# Operating in a deep underground facility improves the locking of
gradiometric fluxonium qubits at the sweet spots

Daria Gusenkova  IQMT, Karlsruher Institute of Technology, Eggenstein-
Leopoldshafen, Germany  PHI, Karlsruhe Institute of Technology, Karlsruhe,
Germany  Francesco Valenti  IQMT, Karlsruher Institute of Technology,
Eggenstein-Leopoldshafen, Germany  Martin Spiecker  IQMT, Karlsruher Institute
of Technology, Eggenstein-Leopoldshafen, Germany  PHI, Karlsruhe Institute of
Technology, Karlsruhe, Germany  Simon Günzler  IQMT, Karlsruher Institute of
Technology, Eggenstein-Leopoldshafen, Germany  Patrick Paluch  IQMT,
Karlsruher Institute of Technology, Eggenstein-Leopoldshafen, Germany  PHI,
Karlsruhe Institute of Technology, Karlsruhe, Germany  Dennis Rieger  PHI,
Karlsruhe Institute of Technology, Karlsruhe, Germany  Larisa-Milena Pioraş-
Ţimbolmaş  CETATEA, National Institute for Research and Development of
Isotopic and Molecular Technologies, Cluj-Napoca, Romania  Faculty of Physics,
Babeş-Bolyai University, Cluj-Napoca, Romania  Liviu P. Zârbo  CETATEA,
National Institute for Research and Development of Isotopic and Molecular
Technologies, Cluj-Napoca, Romania  Nicola Casali  INFN, Sezione di Roma,
Roma, Italy  Ivan Colantoni  INFN, Sezione di Roma, Roma, Italy  CNR, Istituto
di Nanotecnologia, c/o Dip. Fisica, Sapienza Università di Roma, Roma, Italy
Angelo Cruciani  INFN, Sezione di Roma, Roma, Italy  Stefano Pirro  INFN
Laboratori Nazionali del Gran Sasso, Assergi (AQ), Italy  Laura Cardani  INFN,
Sezione di Roma, Roma, Italy  Alexandru Petrescu  Institut Quantique and
Département de Physique, Université de Sherbrooke, Sherbrooke, Canada
Wolfgang Wernsdorfer  IQMT, Karlsruher Institute of Technology, Eggenstein-
Leopoldshafen, Germany  PHI, Karlsruhe Institute of Technology, Karlsruhe,
Germany  Patrick Winkel  IQMT, Karlsruher Institute of Technology, Eggenstein-
Leopoldshafen, Germany  Ioan M. Pop<EMAIL_ADDRESS>IQMT, Karlsruher
Institute of Technology, Eggenstein-Leopoldshafen, Germany  PHI, Karlsruhe
Institute of Technology, Karlsruhe, Germany

(August 29, 2024)

###### Abstract

We demonstrate flux-bias locking and operation of a gradiometric fluxonium
artificial atom using two symmetric granular aluminum (grAl) loops to
implement the superinductor. The gradiometric fluxonium shows two orders of
magnitude suppression of sensitivity to homogeneous magnetic fields, which can
be an asset for hybrid quantum systems requiring strong magnetic field
biasing. By cooling down the device in an external magnetic field while
crossing the metal-to-superconductor transition, the gradiometric fluxonium
can be locked either at $0$ or $\Phi_{0}/2$ effective flux bias, corresponding
to an even or odd number of trapped fluxons, respectively. At mK temperatures,
the fluxon parity prepared during initialization survives to magnetic field
bias exceeding $100\,\Phi_{0}$. However, even for states biased in the
vicinity of $1\,\Phi_{0}$, we observe unexpectedly short fluxon lifetimes of a
few hours, which cannot be explained by thermal or quantum phase slips. When
operating in a deep-underground cryostat of the Gran Sasso laboratory, the
fluxon lifetimes increase to days, indicating that ionizing events activate
phase slips in the grAl superinductor.

The unique properties of the superconducting state emerging in a select list
of materials below the critical temperature $T_{\mathrm{c}}$ have already been
used for particle detection [1, 2, 3, 4], quantum-limited amplification [5, 6,
7], quantum information processing [8, 9] and hybrid mesoscopic hardware [10].
While the main benefits offered by the superconducting state are unarguably
its intrinsically low dissipation and the possibility to engineer strongly
non-linear elements such as Josephson junctions (JJs), another potential
resource emerges as a consequence of the magnetic field quantization in
superconducting loops and the associated long-lived persistent currents [11,
12]. In classical superconducting circuits, trapped flux quanta called fluxons
have been used for more than two decades in so-called rapid single flux-
quantum electronics [13, 14], and might constitute a valuable resource for
local magnetic field biasing in superconducting quantum processors [15]. In
the quantum regime, fluxons have recently been proposed as a resource for
quantum simulators [16, 17].

A prominent example for fluxon dynamics is the fluxonium qubit, in which the
tunneling of a fluxon through a JJ shunted by a superinductor [18, 19, 20]
determines the eigenenergies and wavefunctions of the system [21]. Besides the
large anharmonicity of its energy spectrum, enabling fast operation, the
fluxonium exhibits a so-called sweet spot with a long energy relaxation time
and slow dephasing when the magnetic flux enclosed in the loop is half a flux
quantum [22, 23].

Figure 1: Gradiometric fluxonium. (a) Optical microscopy images of our device,
consisting of a pair of fluxonium artificial atoms (central panel) - a non-
gradiometric fluxonium with a single loop (left panel) used to calibrate the
external magnetic field, roughly $1\,\mathrm{mm}$ apart from a fluxonium with
gradiometric loop (right panel). In both artificial atoms, a single JJ is
shunted by a superinductor made out of grAl similar to Ref. [24]. The false
colors indicate regions in which pure Al (blue), grAl (red), and a stack of
both (purple) are deposited on a sapphire substrate (grey). For readout, the
atoms are dispersively coupled to dedicated linear modes via a shared
inductance. (b) Effective circuit diagram of the non-gradiometric fluxonium,
where $C_{\mathrm{r}}$ and $L_{\mathrm{r}}$ are the capacitance and inductance
associated with the linear readout mode, respectively, $L_{\mathrm{s}}$ is the
shared inductance, $L_{\mathrm{q}}$ is the atom’s loop inductance, and
$L_{\mathrm{J}}$ and $C_{\mathrm{J}}$ are the Josephson inductance and
capacitance, respectively. (c) Effective circuit diagram of the gradiometric
fluxonium. The JJ is shunted by two superinductors, forming three loops in
total. The flux enclosed in the two inner loops is $\Phi_{\mathrm{ext,1}}$ and
$\Phi_{\mathrm{ext,2}}$. (d) Both implementations can be mapped onto a
simplified circuit model. While the mapping of the non-gradiometric design is
trivial, we find non-trivial expressions for the effective flux bias
$\bar{\Phi}_{\mathrm{ext}}$ and the effective loop inductance
$\bar{L}_{\mathrm{q}}$ for the gradiometric case (see supplementary material
S1).

In our device, we use an additional superinductor made from a superconducting
granular aluminum (grAl) [25, 26] thin film shunting the single Al-AlOx-Al JJ
to build a fluxonium artificial atom with a gradiometric loop geometry, as
shown in Fig. 1. As a result, our device has three loops in total: an outer
loop entirely formed by superinductors and two inner loops which are connected
by a JJ weak link enabling quantum tunneling of fluxons between them. Similar
to other gradiometric devices [27, 28], this loop geometry highly reduces the
circuit’s sensitivity to global magnetic fields, in our device by two orders
of magnitude. This feature opens the way for its use in hybrid systems [29,
30] where a large magnetic field is required to bias other quantum degrees of
freedom, for instance electronic spins in semiconducting heterostructures [31,
32] or molecular qubits [33, 34].

The ground state for a superconducting loop threaded by a perpendicular
external magnetic field involves a non-zero current, also known as persistent
current, if the magnetic flux enclosed in the loop is not an integer multiple
of the magnetic flux quantum $\Phi_{0}=h/(2e)$. Similar to the Meissner
effect, persistent currents can be induced by a static magnetic field if the
superconducting loop is cooled below $T_{\mathrm{c}}$ and crosses the metal-
to-superconductor phase transition. When the magnetic field is ramped down at
temperatures well below $T_{\mathrm{c}}$, the changing magnetic field induces
a screening current such that the number of trapped flux quanta in the loop
remains constant [19].

We demonstrate that our grAl gradiometric fluxonium can be initialized at the
half-flux sweet spot by cooling down through $T_{\mathrm{c}}$ in a static
magnetic field corresponding to $1\,\Phi_{0}$ in the outer loop. From pulsed
time-domain measurements we find an average energy relaxation time of
$T_{1}=10.0\pm 0.3\,$\mathrm{\SIUnitSymbolMicro s}$$ and a coherence time
$T_{2}^{\star}=0.59\pm 0.02\,$\mathrm{\SIUnitSymbolMicro s}$$. Since the echo
time $T_{2}=5.3\pm$0.3\text{\,}\mathrm{\SIUnitSymbolMicro s}$$ is roughly an
order of magnitude larger, we infer that our device is limited by local low-
frequency noise of unknown origin, qualitatively consistent with previous
observations [35].

Although the grAl superinductor is expected to have an extremely low phase-
slip rate $\sim 10^{-20}\,$\mathrm{Hz}$$, we only observe a lifetime of the
persistent current on the order of hours in a typical setup not shielded from
ionizing radiation. The measured extinction of persistent current in our
$50\times$160\text{\,}\mathrm{nm}$^{2}$ cross-section grAl wire is reminiscent
of the operating principle of transition edge sensors [36] and superconducting
nanowire single-photon detectors [37, 38], which, when DC current biased, can
transition to a dissipative state due to a sudden burst of quasiparticles
following an energy absorbing event. We confirm that the escape of trapped
flux from the gradiometric loop is related to radioactivity by moving samples
to the Gran Sasso National Laboratory (LNGS) underground facility. Here, we
measure a significant fluxon lifetime increase, from hours (above ground) to
days. In the presence of a $\mathrm{ThO_{2}}$ radioactive source (same stup as
in Ref. [39]) this time reduces again to $\sim$30 minutes.

Figure 2: (a) Calibration of the external field using the periodicity of the
non-gradiometric fluxonium spectrum. The colorplot shows the phase of the
reflection coefficient $\arg(S_{11})$ of the linear readout mode as a function
of the external magnetic field $B_{\perp}$. The fundamental transition
frequency of the fluxonium $f_{01}(\bar{\Phi}_{\mathrm{ext}})$ crosses the
readout mode several times, resulting in repeated avoided crossings with a
period $B_{0}=$280\text{\,}\mathrm{nT}$$ corresponding to a flux quantum
$\Phi_{0}$ enclosed in the fluxonium loop. (b) Left panel: gradiometric
fluxonium initialized at the effective half-flux bias by cooling down in
$B_{\mathrm{init}}=B_{0}$. Notice the factor 120 reduced sensitivity of the
gradiometric device to $B_{\perp}$ in comparison to panel (a). Central panel:
the time trace of the phase response measured at $B_{\perp}=0$. The
corresponding cut is indicated in left panel by a vertical dashed line. The
jump of the frequency of the readout mode detected at $\approx 85$ minutes
after crossing $T_{\mathrm{c,grAl}}\approx 2\,\mathrm{K}$ corresponds to an
escape of the trapped flux. Right panel: gradiometric device after the flux
escape. The direction of the avoided crossings demonstrates that the
fundamental fluxonium transition is found above (left) and below (right) the
readout mode frequency in applied zero-field $B_{\perp}=0$. The small avoided
crossings visible in the vicinity of $B_{\perp}=0$ in the right panel
correspond to two-photon transitions. (c) Coherence of the gradiometric
fluxonium after half-flux initialization: the qubit population inversion as
function of time for energy relaxation (left), Ramsey fringes (center) and
Hahn-echo experiment (right). Zero inversion corresponds to the finite
population caused by thermal excitations at the fridge temperature of
$20\text{\,}\mathrm{mK}$ and other non-equilibrium processes. The black lines
indicate the numerical fit of the data (markers). Error bars in left panel
show the measured standard deviation.  Figure 3: Fluxon dynamics measured
deep-underground in LNGS. The LNGS cryostat is located under a 1.4 km granite
overburden (3.6 km water equivalent) and is additionally protected from
ionizing radiation with lead shields located both inside and outside the
refrigerator. We measured a chip with three gradiometric devices (labeled A, B
and C) to check correlations between flux tunneling events. Top panels: the
left-hand panels in (a) and (b) show the field dependence of device A in two
separate cooldowns demonstrating odd and even state initialization,
respectively. The right-hand panels show time traces measured at
$B_{\perp}=0$. Notice the stability of the trapped flux on timescales of days,
before exposing the cryostat to a $\mathrm{ThO_{2}}$ radioactive source (red
vertical line), which activates fluxon dynamics. The blue vertical line
indicates source removal. The bottom panels show measured switching dynamics
between odd and even states for all devices during $\mathrm{ThO_{2}}$
exposure.

The sample design, shown in Fig. 1 (a), consists of a pair of fluxonium
artificial atoms, one with a non-gradiometric and the other with a
gradiometric loop geometry, respectively. Both devices are fabricated on a
$0.33\times 10\times$15\text{\,}\mathrm{mm}$^{3}$ c-plane sapphire substrate
by means of a three-angle shadow evaporation, similar to Ref. [24] (see S2).
The modulation periodicity of the non-gradiometric atom is used to calibrate
the external magnetic flux created by a superconducting field coil. Although
the devices are around $1\,\mathrm{mm}$ apart to reduce electromagnetic
interaction, the diameter of the field coil is more than one order of
magnitude larger, ensuring a homogeneous field $B_{\perp}$. For readout, both
fluxonium atoms are dispersively coupled to dedicated readout modes by sharing
a small fraction of their loop inductance. The capacitor of these two readout
modes is designed in the form of a microwave antenna and couples them to the
electric field of a 3D copper waveguide sample holder similar to Ref. [24].

For both device geometries we derive effective lumped-element circuit models
(see Fig. 1 panels (b) and (c)). Since the readout is implemented similarly,
the capacitance and inductance of the readout modes are denoted
$C_{\mathrm{r}}$ and $L_{\mathrm{r}}$, respectively, and $L_{\mathrm{s}}$ is
the shared inductance. The non-gradiometric design has a single loop with a
superinductance $L_{\mathrm{q}}$ shunting the JJ (blue crossed-box symbol).
The gradiometric design has two shunt inductances forming three loops: an
outer loop with surface area
$A=50\times$150\text{\,}\mathrm{\SIUnitSymbolMicro m}$^{2}$, and two inner
loops with surface area $A/2$. The inductance in each loop branch is denoted
$L_{i}$, with the index $i\in\\{1,2,3\\}$ indicating the corresponding branch.
The gradiometric atom can be mapped onto the standard fluxonium circuit
diagram shown in Fig. 1 (d) using an effective flux bias
$\bar{\Phi}_{\mathrm{ext}}$ and an effective shunting inductance
$\bar{L}_{\mathrm{q}}$ (see S1).

The superconducting field coil is calibrated by measuring the spectrum of the
non-gradiometric device, designed with the same loop area $A$, located on the
same substrate. Figure 2 (a) depicts the phase response $\arg(S_{11})$ of the
readout mode coupled to the non-gradiometric fluxonium atom as a function of
the probe frequency $f_{\mathrm{d}}$ and the external magnetic field
$B_{\perp}$, measured in close vicinity of the readout frequency
$f_{\mathrm{r}}=7.445\,\mathrm{GHz}$. The fundamental transition frequency of
the fluxonium $f_{01}(\bar{\Phi}_{\mathrm{ext}})$ oscillates between values
below and above the readout frequency, resulting in avoided-level-crossings
repeated with periodicity of $B_{0}=$0.28\text{\,}\mathrm{\SIUnitSymbolMicro
T}$$.

The gradiometric fluxonium can be initialized at the half-flux effective bias
by cooling the device down through the metal-to-superconductor phase
transition in a static magnetic field $B_{\mathrm{init}}=B_{0}$ corresponding
to a single flux quantum enclosed in the outer fluxonium loop (see S3). The
magnetic field is ramped down at the base temperature of the cryogenic
refrigerator ($20\text{\,}\mathrm{mK}$), well below the critical temperature
$T_{\mathrm{c,grAl}}\approx 2\,\mathrm{K}$ of the grAl film. However, the
enclosed flux is now trapped in the gradiometric loop. In case of perfectly
symmetric inner loops and zero field gradient the phase difference across the
JJ equals $\pi$, pinning the atom at the half-flux bias. Fig. 2 (b) shows the
gradiometric fluxonium after initialization at the effective half-flux bias
(left panel). Wide range flux sweeps of the gradiometric device are shown in
S5. The difference in field range covered in Fig. 2 (a) and Fig. 2 (b)
illustrates the suppression of global magnetic field sensitivity by roughly a
factor of 120 for the gradiometric fluxonium. According to our effective
circuit model, the remaining field sensitivity could be either caused by an
asymmetry of the outer loop inductances, or by a small field gradient.

Figure 2 (c) depicts time-domain characterization of the coherence properties
of the gradiometric atom. For the gradiometric fluxonium initialized at the
effective half-flux bias we find a Ramsey coherence time of
$T_{2}^{\star}=0.59\pm 0.02\,$\mathrm{\SIUnitSymbolMicro s}$$, which is not
limited by the energy relaxation time $T_{1}=10.0\pm
0.3\,$\mathrm{\SIUnitSymbolMicro s}$$. We measured $T_{1}$ fluctuations of
$10\%$ on a timescale of two hours. Notably, the non-gradiometric fluxonium
located on the same chip exhibits similar coherence times $T_{1}=2.5\pm
0.3\,$\mathrm{\SIUnitSymbolMicro s}$$ and $T_{2}^{\star}=0.76\pm
0.04\,$\mathrm{\SIUnitSymbolMicro s}$$, which excludes the gradiometric
geometry as the cause of the much smaller coherence compared to previous
fluxonium implementations based on similar grAl superinductors [24]. Moreover,
in both devices we do not observe an improvement in coherence around the half-
flux sweet spot (see S4). While the sensitivity to homogeneous fields is
decreased for the gradiometric device, this is not the case for local flux
noise, which might even increase due to larger length of the shunting
inductance [40]. A single spin echo pulse improves the coherence by almost an
order of magnitude for the gradiometric fluxonium, up to
$T_{2}=5.3\pm$0.3\text{\,}\mathrm{\SIUnitSymbolMicro s}$$, and by factor of
3.5 for the non-gradiometric fluxonium, up to
$T_{2}=2.6\pm$0.4\text{\,}\mathrm{\SIUnitSymbolMicro s}$$. Therefore, we
conclude that Ramsey coherence of all devices on this chip is limited by local
and low-frequency noise of unknown origin.

The time stability of the half-flux initialization is determined by fluxon
escape rate, which becomes apparent by an abrupt change of persistent current
under constant or zero magnetic field bias. To suppress fluxon dynamics the
outer loop of gradiometric devices needs to be implemented using a
superconducting wire with low phase slip rate. The expected phase slip rate in
our grAl superinductance can be found by modeling the material as an effective
array of JJs [41]. The calculated phase-slip rate is $\sim
10^{-20}\,$\text{\,}\mathrm{Hz}$$ (see S5). In strong contrast, in all four
cooldowns in the cryostat located in Karlsruhe (not shielded from ionizing
radiation) we observe an escape of the trapped flux once in a few hours,
similar to the phase slip rate found in conventional JJ array superinductors
[19]. The time evolution of the readout mode in Fig. 2 (b) shows a detected
flux escape event, manifesting as a frequency jump at $\approx 85$ minutes
after crossing $T_{\mathrm{c,grAl}}$. In order to test whether these jumps are
caused by ionizing radiation [42, 43, 44, 45, 46] we measure three similar
gradiometric devices in the LNGS deep-underground facility (Fig. 3), which was
previously used to quantify non-equilibrium quasiparticle poisoning in
superconducting quantum circuits [39]. For all devices measured in LNGS the
trapped flux remains stable on a time scale of days. Exposing the cryostat to
a $\mathrm{ThO_{2}}$ radioactive source leads to uncorrelated flux tunneling
events, and reduces fluxon lifetime to approximately half an hour. The fluxon
stability is restored after removal of the source.

In summary, we have demonstrated the implementation of a superconducting
fluxonium artificial atom with a gradiometric loop geometry, which is two
orders of magnitude less sensitive to global magnetic fields compared to a
standard, non-gradiometric device with similar loop area. We can initialize
our device at the half-flux sweet spot by inducing a persistent current into
the outer loop when cooling it down through the metal-to-superconductor
transition in a static external magnetic field of
$B_{\mathrm{init}}=$0.28\text{\,}\mathrm{\SIUnitSymbolMicro T}$$, equivalent
to a single flux quantum enclosed in the outer loop. From pulsed time-domain
measurements we find that the coherence of the gradiometric device is
comparable to that of regular fluxoniums on the same chip, and it is limited
by local, low-frequency noise, which can be filtered by a single spin echo
pulse.

Although the superinductor in our device is implemented using superconducting
grAl, which is expected to have a significantly smaller phase-slip rate
compared to conventional JJ arrays [19], we observe a similar escape rate of
the flux after half-flux initialization, which is indicative of catastrophic
events, for instance caused by radioactive or cosmic impacts locally weakening
superconductivity in the outer loop wire. Indeed, we confirm this hypothesis
by measuring order of magnitude increased lifetimes of trapped fluxons in the
LNGS deep-underground facility. Our results add another item to the list of
detrimental effects of ionizing radiation in superconducting hardware, and
provide additional motivation to implement radiation mitigation [47, 48, 39,
49, 45].

See supplementary material for the Hamiltonian of gradiometric fluxonium,
sample fabrication, gradiometric fluxonium initialization at half flux bias,
measured spectrum and coherence of the gradiometric fluxonium, and escape of
the trapped flux from grAl loop.

We are grateful to L. Grünhaupt for fruitful discussions, and we acknowledge
technical support from S. Diewald, A. Lukashenko and L. Radtke. Funding is
provided by the Alexander von Humboldt foundation in the framework of a Sofja
Kovalevskaja award endowed by the German Federal Ministry of Education and
Research, and by the European Union’s Horizon 2020 programme under Nº 899561
(AVaQus). P.P. and I.M.P. acknowledge support from the German Ministry of
Education and Research (BMBF) within the QUANTERA project SiUCs (FKZ:
13N15209). L.P.Z and L-M. P-Ţ. acknowledge the Romanian National Authority for
Scientific Research and Innovation, CNCS-UEFISCDI for funding through the
project PCCF Project No. PN-III-P4-ID-PCCF-2016-0047 and the project Quantum
Computation with Schrödinger cat states, contract ERANET-QUANTERA-QuCos
120/16.09.2019. S.G., D.R. and W.W. acknowledge support from the European
Research Council advanced grant MoQuOS (N. 741276). Facilities use was
supported by the KIT Nanostructure Service Laboratory (NSL). We acknowledge
qKit for providing a convenient measurement software framework. We thank the
director and technical staff of INFN-LNGS and the LNGS Computing and Network
Service for their support. We are grateful to M. Guetti and M. Iannone for the
support with construction of the cryogenic setup at LNGS. The work was
supported by INFN and by the U.S. Department of Energy, Office of Science,
National Quantum Information Science Research Centers, Superconducting Quantum
Materials and Systems Center (SQMS) under the contract No. DE-AC02-07CH11359.

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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## Supplementary Material

In this supplementary material we provide details on the Hamiltonian of
gradiometric fluxonium, sample fabrication, the initialization of the
gradiometric fluxonium at the half-flux bias point, measured spectrum and
coherence of the gradiometric fluxonium, and escape of the trapped flux from
grAl loop.

###  S1 Fluxonium equivalent circuit model

Using the lumped-element circuit in Fig. 1 (c) in the main text as a starting
point for obtaining a quantum theory, we arrive at the following Hamiltonian
describing readout resonator, gradiometric fluxonium qubit, and their linear
inductive coupling:

$\displaystyle\begin{split}\bm{H}=&\frac{1}{2C_{\mathrm{r}}}\bm{Q}_{\mathrm{r}}^{2}+\frac{1}{2\bar{L}_{\mathrm{r}}}\bm{\Phi}_{\mathrm{r}}^{2}\\\
&+\frac{1}{2C_{\mathrm{q}}}\bm{Q}_{\mathrm{q}}^{2}+\frac{1}{2\bar{L}_{\mathrm{q}}}\bm{\Phi}_{\mathrm{q}}^{2}-E_{\mathrm{J}}\cos\left(\bm{\varphi}_{\mathrm{q}}+\frac{2\pi}{\Phi_{0}}\bar{\Phi}_{\text{ext}}\right)\\\
&-\frac{1}{2\bar{L}_{\mathrm{rq}}}\bm{\Phi}_{\mathrm{r}}\bm{\Phi}_{\mathrm{q}},\end{split}$
(S1.1)

Here, we have introduced two pairs of canonically-conjugate charge and flux
variables, $\left[\bm{\Phi}_{i},\bm{Q}_{j}\right]=i\hbar\delta_{ij}$ for
$i,j=q,r$ corresponding to the qubit and resonator, respectively. For the
qubit, we defined the superconducting phase
$\bm{\varphi}_{\mathrm{q}}=2\pi\bm{\Phi}_{\mathrm{q}}/\Phi_{0}$, with
$\Phi_{0}$ the superconducting flux quantum.

Figure S1.1: Comparison between the spectra corresponding to a single-loop
fluxonium qubit and the gradiometric fluxonium, for the parameters stated in
the main text. (a) The fluxonium spectrum as function of the external flux
$\Phi_{\text{ext}}$ (solid lines) maps to that of a gradiometric fluxonium
qubit (dashed lines) as a function of the relative flux
$(\Phi_{\text{ext,2}}-\Phi_{\text{ext,1}})/2$ in the two smaller loops. (b)
Lowest energy levels $\left|n_{\mathrm{r}}m_{\mathrm{q}}\right\rangle$ in the
combined spectra of the qubit-resonator system. The transitions primarily
associated to the linear readout mode are bias field independent and visible
as horizontal lines. Thanks to the finite shared inductance, avoided-level-
crossings are visible whenever a fluxonium transition crosses the readout
mode. (c) Dispersive shift $\chi$ as extracted from the curves shown in panel
(b); excluded are areas near avoided crossings where a proper distinction
between linear readout mode and non-linear qubit mode becomes ambiguous.

The gradiometric circuit is defined in terms of effective fluxonium shunt,
resonator, and coupling inductances

$\displaystyle\begin{split}\bar{L}_{\mathrm{q}}&=\left[\dfrac{L_{3}L_{\sigma}^{2}(L_{s}+L_{\mathrm{r}})+L_{\mathrm{r}}L_{s}L_{2}L_{3}}{L_{\sigma}^{2}(L_{\mathrm{r}}L_{a}^{2}+L_{s}L_{b}^{2})}+\frac{L_{1}}{L_{\sigma}^{2}}\right]^{-1},\\\
\bar{L}_{\mathrm{r}}&=\frac{L_{\mathrm{r}}L_{a}^{2}+L_{s}L_{b}^{2}}{L_{\epsilon}^{2}},\\\
\bar{L}_{\text{rq}}&=\frac{L_{\mathrm{r}}L_{a}^{2}+L_{s}L_{b}^{2}}{2L_{s}L_{3}},\end{split}$
(S1.2)

where $L_{\sigma}^{2}=L_{1}L_{2}+L_{2}L_{3}+L_{1}L_{3}$,
$L_{\epsilon}^{2}=L_{s}L_{2}+L_{s}L_{3}+L_{\sigma}^{2}$,
$L_{a}^{2}=L_{s}L_{2}+L_{\sigma}^{2}$, and
$L_{b}^{2}=L_{\mathrm{r}}L_{3}+L_{\sigma}^{2}$. In the system Hamiltonian
$\bm{H}$, the external fluxes enter via the linear combination

$\displaystyle\begin{split}\bar{\Phi}_{\text{ext}}=&\Phi_{\Delta}+\alpha\Phi_{\Sigma},\;\alpha=\frac{L_{3}-L_{1}-L_{s}}{L_{1}+L_{s}+L_{3}}\end{split}$
(S1.3)

where $\Phi_{\Sigma/\Delta}=(\Phi_{\text{ext},1}\pm\Phi_{\text{ext},2})/2$,
and $\alpha$ is the inductance asymmetry. A symmetric configuration,
$\alpha=0$, leads to insensitivity to variations in the global magnetic flux.
Nevertheless, the spectrum always depends on the flux imbalance
$\Phi_{\Delta}$.

Figure S3.1: Flux bias initialization sequence. (a) The initialization of the
gradiometric fluxonium at the effective half-flux bias is realised by cooling
down the device through the metal-to-superconductor phase transition in an
external magnetic field $B_{\mathrm{init}}=B_{0}$ corresponding to single flux
quantum in the outer fluxonium loop. Magnetic field is ramped down at the base
temperature of the cryogenic refrigerator mixing chamber plate
($T_{\mathrm{mix}}=$20\text{\,}\mathrm{mK}$$), well below the critical
temperature of grAl $T_{\mathrm{c}}\approx 2\,\mathrm{K}$ of the grAl film.
(b) Schematic drawing of the external flux bias above $T_{\mathrm{c}}$. (c)
Circulating current (purple arrow) induced after ramping down external
magnetic field corresponds to the effective half-flux bias
$\bar{\Phi}_{\mathrm{ext}}=0.5\Phi_{0}$.

As noted before in the main text, the qubit Hamiltonian in eq. S1.1 can be
made exactly equivalent to the Hamiltonian of a single-loop fluxonium.
Explicitly, setting the shared inductance $L_{s}=0$ to decouple the qubit from
the resonator, and putting further $L_{2}=0$ and $L_{1}=L_{3}=2L_{q}$, we
arrive at the Hamiltonian of a single-loop fluxonium qubit with effective flux
equal to the flux imbalance $\Phi_{\Delta}$ and shunt inductance
$\bar{L}_{\mathrm{q}}=L_{q}$. In Fig. S1.1 (a), we exemplify this exact
agreement for the effective fluxonium parameters obtained in the main text.

Turning our attention to the circuit QED setup at $L_{s}\neq 0$, to
numerically diagonalize the two-mode Hamiltonian $\bm{H}$, we recast it into
the Fock basis determined by the terms [50] in the first two rows of eq. S1.1.
The lowest energy states in the spectrum of the combined qubit-resonator
Hamiltonian $\left|n_{\mathrm{r}}m_{\mathrm{q}}\right\rangle$ and the
resonator dispersive shift $\chi$ are shown in Fig. S1.1, panels (b) and (c),
respectively. For the sweep over external flux for the qubit-resonator
Hamiltonian, we have employed $m=25$ Fock states for the qubit, and $n=15$
states for the resonator mode, and have used larger-scale simulations of up to
a total Hilbert space dimension of 3500 in order to obtain a numerically-
precise value for the dispersive shift at the half-flux sweet spot,
$\chi/2\pi=-7.683\,\mathrm{MHz}$, which is in good agreement with the
experimentally obtained shift $\chi_{\mathrm{exp}}/2\pi=-7.8\,\mathrm{MHz}$.

###  S2 Fabrication

We use a three-angle shadow evaporation technique [24] to fabricate our
fluxonium devices on a c-plane sapphire substrate. First, a $20\,\mathrm{nm}$
thick Al film is evaporated under an angle of $+30^{\circ}$, measured from the
normal, which is then oxidized for 340 seconds under an oxygen partial
pressure of $30\text{\,}\mathrm{mbar}$. Second, another $30\,\mathrm{nm}$
thick Al film is evaporated under an angle $-30^{\circ}$, overlapping with the
first layer and forming the JJ, followed by a $50\,\mathrm{nm}$ thick grAl
film deposited at zero-angle. Thanks to the undercut in the double resist
stack, most of the grAl loop wire remains unshunted by the pure Al films,
while the JJ is formed entirely between the two Al films.

###  S3 Gradiometric fluxonium initialization at half flux bias

The initialization sequence of the gradiometric fluxonium at the half-flux
bias point is illustrated in Fig. S3.1. First, the system is cooled down
through the metal-to-superconductor phase transition in an external magnetic
field $B_{0}$. Thanks to the diameter of the superconducting field coil used
in this experiment, the magnetic field can be approximated as homogeneous on
the length-scale of our device. Notably, prior to this sequence the external
magnetic bias field $B_{0}$ is calibrated using a non-gradiometric device on
the same wafer (see Fig. 1 and Fig. 2 in the main text). Well below the
critical temperature of our grAl film, which is on the order of
$2\,\mathrm{K}$, the external magnetic field is ramped down, which induces a
persistent current $J$ in the outer loop, indicated by the purple arrows in
Fig. S3.1 (c). The total flux enclosed in the loop corresponds to a single
flux quantum.

###  S4 Spectrum and coherence of the gradiometric fluxonium

Figure S4.1: Spectrum and coherence of the gradiometric fluxonium. (a)
Spectrum of the gradiometric fluxonium measured as a function of the effective
magnetic flux $\bar{\Phi}_{\mathrm{ext}}$ and the applied magnetic field
$B_{\perp}$, when initialized in zero-field $B_{\mathrm{init}}=0$. The red
markers indicate the fundamental transition frequency $f_{01}$ extracted from
a continuous-wave two-tone measurement, by monitoring the readout resonator
response while sweeping the frequency of an additional drive tone. The blue
markers correspond to transition to the second excited state. The black solid
lines indicate a fit to the fluxonium spectrum obtained by diagonalizing the
uncoupled fluxonium Hamiltonian numerically. From the fit, we extract the
effective circuit parameters $\bar{L}_{\mathrm{q}}=172\,\mathrm{nH}$,
$C_{\mathrm{J}}=3.4\,\mathrm{fF}$, and $E_{\mathrm{J}}/h=5.1\,\mathrm{GHz}$.
Notably, since we change the spectrum only by applying a global magnetic
field, the value of the magnetic field at which we reach the half-flux sweet
spot corresponds to more than $50\,\Phi_{0}$ enclosed in the outer loop. (b)
Ramsey coherence time $T_{2}^{\star}$ (black markers) measured around spectrum
minimum for the gradiometric fluxonium initialized at the half-flux bias
($B_{\mathrm{init}}=B_{0}$). Notice that after initialization the minimum of
the qubit fundamental frequency corresponds to zero applied field. Orange
markers show the detuning of the qubit fundamental transition frequency
$\Delta_{01}=f_{01}(\bar{\Phi}_{\mathrm{ext}})-f_{01}(\bar{\Phi}_{\mathrm{ext}}=0)$
from its frequency at the degeneracy point
$f_{01}(\bar{\Phi}_{\mathrm{ext}}=0)$. The black line is a parabolic fit.
Figure S5.1: Current-activated phase slips in the gradiometric fluxonium.
Color plots show wide flux sweeps of the phase response of the readout mode
coupled to the gradiometric fluxonium with even (a) and odd (b) flux quanta
trapped in the grAl loop. The parabolic change of the frequency (black dashed
lines) is due to the dependence of the grAl kinetic inductance on the
circulating current induced by applied magnetic field in the fluxonium loop.
This circulating current also flows through the coupling segment of the
readout resonator. The top axes shows the applied magnetic field in units of
flux quantum in the fluxonium outer loop. The jumps of frequency observed at
$\gtrsim 130\Phi_{0}$ correspond to current-activated phase slips. One flux
sweep takes approximately 5 minutes.

We measure the spectrum of the gradiometric fluxonium using two-tone
spectroscopy (see Fig. S4.1 (a)), meaning we continuously probe the readout
resonator with a weak tone on resonance while applying a second tone with
varying frequency to the qubit. In the investigated frequency range between
$3\,\mathrm{GHz}$ and $15\,\mathrm{GHz}$ we find two field dependent
transitions associated with the gradiometric fluxonium, a single photon
transition into the first excited state $0\rightarrow 1$ and the second
excited state $0\rightarrow 2$ shown in red and blue, respectively. The
fluxonium circuit parameters $\bar{L}_{\mathrm{q}}=172\,\mathrm{nH}$,
$C_{\mathrm{J}}=3.4\,\mathrm{fF}$, and $E_{\mathrm{J}}/h=5.1\,\mathrm{GHz}$
are extracted by fitting the spectrum to the numerically diagonalized
fluxonium Hamiltonian. For the linear readout mode we find
$C_{\mathrm{r}}=$20.2\text{\,}\mathrm{fF}$$ and
$L_{\mathrm{r}}=$21.6\text{\,}\mathrm{nH}$$. The value of the shared
inductance $L_{\mathrm{s}}=$2.8\text{\,}\mathrm{nH}$$ is extracted from the
dispersive shift $\chi_{\mathrm{ge}}(0.5\Phi_{0})/(2\pi)=-7.8\,\mathrm{MHz}$
measured at the half-flux sweet spot.

Fig. S4.1 (b) shows the Ramsey coherence time $T_{2}^{\star}$ of the
gradiometric fluxonium measured after the half-flux bias initialization around
the spectrum minimum (black markers). The detuning of the fundamental
transition frequency of the gradiometric fluxonium is shown in orange markers.
In contrast to previous fluxonium implementations [24, 22], the device does
not exhibit an improvement in coherence around the half-flux sweet spot.

###  S5 Escape of the trapped flux from grAl loop

Quantum phase slips of the superconducting phase in the granular aluminum
superinductor are a potential decay mechanism for the trapped flux. The
expected phase slip rate can be calculated by modeling grAl superinductor as
an effective array of JJs [41]. GrAl consists of pure Al grains embedded in an
insulating $\mathrm{AlO_{x}}$ matrix. With a typical grain size of
$\sim$4\text{\,}\mathrm{nm}$$ we find the number of effective junctions in the
outer loop $N\approx 75\times 10^{3}$, with
$E_{\mathrm{J}}=(\Phi_{0}/2\pi)^{2}/L_{\mathrm{J0}}\approx$53\text{\,}\mathrm{THz}$$
and $E_{\mathrm{C}}=e^{2}/2C_{\mathrm{J}}\approx$48\text{\,}\mathrm{GHz}$$.
The phase slip rate in the Josephson junction array [51, 52] is

$v=\frac{1}{h}N\frac{4}{\sqrt{\pi}}(8E_{\mathrm{J}}^{3}E_{\mathrm{C}})^{1/4}\exp\left(-\sqrt{8\frac{E_{\mathrm{J}}}{E_{\mathrm{C}}}}\right),$
(S5.1)

which gives $v\lesssim 10^{-20}\,$\text{\,}\mathrm{Hz}$$ for the grAl
superinductor in our device. We would like to point out that the expected rate
is orders of magnitude smaller compared to conventional JJ arrays [19]. The
phase slip rate is enhanced by the induction of circulating currents because
the effective Josephson barrier between the grAl grains is reduced. For the
gradiometric fluxonium measured in a wide flux range we observe current-
activated phase slips at an external magnetic field bias corresponding to
$\gtrsim 130\Phi_{0}$ on the outer loop (see Fig. S5.1). The fact that we can
prepare such a highly metastable state in the fluxonium loop confirms the
stiffness of the condensate in grAl.