise spectrum around the readout resonator frequency $f_{r}=6.990~{}\rm{GHz}$
with a peak at the set LO frequency of $7.4~{}\rm{GHz}$. The background noise
power spectral density increases with the LO power setting and leads to a
population of the readout resonator. Note, that measurements using different
LO frequency settings, which are not presented here, show that the resonator
population is indeed caused by the background noise spectrum and not by the LO
resonance. To calibrate the average number of noise photons, we measure the
shift of the qubit frequency
$\Delta\omega=\omega_{\tilde{\mathcal{A}}}-\omega_{\tilde{\mathcal{A}},\rm
noise}$ of mode $\tilde{\mathcal{A}}$ at the readout point $\phi_{\rm r}$
($\phi_{\rm ext}=0$) for different LO power settings [Fig. 7(b)], with
$\omega_{\tilde{\mathcal{A}},\rm noise}$ denoting the shifted frequency in the
presence of the noise. The error bars are given by the fit of the qubit
frequency, which is measured in spectroscopy, and are small compared to the
size of the data points. The average noise photon number is calculated via
$\langle n\rangle=\Delta\omega/(2\chi_{\tilde{\mathcal{A}}})$ from the
measured dispersive shift
$\chi_{\tilde{\mathcal{A}}}/2\pi=-0.32(1)~{}\rm{MHz}$ at the readout point
$\phi_{\rm{r}}$, as shown in Fig. 7(c). By measuring the coherence time
$T_{2}$ for different LO power settings, we obtain the $T_{2}$-time dependent
on the noise photon number, as shown in Fig. 3(b) in the main text.

Figure 7: (a) Noise power spectrum of the output signal of the Rhode & Schwarz
SGS100A LO at room temperature for different power settings. (b) Frequency
shift $\Delta\omega$ of mode $\tilde{\mathcal{A}}$ at the readout point
$\phi_{\rm{r}}$ in dependence of the LO power setting. (c) Average noise
photon number $\langle n\rangle$ as a function of the LO power setting.

## References

  * Leibfried  _et al._ [2003] D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Barrett, J. Britton, W. M. Itano, B. Jelenković, C. Langer, T. Rosenband, and D. J. Wineland, Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate, Nature 422, 412 (2003).
  * Häffner  _et al._ [2008] H. Häffner, C. Roos, and R. Blatt, Quantum computing with trapped ions, Physics Reports 469, 155 (2008).
  * Raimond  _et al._ [2001] J. M. Raimond, M. Brune, and S. Haroche, Manipulating quantum entanglement with atoms and photons in a cavity, Rev. Mod. Phys. 73, 565 (2001).
  * Gaëtan  _et al._ [2009] A. Gaëtan, Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, Observation of collective excitation of two individual atoms in the rydberg blockade regime, Nature Physics 5, 115 (2009).
  * Evered  _et al._ [2023] S. J. Evered, D. Bluvstein, M. Kalinowski, S. Ebadi, T. Manovitz, H. Zhou, S. H. Li, A. A. Geim, T. T. Wang, N. Maskara, H. Levine, G. Semeghini, M. Greiner, V. Vuletić, and M. D. Lukin, High-fidelity parallel entangling gates on a neutral-atom quantum computer, Nature 622, 268 (2023).
  * Blais  _et al._ [2021] A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, Circuit quantum electrodynamics, Rev. Mod. Phys. 93, 025005 (2021).
  * Krinner  _et al._ [2020] S. Krinner, S. Lazar, A. Remm, C. Andersen, N. Lacroix, G. Norris, C. Hellings, M. Gabureac, C. Eichler, and A. Wallraff, Benchmarking coherent errors in controlled-phase gates due to spectator qubits, Phys. Rev. Appl. 14, 024042 (2020).
  * Google Quantum AI [2023] Google Quantum AI, Suppressing quantum errors by scaling a surface code logical qubit, Nature 614, 676 (2023).
  * Kim  _et al._ [2023] Y. Kim, A. Eddins, S. Anand, K. X. Wei, E. van den Berg, S. Rosenblatt, H. Nayfeh, Y. Wu, M. Zaletel, K. Temme, and A. Kandala, Evidence for the utility of quantum computing before fault tolerance, Nature 618, 500 (2023).
  * Place  _et al._ [2021] A. P. Place, L. V. Rodgers, P. Mundada, B. M. Smitham, M. Fitzpatrick, Z. Leng, A. Premkumar, J. Bryon, A. Vrajitoarea, S. Sussman, G. Cheng, T. Madhavan, H. K. Babla, X. H. Le, Y. Gang, B. Jäck, A. Gyenis, N. Yao, R. J. Cava, N. P. de Leon, and A. A. Houck, New material platform for superconducting transmon qubits with coherence times exceeding 0.3 milliseconds, Nature Communications 12, (2021).
  * Somoroff  _et al._ [2023] A. Somoroff, Q. Ficheux, R. A. Mencia, H. Xiong, R. Kuzmin, and V. E. Manucharyan, Millisecond coherence in a superconducting qubit, Phys. Rev. Lett. 130, 267001 (2023).
  * Kono  _et al._ [2023] S. Kono, J. Pan, M. Chegnizadeh, X. Wang, A. Youssefi, M. Scigliuzzo, and T. J. Kippenberg, Mechanically induced correlated errors on superconducting qubits with relaxation times exceeding 0.4 milliseconds (2023), arXiv:2305.02591 [quant-ph] .
  * Biznárová  _et al._ [2023] J. Biznárová, A. Osman, E. Rehnman, L. Chayanun, C. Križan, P. Malmberg, M. Rommel, C. Warren, P. Delsing, A. Yurgens, J. Bylander, and A. F. Roudsari, Mitigation of interfacial dielectric loss in aluminum-on-silicon superconducting qubits (2023), arXiv:2310.06797 [quant-ph] .
  * Houck  _et al._ [2008] A. A. Houck, J. A. Schreier, B. R. Johnson, J. M. Chow, J. Koch, J. M. Gambetta, D. I. Schuster, L. Frunzio, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Controlling the spontaneous emission of a superconducting transmon qubit, Phys. Rev. Lett. 101, 080502 (2008).
  * Schuster  _et al._ [2005] D. I. Schuster, A. Wallraff, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. M. Girvin, and R. J. Schoelkopf, ac stark shift and dephasing of a superconducting qubit strongly coupled to a cavity field, Phys. Rev. Lett. 94, 123602 (2005).
  * Clerk and Utami [2007] A. A. Clerk and D. W. Utami, Using a qubit to measure photon-number statistics of a driven thermal oscillator, Phys. Rev. A 75, 042302 (2007).
  * Rigetti  _et al._ [2012] C. Rigetti, J. M. Gambetta, S. Poletto, B. L. T. Plourde, J. M. Chow, A. D. Córcoles, J. A. Smolin, S. T. Merkel, J. R. Rozen, G. A. Keefe, M. B. Rothwell, M. B. Ketchen, and M. Steffen, Superconducting qubit in a waveguide cavity with a coherence time approaching 0.1 ms, Phys. Rev. B 86, 100506 (2012).
  * Sears  _et al._ [2012] A. P. Sears, A. Petrenko, G. Catelani, L. Sun, H. Paik, G. Kirchmair, L. Frunzio, L. I. Glazman, S. M. Girvin, and R. J. Schoelkopf, Photon shot noise dephasing in the strong-dispersive limit of circuit qed, Phys. Rev. B 86, 180504 (2012).
  * Sheldon  _et al._ [2017] S. Sheldon, M. Sandberg, H. Paik, B. Abdo, J. M. Chow, M. Steffen, and J. M. Gambetta, Characterization of hidden modes in networks of superconducting qubits, Applied Physics Letters 111, 222601 (2017).
  * Yan  _et al._ [2018] F. Yan, D. Campbell, P. Krantz, M. Kjaergaard, D. Kim, J. L. Yoder, D. Hover, A. Sears, A. J. Kerman, T. P. Orlando, S. Gustavsson, and W. D. Oliver, Distinguishing coherent and thermal photon noise in a circuit quantum electrodynamical system, Phys. Rev. Lett. 120, 260504 (2018).
  * Swiadek  _et al._ [2023] F. Swiadek, R. Shillito, P. Magnard, A. Remm, C. Hellings, N. Lacroix, Q. Ficheux, D. C. Zanuz, G. J. Norris, A. Blais, S. Krinner, and A. Wallraff, Enhancing dispersive readout of superconducting qubits through dynamic control of the dispersive shift: Experiment and theory (2023), arXiv:2307.07765 [quant-ph] .
  * Lu  _et al._ [2023] Y. Lu, A. Maiti, J. W. Garmon, S. Ganjam, Y. Zhang, J. Claes, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, High-fidelity parametric beamsplitting with a parity-protected converter, Nature Communications 14, (2023).
  * [23] G. Huber, in preperation.
  * Reed  _et al._ [2010] M. D. Reed, B. R. Johnson, A. A. Houck, L. DiCarlo, J. M. Chow, D. I. Schuster, L. Frunzio, and R. J. Schoelkopf, Fast reset and suppressing spontaneous emission of a superconducting qubit, Applied Physics Letters 96, 203110 (2010).
  * Jeffrey  _et al._ [2014] E. Jeffrey, D. Sank, J. Y. Mutus, T. C. White, J. Kelly, R. Barends, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. Megrant, P. J. J. O’Malley, C. Neill, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland, and J. M. Martinis, Fast accurate state measurement with superconducting qubits, Phys. Rev. Lett. 112, 190504 (2014).
  * Bronn  _et al._ [2015] N. T. Bronn, Y. Liu, J. B. Hertzberg, A. D. Córcoles, A. A. Houck, J. M. Gambetta, and J. M. Chow, Broadband filters for abatement of spontaneous emission in circuit quantum electrodynamics, Applied Physics Letters 107, 172601 (2015).
  * Heinsoo  _et al._ [2018] J. Heinsoo, C. K. Andersen, A. Remm, S. Krinner, T. Walter, Y. Salathé, S. Gasparinetti, J.-C. Besse, A. Potočnik, A. Wallraff, and C. Eichler, Rapid high-fidelity multiplexed readout of superconducting qubits, Phys. Rev. Appl. 10, 034040 (2018).
  * Sunada  _et al._ [2022] Y. Sunada, S. Kono, J. Ilves, S. Tamate, T. Sugiyama, Y. Tabuchi, and Y. Nakamura, Fast readout and reset of a superconducting qubit coupled to a resonator with an intrinsic purcell filter, Phys. Rev. Appl. 17, 044016 (2022).
  * Sunada  _et al._ [2023] Y. Sunada, K. Yuki, Z. Wang, T. Miyamura, J. Ilves, K. Matsuura, P. A. Spring, S. Tamate, S. Kono, and Y. Nakamura, Photon-noise-tolerant dispersive readout of a superconducting qubit using a nonlinear purcell filter (2023), arXiv:2309.04315 [quant-ph] .
  * Lidar  _et al._ [1998] D. A. Lidar, I. L. Chuang, and K. B. Whaley, Decoherence-free subspaces for quantum computation, Phys. Rev. Lett. 81, 2594 (1998).
  * Dicke [1954] R. H. Dicke, Coherence in spontaneous radiation processes, Phys. Rev. 93, 99 (1954).
  * DeVoe and Brewer [1996] R. G. DeVoe and R. G. Brewer, Observation of superradiant and subradiant spontaneous emission of two trapped ions, Phys. Rev. Lett. 76, 2049 (1996).
  * Casabone  _et al._ [2015] B. Casabone, K. Friebe, B. Brandstätter, K. Schüppert, R. Blatt, and T. E. Northup, Enhanced quantum interface with collective ion-cavity coupling, Phys. Rev. Lett. 114, 023602 (2015).
  * Guerin  _et al._ [2016] W. Guerin, M. O. Araújo, and R. Kaiser, Subradiance in a large cloud of cold atoms, Phys. Rev. Lett. 116, 083601 (2016).
  * Filipp  _et al._ [2011] S. Filipp, M. Göppl, J. M. Fink, M. Baur, R. Bianchetti, L. Steffen, and A. Wallraff, Multimode mediated qubit-qubit coupling and dark-state symmetries in circuit quantum electrodynamics, Phys. Rev. A 83, 063827 (2011).
  * Loo  _et al._ [2013] A. F. V. Loo, A. Fedorov, K. Lalumieŕe, B. C. Sanders, A. Blais, and A. Wallraff, Photon-mediated interactions between distant artificial atoms, Science 342, 1494 (2013).
  * Wang  _et al._ [2020] Z. Wang, H. Li, W. Feng, X. Song, C. Song, W. Liu, Q. Guo, X. Zhang, H. Dong, D. Zheng, H. Wang, and D.-W. Wang, Controllable switching between superradiant and subradiant states in a 10-qubit superconducting circuit, Phys. Rev. Lett. 124, 013601 (2020).
  * Zanner  _et al._ [2022] M. Zanner, T. Orell, C. M. Schneider, R. Albert, S. Oleschko, M. L. Juan, M. Silveri, and G. Kirchmair, Coherent control of a multi-qubit dark state in waveguide quantum electrodynamics, Nature Physics 18, 538 (2022).
  * Zhu and Scully [1996] S.-Y. Zhu and M. O. Scully, Spectral line elimination and spontaneous emission cancellation via quantum interference, Phys. Rev. Lett. 76, 388 (1996).
  * Gyenis  _et al._ [2021] A. Gyenis, P. S. Mundada, A. Di Paolo, T. M. Hazard, X. You, D. I. Schuster, J. Koch, A. Blais, and A. A. Houck, Experimental realization of a protected superconducting circuit derived from the $0$–$\pi$ qubit, PRX Quantum 2, 010339 (2021).
  * Gambetta  _et al._ [2011] J. M. Gambetta, A. A. Houck, and A. Blais, Superconducting qubit with purcell protection and tunable coupling, Phys. Rev. Lett. 106, 030502 (2011).
  * Roy  _et al._ [2017] T. Roy, S. Kundu, M. Chand, S. Hazra, N. Nehra, R. Cosmic, A. Ranadive, M. P. Patankar, K. Damle, and R. Vijay, Implementation of pairwise longitudinal coupling in a three-qubit superconducting circuit, Phys. Rev. Appl. 7, 054025 (2017).
  * Roy  _et al._ [2018] T. Roy, M. Chand, A. Bhattacharjee, S. Hazra, S. Kundu, K. Damle, and R. Vijay, Multimode superconducting circuits for realizing strongly coupled multiqubit processor units, Phys. Rev. A 98, 052318 (2018).
  * Hazra  _et al._ [2020] S. Hazra, K. V. Salunkhe, A. Bhattacharjee, G. Bothara, S. Kundu, T. Roy, M. P. Patankar, and R. Vijay, Engineering cross resonance interaction in multi-modal quantum circuits, Applied Physics Letters 116, 152601 (2020).
  * Roy  _et al._ [2020] T. Roy, S. Hazra, S. Kundu, M. Chand, M. P. Patankar, and R. Vijay, Programmable superconducting processor with native three-qubit gates, Phys. Rev. Appl. 14, 014072 (2020).
  * Diniz  _et al._ [2013] I. Diniz, E. Dumur, O. Buisson, and A. Auffèves, Ultrafast quantum nondemolition measurements based on a diamond-shaped artificial atom, Phys. Rev. A 87, 033837 (2013).
  * Dassonneville  _et al._ [2020] R. Dassonneville, T. Ramos, V. Milchakov, L. Planat, E. Dumur, F. Foroughi, J. Puertas, S. Leger, K. Bharadwaj, J. Delaforce, C. Naud, W. Hasch-Guichard, J. J. García-Ripoll, N. Roch, and O. Buisson, Fast high-fidelity quantum nondemolition qubit readout via a nonperturbative cross-kerr coupling, Phys. Rev. X 10, 011045 (2020).
  * Dassonneville  _et al._ [2023] R. Dassonneville, T. Ramos, V. Milchakov, C. Mori, L. Planat, F. Foroughi, C. Naud, W. Hasch-Guichard, J. García-Ripoll, N. Roch, and O. Buisson, Transmon-qubit readout using an in situ bifurcation amplification in the mesoscopic regime, Phys. Rev. Appl. 20, 044050 (2023).
  * Finck  _et al._ [2021] A. Finck, S. Carnevale, D. Klaus, C. Scerbo, J. Blair, T. McConkey, C. Kurter, A. Carniol, G. Keefe, M. Kumph, and O. Dial, Suppressed crosstalk between two-junction superconducting qubits with mode-selective exchange coupling, Phys. Rev. Appl. 16, 054041 (2021).
  * Chávez-Garcia  _et al._ [2022] J. M. Chávez-Garcia, F. Solgun, J. B. Hertzberg, O. Jinka, M. Brink, and B. Abdo, Weakly flux-tunable superconducting qubit, Phys. Rev. Appl. 18, 034057 (2022).
  * Wills  _et al._ [2022] J. Wills, G. Campanaro, S. Cao, S. Fasciati, P. Leek, and B. Vlastakis, Spatial charge sensitivity in a multimode superconducting qubit, Phys. Rev. Appl. 17, 024058 (2022).
  * Srinivasan  _et al._ [2011] S. J. Srinivasan, A. J. Hoffman, J. M. Gambetta, and A. A. Houck, Tunable coupling in circuit quantum electrodynamics using a superconducting charge qubit with a $v$-shaped energy level diagram, Phys. Rev. Lett. 106, 083601 (2011).
  * Hoffman  _et al._ [2011] A. J. Hoffman, S. J. Srinivasan, J. M. Gambetta, and A. A. Houck, Coherent control of a superconducting qubit with dynamically tunable qubit-cavity coupling, Phys. Rev. B 84, 184515 (2011).
  * Zhang  _et al._ [2017] G. Zhang, Y. Liu, J. J. Raftery, and A. A. Houck, Suppression of photon shot noise dephasing in a tunable coupling superconducting qubit, npj Quantum Information 3, (2017).
  * Koch  _et al._ [2007] J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Charge-insensitive qubit design derived from the cooper pair box, Phys. Rev. A 76, 042319 (2007).
  * Bylander  _et al._ [2011] J. Bylander, S. Gustavsson, F. Yan, F. Yoshihara, K. Harrabi, G. Fitch, D. G. Cory, Y. Nakamura, J. S. Tsai, and W. D. Oliver, Noise spectroscopy through dynamical decoupling with a superconducting flux qubit, Nature Physics 7, 565 (2011).
  * Hutchings  _et al._ [2017] M. D. Hutchings, J. B. Hertzberg, Y. Liu, N. T. Bronn, G. A. Keefe, M. Brink, J. M. Chow, and B. L. T. Plourde, Tunable superconducting qubits with flux-independent coherence, Phys. Rev. Appl. 8, 044003 (2017).
  * Ganzhorn  _et al._ [2020] M. Ganzhorn, G. Salis, D. J. Egger, A. Fuhrer, M. Mergenthaler, C. Müller, P. Müller, S. Paredes, M. Pechal, M. Werninghaus, and S. Filipp, Benchmarking the noise sensitivity of different parametric two-qubit gates in a single superconducting quantum computing platform, Phys. Rev. Res. 2, 033447 (2020).
  * Zhu  _et al._ [2013] G. Zhu, D. G. Ferguson, V. E. Manucharyan, and J. Koch, Circuit qed with fluxonium qubits: Theory of the dispersive regime, Phys. Rev. B 87, 024510 (2013).
  * Blais  _et al._ [2007] A. Blais, J. Gambetta, A. Wallraff, D. I. Schuster, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, Quantum-information processing with circuit quantum electrodynamics, Phys. Rev. A 75, 032329 (2007).
  * Motzoi  _et al._ [2009] F. Motzoi, J. M. Gambetta, P. Rebentrost, and F. K. Wilhelm, Simple pulses for elimination of leakage in weakly nonlinear qubits, Phys. Rev. Lett. 103, 110501 (2009).
  * Werninghaus  _et al._ [2021] M. Werninghaus, D. J. Egger, F. Roy, S. Machnes, F. K. Wilhelm, and S. Filipp, Leakage reduction in fast superconducting qubit gates via optimal control, npj Quantum Information 7, (2021).
  * Klimov  _et al._ [2018] P. V. Klimov, J. Kelly, Z. Chen, M. Neeley, A. Megrant, B. Burkett, R. Barends, K. Arya, B. Chiaro, Y. Chen, A. Dunsworth, A. Fowler, B. Foxen, C. Gidney, M. Giustina, R. Graff, T. Huang, E. Jeffrey, E. Lucero, J. Y. Mutus, O. Naaman, C. Neill, C. Quintana, P. Roushan, D. Sank, A. Vainsencher, J. Wenner, T. C. White, S. Boixo, R. Babbush, V. N. Smelyanskiy, H. Neven, and J. M. Martinis, Fluctuations of energy-relaxation times in superconducting qubits, Phys. Rev. Lett. 121, 090502 (2018).
  * Schlör  _et al._ [2019] S. Schlör, J. Lisenfeld, C. Müller, A. Bilmes, A. Schneider, D. P. Pappas, A. V. Ustinov, and M. Weides, Correlating decoherence in transmon qubits: Low frequency noise by single fluctuators, Phys. Rev. Lett. 123, 190502 (2019).
  * Carroll  _et al._ [2022] M. Carroll, S. Rosenblatt, P. Jurcevic, I. Lauer, and A. Kandala, Dynamics of superconducting qubit relaxation times, npj Quantum Information 8, (2022).
  * Osman  _et al._ [2021] A. Osman, J. Simon, A. Bengtsson, S. Kosen, P. Krantz, D. P. Lozano, M. Scigliuzzo, P. Delsing, J. Bylander, and A. Fadavi Roudsari, Simplified Josephson-junction fabrication process for reproducibly high-performance superconducting qubits, Applied Physics Letters 118, 064002 (2021).
  * Hertzberg  _et al._ [2021] J. B. Hertzberg, E. J. Zhang, S. Rosenblatt, E. Magesan, J. A. Smolin, J. B. Yau, V. P. Adiga, M. Sandberg, M. Brink, J. M. Chow, and J. S. Orcutt, Laser-annealing josephson junctions for yielding scaled-up superconducting quantum processors, npj Quantum Information 7, (2021).
  * Chen  _et al._ [2014] Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, M. R. Geller, A. N. Cleland, and J. M. Martinis, Qubit architecture with high coherence and fast tunable coupling, Phys. Rev. Lett. 113, 220502 (2014).
  * McKay  _et al._ [2016] D. C. McKay, S. Filipp, A. Mezzacapo, E. Magesan, J. M. Chow, and J. M. Gambetta, Universal gate for fixed-frequency qubits via a tunable bus, Phys. Rev. Appl. 6, 064007 (2016).
  * Mundada  _et al._ [2019] P. Mundada, G. Zhang, T. Hazard, and A. Houck, Suppression of qubit crosstalk in a tunable coupling superconducting circuit, Phys. Rev. Appl. 12, 054023 (2019).
  * Billangeon  _et al._ [2015] P.-M. Billangeon, J. S. Tsai, and Y. Nakamura, Scalable architecture for quantum information processing with superconducting flux qubits based on purely longitudinal interactions, Phys. Rev. B 92, 020509 (2015).
  * Richer and DiVincenzo [2016] S. Richer and D. DiVincenzo, Circuit design implementing longitudinal coupling: A scalable scheme for superconducting qubits, Phys. Rev. B 93, 134501 (2016).
  * Cross and Gambetta [2015] A. W. Cross and J. M. Gambetta, Optimized pulse shapes for a resonator-induced phase gate, Phys. Rev. A 91, 032325 (2015).
  * Virtanen  _et al._ [2020] P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, İ. Polat, Y. Feng, E. W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Henriksen, E. A. Quintero, C. R. Harris, A. M. Archibald, A. H. Ribeiro, F. Pedregosa, P. van Mulbregt, and SciPy 1.0 Contributors, SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python, Nature Methods 17, 261 (2020).
  * Efron and Tibshirani [1994] B. Efron and R. Tibshirani,  _An Introduction to the Bootstrap_ (Chapman and Hall/CRC, 1994).