# Tunable coupling scheme for implementing two-qubit gates on fluxonium qubits

I. N. Moskalenko  National University of Science and Technology ”MISIS”,
119049 Moscow, Russia  Russian Quantum Center, 143025 Skolkovo, Moscow, Russia
I. S. Besedin<EMAIL_ADDRESS>National University of Science and
Technology ”MISIS”, 119049 Moscow, Russia  Russian Quantum Center, 143025
Skolkovo, Moscow, Russia  I. A. Simakov  National University of Science and
Technology ”MISIS”, 119049 Moscow, Russia  Russian Quantum Center, 143025
Skolkovo, Moscow, Russia  Skolkovo Institute of Science and Technology, 143026
Moscow, Russia  Moscow Institute of Physics and Technology, 141701
Dolgoprundy, Russia  A. V. Ustinov  National University of Science and
Technology ”MISIS”, 119049 Moscow, Russia  Russian Quantum Center, 143025
Skolkovo, Moscow, Russia  Physikalisches Institut, Karlsruhe Institute of
Technology, Karlsruhe, Germany

###### Abstract

The superconducting fluxonium circuit is an RF-SQUID-type flux qubit that uses
a large inductance built from an array of Josephson junctions or a high
kinetic inductance material. This inductance suppresses charge sensitivity
exponentially and flux sensitivity quadratically. In contrast to the transmon
qubit, the anharmonicity of fluxonium can be large and positive, allowing for
better separation between the low energy qubit manifold of the circuit and
higher-lying excited states. Here, we propose a tunable coupling scheme for
implementing two-qubit gates on fixed-frequency fluxonium qubits, biased at
half flux quantum. In this system, both qubits and coupler are coupled
capacitively and implemented as fluxonium circuits with an additional harmonic
mode. We investigate the performance of the scheme by simulating a universal
two-qubit fSim gate. In the proposed approach, we rely on a planar on-chip
architecture for the whole device. Our design is compatible with existing
hardware for transmon-based devices, with the additional advantage of lower
qubit frequency facilitating high-precision gating.

Quantum superconducting circuits based on Josephson tunnel junctions are a
flexible platform for building artificial atoms. Rapid progress has been made
in the last decade due to appearance of new types of qubits [1, 2] and
improvements in coherence properties [3]. Successful prototypes of
superconducting quantum processors developed by different research groups [4,
5, 6] to date are based on transmons, which have shown the best gate
fidelities among superconducting qubits. Despite the relatively high values of
coherence times of transmons in the order $100\ $\mathrm{\SIUnitSymbolMicro
s}$$ they are outperformed by an order magnitude in $T_{1}$ coherence times by
fluxonium qubits [2, 4]. The spectra of transmon qubits are similar to those
of weakly anharmonic oscillators. Although multiqubit processors with
efficient two-qubit gates[4, 5, 6] have already been demonstrated, weak
anharmonicity of their base elements presents a significant challenge for
further scaling them up and improving gate fidelities.

A changeover to fluxonium qubits could provide a possible upgrade path towards
large-scale superconducting quantum processors [2, 9, 3, 11, 4, 5] as
fluxoniums have millisecond energy relaxation times at flux degeneracy point.
Such long lifetime of the first excited state is partially due to its very low
(hundreds of megahertz) transition frequency from the ground state. This leads
to lower decay rates, since single-photon dielectric loss tangents only weakly
depend on frequency[13]. Low transition frequencies, however, lead to
operation of the qubit in a relatively “hot” environment. Because of this,
qubits can’t be initialized in the ground state by passive thermalization.
However, in a practical quantum processor qubit state initialization can be
realized by fast active reset [14]. Promising coherence times ($>200\
$\mathrm{\SIUnitSymbolMicro s}$$) have already been obtained in chip-
integrated fluxoniums [15], while in 3D cavities coherence times exceed even 1
ms [16]. In a recent work [7] first microwave-activated CZ gates have been
demonstrated also in a 3D cavity. Recently, another type of microwave-
activated two-qubit gate has been proposed, the bSWAP gate [18]. However,
high-fidelity two-qubit gates in planar geometry are yet to be demonstrated.
Moreover, scaling up beyond two qubits is extremely challenging in a 3D
architecture.

Figure 1: (color online) (a) Modified fluxonium circuit diagram, consisting of
one Josephson junction, two large inductors and three capacitors. (b) Concept
layout with readout resonator and bias line for magnetic flux control. (c)
Energy levels of the modified fluxonium system vs external magnetic flux
${\Phi^{\textnormal{x}}}$ for $E_{\textnormal{J}}=2.24\ $\mathrm{GHz}$$,
$E_{\textnormal{L}}=1.64\ $\mathrm{GHz}$$, $C_{1,2}=70.1\ $\mathrm{fF}$$,
$C_{\textnormal{J}}=1.3\ $\mathrm{fF}$$

In this work, we consider a specific parameter regime of fluxonium which
allows strong capacitive coupling to the qubit transition. In terms of
frequency and anharmonicity it is close to conventional fluxonium [2, 9, 3],
while the ratio between the Josephson and shunt inductance is close to the
quarton regime [1]. At the same time, the charging energy is relatively high:
$E_{J}\sim E_{L}\sim 4E_{C}$. A detailed comparison is given in the
Supplementary Information. The circuit consists of two superconducting islands
connected with a small Josephson junction, and inductively shunted to the
ground electrode (Fig. 1a). The proposed fluxonium can be utilized as the unit
cell (both qubit and coupler) for a scalable quantum processor. A possible
layout corresponding to realistic capacitances and inductances is shown in
Fig. 1b. Neighboring qubits can be capacitively coupled, allowing to adapt the
simple and broadly applicable capacitive tunable coupling scheme [20, 21, 5].

The scheme that we propose here consists of two fluxonium qubits with a
tunable coupler between them, which by itself is also a fluxonium qubit. Both
computational qubits are biased at the flux degeneracy point. The interaction
strength between the qubits is controlled by the central “coupler” fluxonium
flux bias. At the flux degeneracy point, all three qubits are close to
resonance and exhibit a strong $XX$-type interaction. Away from it, only a
small residual $ZZ$-type interaction between the qubits is left. A
$\sqrt{\mathrm{iSWAP}}$-like gate is performed by tuning the coupler from the
upper flux sweet stop to the lower sweet spot, waiting quarter of a vacuum
Rabi cycle, and tuning back. Using numerical simulation, we demonstrate how
decoherence, leakage and coherent errors can affect the gate performance.

The proposed scheme is compatible with existing hardware, moreover, the
additional advantage of this approach is the ability to use lower frequency
electronics for qubit and coupler control. Switching to sub-gigahertz controls
could drastically reduce the cost and complexity of the control electronics
and wiring.

A modified fluxonium circuit and a possible layout are shown in Fig. 1. It
consists of a Josephson junction with energy $E_{\textnormal{J}}$ shunted by a
capacitance $C_{\textnormal{J}}$ and two large (super-) inductors $L_{1}$ and
$L_{2}$ linked to form a loop. Superinductances $L_{1,2}$ can be built from
long arrays ($>50$) of large identical Josephson junctions. Both nodes $1;2$
have a distributed mutual capacitance with the ground node $C_{1;2}$. External
magnetic flux $\Phi^{\mathrm{x}}$ can be applied with a current bias line,
which is grounded through a part of the fluxonium loop. The inductance of that
wire $M$ determines how much current is required to tune the qubit frequency
from maximum to minimum. We neglect the influence of this inductance for the
qubit Hamiltonian, as it is several orders of magnitude smaller that the large
inductances $L_{1}$ and $L_{2}$.

The circuit has two degrees of freedom. We denote the nodal phases as
$\varphi_{1}$ and $\varphi_{2}$. Due to the circuit’s symmetry, the normal
mode coordinates of the circuit are defined as:

$\vartheta^{+}=\varphi_{1}+\varphi_{2};\ \ \ \
\vartheta^{-}=\varphi_{1}-\varphi_{2}.\ \ \ \ \ \ \ \ \ \ $ (1)

The $\vartheta^{-}$-mode is associated with a phase differences across the
Josephson junction and is thus nonlinear, the $\vartheta^{+}$-mode does not
bias the junction and is therefore a fully harmonic mode. In the absence of
disorder among circuit elements $L_{1}=L_{2}=L$, $C_{1}=C_{2}=C$ the modes are
decoupled, and the Hamiltonian is

$\hat{H}=\hat{H}_{\textnormal{h}}+\hat{H}_{\textnormal{f}},\ \ \ \ \ \ \ \ \ \
\ \ \ \ $ (2)
$\hat{H}_{\textnormal{h}}=4E_{\textnormal{Ch}}(\hat{n}^{+})^{2}+\frac{1}{2}E_{\textnormal{L}}(\hat{\vartheta}^{+}-{\varphi}^{x})^{2},\
\ \ \ \ \ \ \ $ (3)
$\hat{H}_{\textnormal{f}}=4E_{\textnormal{Cf}}(\hat{n}^{-})^{2}+\frac{1}{2}E_{\textnormal{L}}(\hat{\vartheta}^{-}-{\varphi}^{x})^{2}+E_{\textnormal{J}}[1-\cos(\hat{\vartheta^{-}})],$
(4)

where $\hat{n}^{-}$ and $\hat{n}^{+}$ are the canonically conjugate Cooper
pair numbers to $\hat{\vartheta}^{-}$ and $\hat{\vartheta}^{+}$, respectively.
Here we also introduce a dimensionless variable for external flux
$\varphi^{\textnormal{x}}=\frac{2\pi{\Phi}^{\textnormal{x}}}{\Phi_{0}}$, and
convert the circuit element parameters to energy units
$E_{\textnormal{L}}=(\Phi_{0}/2\pi)^{2}/2L$,
$E_{\textnormal{Cf}}=e^{2}/2C_{\textnormal{f}}$, where
$C_{\textnormal{f}}=(C+C_{\textnormal{J}})/2$,
$E_{\textnormal{Ch}}=e^{2}/2C_{\textnormal{h}}$, where
$C_{\textnormal{h}}=C/2$.

Mutual capacitance between the fluxonium mode and other circuit elements is a
scarce resource. Increasing the absolute value of a mutual capacitance also
increases the total capacitance of the fluxonium mode, which drives down the
qubit frequency and decreases the coupling strength of the fluxonium to
everything else. This contrasts with inductively coupled fluxonium qubits,
where the coupling strength does not directly depend on the qubit frequency.
The two-island configuration of the fluxonium qubit can utilize either of the
two islands to couple to other elements, while the total effective capacitance
is half of the total capacitance of each of the islands relative to the ground
electrode. This configuration allows us to work in the $300-700\
$\mathrm{MHz}$$ qubit frequency range at the operating point and still have
large coupling strengths between neighboring fluxoniums.

The computed energy spectrum for our qubit as a function of external flux
$\Phi^{\textnormal{x}}$ is plotted in Fig. 1 (c). The circuit parameters are
$E_{\textnormal{J}}=2.24\ $\mathrm{GHz}$$, $E_{\textnormal{L}}=1.64\
$\mathrm{GHz}$$, $C=70.1\ $\mathrm{fF}$$, $C_{\textnormal{J}}=1.3\
$\mathrm{fF}$$. These circuit parameters will be further used for the tunable
coupler. The eigenstates are labeled as
$\ket{n_{\textnormal{h}},n_{\textnormal{f}}}$, where $n_{\textnormal{h}}$ is
the harmonic mode occupancy and $n_{\textnormal{f}}$ is the fluxonium mode
occupancy. The harmonic mode frequency is $2.0\ $\mathrm{GHz}$$. The fluxonium
mode fundamental transition frequency $f_{\textnormal{Q}}$ spans from $625\
$\mathrm{MHz}$$ at the flux degeneracy point to $3.31\ $\mathrm{GHz}$$ at zero
flux bias. The fluxonium mode anharmonicity $\delta f_{\textnormal{Q}}$ at the
flux degeneracy point is around $1.911\ $\mathrm{GHz}$$.

The flux bias line is coupled to the fluxonium mode of the qubit, allowing to
perform both excitation and qubit frequency control with a single wire. This
approach has been used to reduce wiring complexity in large NISQ devices [24].
However, if the inductance $M$ is too large, it becomes a significant decay
channel for the qubit excitation. The decay rate of this process can be
obtained through Fermi’s Golden rule:

$\gamma=\omega\frac{R_{Q}}{2Z_{0}}\left(\frac{M}{L_{1}+L_{2}}\right)^{2}\left|\langle
0|\hat{\vartheta^{-}}|1\rangle\right|^{2},$ (5)

where $\omega$ is the qubit frequency,
$Z_{0}=$50\text{\,}\mathrm{\SIUnitSymbolOhm}$$ is the control line impedance,
$R_{Q}$ is the von Klitzing constant, and $\langle
0|\hat{\vartheta^{-}}|1\rangle$ is the matrix element of the fluxonium mode
phase operator for the fundamental transition. We choose $M=12~{}\mathrm{pH}$
for the control wire inductance, which corresponds to a relaxation time of 1
ms in the flux degeneracy point. Inducing half a flux quantum in the SQUID
loop requires $83\text{\,}\mathrm{\SIUnitSymbolMicro A}$ of current. Due to
the lower frequency of the fluxonium, this current is lower than the current
required to induce the same flux in the SQUID of a typical transmon with the
same decay rate into the flux line. Lower control signal amplitudes are
beneficial because they help reducing RF crosstalk and give more flexibility
in signal chain attenuation and filtering at low temperatures.

A simplified scheme of the two qubit coupling design is shown in Fig. 2(a).
The system has three qubit-qubit coupling channels: direct capacitive
coupling, fluxonium mode-mediated coupling and harmonic mode-mediated
coupling. Due to the different symmetries of the harmonic mode and the
fluxonium mode, the coupling constants resulting from them have different
signs. By carefully choosing the mutual capacitances and mode frequencies, we
aim to utilize the destructive interference between the coupling channels and
minimize the static ZZ interaction between the qubits near the zero flux bias
point of the coupler.

The harmonic modes of the qubits also interact with the coupler. Since these
modes are out of resonance with the computational subspace, we exclude them in
the simulation of gate dynamics for the sake of computational efficiency.
However, due to their non-negligable contribution to the crosstalks, coupling
to these modes is accounted for in the calculation of static coupling terms.

The electric circuit schematic is shown in Fig. 2b. It consists of two
computational fluxonium qubits ($f_{1}$, $f_{2}$) each coupled to a tunable
coupler with fluxonium ($f_{\textnormal{C}}$) and harmonic
($h_{\textnormal{C}}$) modes with a coupling strength $g_{j\textnormal{f}}$
and $g_{j\textnormal{h}}$ (j = 1, 2), as well as to each other with a coupling
strength $g_{12}$. The Hamiltonian for the circuit is:

$\hat{H}_{\textnormal{full}}=\hat{H}_{\textnormal{f1}}+\hat{H}_{\textnormal{hc}}+\hat{H}_{\textnormal{fc}}+\hat{H}_{\textnormal{f2}}+\hat{H}_{\textnormal{V}}$
(6)

where first four terms describe the independent Hamiltonians for qubit and
coupler modes and $\hat{H}_{V}$ is responsible for the effective qubit-qubit
interaction. The interaction term has five contributions (see Supplementary
Information for the derivation): one term due to direct qubit-qubit coupling
(capacitive connection between the blue and green nodes), and four terms
corresponding to the interaction of either of the qubits to either of the
coupler modes (capacitive connection to red nodes in Fig. 2b). Due to the
different symmetries of the harmonic and fluxonium modes of the coupler,
effective couplings mediated by these modes interfere destructively, allowing
to cancel out either the XX or the ZZ coupling completely [22].

Figure 2: (color online) (a) Simplified system schematic. Two fluxonium qubits
($f_{1;2}$) are capacitively coupled via a coupler with harmonic
($h_{\textnormal{C}}$) and tunable fluxonium ($f_{\textnormal{C}}$) modes. The
plus and minus signs denote the sign of the $XX$ coupling constant between the
corresponding modes. (b) Electric circuit schematic. Each mode is highlighted
in different colours (qubit mode 1 (blue), qubit mode 2 (green), and coupler
mode c (red)). The computational qubits are biased at the flux degeneracy
point.

The natural gate available for this device is an iSWAP-like fSim gate[23]. In
our simulation, the gate is executed by applying a time-dependent flux to the
coupler, changing the coupler’s fluxonium mode frequency $f_{\textnormal{C}}$.
As the coupler’s fluxonium mode frequency gets close to the qubit frequencies,
the mediated interaction becomes resonant and energy exchange occurs. Due to
the finite anharmonicity of the fluxonium qubits, the interaction is not
purely transverse.

The effective interaction strength between the qubits can be obtained by
diagonalizing the full system Hamiltonian, eliminating the coupler degrees of
freedom, and building an effective low-energy Hamiltonian:

$\hat{H}_{\textnormal{eff}}/\hbar=-\frac{1}{2}\omega_{1}\sigma^{\textnormal{z}}_{1}-\frac{1}{2}\omega_{2}\sigma^{\textnormal{z}}_{2}+g_{\textnormal{xx}}\sigma^{\textnormal{x}}_{1}\sigma^{\textnormal{x}}_{2}+\frac{1}{4}\zeta_{\textnormal{zz}}\sigma^{\textnormal{z}}_{1}\sigma^{\textnormal{z}}_{2}.$
(7)

Details of the numerical calculations are presented in the Supplementary
Information. For equal-frequency data qubits, the energy gap between symmetric
and antisymmetric modes corresponds to the effective coupling
$2g_{\textnormal{xx}}(\Phi^{\textnormal{x}}_{\textnormal{C}})$ (Fig. 3a). The
parasitic ZZ crosstalk between $f_{1}$ and $f_{2}$ (Fig. 3b) is defined as
$\zeta_{ZZ}=\omega_{11}-\omega_{10}-\omega_{01}$.

Figure 3: (color online) Effective couplings as a functions of the magnetic
flux threading the coupler loop. (a) Effective transverse coupling strength
$2g_{\textnormal{XX}}(\Phi^{\textnormal{x}}_{\textnormal{C}})$. (b) ZZ
crosstalk $\zeta_{\textnormal{ZZ}}(\Phi^{\textnormal{x}}_{\textnormal{C}})$
Figure 4: Shape of drive flux signal and corresponding frequency of the
coupler fluxonium mode (inserted plots). (a) Data qubits have the same
frequencies. The gate can be optimized over the control flux pulse rise and
fall time and flat top duration. (b) Data qubits with different frequencies.
Here we can also optimize the control flux pulse edges, frequency and duration
of modulation.

Magnetic flux in the coupler can be used to turn on and off the effective
transverse qubit-qubit interaction. Near the zero flux bias point the
effective coupling is $40\ $\mathrm{kHz}$$ and increases to $13\
$\mathrm{MHz}$$ at the flux degeneracy point. At the same time, the parasitic
ZZ crosstalk can be reduced to around $5\ $\mathrm{kHz}$$ near the zero flux
bias point. Switching between coupling on and coupling off using flux bias may
induce resonant leakage into the fluxonium coupler mode, when its frequency
crosses the sum of the qubit frequencies, as shown in the Supplementary
Information. This resonance also gives rise in the singularity in the
$\zeta_{\textnormal{zz}}$ dependence on flux. In the operating point
($\Phi^{x}_{C}=0.5\Phi_{0}$) the parasitic ZZ crosstalk reaches
$\zeta_{ZZ}/2\pi=-1.5\ $\mathrm{MHz}$$ and causes phase accumulation of the
doubly excited state. In applications this phase accumulation can be
eliminated using an echo protocol.

The fSim family of two-qubit gates [5, 23] describes the set of excitation
number-preserving quantum logic operations on two qubits up to single-qubit
phase rotations. Its matrix representation in the $\ket{00}$, $\ket{01}$,
$\ket{10}$, $\ket{11}$ basis is given by:

$\operatorname{fSim}(\theta,\varphi)=\left(\begin{array}[]{cccc}1&0&0&0\\\
0&\cos\theta&-i\sin\theta&0\\\ 0&-i\sin\theta&\cos\theta&0\\\
0&0&0&e^{-i\varphi}\end{array}\right).$ (8)

Here we focus on the implementation of an $\sqrt{\mathrm{iSWAP}}$-like gate,
with $\theta=-\pi/4$. Due to the non-negligible ZZ crosstalk, our gate also
accumulates some small conditional phase $\varphi$. An important feature of
this gate is that its entangling power does not depend on $\varphi$, and two
such gates can be used to construct the maximally entangling CPHASE gate (see
Supplementary Material for the gate sequence). In combination with single-
qubit gates, $\operatorname{fSim}\left(-\pi/4,\varphi\right)$ gates can be
used to build any arbitrary two-qubit gate.

The interaction between the computational qubits can be adiabatically turned
on by slowly tuning the external magnetic flux in the coupler loop to the flux
degeneracy point ($\Phi^{\textnormal{x}}_{\textnormal{C}}=0.5\Phi_{0}$). Once
the coupler fluxonium mode frequency is close to the frequency of the data
qubits, their effective transverse coupling strength increases, inducing
vacuum Rabi oscillations between them. After half of a Rabi cycle, we
similarly turn off the coupler flux bias.

The pulse should be as short as possible while remaining adiabatic with
respect to leakage outside the computational subspace. The most probable
leakage scenarios involve populating the coupler fluxonium mode. To avoid
these transitions, we use a smooth pulse shape
$\Phi_{\mathrm{C}}^{\mathrm{x}}(t)$ with slow ramp close to the flux sweet
spot.

Figure 5: Time evolution of populations for four initial computational states
during the gate: (a-d) qubits with the same frequency; (e-h) frequency
difference of data qubits around 28 MHz. Obtained fidelities are $F\approx
0.9999$ and $F\approx 0.9996$, conditional phase $\varphi$ in the fSim gate is
$-0.07\pi$ and $-0.20\pi$ respectively. The state notation corresponds to the
mode occupations of the Hamiltonian (6) as follows:
$|f_{1}h_{C}f_{C}f_{2}\rangle$, where $f_{1}$, $f_{2}$ relate to computational
qubits, $h_{\textnormal{C}}$ and $f_{\textnormal{C}}$ are harmonic and
fluxonium modes of the tunable coupler.

The Hamiltonian of the system is given by the formula (6). In each mode of
excitation, the first three energy levels are taken into account. This
approximation captures the main effects of the system’s evolution. We simulate
the time evolution of the system by numerically solving the Schrödinger
equation with the computational stationary states as the initial conditions,
and compute the projections of the resulting states onto the computational
stationary states. The simulation accounts for leakage outside the
computational subspace, which can occur, for example, due to excitation of the
coupler degree of freedom, which results in the non-unitarity of the resulting
matrix. To simplify further analysis, we remove the single-qubit rotations
about the $z$-axis. We optimize the gate duration to get $\theta$ equal to
$-\pi/4$. The resulting 35-ns long pulse corresponds to an fSim gate with
$\varphi\approx-0.07\pi$ with fidelity $F\approx 0.9999$. We use the standard
expression for the two-qubit gate fidelity [25]:

$F=\frac{\text{Tr}(R_{\text{ideal}}^{\dagger}R)+4}{20}.$ (9)

Here, $R_{\text{ideal}}$ and $R$ are Pauli transfer matrices corresponding to
the actions of the closest ideal fSim gate and our simulated gate,
respectively. Time evolution of the computational states during the gate
operation are presented in Fig. 5(a-d).

In real devices, qubits may be detuned from each other. In that case, one can
use a parametric modulation approach and implement the very same gate by
replacing the flat-top pulse by a periodic modulation of the tunable coupler.

Here we suggest to modulate the drive flux near the operating point
($0.5\Phi_{0}$) with a sine wave profile at a frequency close to the energy
difference between the fundamental transitions of the computational qubits as
shown in Fig. 4(b). In this case we get $F\approx 0.9996$ with
$\varphi\approx-0.20\pi$ and the dynamics of the population of the
computational states is presented in Fig. 5(e-h). In this case we have also
optimized the drive pulse rise and fall times, as well as frequency and
duration of the flux modulation. The entire parametric gate duration is 67 ns
and can be reduced further by advanced flux pulse shaping.

Finally, we perform a decoherence-aware simulation of the gate by numerically
integrating the Lindblad equation with the fourth order Runge-Kutta method
with different collapse operators. The gate error is calculated as
$\epsilon=1-F$ where $F$ denotes the gate fidelity, see Eq. (9). We take into
account decoherence mechanisms involving only the ground and first excited
levels of each mode because the other levels are practically unoccupied during
the gate time (Fig. 5b) and hardly contribute to the resulting gate error. The
collapse operators corresponding to relaxation and dephasing are defined as:

$\displaystyle L_{1}=\frac{1}{\sqrt{T_{1}}}\left(\begin{array}[]{ccc}0&1&0\\\
0&0&0\\\
0&0&0\end{array}\right)\,L_{\varphi}=\frac{1}{\sqrt{2T_{\varphi}}}\left(\begin{array}[]{ccc}1&0&0\\\
0&-1&0\\\ 0&0&0\end{array}\right)$ (10)

The gate errors introduced by each decoherence channel are presented in Table
1. Apart from white noise that that can be modeled with the Lindblad equation,
gates based on flux tuning of SQUIDs are susceptible to low-frequency flux
noise. The characteristic time scales of this noise are usually significantly
longer than the gate duration and they can be approximated by a random static
flux shift during the gate. In the flux sweet spots the circuit is to first
order insensitive to flux noise, leaving the rising and falling edges of the
flux pulse most vulnerable to such noise.  For the simulations we use
estimates of the coherence times
$T_{1}=$300\text{\,}\mathrm{\SIUnitSymbolMicro s}$$ and
$T_{\varphi}=$300\text{\,}\mathrm{\SIUnitSymbolMicro s}$$ [15].  In the small-
error limit, errors are linear with respect to the decoherence rates. Our
simulation shows that the effect of decoherence on the data qubits contributes
on the level of $\sim 10^{-5}$ to the gate error, while the effect of coupler
decoherence is by a further order of magnitude smaller. Taking into account
the latest coherence results for fluxonium qubits in a 3D cavity [3], we
believe that improvements in fabrication techniques will likely continue to
enhance the coherence of planar devices. All time-domain simulations have been
carried out using the open-source packages TensorFlow and NumPy.

| Unitary | Relaxation | Dephasing
---|---|---|---
| errors | $T_{1}=$300\text{\,}\mathrm{\SIUnitSymbolMicro s}$$ | $T_{\varphi}=$300\text{\,}\mathrm{\SIUnitSymbolMicro s}$$
|  | $f_{1}$ | $f_{2}$ | $h_{C}$ | $f_{C}$ | $f_{1}$ | $f_{2}$ | $h_{C}$ | $f_{C}$
$\epsilon,\ 10^{-4}$ | 3.6 | 0.6 | 0.6 | 0.0 | 0.0 | 0.2 | 0.2 | 0.0 | 0.0
Table 1: Error budget. In the “unitary errors” column we show infidelity of
the gate due to leakage and non-excitation-number preserving processes, and in
the next eight columns we perform infidelity calculation for each decoherence
channel separately.

In conclusion, we have proposed an experimentally realizable tunable coupling
scheme for implementing scalable two-qubit fSim-type gates between fluxonium
qubits. The scheme is based on a simple base element with experimentally
accessible circuit parameters. The performance and properties of the circuit
have been simulated using numerical diagonalization of the circuit
Hamiltonian.

The gate fidelity in our scheme is mainly limited by unitary errors. The
largest contributions to non-unitary errors come from $T_{1}$ and
$T_{\varphi}$ times of the data qubits. These coherence times have been shown
to routinely exceed hundreds of microseconds in fluxonium devices. Our
proposed iSWAP-like parametrically driven gate provides a promising
alternative pathway towards high fidelity two-qubit gates using the existing
transmon-based designs. We emphasize that the low frequency of fluxonium
qubits opens the possibility of using sub-gigahertz wiring and electronics for
gate operations.

## Data availablity

The data that supports the findings of this study are available within the
article.

###### Acknowledgements.

Development of theoretical model was supported by the Russian Science
Foundation, Project (21-72-30026). Numerical simulations were supported by the
Ministry of Science and Higher Education of the Russian Federation (project
no. K2A-2018-048). This work was partially supported by Rosatom.

## References

  * [1] W. D. Oliver, P. B. Welander, Materials in superconducting quantum bits, MRS Bulletin, 38(10), 816 (2013).
  * [2] Z. L. Xiang, S. Ashhab, J. Q. You, F. Nori, Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems, Reviews of Modern Physics, 85(2), 623 (2013).
  * [3] Place, A.P.M., Rodgers, L.V.H., Mundada, P. et al. New material platform for superconducting transmon qubits with coherence times exceeding 0.3 milliseconds. Nat Commun 12, 1779 (2021).
  * [4] Petar Jurcevic, Ali Javadi-Abhari, Lev S. Bishop, and all. Demonstration of quantum Vol. 64 on a superconducting quantum computing system. Quantum Sci. Technol. 6 025020, (2021).
  * [5] Arute, F., Arya, K., Babbush, R. et al. Quantum supremacy using a programmable superconducting processor. Nature 574, 505–510 (2019). doi: 10.1038/s41586-019-1666-5
  * [6] Hong, Sabrina S. and Papageorge, Alexander T. and Sivarajah, Prasahnt and Crossman, Genya and Didier, Nicolas and Polloreno, Anthony M. and Sete, Eyob A. and Turkowski, Stefan W. and da Silva, Marcus P. and Johnson, Blake R. Demonstration of a parametrically activated entangling gate protected from flux noise. Phys. Rev. A, 101, 012302, 6 Jan 2020. doi: 10.1103/PhysRevA.101.012302
  * [7] Vladimir E. Manucharyan, Jens Koch, Leonid I. Glazman, Michel H. Devoret Fluxonium: Single Cooper-Pair Circuit Free of Charge Offsets Science, Vol. 326, pp. 113-116, (2009)
  * [8] Pop, I., Geerlings, K., Catelani, G. et al. Coherent suppression of electromagnetic dissipation due to superconducting quasiparticles. Nature 508, 369–372 (2014). https://doi.org/10.1038/nature13017
  * [9] V. E. Manucharyan, Superinductance, PhD thesis, Yale University (2012).
  * [10] Long B. Nguyen, Yen-Hsiang Lin, Aaron Somoroff, Raymond Mencia, Nicholas Grabon, and Vladimir E. Manucharyan. High-Coherence Fluxonium Qubit. Phys. Rev. X 9, 041041, 25 Nov 2019. doi: 10.1103/PhysRevX.9.041041.
  * [11] N. A. Masluk, Reducing the losses of the fluxonium artificial atom, PhD thesis, Yale University (2012).
  * [12] Helin Zhang, Srivatsan Chakram, Tanay Roy, Nathan Earnest, Yao Lu, Ziwen Huang, D.K. Weiss, Jens Koch, and David I. Schuster. Universal Fast-Flux Control of a Coherent, Low-Frequency Qubit. Phys. Rev. X 11, 011010.15 January 2021, doi: 10.1103/PhysRevX.11.011010.
  * [13] S. T. Skacel, C. Kaiser, S. Wuensch, H. Rotzinger, A. Lukashenko, M. Jerger, G. Weiss, M. Siegel, and A. V. Ustinov. Appl. Phys. Lett. 106, 022603 (2015).
  * [14] Richard Gebauer, Nick Karcher, Daria Gusenkova, Martin Spiecker, Lukas Grünhaupt, Ivan Takmakov, Patrick Winkel, Luca Planat, Nicolas Roch, Wolfgang Wernsdorfer, Alexey V. Ustinov, Marc Weber, Martin Weides, Ioan M. Pop, Oliver Sander, ”State preparation of a fluxonium qubit with feedback from a custom FPGA-based platform.” https://arxiv.org/abs/1912.06814
  * [15] Helin Zhang, Srivatsan Chakram, Tanay Roy, Nathan Earnest, Yao Lu, Ziwen Huang, D. K. Weiss, Jens Koch, and David I. Schuster Universal Fast-Flux Control of a Coherent, Low-Frequency Qubit. Phys. Rev. X 11, 011010 (2021).
  * [16] Aaron Somoroff, Quentin Ficheux, Raymond A. Mencia, Haonan Xiong, Roman Kuzmin, and Vladimir E. Manucharyan. Millisecond coherence in a superconducting qubit. https://arxiv.org/abs/2103.08578v1
  * [17] Quentin Ficheux, Long B. Nguyen, Aaron Somoroff, Haonan Xiong, Konstantin N. Nesterov, Maxim G. Vavilov, and Vladimir E. Manucharyan. Fast logic with slow qubits: microwave-activated controlled-Z gate on low-frequency fluxoniums. Phys. Rev. X 11, 021026,3 May 2021. doi: 10.1103/PhysRevX.11.021026.
  * [18] Nesterov, K. N., Ficheux, Q., Manucharyan, V. E., and Vavilov, M. G. Proposal for Entangling Gates on Fluxonium Qubits via a Two-Photon Transition. PRX Quantum (2021). doi:10.1103/prxquantum.2.020345
  * [19] Fei Yan, Youngkyu Sung, Philip Krantz, Archana Kamal, David K. Kim, Jonilyn L. Yoder, Terry P. Orlando, Simon Gustavsson, and William D. Oliver. Engineering Framework for Optimizing Superconducting Qubit Designs. arXiv:2006.04130v1 (2020)
  * [20] Fei Yan, Philip Krantz, Youngkyu Sung, Morten Kjaergaard, Daniel L. Campbell, Terry P. Orlando, Simon Gustavsson, and William D. Oliver. Tunable Coupling Scheme for Implementing High-Fidelity Two-Qubit Gates. Phys. Rev. Applied 10, 054062, 28 Nov 2018. doi: 10.1103/PhysRevApplied.10.054062.
  * [21] X. Li, T. Cai, H. Yan, Z. Wang, X. Pan, Y. Ma, W. Cai, J. Han, Z. Hua, X. Han, Y. Wu, H. Zhang, H. Wang, Yipu Song, Luming Duan, and Luyan Sun. Tunable Coupler for Realizing a Controlled-Phase Gate with Dynamically Decoupled Regime in a Superconducting Circuit. Phys. Rev. Applied 14, 024070, 25 Aug 2020
  * [22] Mundada, P., Zhang, G., Hazard, T., and Houck, A. Suppression of Qubit Crosstalk in a Tunable Coupling Superconducting Circuit. Phys. Rev. Appl. (2019). doi:10.1103/PhysRevApplied.12.054023
  * [23] B. Foxen, C. Neill, A. Dunsworth et al. Demonstrating a Continuous Set of Two-Qubit Gates for Near-Term Quantum Algorithms. Phys. Rev. Lett. 125, 120504, 15 Sep 2020 doi: 10.1103/PhysRevLett.125.120504,
  * [24] Arute, F., Arya, K., Babbush, R. et al. Quantum supremacy using a programmable superconducting processor. Nature 574, 505–510 (2019). https://doi.org/10.1038/s41586-019-1666-5
  * [25] Michael A Nielsen, A simple formula for the average gate fidelity of a quantum dynamical operation, Physics Letters A, Vol. 303, Issue 4, 2002, Pages 249-252, ISSN 0375-9601,

##  Appendix A COMPARISON WITH OTHER PARAMETER REGIMES

Fluxonium qubits can be described in the framework of the generalized flux
qubit [1] system. For sufficiently long chains of junctions used to implement
shunt inductance, generalized flux qubits are essentially RF SQUIDs and can be
described by three parameters: charging energy $E_{C}$, Josephson energy
$E_{J}$ and shunt inductance energy $E_{L}$. In compare to previous RF-SQUID
type qubits, fluxonium[2] utilizes a chain of Josephson junctions, which
allows to exceed the vacuum impedance and operate in the $E_{J}\gg E_{C}\gg
E_{C}$ regime. Additional capacitive shunting of the phase slip junction and
reduction of energy participation ratios of interfaces and improves coherence
times[3, 4]. Extreme shunting of the phase slip junction, both inductive and
capacitive, significantly lowers the qubit frequency and reduces sensitivity
to AC voltage; the corresponding parameter regime has been dubbed heavy
fluxonium [5, 6]. Between fluxonium-type qubits with coherent tunneling in a
double-well potential and transmon qubits with plasma oscillations in a weakly
anharmonic potential lies the quarton[1] which is characterized by
$E_{J}=E_{L}$. We compare anharmonicity, qubit frequency and coupling strength
between two identical capacitively coupled qubits biased at half flux quantum
for different ratios of $E_{L}/E_{C}$ and $E_{J}/E_{C}$. For this purpose we
consider the Hamiltonian

$\hat{H}=\sum\limits_{\alpha=1,2}4E_{C}\hat{n}_{\alpha}^{2}+E_{J}\cos\hat{\varphi}_{\alpha}+\frac{1}{2}E_{L}\hat{\varphi}_{\alpha}^{2}\\\
+4\kappa E_{C}\hat{n}_{1}\hat{n}_{2},$ (11)

which corresponds to two capacitively coupled fluxonium qubits (Fig. 6). The
charging energy $E_{C}=e^{2}/(2C_{\Sigma})$ is defined by the effective
fluxonium capacitance $C_{\Sigma}=(C+2C_{C})/(1+C/C_{Q})$, and the effective
capacitive coupling ratio $\kappa=C_{C}/(C+C_{C})$ cannot exceed 1.

Figure 6: (color online) Equivalent lumped-element circuit for the two
capacitively coupled generalized flux qubits. Each qubit circuit is
highlighted in different colours (qubit 1 (blue), qubit 2 (green)). $L_{i}$
stand for inductors, $C_{i}$ stand for the capacitances with respect to the
ground electrode, $C_{C}$ are the mutual capactitances between nodes $1$ and
$2$ that facilitate coupling between the qubits.

Results of the comparison are shown in Fig. 7. For presentation purposes, the
parameter regimes demonstrated for regular fluxoniums [2, 3, 4], heavy
fluxoniums [5, 6], and quarton qubits [1], as well as our proposed design, are
shown with solid markers.

Figure 7: (color online) Dependence of the two-qubit system parameters on the
qubit Josephson junction energy and inductive energy. a) Qubit frequency; b)
anharmonicity; c) the effective coupling strength between two capacitively
coupled qubits.

The frequency and coupling ratio-normalized capacitive coupling strength shown
in Fig. 7b is limited to $0.5$. This maximal normalized coupling is realized
in the harmonic oscillator limit $E_{L}\gg E_{J}$ and in transmon qubits. We
propose to operate capacitively coupled fluxoniums in the frequency regime
typical for regular fluxoniums $\sim$0.5\text{\,}\mathrm{GHz}$$, while
maintaining a $E_{J}/E_{L}$ ratio close to unity characteristic to quarton
qubits, which does not significanlty degrade coupling strength. At the same
time, the relative qubit anharmonicity is significantly larger than the
asympototical 0.3 value of $E_{J}\gg E_{C}$ quartons.

It should be noted that the coupling strength degradation only applies to the
fundamental qubit transition. Capacitive coupling to other transitions of
fluxoniums can be effective even for $E_{J}/E_{L}\sim 10$, allowing fast two-
qubit gates as shown in the work [7].

##  Appendix B FULL-CIRCUIT HAMILTONIAN AND QUANTIZATION

The extended circuit model implementing our proposal is shown in Fig. 8. Each
of the three elements is treated as a modified heavy fluxonium formed by two
capacitors $C_{i}$, two inductors $L_{i}$, where $i=1,\dots,6$, and a
Josephson junction $J_{\lambda}$, where $\lambda=1,C,2$. The external fluxes
$\Phi^{\textnormal{x}}_{\lambda}$ are applied to loops of the computational
qubits and coupler.

Figure 8: (color online) Equivalent lumped-element circuit for the proposed
two qubit scheme with a tunable coupler. Each heavy fluxonium circuit is
highlighted in different colours (qubit 1 (blue), qubit 2 (green), and coupler
C (red)). $L_{i}$ stand for superinductors, $C_{i}$ stand for the electrode
capacitances with respect to the ground electrode, $C_{J\lambda}$
($\lambda=1,C,2$) are the capacitance of Josephson junctions, $C_{ij}$ are the
mutual capactitances between nodes $i$ and $j$ that facilitate coupling
between the qubits.

We choose node fluxes $\phi_{i}$, corresponding to nodes $i$ in Fig. 8, as the
generalized coordinates of the system. We can write down the circuit
Lagrangian $L(\phi_{i},\dot{\phi_{i}})$ using node fluxes together with the
voltages $\dot{\phi}_{i}$:

$L=T-U,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ (12)
$T=\frac{1}{2}\big{[}C_{1}\dot{\phi}_{1}^{2}+C_{2}\dot{\phi}_{2}^{2}+C_{\textnormal{J1}}(\dot{\phi}_{2}-\dot{\phi}_{1})^{2}+C_{3}\dot{\phi}_{3}^{2}+\\\
C_{4}\dot{\phi}_{4}^{2}+C_{\textnormal{JC}}(\dot{\phi}_{4}-\dot{\phi}_{3})^{2}+C_{5}\dot{\phi}_{5}^{2}+C_{6}\dot{\phi}_{6}^{2}+\\\
C_{\textnormal{J2}}(\dot{\phi}_{6}-\dot{\phi}_{5})^{2}+C_{13}(\dot{\phi}_{3}-\dot{\phi}_{1})^{2}+C_{23}(\dot{\phi}_{3}-\dot{\phi}_{2})^{2}+\\\
C_{45}(\dot{\phi}_{5}-\dot{\phi}_{4})^{2}+C_{46}(\dot{\phi}_{6}-\dot{\phi}_{4})^{2}+C_{24}(\dot{\phi}_{4}-\dot{\phi}_{2})^{2}+\\\
C_{35}(\dot{\phi}_{5}-\dot{\phi}_{3})^{2}C_{25}(\dot{\phi}_{5}-\dot{\phi}_{2})^{2}\big{]},$
(13)
$U=E_{\textnormal{J1}}[1-\cos(\frac{2\pi(\phi_{2}-\phi_{1})}{\Phi_{0}})]+\\\
E_{\textnormal{JC}}[1-\cos(\frac{2\pi(\phi_{4}-\phi_{3})}{\Phi_{0}})]+E_{\textnormal{J2}}[1-\cos(\frac{2\pi(\phi_{6}-\phi_{5})}{\Phi_{0}})]+\\\
\frac{1}{2L_{1}}\phi_{1}^{2}+\frac{1}{2L_{2}}(\phi_{2}-\phi^{\textnormal{x}}_{1})^{2}+\frac{1}{2L_{3}}\phi_{3}^{2}+\\\
\frac{1}{2L_{4}}(\phi_{4}-\phi^{\textnormal{x}}_{C})^{2}+\frac{1}{2L_{5}}\phi_{5}^{2}+\frac{1}{2L_{6}}(\phi_{6}-\phi^{\textnormal{x}}_{2})^{2},$
(14)

where $T$ and $U$ are, respectively, the kinetic and potential energy.

The kinetic energy term can be rewritten in matrix form
$T=\frac{1}{2}\vec{\dot{\phi}}^{T}C_{\textnormal{mat}}\vec{\dot{\phi}}$, where
$\vec{\dot{\phi}}=[\dot{\phi}_{1},\dot{\phi}_{2},\dot{\phi}_{3},\dot{\phi}_{4},\dot{\phi}_{5},\dot{\phi}_{6}]$
and $C_{\textnormal{mat}}$ is a $6\times 6$ capacitance matrix:

$C_{\textnormal{mat}}=\begin{bmatrix}C_{\textnormal{f1}}&-C_{\textnormal{J1}}&-C_{13}&0&0&0\\\
-C_{\textnormal{J1}}&C_{\textnormal{f2}}&-C_{23}&-C_{24}&-C_{25}&0\\\
-C_{13}&-C_{23}&C_{\textnormal{f3}}&-C_{\textnormal{JC}}&-C_{35}&0\\\
0&-C_{24}&-C_{\textnormal{JC}}&C_{\textnormal{f4}}&-C_{45}&-C_{46}\\\
0&-C_{25}&-C_{35}&-C_{45}&C_{\textnormal{f5}}&-C_{\textnormal{J2}}\\\
0&0&0&-C_{46}&-C_{\textnormal{J2}}&C_{\textnormal{f6}}\end{bmatrix},\\\ $ (15)

where

$C_{\textnormal{f1}}=C_{1}+C_{\textnormal{J1}}+C_{13},\ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \\\
C_{\textnormal{f2}}=C_{2}+C_{\textnormal{J1}}+C_{23}+C_{24}+C_{25},\ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\\
C_{\textnormal{f3}}=C_{3}+C_{\textnormal{JC}}+C_{13}+C_{23}+C_{35},\ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\\
C_{\textnormal{f4}}=C_{4}+C_{\textnormal{JC}}+C_{24}+C_{45}+C_{46},\ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\\
C_{\textnormal{f5}}=C_{5}+C_{\textnormal{J2}}+C_{45}+C_{35}+C_{25},\ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\\
C_{\textnormal{f6}}=C_{6}+C_{\textnormal{J2}}+C_{46}.\ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ (16)

To simplify further calculations, the superinductances and capacitances in
each fluxonium are set equal, $L_{1}=L_{2}=L_{\textnormal{Q1}}$,
$L_{3}=L_{4}=L_{\textnormal{QC}}$, $L_{5}=L_{6}=L_{\textnormal{Q2}}$,
$C_{f1}=C_{f2}=C_{\textnormal{Q1}}$, $C_{f3}=C_{f4}=C_{\textnormal{QC}}$,
$C_{f5}=C_{f6}=C_{\textnormal{Q2}}$.

Neglecting capacitive interactions between the qubits, the circuit normal
modes can be defined as

$\theta^{+}_{1}=\phi_{1}+\phi_{2};\ \ \ \ \ \theta^{-}_{1}=\phi_{1}-\phi_{2};\
\ \ \ \\\ \theta^{+}_{C}=\phi_{3}+\phi_{4};\ \ \ \ \
\theta^{-}_{C}=\phi_{3}-\phi_{4};\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \\\ \theta^{+}_{2}=\phi_{5}+\phi_{6};\ \ \ \ \
\theta^{-}_{2}=\phi_{5}-\phi_{6}.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ (17)

Appling this coordinate transformation to the capacitance matrix yields

$C_{\textnormal{new}}=T_{r}^{T}\times C_{\textnormal{mat}}\times T_{r},\ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ (18)

where the transformation matrix $T_{r}$ is defined as:

$T_{r}=\frac{1}{2}\begin{bmatrix}1&1&0&0&0&0\\\ 1&-1&0&0&0&0\\\ 0&0&1&1&0&0\\\
0&0&1&-1&0&0\\\ 0&0&0&0&1&1&\\\ 0&0&0&0&1&-1&\end{bmatrix}.\ \ \ \ \ \ \ \ \ \
\ \ \ \ \ $ (19)

The potential energy becomes

$U=\sum_{i=1,C,2}\bigg{[}E_{\textnormal{J}i}[1-\cos(\frac{2\pi\theta^{-}_{i}}{\Phi_{0}})]+\\\
\frac{1}{4L_{\textnormal{Qi}}}(\theta^{+}_{i}-\phi^{\textnormal{x}}_{i})^{2}+\frac{1}{4L_{\textnormal{Q}i}}(\theta^{-}_{i}-\phi^{\textnormal{x}}_{i})^{2}\bigg{]}.$
(20)

We define the canonically conjugate momenta ${q^{\pm}}_{i}$ corresponding to
the variables introduced in Eq. (17) as

$q^{\pm}_{i}=\frac{\partial L}{\partial{\dot{\theta}^{\pm}}_{i}},\ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ $ (21)

and the canonical momentum vector
$\vec{q}=[q^{+}_{1},q^{-}_{1},q^{+}_{C},q^{-}_{C},q^{+}_{2},q^{-}_{2}]$.

The system Hamiltonian in terms of the first-order normal modes is defined as

$H=\sum_{i,\alpha}q^{\alpha}_{i}{\dot{\theta}^{\alpha}}_{i}-L=\frac{1}{2}\vec{q}^{T}C^{-1}_{\textnormal{new}}\vec{q}+U,\
\ \ \ \ $ (22)

where $C^{-1}_{\textnormal{new}}$ is the inverse capacitance matrix.

Finally, promoting classical degrees of freedom to quantum operators, we
obtain

$\hat{H}=\sum_{\alpha}\hat{H}_{\alpha}+\sum_{\alpha\not=\beta}\hat{H}_{\alpha\beta},\
\
\\{\alpha,\beta\\}\in\\{\textnormal{h}_{1},\textnormal{f}_{1},\textnormal{h}_{\textnormal{C}},\textnormal{f}_{\textnormal{C}},\textnormal{h}_{2},\textnormal{f}_{2}\\}.$
(23)

The indeces $\textnormal{h}_{i}$ and $\textnormal{f}_{j}$ correspond to the
Hamiltonian terms associated with the symmetric $\theta^{+}_{i}$ and
antisymmetric $\theta^{-}_{i}$ mode coordinates. The symmetric modes are
described by harmonic oscillator-type Hamiltonians

$\hat{H}_{\textnormal{h}i}=4E_{\textnormal{C}{\textnormal{h}i}}({\hat{n}^{+}}_{i})^{2}+\frac{1}{2}E_{L{\textnormal{h}i}}(\vartheta^{+}_{i}-\varphi^{\textnormal{x}}_{i})^{2},$
(24)

while the antisymmetric modes are described by fluxonium-type Hamiltonians

$\hat{H}_{\textnormal{f}i}=4E_{\textnormal{C}{\textnormal{f}i}}({\hat{n}^{-}}_{i})^{2}+E_{\textnormal{J}i}[1-\cos(\vartheta^{-}_{i})]+\frac{1}{2}E_{\textnormal{L}{\textnormal{f}i}}(\vartheta^{-}_{i}-\varphi^{\textnormal{x}}_{i})^{2}.$
(25)

where the dimensionless variables for the flux
${\hat{\vartheta}^{\alpha}}_{i}=2\pi{\hat{\theta}^{\alpha}}_{i}/\Phi_{0}$ and
their canonically conjugate Cooper pair numbers
${\hat{n}^{\alpha}}_{i}={\hat{q}^{\alpha}}_{i}/2e$ are introduced. The
inductive and capacitive energies are defined as

$E_{L{\textnormal{h}i}}=E_{L{\textnormal{f}i}}=\frac{[\Phi_{0}/(2\pi)]^{2}}{2L_{\textnormal{Q}i}},\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ (26)
$E_{C\alpha}=\frac{e^{2}}{2}\left(C_{\text{new}}^{-1}\right)_{\alpha\alpha}=\frac{[\Phi_{0}/(2\pi)]^{2}}{2L_{\textnormal{Q}i}},\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ (27)

where $\left(C_{\text{new}}^{-1}\right)_{\alpha\alpha}$ is the diagonal matrix
element of the inverse capacitance matrix corresponding to the variable
$\alpha$,
$\alpha\in\\{\text{h}_{1},\text{f}_{1},\text{h}_{C},\text{f}_{C},\text{h}_{2},\text{f}_{2}\\}$
and the dimensionless external fluxes are defined as

$\varphi^{\textnormal{x}}_{i}=\frac{2\pi}{\Phi_{0}}\phi^{\textnormal{x}}_{i}.\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ (28)

The double-indexed terms $\hat{H}_{\alpha\beta}$ in Eq.(23) describe the
capacitive coupling between different modes. In a symmetric circuit, direct
interaction between the harmonic and fluxonium modes on the same node vanish:

$\hat{H}_{\textnormal{h1}\textnormal{f1}}=0,\ \ \
\hat{H}_{\textnormal{hc}\textnormal{fc}}=0,\ \ \
\hat{H}_{\textnormal{h2}\textnormal{f2}}=0.\ \ \ $ (29)

The simplified Hamiltonian in the main text of the article Eq. 5 can be
obtained by dropping the harmonic mode terms of the computational qubits,
yielding

$\hat{H}_{\textnormal{full}}=\hat{H}_{\textnormal{f1}}+\hat{H}_{\textnormal{hc}}+\hat{H}_{\textnormal{fc}}+\hat{H}_{\textnormal{f2}}+\hat{H}_{\textnormal{V}},\
\ \ \ \ $ (30)

where the interaction $\hat{H}_{\textnormal{V}}$ of two qubits consists of
five terms: the direct coupling ($\hat{H}_{\textnormal{f1}\textnormal{f2}}$),
the indirect coupling via the coupler harmonic mode
($\hat{H}_{\textnormal{f1}\textnormal{hc}}$ and
$\hat{H}_{\textnormal{hc}\textnormal{f2}}$) and the indirect coupling via the
coupler fluxonium mode ($\hat{H}_{\textnormal{f1}\textnormal{fc}}$ and
$\hat{H}_{\textnormal{fc}\textnormal{f2}}$).

Note that this description is not entirely accurate, as the harmonic modes do
interact with the fluxonium modes of the computational qubit due to their
coupling to the coupler’s modes. Moreover, circuit asymmetry and nonlinearity
in the superinductor can also contribute to the interaction between the
fluxonium and harmonic modes on a single node. The contribution of the
harmonic modes of the qubits to the effective qubit-qubit interactions leads
to a small renormalization of the low-energy Hamiltonian. We include these
modes in our static Hamiltonian simulations, specifically for the static ZZ-
interaction, and neglect them in the gate simulations.

The circuit parameters used for the following calculations are
$C_{1}=C_{6}=70.53\ $\mathrm{fF}$$, $C_{2}=C_{5}=51.17\ $\mathrm{fF}$$,
$C_{3}=C_{4}=49.17\ $\mathrm{fF}$$, $C_{J1}=C_{JC}=C_{J2}=1.056\
$\mathrm{fF}$$, $C_{25}=0.167\ $\mathrm{fF}$$, $C_{23}=C_{45}=19.20\
$\mathrm{fF}$$, $C_{13}=C_{46}=0.176\ $\mathrm{fF}$$, $C_{24}=C_{35}=0.234\
$\mathrm{fF}$$,
$E_{\textnormal{J1}}=E_{\textnormal{JC}}=E_{\textnormal{J2}}=2.14\
$\mathrm{GHz}$$, $E_{L1}=E_{L2}=E_{L5}=E_{L6}=1.514\ $\mathrm{GHz}$$,
$E_{L3}=E_{L4}=1.634\ $\mathrm{GHz}$$. This choice of capacitances allowed us
to reach the desired values of qubit frequencies and effective qubit-qubit
coupling. The Josephson junction energies and inductive energies are
accessible within the fabrication techniques used in our previous work [8].
For the phase slip element we propose to use a $S_{1}\approx 100\times 90\
$\mathrm{nm}$^{2}$ Josephson junction, and for the superinductance an array
($N\approx 80$) of series-connected of big Josephson junctions ($S_{2}\approx
1000\times 500\ $\mathrm{nm}$^{2}$). All junctions can be fabricated by the
shadow evaporation technique with critical current density $j=0.5\
$\mathrm{\SIUnitSymbolMicro A}$/$\mathrm{\SIUnitSymbolMicro m}$^{2}$.

##  Appendix C NUMERICAL RESULTS

In this Appendix we present the results of numerical calculation of the full
system Hamiltonian. We found the eigenvalues and charge matrix elements for
all independent fluxonium and harmonic modes from Eqs. (24),(25) using
numerical diagonalization. The data qubits are design to be kept in the lower
flux sweet spot ($\varphi_{1,2}^{\text{x}}=\pi$), while the magnetic flux in
the coupler loop is varied between zero flux and half flux quantum
($\varphi_{C}^{\text{x}}\in\left[0,\pi\right]$).

Figure 9: (color online) a) Energy levels of the tunable system vs magnetic
flux in the coupler ${\Phi}^{\textnormal{x}}_{\textnormal{C}}$. b) The red
dotted rectangle outlines eigenenergies of the data qubits one-excitation
manifold.

To specify the complete Hamiltonian we used the open-source QuTiP[9] package.
In each fluxonium-type mode we took the first five levels, and in each
harmonic mode we took the first three levels and used corresponding matrix
elements to take into account the terms responsible for the interaction (30).
Finally, we numerically diagonalized the full Hamiltonian. The computed energy
spectrum as a function of magnetic flux
${\Phi}^{\textnormal{x}}_{\textnormal{C}}$ is plotted in Fig. 9a.

Full system eigenstates are labeled as
$\ket{n_{\textnormal{h1}},n_{\textnormal{f1}},n_{\textnormal{hc}},n_{\textnormal{fc}},n_{\textnormal{h2}},n_{\textnormal{f2}}}$,
where $n_{\alpha}$ is the occupancy of the $\alpha$-mode,
$\alpha\in\\{\text{h}_{1},\text{f}_{1},\text{h}_{C},\text{f}_{C},\text{h}_{2},\text{f}_{2}\\}$.
The five lowest-lying levels are labeled in Fig. 9a. These levels play a key
role in the two-qubit gates. Since the computational levels of first qubit
$\ket{010000}$ and second qubit $\ket{000001}$ are degenerate (Fig. 9b), the
eigenstates are their symmetric (green line) and antisymmetric (orange line)
combinations, and the energy gap between these states corresponds to the
effective $XX$ coupling.

Figure 10: (color online) Dependence of the low-energy effective Hamiltonian
parameters on the critical current of small and large Josephson junctions. a)
The effective coupling at the zero flux bias point
$g^{\textnormal{off}}_{\textnormal{xx}}=g_{\textnormal{xx}}(\Phi_{\textnormal{C}}=0)$;
b) the effective coupling at the flux degeneracy point
$g^{\textnormal{on}}_{\textnormal{xx}}=g_{\textnormal{xx}}(\Phi_{\textnormal{C}}=\Phi_{0})$;
d) parasitic ZZ crosstalk at the zero flux bias point
$\zeta^{\textnormal{off}}_{\textnormal{zz}}$; e) parasitic ZZ crosstalk at the
flux degeneracy point $\zeta^{\textnormal{on}}_{\textnormal{zz}}$. c),f),g),h)
Qubit and coupler frequencies $f^{\textnormal{off}}_{\textnormal{Q}}$ and
$f^{\textnormal{on}}_{\textnormal{Q}}$,
$f^{\textnormal{off}}_{\textnormal{C}}$ and
$f^{\textnormal{on}}_{\textnormal{C}}$ at the zero flux bias point and at the
flux degeneracy point of the coupler. i) Data qubit anharmonicity $\delta
f^{\textnormal{off}}_{\textnormal{Q}}$.

##  Appendix D CRITICAL CURRENT DEPENDENCE

A crucial issue for large scale Josephson junction based circuits is
robustness with respect to critical current deviations of small junctions. The
aim of this section is to identify how these deviations affect the effective
low-energy Hamiltonian parameters. We sweep the critical current value of
small Josephson junctions used as the nonlinear element for data qubits and
coupler (for simplicity we consider them the same) and large Josephson
junctions used in superinductances arrays. The data qubits’ superinductances
consist of 41 junctions, while the coupler’s superindutances have 38 junctions
each, which results in the coupler frequency being $\approx 100\ \mathrm{MHz}$
higher in the flux degeneracy point. The result of this calculation are shown
in Fig. 10.

Figure 11: Suitable critical current values. Black area indicates the range of
critical currents values allowing one to implement the proposed scheme of two
fluxonium qubits in the desired range of low energy effective Hamiltonian
parameters.

Here we found the effective coupling at the zero flux bias point and the flux
degeneracy point in the coupler loop ($g^{\textnormal{off}}_{\textnormal{xx}}$
and $g^{\textnormal{on}}_{\textnormal{xx}}$ respectively) as well as parasitic
ZZ crosstalk ($\zeta^{\textnormal{off}}_{\textnormal{zz}}$ and
$\zeta^{\textnormal{on}}_{\textnormal{zz}}$ respectively). We also defined
data qubits frequencies $f^{\textnormal{off}}_{\textnormal{Q}}$ and
$f^{\textnormal{on}}_{\textnormal{Q}}$ and coupler frequencies
$f^{\textnormal{off}}_{\textnormal{C}}$ and
$f^{\textnormal{on}}_{\textnormal{C}}$ at the coupler zero flux bias point and
the flux degeneracy point. For the sake of completeness we also present here
data qubit anharmonicity $\delta f^{\textnormal{off}}_{\textnormal{Q}}$. Fig.
11 shows the region (black area) with suitable critical current values, at
which the proposed tunable coupling scheme can be physically implemented. This
region was defined from the conditions: $8\
\mathrm{MHz}<g^{\textnormal{on}}_{\textnormal{xx}}<30\ \mathrm{MHz}$,
$g^{\textnormal{off}}_{\textnormal{xx}}<0.5\ \mathrm{MHz}$,
$|\zeta^{\textnormal{off}}_{\textnormal{zz}}|<5\ \mathrm{kHz}$,
$|\zeta^{\textnormal{on}}_{\textnormal{zz}}|<1.5\ \mathrm{MHz}$, $200\
\mathrm{MHz}<f^{\textnormal{off}}_{\textnormal{Q}}<600\ \mathrm{MHz}$, $\delta
f^{\textnormal{off}}_{\textnormal{Q}}>1.2\ \mathrm{GHz}$. It should be noted
that the Fig. 11 is shown as an example and the selected conditions are not
strict.

##  Appendix E CONSTRUCTION OF THE CPHASE GATE

The control parameter used to implement the two-qubit gates, the coupler flux,
changes both qubit frequencies, XX and ZZ couplings at the same time. As a
result, the two-qubit gate family that can be implemented using this method is
equivalent to $\operatorname{fSim}(\theta,\varphi)$, with both $\theta$ and
$\phi$ somehow depending on the control signal $\Phi_{C}^{x}(t)$ applied to
the coupler flux line.

& []H []U_1(-φ) [2]fSim(π4, φ) []X [2]fSim(π4, φ) []U_1(-φ) []H []S
[]H []U_1(-φ) []U_1(-φ) []H []S^† .

Figure 12: Construction of the CPHASE gate from two fSim gates with
$\theta=\pi/4$ and arbitrary conditional phase angle $\varphi$

A wide range of quantum algorithms relies on the CPHASE gate. To construct the
CPHASE we gate using our proposed two-qubit scheme, we propose the spin-echo
technique initially devised to remove the conditional phase from cross-
resonance gates [10]. The gate sequence implementing a CPHASE gate is shown in
Fig. 12. The gate sequence consists of two two-qubit fSim gates interleaved by
single-qubit gates. In applications the single-qubit gates before and after
the fSim gates can be merged together with other gates for better fidelity.

##  Appendix F COUPLING OF HARMONIC AND FLUXONIUM MODE

The presence of finite asymmetry in the qubit capacitances and inductances
translates into coupling between the harmonic and fluxonium mode. For a single
qubit circuit, we introduce the capacitance and inductance asymmetries $\delta
C,\delta L$, with $C_{1}=C+\delta C/2$, $C_{2}=C-\delta C/2$, $L_{1}=L+\delta
L/2$, $L_{2}=L-\delta L/2$.

For small asymmetries the Hamiltonian perturbation is defined as

$\hat{V}=\frac{4e^{2}\delta C}{C(C+2C_{J})-\delta
C^{2}/4}\hat{n}_{f}\hat{n}_{h}+\\\ \frac{\hbar^{2}}{4e^{2}}\frac{\delta
L}{L^{2}-\delta L^{2}/4}\hat{\varphi}_{f}\hat{\varphi}_{h},$ (31)

which is Jaynes-Cummings type Hamiltonian for a fluxonium qubit coupled to a
resonator. When the qubit is biased at half flux quantum, the frequency
detuning between the two modes is large. In this dispersive regime excitations
in the resonator mode induce a dispersive shift $\chi$ in the qubit frequency
which is quadratic in $\delta L$ and $\delta C$. For relative asymmetries
$\delta C/C$ and $\delta L/L$ of 5% in both capacitance and inductance the
dispersive shift arising from this coupling is $\chi=23~{}\mathrm{MHz}$.

Another source of dispersive shifts is the nonlinearilty of the
superinductors. In the proposed design with $N=80$ junction in each inductor,
the first nonlinear correction to the superinductance Hamiltonian is given by

$\hat{V}=-\frac{E_{L}}{24N^{2}}\left(\hat{\varphi}_{f}+\hat{\varphi}_{h}\right)^{4}.$
(32)

From first-order perturbation theory we obtain a cross-Kerr coefficient of
$\chi=0.5~{}\mathrm{MHz}$.

Similar to the case of 0-$\pi$ qubits, thermal excitation may degrade qubit
coherence times [11]. The pure dephasing rate associated with this process can
be estimated for low thermal harmonic mode occupancies $n_{\mathrm{th}}$ by
the formula[12]

$\gamma_{\varphi}=\frac{n_{\mathrm{th}}\kappa\chi^{2}}{\chi^{2}+\kappa^{2}},$
(33)

where $n_{\mathrm{th}}$ is the photon number, $\kappa$ is the harmonic mode
decay rate and $\chi$ is the dispersive shift. We expect that in real devices
$\chi\gg\kappa$. The thermal population of the harmonic mode can be estimated
as $n_{\mathrm{th}}\approx 10^{-4}$ for $T=10~{}\mathrm{mK}$.

The decay rate for the harmonic mode $\kappa$ can be obtained through Fermi’s
Golden rule:

$\kappa=\omega\frac{2\pi
Z_{0}}{R_{Q}}\left(\frac{C_{\mathrm{an}}}{C+C_{\mathrm{an}}}\right)^{2}\left|\langle
0|\hat{n}^{+}|1\rangle\right|^{2},$ (34)

where $\omega$ is the harmonic mode frequency,
$Z_{0}=$50\text{\,}\mathrm{\SIUnitSymbolOhm}$$ is the control line impedance,
$R_{Q}$ is the von Klitzing constant, and $\langle 0|\hat{n}^{+}|1\rangle$ is
the matrix element of the harmonic mode charge operator for the fundamental
transition. We choose $C_{\mathrm{an}}=0.34~{}\mathrm{fF}$ for the coupling
capacitance with microwave antenna (Fig. 1 from the main article), which
corresponds to a decay rate $\kappa=0.01~{}\mathrm{MHz}$. From Eq. (33) we
obtain $T_{\varphi}>1~{}\mathrm{s}$ for $T=10~{}\mathrm{mK}$.

## References

  * [1] Fei Yan, Youngkyu Sung, Philip Krantz, Archana Kamal, David K. Kim, Jonilyn L. Yoder, Terry P. Orlando, Simon Gustavsson, and William D. Oliver. Engineering Framework for Optimizing Superconducting Qubit Designs. arXiv:2006.04130v1 (2020)
  * [2] Vladimir E. Manucharyan, Jens Koch, Leonid I. Glazman, Michel H. Devoret Fluxonium: Single Cooper-Pair Circuit Free of Charge Offsets Science, Vol. 326, pp. 113-116, (2009)
  * [3] Long B. Nguyen, Yen-Hsiang Lin, Aaron Somoroff, Raymond Mencia, Nicholas Grabon, and Vladimir E. Manucharyan. High-Coherence Fluxonium Qubit. Phys. Rev. X 9, 041041, 25 Nov 2019. doi: 10.1103/PhysRevX.9.041041.
  * [4] Pop, I., Geerlings, K., Catelani, G. et al. Coherent suppression of electromagnetic dissipation due to superconducting quasiparticles. Nature 508, 369–372 (2014). https://doi.org/10.1038/nature13017
  * [5] Helin Zhang, Srivatsan Chakram, Tanay Roy, Nathan Earnest, Yao Lu, Ziwen Huang, D.K. Weiss, Jens Koch, and David I. Schuster. Universal Fast-Flux Control of a Coherent, Low-Frequency Qubit. Phys. Rev. X 11, 011010 (2021). doi: 10.1103/PhysRevX.11.011010.
  * [6] N.Earnest, S.Chakram, Y.Lu,N. Irons, R.K.Naik, N.Leung, L.Ocola, D.A.Czaplewski, B.Baker, Jay Lawrence, Jens Koch, and D.I. Schuster. Realization of a $\lambda$ System with Metastable States of a Capacitively Shunted Fluxonium. Phys. Rev. Lett. 120, 150504 (2018). doi: 10.1103/PhysRevX.11.011010.
  * [7] Quentin Ficheux, Long B. Nguyen, Aaron Somoroff, Haonan Xiong, Konstantin N. Nesterov, Maxim G. Vavilov, and Vladimir E. Manucharyan. Fast logic with slow qubits: microwave-activated controlled-Z gate on low-frequency fluxoniums. Phys. Rev. X 11, 021026,3 May 2021. doi: 10.1103/PhysRevX.11.021026.
  * [8] Moskalenko, I.N., Besedin, I.S., Tsitsilin, I.A. et al. Planar Architecture for Studying a Fluxonium Qubit. JETP Lett. 110, 574–579 (2019) doi: 10.1134/S0021364019200074
  * [9] J.R. Johansson, P.D. Nation, Franco Nori, QuTiP 2: A Python framework for the dynamics of open quantum systems, Computer Physics Communications, Vol. 184, Issue 4, 2013, Pages 1234-1240,
  * [10] Córcoles, A.D., J.M. Gambetta, J.M. Chow, J.A. Smolin, M. Ware, J. Strand, B.L.T. Plourde, and M.Steffen, Phys. Rev. A 87, 030301 (2013).
  * [11] Peter Groszkowski, A. Di Paolo, A.L. Grimsmo, A. Blais, D.I. Schuster, A.A. Houck, and Jens Koch, New J. Phys. 20, 043053 (2018).
  * [12] Z. Wang, S. Shankar, Z.K. Minev, P. Campagne-Ibarcq, A. Narla, and M.H. Devoret, Cavity Attenuators for Superconducting Qubits. Phys. Rev. Applied 11, 014031 (2019).