# Fast universal control of a flux qubit via exponentially tunable wave-
function overlap

Svend Krøjer  Center for Quantum Devices, Niels Bohr Institute, University of
Copenhagen, DK-2100 Copenhagen, Denmark  Anders Enevold Dahl  Center for
Quantum Devices, Niels Bohr Institute, University of Copenhagen, DK-2100
Copenhagen, Denmark  Kasper Sangild Christensen  Center for Quantum Devices,
Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark
Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, 8000
Aarhus C, Denmark  Morten Kjaergaard  Center for Quantum Devices, Niels Bohr
Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark  Karsten
Flensberg  Center for Quantum Devices, Niels Bohr Institute, University of
Copenhagen, DK-2100 Copenhagen, Denmark

###### Abstract

Fast, high fidelity control and readout of protected superconducting qubits
are fundamentally challenging due to their inherent insensitivity. We propose
a flux qubit variation which enjoys a tunable level of protection against
relaxation to resolve this outstanding issue. Our qubit design, the double-
shunted flux qubit (DSFQ), realizes a generic double-well potential through
its three junction ring geometry. One of the junctions is tunable, making it
possible to control the barrier height and thus the level of protection. We
analyze single- and two-qubit gate operations that rely on lowering the
barrier. We show that this is a viable method that results in high fidelity
gates as the non-computational states are not occupied during operations.
Further, we show how the effective coupling to a readout resonator can be
controlled by adjusting the externally applied flux while the DSFQ is
protected from decaying into the readout resonator. Finally, we also study a
double-loop gradiometric version of the DSFQ which is exponentially
insensitive to variations in the global magnetic field, even when the loop
areas are non-identical.

##  I Introduction

Qubits based on superconducting junctions form a promising platform for
quantum computation (QC) architectures [1, 2, 3]. In order to scale up fault-
tolerant QC, it is crucial that gate and readout infidelities must be lower
than the threshold for quantum error correction (QEC), which for the surface
code is about 1% [4, 5]. A number of experiments using transmon-based multi-
qubit chips have demonstrated surface code QEC close to the threshold [6, 7,
8].

To go beyond the capabilities of contemporary transmon-based architectures, a
number of $T_{1}$-protected qubit designs have appeared [9, 10, 11, 12, 2].
The general idea of a $T_{1}$-protected superconducting qubit is that the
computational states are localized in different quantum wells, leading to
exponentially suppressed noise-induced transitions, enhancing the relaxation
time significantly [2]. Additionally, the double-well potential realizes low-
frequency qubits resulting in less sensitivity to dielectric loss and Ohmic
noise channels [13, 14].

In the flux qubit modality, this kind of double-well protection can be reached
by biasing the superconducting loop with an external flux close to half a flux
quantum [15, 16]. Here, the low-energy computational states corresponds to
supercurrent flowing in opposite directions in the loop. At a bias of half a
flux quantum, the fluxon states are degenerate up to the exponentially small
splitting due to overlap of the evanescent part of the wave functions across
the barrier separating the two wells. Below we refer to this small splitting
as the wave-function overlap. The fluxon states are sensitive to the external
magnetic flux as it picks out a preferred current direction and determines the
energy splitting. The strong flux dependence leads to a linear sensitivity of
the qubit frequency to flux noise, causing dephasing of the qubit and limiting
coherence [17, 10].

Despite the enhanced relaxation time of low-frequency qubits (e.g. heavy
fluxonium [10, 18], $0-\pi$ qubit [9, 19], etc.), a general disadvantage is
that gate times typically also increase due to the vanishing wave-function
overlap of the computational states. One way of circumventing this limitation
is to use higher lying non-computational states [10, 2, 19]. In this manner,
single and two qubit gates can be activated through multi-tone driving [20].
The downside of such an approach, however, is that the momentary occupancy of
the non-computational states leads to increased decoherence, limiting gate
fidelities [21]. Another possibility is to rely on diabatic single qubit
control [18].

In this paper, we explore an alternative approach to perform gates on
$T_{1}$-protected qubits that rely on adiabatically adjusting the level of
protection by lowering the barrier between the two wells. We propose a qubit
design, the double-shunted flux qubit (DSFQ), which aims to be a relatively
simple modification of a flux qubit with exponentially tunable wave-function
overlap. The DSFQ is related to the persistent current flux qubit (PCFQ) [15,
17] and the capacitively shunted flux qubit (CSFQ) [22] as they all share the
same circuit layout of three Josephson junctions (JJs) connected in a loop,
see Fig. 1. While the PCFQ realizes a large $E_{J}/E_{C}$ via three large
junctions, the CSFQ uses smaller junctions with one large capacitive shunt
such that one mode is heavy (large $E_{J}/E_{C}$) and one mode is light
(smaller $E_{J}/E_{C}$). The DSFQ finds the middle ground between these
designs by using small junctions and two large capacitive shunts such that
both modes are heavy, similar to the PCFQ. Since both modes are heavy, the
lowest energy wave functions are localized in separate wells, protecting the
qubit from relaxation. Other designs, namely the super-semi $\cos(2\phi)$
qubit and the bifluxon, have successfully shown an order-of-magnitude
improvement of the relaxation time in the protected regime [11, 12]. However,
both qubits are challenging to fabricate and tune to the ideal regime and two-
qubit gates have not yet been realized [2]. The DSFQ offers a comparatively
simple platform for studying universal gate sets for qubits with variable
wave-function overlap. In addition to the universal gate scheme, we also
propose a noise-insensitive readout method for the DSFQ.

We imagine tuning the barrier height by a tunable junction, implemented either
in a SQUID-loop as in previous PCFQ experiments [23, 24, 25, 26] or in a
hybrid version where the tunable junction is a superconductor-semiconductor-
superconductor junction. This type of junction has been demonstrated earlier
to be stable and having coherence times longer than the anticipated gate times
[27, 28, 29, 11, 30]. However, we note that the coherence times for the
semiconductor-based junctions are still shorter than the more standard
insulator-barrier junctions. The physics of this is still not understood and
the coherence times could improve with future devices [31].

We calculate the coherence properties of the DSFQ and discuss the flux-noise
sensitivity. In order to reduce the flux dephasing, we propose a double-loop
gradiometric version of the DSFQ which gives exponential protection against
global flux noise. Gradiometric qubit designs have been proposed previously
but rely on identical areas in the two loops [23, 26, 32]. We show that small
area variations can be compensated for by adjusting the tunable junction
without introducing sensitivity to the junction control line. The main focus
of our study is a set of one- and two-qubit gates where the idea is to tune
the qubit out of the protected regime by adiabatically lowering the barrier
between the two wells and thereby hybridize the computational states. Two-
qubit gates can be performed by simultaneously lowering the barriers for two
capacitively coupled DSFQ’s while single qubit gates require a fast single-
tone microwave pulse in the an intermediate regime. Advantages of variable-
protection gates are that fast-decaying non-computational states do not
participate in gate operations and that two-qubit interactions can be turned
off with exponential on/off ratio while maintaining the ability to perform
one-qubit gates. Finally, we show how the effective coupling to a readout
resonator can be adjusted with a simple flux control of the qubit, leading to
an order-of-magnitude on/off ratio while decay to the readout resonator is
suppressed.

##  II The double-shunted flux qubit

Figure 1: (a) Circuit layout for the DSFQ with a variable junction by either a
SQUID or gate voltage tunable nanowire junction. (b) Potential landscape of
the DSFQ with the two lowest energy eigenstates shown in red and blue with
$E_{J}/E_{C}=100,\alpha=1$ and $\phi_{\mathrm{ext}}=0.997\pi$. (c) One
dimensional cut of (b) along $\phi=(\phi_{1}-\phi_{2})/2$ with wave functions
showing their exponential separation at $\alpha=1$. The potential at
$\alpha=0.7$ is shown in gray dashed. (d) Energy splitting of the qubit as a
function of the barrier height controlled by $\alpha$. The value of $\alpha$
corresponding to the CSFQ/PCFQ is indicated with a star/bullet
($\alpha=0.5/0.8$). Energies are in units of the Josephson energy, $E_{J}$.

We consider a system of three Josephson junctions connected in a ring. The
circuit is illustrated in Fig. 1(a) where the Josephson energy of the tunable
junction is denoted by $\alpha E_{J}$. The two other junctions have Josephson
energy $E_{J}$, but they do not have to be identical for our proposal to work.
In the phase variables $\phi=(\phi_{1}-\phi_{2})/2$ and
$\theta=(\phi_{1}+\phi_{2})/2$, the potential energy of the qubit is thus
given by

$H_{J}=-2E_{J}\cos(\phi)\cos(\theta)-\alpha
E_{J}\cos(2\phi+\phi_{\mathrm{ext}}),$ (1)

where $\phi_{\mathrm{ext}}=2\pi\Phi/\Phi_{0}$ and $\Phi$ is the flux through
the loop, controlled by an external magnetic field whose value is typically
set to $\phi_{\mathrm{ext}}=0.997\pi$ unless other stated. At $\alpha=0.5$,
the barrier is completely lowered, making the potential along the
$\phi$-direction approximately quartic as for the CSFQ [22]. At a value of
$\alpha=0.8$, the barrier is significant and the potential of the PCFQ [15,
17] is recovered. Controlling the barrier height of the DSFQ through $\alpha$
thus interpolates between the PCFQ and the CSFQ. Note that in the flux-tunable
PCFQ, the barrier height can be controlled via an external flux in a slightly
different geometry [23, 25, 26].

The charging energy is determined by the capacitances $C$ shown in Fig. 1(a)
and gives rise to the kinetic energy [33]

$H_{C}=2E_{C}\left(-i\partial_{\phi}-n_{g\phi}\right)^{2}+2E_{C}\left(-i\partial_{\theta}-n_{g\theta}\right)^{2}.$
(2)

Here we have included offset charges $n_{g\phi}$ and $n_{g\theta}$ (the
$4E_{C}$ typically found as the prefactor is reduced due to the change of
variables $n_{\theta/\phi}=n_{1}\pm n_{2}$). The qubit will be operated in the
regime of small $E_{C}=e^{2}/2C$ (i.e., both $\phi$ and $\theta$ being heavy
modes). Realistically, the Josephson capacitances are about two orders of
magnitude smaller than the large shunting capacitances and thus merely
renormalizes $E_{C}$ without affecting the results presented in this work.

The potential landscape and the ground-state wave functions are shown in Fig.
1(b) in the heavy-modes regime ($E_{J}/E_{C}=100$). The external flux is tuned
to a value close to half a flux quantum. The two wave function shown in red
and blue ($\psi_{0}$ and $\psi_{1}$) are clearly well separated and localized
in the two wells. They represent the qubit states $\ket{0}$ and $\ket{1}$. The
state separation is most easily seen in Fig. 1(c) which is a cut along the
$\phi$-direction. Due to their separation, the tunneling between the two wells
is suppressed. It results in a small qubit splitting near $\alpha=1$
determined by the external flux and also a large anharmonicity, see Fig. 1(d).
Lowering the barrier by reducing $\alpha$, increases the qubit frequency and
decreases the anharmonicity
$\alpha_{\text{an}}=(\omega_{02}-\omega_{01})-\omega_{01}$, as the states
hybridize and change significantly. This fact is used below to perform fast
gates by lowering the value of $\alpha$ to $\alpha\approx 0.7$ where the
logical states partially overlap.

Figure 2: (a) Circuit layout of the single-loop DSFQ. (b) Dispersion of the
qubit frequency with respect to the reduced external flux through the loop,
showing the linear dependence in the region
$\phi_{\text{ext}}/2\pi=0.47-0.53$. (c-d) Relation between the
relaxation/dephasing time ($T_{1}/T_{\varphi}$) and the barrier height
controlled by $\alpha$ for the single loop DSFQ. (e) Circuit layout of the
gradiometric DSFQ with two tunable junctions. The inconvenient placement of
large capacitors in the loops can be worked around by using cross-over
junctions. (f) Dispersion of the qubit frequency with respect to the redcued
external global flux, $\phi_{G}=\frac{2\pi\Phi_{G}}{\Phi_{0}}$. We display the
cases where the loop areas are identical (solid line), non-identical (dot-
dashed line) and non-identical with compensating asymmetric junctions (dashed
line). (g-h) Relation between the relaxation/dephasing time
($T_{1}/T_{\varphi}$) and the barrier height controlled by $\alpha$ for the
gradiometric DSFQ. Note the insensitivity to noise in the global magnetic
field and sensitivity to local magnetic field noise. The noise amplitudes in
all figures are $A_{\Phi}=10^{-6}\Phi_{0}/\sqrt{\text{Hz}}$,
$A_{n_{g}}=10^{-4}e/\sqrt{\text{Hz}}$ [13], $B_{n_{g}}=5.2\times
10^{-9}e/\sqrt{\text{Hz}}$ [22] and $\tan\delta_{\text{diel}}=2\crossproduct
10^{-7}$ [14]. The Josephson energy is $E_{J}=10\,h\,$\mathrm{GHz}$$ and
external flux is $\phi_{ext}=0.997\pi$ where relevant.

###  II.1 Gradiometric DSFQ

The qubit discussed above is designed to have a large relaxation time due to
the exponential suppression of inter-well coupling. However, it is likely to
have a poor dephasing coherence time because of the sensitivity of the energy
difference of the two wells to flux noise. To improve the dephasing time, we
propose a double-loop variation as in Fig. 2(e) which is designed to cancel
out any fluctuations in the global flux. In the double-loop design, we picture
the variable junctions as tunable nanowire junctions. Alternatively, these
could be SQUIDs controlled by individual flux lines without defeating the
purpose of the gradiometric setup. However, the additional flux loops will
complicate the control of the qubit because there will be flux lines to each
SQUID and one to control the global flux. The tunable Josephson junctions give
an advantage with fewer flux control lines compared to using SQUIDs at the
potential expense of reduced coherence due to semiconducting junctions. To
understand the double-loop cancellation better, we consider the situation
where half a flux quantum threads through each loop. This gives rise to two
lowest-energy combinations of current flowing in the circuit;
$\ket{\circlearrowleft\circlearrowright},\ket{\circlearrowright\circlearrowleft}$,
where an arrow indicates the direction of the current in each loop. Thus, the
two lowest energy states correspond to the situation where current flows in
opposite directions, making them indifferent to variations in the external
flux. Said differently, the magnetic dipole moment vanishes and the
computational states are only affected by magnetic field gradients through the
magnetic quadrupole moment as verified in Refs. [26, 34, 32]. In Fig. 2(b,f),
we show the dependence of the qubit splitting on the global flux for both
single- and double-loop DSFQs.

For a symmetric situation where the areas of the two loops and the Josephson
energies of two outer junctions are identical, the dependence of the global
flux $\Phi_{G}$ (proportional to a global magnetic field) has zero slope when
$\Phi_{G}$ is at half flux quantum (see Fig. 2(f), blue solid line). In an
experimental situation, the loop areas will be slightly different, leading to
a sensitivity to the global magnetic field (blue dash-dotted line). However,
by appropriately choosing the ratio of the tunable junctions, the dispersion
with $\Phi_{G}$ can become exponentially flat again at the expense of
splitting the degeneracy (blue dashed line). If the flux through the two non-
identical loops is controlled by a single global field, and the tunable
junctions can be tuned to be asymmetric, $\alpha_{2}=(1+\delta)\alpha_{1}$,
then the sweet-spot simply shifts to

$\delta=-1+\frac{1+r}{1-r}\cos\left(\frac{2\pi r}{1-r}\right)\approx 2r,\quad
r=\frac{A_{1}-A_{2}}{A_{1}+A_{2}}.$ (3)

where $r$ is a measure of the loop area asymmetry and assumed small, see also
Appendix A where the condition on $\delta$ is derived. Here, it is also shown
that the fluctuations in $\delta$ has very little effect on the energies near
half a flux quantum as can also be seen by comparing the dashed ($\delta=2r$)
and dash-dotted blue line ($\delta=0$). Fig. 2(f) summaries how the
sensitivity to the external global magnetic field and how choosing the value
of the Josephson energy of the second junction can make the spectrum
practically insensitive to the global field. As detailed in Appendix A, the
slope and height of the curve is set by the area and junction asymmetry. While
being insensitive to variations in the global magnetic field, the qubit
frequency is still linearly sensitive to the local fluxes in the individual
loops, see Fig. 2(h) and discussion below.

###  II.2 Decoherence times

The decoherence of the DSFQ is estimated by calculating relaxation and
dephasing rates for different noise sources. The relaxation time
$T_{1}=\left(\sum_{\lambda}\Gamma_{1}^{\lambda}\right)^{-1}$ is computed
through the relaxation rates which are given by Fermi’s Golden rule [35, 14,
13]

$\displaystyle\Gamma_{1}^{\lambda}$
$\displaystyle=\frac{1}{\hbar^{2}}\left|\bra{1}\partial_{\lambda}H\ket{0}\right|^{2}S_{\lambda}(\omega),$
$\displaystyle\Gamma_{1}^{\text{diel}}$
$\displaystyle=\hbar\left|\bra{1}\phi\ket{0}\right|^{2}S^{\text{diel}}(\omega)$
(4)

where $\lambda$ is an external noise source and $S_{\lambda}(\omega)$ is the
power spectral function for a given noise source. We consider $1/f$ and ohmic
noise which were the limiting noise channels for flux and charge noise
respectively for the CSFQ [22] in addition to dielectric loss, the limiting
factor for fluxonium relaxation time [36, 14]. The associated spectral
functions are

$\displaystyle S_{\lambda}^{\frac{1}{f}}(\omega)$ $\displaystyle=\frac{2\pi
A_{\lambda}^{2}\text{Hz}}{|\omega|},\quad
S_{\lambda}^{\Omega}(\omega)=\frac{B_{\lambda}^{2}\omega}{2\pi\times
1\text{GHz}},$
$S^{\text{diel}}(\omega)=\frac{\omega^{2}\tan\delta_{\text{diel}}}{4E_{C}}\left[\coth(\frac{\omega}{k_{B}T})+1\right],$

where $A_{\lambda}$ and $B_{\lambda}$ are noise amplitudes for $1/f$ and ohmic
noise respectively, $\tan\delta_{\text{diel}}=2\crossproduct 10^{-7}$ is the
loss tangent and $T=20$mK is the temperature [14]. We use typical noise
amplitudes $A_{\Phi}=10^{-6}\Phi_{0}/\sqrt{\text{Hz}}$ [13],
$A_{n_{g}}=10^{-4}e/\sqrt{\text{Hz}}$ [13] and $B_{n_{g}}=5.2\times
10^{-9}e/\sqrt{\text{Hz}}$ [22].

In Fig. 2(c, g), we display the computed relaxation times for the single loop
and double loop (gradiometric) versions of the DSFQ. Both panels show
exponentially enhanced $T_{1}$ in the protected regime ($\alpha=1$) with
$T_{1}=603$ $\mu$s in the single loop and $T_{1}=733$ $\mu$s in the
gradiometric setup, the limiting factor being dielectric loss. In the
unprotected regime ($\alpha=0.5$), the relaxation time is reduced to
$T_{1}=0.35$ $\mu$s in the single loop and $T_{1}=0.35$ $\mu$s in the
gradiometric equivalent to 3 orders of magnitude.

We can compare the relaxation times to the dephasing times shown in Fig. 2(d,
h). The first order dephasing rates for $1/f$ noise are computed through [13],

$\displaystyle\Gamma_{\varphi}^{\frac{1}{f},\lambda}=\sqrt{2A_{\lambda}\left(\partial_{\lambda}\omega_{q}\right)^{2}\ln|\omega_{\mathrm{ir}}t|},$

where we have introduced an infrared cutoff and a characteristic time with the
product $\omega_{\mathrm{ir}}t=2\pi\times 10^{-6}$ as in Ref. [13]. The
dephasing times shown in Fig. 2(d, h) are limiting the coherence time
$\frac{1}{T_{2}}=\frac{1}{2T_{1}}+\frac{1}{T_{\varphi}}$ compared to the
relaxation time due to the linear sensitivity to (local) flux noise in the
$T_{1}$-protected regime. Conversely, in the unprotected regime, the coherence
is limited by relaxation through dielectric loss, illustrating the trade-off
between $T_{1}$-protection and dephasing due to flux noise is general to flux
qubits. Note that the sensitivity to global flux noise in Fig. 2(g, h) is
reduced due to the gradiometric construction of the device. In the $T_{1}$
protected regime ($\alpha=1$) the dephasing time is $T_{\varphi}=0.12$ $\mu$s
in the single loop and $T_{\varphi}=0.74$ $\mu$s in the gradiometric setup. In
the unprotected regime ($\alpha=0.5$) the dephasing time is enhanced to
$T_{\varphi}=7.6$ $\mu$s in the single loop and $T_{\varphi}=98$ $\mu$s in the
gradiometric setup. The CSFQ has relaxations times reported in the range
$T_{1}=20-60$ $\mu$s [22]. State of the art transmon qubit report relaxations
times up to $T_{1}=0.5$ ms [37].

In total, the DSFQ does not exceed the relaxation time of state of the art
transmon qubits but offers a platform with adjustable and strong noise bias
and a tunable degree of $T_{1}$-protection, which can be used to study optimum
strategies for gate operations on protected qubits. While the noise bias, in
principle, opens up paths towards efficient noise biased error correcting
codes, the linear sensitivity to (local) flux noise is a limiting factor. This
could be suppressed by choosing a larger qubit splitting, creating a wider
sweet spot at half flux quantum. However, we have chosen to focus on the
$T_{1}$ protected regime here. We note that such compromise is relevant for
other qubit proposals such as the heavy fluxonium and the bifluxon [10, 18,
12].

##  III Qubit control

To control the DSFQ, we leave the protected regime ($\alpha=1$) and lower the
barrier between the two wells ($\alpha\simeq 0.5-0.7$). When the barrier is
lowered, traditional techniques in microwave control such as DRAG and IQ-
mixing can be used for the DSFQ [38, 39]. As detailed in the sections below,
the height of the barrier at the operating point and the rate at which it is
lowered depends on whether single or two-qubit gates are performed. We
continue in the following section by implementing an $\sigma_{x}$ gate
numerically to illustrate how single qubit gates can be performed on qubits
with variable-protection using single-tone driving.

###  III.1 Variable-protection single qubit gates

Figure 3: (a) The coupling of computational states through the charge operator
as a function of $\alpha$, showing when transitions can be stimulated through
a capacitively coupled drive-line. (b) The pulse profile for the $\sigma_{x}$
gate displaying the low-frequency $\alpha$ drive (black) and the high-
frequency microwave drive (red). The envelope of the microwave pulse is
$11\,$\mathrm{ns}$$ long with a $1.5\,$\mathrm{ns}$$ cosine ramp up/down. The
drive frequency is slightly detuned from the qubit frequency
$\omega_{d}=0.979\,\omega_{q}$. (c) Numerical data from non-dissipative
simulations showing the time history of the spectral weights during the low-
leakage, high fidelity $\sigma_{x}$ gate. In all panels, the scale of the
Josephson energy is $E_{J}=10\,h\,$\mathrm{GHz}$$ and $E_{J}/E_{C}=100$ with
the flux bias set to $\phi_{\text{ext}}=0.995\pi$.

Our proof-of-concept $\sigma_{x}$-gate has three steps as illustrated in Fig.
3:

  1. I.

Lower the barrier adiabatically, $\alpha=1\to 0.7$.

  2. II.

Apply an appropriate microwave pulse to the qubit.

  3. III.

Raise the barrier adiabatically, $\alpha=0.7\to 1$.

This control sequence is illustrated in Fig. 3 where the lowering and raising
of the barrier takes $7\,$\mathrm{ns}$$ and the microwave drive takes
$11$\mathrm{ns}$$ (including $1.5\,$\mathrm{ns}$$ ramp up/down), totalling a
gate time of $25\,$\mathrm{ns}$$. The microwave drive line is coupled to one
of the nodes of the qubit through a small capacitance $C_{d}\ll C$, giving
rise to the Hamiltonian term $H_{d}=\frac{C_{d}}{C+C_{d}}V_{d}(t)n_{1}$ [39].
As the barrier is lowered, the quantum states changes significantly and a
small subspace of states is insufficient to describe the evolution due to
$H(t)=H_{C}+H_{J}(t)$. We therefore perform simulation in a relatively large
Hilbert space with $625$ states (in the charge basis with cutoff
$n_{\mathrm{cutoff}}=12$ for both the $\phi$\- and $\theta$-mode) and
numerically evaluate $\exp(-i\Delta t\,H(t))$ at each time step to perform the
time-evolution (857 timesteps/nanosecond). When the drive is turned on at
fixed $\alpha$, we instead numerically integrate the time-dependent
Schrödinger equation using the same Hilbert space dimension. At each time-
step, we numerically diagonalize the Hamiltonian and compute the overlap with
the instantaneous qubit states to produce Fig. 6(c).

In our single qubit gate scheme, we choose to lower the barrier only partially
($\alpha=0.7$) to limit the time spent adiabatically adjusting $\alpha$ and to
avoid small, unwanted interactions with neighboring qubits which arise when
the barrier is completely lowered, see also Appendix C. The qubit frequency is
changed from $\omega_{q}(\alpha=1)=0.25\,h\,$\mathrm{GHz}$$ to
$\omega_{q}(\alpha=0.7)=0.39\,h\,$\mathrm{GHz}$$, where the Josephson energy
is $E_{J}=10\,h\,$\mathrm{GHz}$$ and $E_{J}/E_{C}=100$ with the flux bias set
to $\phi_{ext}=0.995\pi$. At the operating point ($\alpha=0.7$) the relaxation
time is reduced to $1.6\,$\mathrm{\SIUnitSymbolMicro s}$$. The speed at which
the barrier is lowered is adiabatic with respect to the energy gap between the
computational states and the non-computational states such that the adiabatic
time is set by the desired leakage bound. The $7\,$\mathrm{ns}$$ lowering time
results in a very small ($\sim 10^{-4}$) leakage but does admit for a small
($\sim 10^{-3}$) probability to transition from one logical state to the
other. This small effect makes it necessary to slightly adapt the microwave
pulse to achieve high fidelity. One possibility is to marginally reduce the
drive amplitude, but the qubit frequency is also shifted due to the AC-Stark
effect. We therefore instead adapt the pulse by a minor frequency shift of the
drive, $\omega_{d}=0.979\,\omega_{q}$, to account for both of these
contributions. The limit to the fidelity imposed by coherent errors (leakage)
during the $\sigma_{x}$ gate is $99.98\,\%$ while the gate time is
$T_{g}=25\,$\mathrm{ns}$$. The single qubit gate fidelity is limited by decay
from the shorter relaxation time at the operating point. We estimate the
$T_{1}$ limited fidelity via $F\approx\exp[-\int_{0}^{T_{g}}\differential
t\,\Gamma_{1}(t)]$, where $\Gamma_{1}(t)$ is the sum of (instantaneous) decay
rates. The resulting $T_{1}$-limited fidelity is $99.1\%$ for the single qubit
X-gate. While the gate is limited by decay in this device, the coherence
limited gate fidelity is comparable to state-of-the-art single qubit gates on
unprotected qubits such as the transmon [7] and potentially faster than
alternative gates on $T_{1}$-protected qubits [20]. The latter makes use of
non-computational states, multi-tone driving and an optimal control algorithm
to optimize gate performance. The comparatively simple variable-protection
gate shows the benefits of tuning in and out of protection, and that the
access to fast, single tone pulse control outweigh the additional overhead
from the adiabatic control of the level of protection. In Appendix B, we
exemplify using standard IQ-mixing how also $\sigma_{y}$ and
$(\sigma_{x}-\sigma_{y})/\sqrt{2}$ gates can be implemented with similar
fidelity as the $\sigma_{x}$ gate. Combined with virtual $\sigma_{z}$ gates,
we have thus demonstrated a compelling scheme for realizing universal single-
qubit control. It is natural to improve upon this proof-of-principle design
using more advanced $\alpha$-profiles combined with microwave pulse shaping
techniques such as DRAG [38] in order to reduce the time spent at low
coherence for smaller $\alpha$. Alternatively, sudden gates or gates where the
flux bias is also controlled may be explored with inspiration from Ref. [18].
Ref. [18] also shows how multi-tone driving can initialize low-frequency
qubits where the qubit frequency is subthermal. Alternatively, our flexible
design also allows for thermal initialization in the unprotected regime.

###  III.2 Variable-protection two-qubit gates

Figure 4: Two qubit setup and gate characteristics from non-dissipative
simulations. (a) Schematic of two capacitively coupled DSFQs with substantial
coupling capacitance $C_{g}=0.3\,C$. (b) The five lowest energy states shown
as the two barriers are lowered simultaneously by decresing
$\alpha_{1}=\alpha_{2}$. (c) The $\alpha_{1}=\alpha_{2}$ profile as a function
of time for the CPHASE gate. (d) The entanglement entropy of the final two-
qubit gate as a function of the waiting time $T_{w}$ and the total adiabatic
control time $T_{a}$. The red markers in this and subsequent panels show the
optimal $\sqrt{\text{iSWAP}}$ (star) and CPHASE (triangle) gates which have
respective fidelities limited by coherent gate errors and gate times of
$\mathcal{F}_{\sqrt{\text{iSWAP}}}=99.96\,\%$,
$T_{\sqrt{\text{iSWAP}}}=32.66\,$\mathrm{ns}$$ and
$\mathcal{F}_{\text{CPHASE}}=99.95\,\%$,
$T_{\text{CPHASE}}=68.76\,$\mathrm{ns}$$. The estimated $T_{1}$-limited
fidelities are $F_{\sqrt{\text{iSWAP}}}=99.7\%$ and
$F_{\text{CPHASE}}=91.4\%$. (e-f) The resulting phase and swap parameters
$\phi_{\text{CPHASE}}$ and $\theta_{\text{SWAP}}$ of the final two-qubit gate
as a function of the waiting time $T_{w}$ and the total adiabatic control time
$T_{a}$. The flux bias is set to $\phi_{\mathrm{ext}}=0.99\pi$.

An advantage of qubits with variable protection is that they can act as their
own tunable couplers with exponential on/off ratio. In the protected idling
regime, the qubit-qubit coupling vanishes due to the exponentially small wave-
function overlap, see also Appendix C. As a result of the exponentially
suppressed coupling between the computational states in the idling regime, a
capacitive qubit-qubit coupling,

$H_{\mathrm{Q-Q}}=4E_{C}\frac{C_{g}}{C+C_{g}}n_{1}n_{3},$ (5)

may be relatively strong $C_{g}\simeq 0.3\,C$ compared to e.g. transmon
qubits, see Appendix C for a derivation of Eq. 5. We can thus implement two-
qubit gates that rely solely on the simultaneous lowering of both barriers of
two capacitively coupled DSFQs.

Our implementation of two-qubit gates has three steps:

  1. I.

Lower both barriers simultaneously in a time $T_{a}/2$,
$\alpha_{1}=\alpha_{2}=1\to\alpha_{\mathrm{min}}$.

  2. II.

Wait for a time $T_{w}$.

  3. III.

Raise the barriers simultaneously in a time $T_{a}/2$,
$\alpha_{1}=\alpha_{2}=\alpha_{\mathrm{min}}\to 1$.

The total gate time thus becomes the sum of the waiting time and the adiabatic
control time, $T_{2Q}=T_{a}+T_{w}$.

When the barriers are lowered, the qubits can exchange excitations through the
capacitive coupling element resulting in an effective
$\sigma_{x}^{(1)}\sigma_{x}^{(2)}+\sigma_{y}^{(1)}\sigma_{Y}^{(2)}$
interaction. Crucially, the adiabatic control time can be adjusted such that
there occurs a transition between the states $\ket{01}$ and $\ket{10}$ due to
their small energy difference and not between other computational states whose
energy difference is large compared to the adiabatic time. As shown in Fig.
4(b), an avoided crossing occurs near $\alpha=0.75$. On the other side of this
avoided crossing, when $\alpha$ is further decreased, the coupling
dramatically increases. See also Fig. 7, where the
$\sigma_{z}^{(1)}\sigma_{z}^{(2)}$-interaction strength is shown. The avoided
crossing shown in Fig. 4(b) is a generic feature of the coupled spectrum as
long as the qubit frequencies of the two interacting qubits are similar at
$\alpha=1$.

To exclude transitions between the other computational states and transitions
out of the computational subspace, the speed at which $\alpha$ is lowered
should be slower compared to the single qubit gate. As a concrete example, we
consider lowering the barriers with a constant speed, meaning that the
adiabatic time is proportional to the minimum value
$\alpha_{\mathrm{min}}=1-\frac{T_{a}/2}{2\cdot 35\,$\mathrm{ns}$}$. Thus, the
barrier can be completely lowered in $35\,$\mathrm{ns}$$ which is three times
slower than for the lowering rate used for the single qubit gate. Adiabatic
lowering/raising times $T_{a}/2$ less than $35\,$\mathrm{ns}$$ results in only
partly lowering the barrier due to the constant lowering/raising speed, see
also Fig. 4(c).

In addition to the
$\sigma_{x}^{(1)}\sigma_{x}^{(2)}+\sigma_{y}^{(1)}\sigma_{y}^{(2)}$
interaction, the energies of the coupled system shifts relative to the bare
energies due to an effective $\sigma_{z}^{(1)}\sigma_{z}^{(2)}$ interaction,
see also Appendix C. Below we simulate the two-qubit gate shown in Fig. 4 and
discuss the types of gates achieved. The two-qubit unitaries can be modelled
by a two-qubit interacting system of the following form

$\displaystyle H_{\mathrm{eff}}$
$\displaystyle=-\frac{\omega_{1}}{2}\sigma_{z}^{(1)}-\frac{\omega_{2}}{2}\sigma_{z}^{(2)}$
(6)
$\displaystyle+\frac{g_{xy}}{2}\left(\sigma_{x}^{(1)}\sigma_{x}^{(2)}+\sigma_{y}^{(1)}\sigma_{y}^{(2)}\right)+\frac{g_{z}}{2}\sigma_{z}^{(1)}\sigma_{z}^{(2)},$

where the $\sigma_{x,y,z}^{(i)}$’s are Pauli matrices acting in the logical
subspace of qubit $i$, $\omega_{i}$ describe the qubit frequencies, and the
swap coupling $g_{xy}$ and $\sigma_{z}^{(1)}\sigma_{z}^{(2)}$ coupling $g_{z}$
are all $\alpha$-dependent. This model Hamiltonian gives rise to the so-called
fSim-gates which interpolate between the iSWAP- and CPHASE-gate [39, 40],

$U_{\text{fSim}}=\begin{pmatrix}1&0&0&0\\\
0&\cos(\theta_{\text{SWAP}})&-i\sin(\theta_{\text{SWAP}})&0\\\
0&-i\sin(\theta_{\text{SWAP}})&\cos(\theta_{\text{SWAP}})&0\\\
0&0&0&e^{-i\phi_{\text{CPHASE}}}\end{pmatrix}$ (7)

which is precisely what we see in the simulation of the full model. By timing
the adiabatic control time and the waiting time to match
$\ket{01}\Longleftrightarrow\ket{10}$ swap oscillations and the rotating
$\sigma_{z}^{(1)}\sigma_{z}^{(2)}$-phase, different gates in the fSim-space
can be targeted as shown in Fig. 4(d-f). Here, we sweep over the adiabatic and
waiting times, $T_{a}$ and $T_{w}$, and in panel (d), we display the
entanglement entropy which is normalized to unity for maximally entangling
gates [41]. The only maximally entangling gates in the fSim-space are CPHASE
and iSWAP. In panels (e) and (f), we decompose the resulting unitary into the
fSim-parameters; the phase angle $\phi_{\text{CPHASE}}$ and the swap angle
$\theta_{\text{SWAP}}$. The red markers show two example gates in the fSim-
space; the CPHASE and $\sqrt{\text{iSWAP}}$ gates. The fidelity limited by
coherent errors (lekage) is well beyond $99.9\%$ (up to single qubit
$\sigma_{z}$-gates) and can be performed in about 69 ns and 33 ns
respectively. Again, the two-qubit gates are limited by decay, with estimated
$T_{1}$-limited fidelities of $F_{\sqrt{\text{iSWAP}}}=99.7\%$ and
$F_{\text{CPHASE}}=91.4\%$. The fidelity of the CPHASE gate is severely
impacted by the low qubit coherence near $\alpha=0.6$ where
$T_{1}=0.6\,$\mathrm{\SIUnitSymbolMicro s}$$ but the $\sqrt{\text{iSWAP}}$
gate is a promising high fidelity alternative. The iSWAP gate cannot be
implemented to high fidelity as it requires both fine-tuning of energies to
achieve a full swap of excitations and zero (mod $2\pi$)
$\sigma_{z}^{(1)}\sigma_{z}^{(2)}$-phase. The combined requirement is
challenging to tune with our parameters, so we instead propose to simply apply
two $\sqrt{\text{iSWAP}}$ gates successively. The $\sqrt{\text{iSWAP}}$ gate
is comparatively easy to perform as a partial swap of excitations happens
before any significant $\sigma_{z}^{(1)}\sigma_{z}^{(2)}$ phase is accrued.
Finally, The CPHASE gate depends to an intermediate degree on the Hamiltonian
parameters as it does not require a transfer of excitations. Our testing finds
that appropriate times $T_{a}$ and $T_{w}$ can be chosen for a range of
parameters to yield a CPHASE gate.

As mentioned, the wave functions change substantially as the barriers are
lowered and complicates the simulation of the qubit interactions. In order to
faithfully simulate the time-evolution, we numerically diagonalize the charge-
basis Hamiltonian ($n_{\mathrm{cutoff}}=9$) at each $\alpha$ and keep the 24
lowest states. Since the diagonalizing unitary,
$V:\,V^{\dagger}HV=\mathrm{diag}(E_{1},E_{2},\ldots)$, is time-dependent, the
Schrödinger equation acquires an additional term,
$-iV^{\dagger}\partial_{t}V$. Finally, using the combined Hamiltonian
$H=H_{1}+H_{2}+H_{\mathrm{Q-Q}}$ (Eqs. (1), (2) and (5)), the time-evolution
operator of the lowest 24 states is evolved by $\exp\left[-i(V^{\dagger}HV-
iV^{\dagger}\partial_{t}V)\Delta t\right]$ at each timestep $\Delta t$ (286
timesteps/nanosecond).

Despite relying only on adiabatic control, the two-qubit gates presented here
are competitive compared to state-of-the-art two-qubit gates for both single-
and double-well qubits [18, 21, 19]. Further advantages include the
exponential on/off coupling ratio, that only the computational states are used
and the possibility of being able to produce different gates in the fSim-
space. Further developments, for example controlling $\alpha_{1}$ and
$\alpha_{2}$ individually as well as the fluxes, will likely provide more
control over what fSim-gates can be reached and reduce the overall gate time
or increase fidelities using optimized strategies. Additionally, recent work
suggests to also use the DSFQ as a transmon-transmon coupler (called the
“double transmon coupler”), which illustrates the exciting flexibility of the
device [42].

###  III.3 Readout

Figure 5:  (a) The qubit coupled to a readout resonator. The qubit induces a
state-dependent shift of the frequency of the resonator, which can be measured
using standard techniques. (b) Dispersive shift as a function of external
flux. By adjusting the flux away from half flux bias, a resonance between one
of the computational states become and a higher energy states comes close to
the frequency of the readout resonator. The resonator shift is increased
resulting in a stronger readout signal. A smaller shift is preferable in the
context of error suppression where it reduces the sensitivity to photon-shot
noise.

Readout of the DSFQ device can be performed using conventional dispersive
readout techniques [39]. However, rather than reading out via the $\phi$-mode,
similar to fluxonium qubits, we instead propose to readout via the
$\theta$-mode. By coupling the qubit capacitively to a readout resonator
through the $\theta$ degree of freedom, as shown in Fig. 5(a), we can achieve
substantial dispersive shifts while remaining in the protected qubit regime to
suppress (Purcell enhanced) relaxation. As we detail below, the plasmon
frequency of the $\theta$-mode depends on which well the $\phi$-mode is
localized in. Further, the difference in plasma frequencies for the two wells
are tuned by the external magnetic flux. In this way, we can use the external
flux to control the state dependent shift of the readout resonator as shown in
Fig. 5(b-c).

We start by considering the Hamiltonian of the combined system which can be
written as [39]

$H=H_{\mathrm{sys}}+g(a+a^{\dagger})n_{\theta}+\omega_{r}a^{\dagger}a,$ (8)

where $H_{\mathrm{sys}}$ is the qubit Hamiltonian, $a(a^{\dagger})$ is the
resonator annihilation(creation) operator, $\omega_{r}$ is the bare resonator
frequency and $g$ is the coupling strength between resonator and qubit. In the
dispersive regime, the resonator frequency is effectively shifted by the state
of the qubit. This can be seen by performing a Schrieffer-Wolff transformation
[43, 44] to second order,

$H_{\mathrm{eff}}=H_{\mathrm{sys}}+\omega_{r}a^{\dagger}a-\left(\frac{\chi}{2}a^{\dagger}a+\delta\right)\sigma_{z},$
(9)

where $\chi$ is the qubit state dependent resonator shift, $\delta$ is a small
shift of the qubit frequency and $\sigma_{z}=\ket{0}\bra{0}-\ket{1}\bra{1}$ is
the qubit Pauli $Z$ operator. To correctly estimate the dispersive resonator
shift it is important to account for higher levels outside of the
computational subspace. Carrying out the perturbation calculation, we find the
dispersive shift as $\chi=\sum_{j}\chi_{1j}-\chi_{0j}$, where

$\chi_{ij}=g^{2}\left|\bra{i}n_{\theta}\ket{j}\right|^{2}\left(\frac{1}{E_{i}-E_{j}-\omega_{r}}+\frac{1}{E_{i}-E_{j}+\omega_{r}}\right).$
(10)

Figure 5(b, c) shows the resonator shift as a function of the externally
applied magnetic flux. For these simulations, we have used a bare resonator
frequency of $\omega_{r}=4.8\,h\,$\mathrm{GHz}$$ and coupling strength of
$g=25\,h\,$\mathrm{MHz}$$.

To explain the working principle of the readout, we briefly adopt a simple,
minimal model of the DSFQ. In this model, we assume that we are away from the
sweet-spot at exactly half flux quantum and write an effective potential for
the $\theta$-degree of freedom by freezing the $\phi$-degree of freedom to one
of the two minima at $\phi_{\pm}=(\pm\pi-\delta\phi_{ext})/3$ for $\alpha=1$
and thus momentarily neglect tunneling between the two wells,

$V_{\pm}=\mp
E_{J}\frac{\delta\phi_{ext}}{2\sqrt{3}}-E_{J}\left(1\pm\frac{\delta\phi_{ext}}{\sqrt{3}}\right)\cos(\theta),$
(11)

where $\phi_{ext}=\pi+\delta\phi$ and $\delta\phi\ll 1$. In this picture, each
minima corresponds to one of the computational states. Close to half flux bias
($\delta\phi_{ext}\approx 0$), $V_{\pm}$ are nearly identical and the readout
resonator cannot discriminate between the computational states as the matrix
elements $|\bra{\pm}n_{\theta}\ket{j}|$ are approximately the same for the two
qubit states $\ket{\pm}$. By increasing the offset from the flux frustration
point, the two terms in Eq. (11) lead to differences between the two wells
that can result in a large dispersive shift if the readout resonator is close
in frequency to the plasma frequency of the $\theta$-mode in one of the wells.
The first term in Eq. (11) contains the simple energy splitting between the
two wells due to the external flux which does not change the plasmon
frequency. The second term in Eq. (11) shows that the plasmon frequency of the
$\theta$-mode in each well
$\omega_{\theta}^{\pm}=\sqrt{8\widetilde{E}_{C}\widetilde{E}_{J}^{\pm}}$,
where $\widetilde{E}_{C}$ and $\widetilde{E}_{J}^{\pm}$ are the effective
charging and Josephson energies of the $\theta$-mode [39], also depends on the
offset from half flux bias. In this way, we may tune the plasmon frequency in
one of the wells close to the readout resonator frequency and thereby achieve
a large dispersive shift, see Fig. 5(b-c). We may now consider what happens at
exactly half flux quantum where the small tunneling between the wells results
in wave functions that are even/odd in $\phi$. In this situation, different
selection rules for the even/odd computational states dictate what matrix
elements can be nonzero and will generally result in a nonzero dispersive
shift. However, as the resonator frequency can be far off the frequency of the
contributing transitions, the dispersive shift remains small.

There are several advantages to performing readout in the proposed scheme:
Suppression of the dispersive shift controlled by the external flux grants us
insensitivity to dephasing through photon shot noise [39]. By coupling the
readout resonator to the $\theta$-mode of the qubit, we also obtain protection
against Purcell decay: The matrix element $\bra{0}n_{\theta}\ket{1}$ (or in
the notation surrounding Eq. (11)), $\bra{+}n_{\theta}\ket{-}$) is zero since
the computational states are both in the even $\theta$-mode ground state in
their respective wells. Via this mechanism, the qubit is protected from the
Purcell effect due to the symmetries of the wave functions. There are no
additional Purcell effect due to $n_{\phi}$ as the readout resonator remains
decoupled from this mode. In total, the dominant source of error during
readout is the direct tunneling between the qubit states. The $T_{1}$-times
computed in Sec. II depends weakly on the external flux and for readout at
$\phi_{\mathrm{ext}}=1.023\pi$ we find $T_{1}=519\,$\mathrm{\SIUnitSymbolMicro
s}$$. For a readout integration time around $1\,$\mathrm{\SIUnitSymbolMicro
s}$$, the $T_{1}$-limited readout fidelity is $F=99.8\%$.

##  IV Conclusions and discussions

In this paper, we have shown how gates and readout can be performed on a new
flux qubit variation with a variable level of $T_{1}$-protection, the DSFQ. By
adiabatically reducing the height of the barrier, the otherwise insensitive
qubit can be made sensitive to a microwave drive. Our implementation of this
variable-protection gate scheme shows that fast, high fidelity single qubits
gates can be performed without involving lossy non-computational states. We
achieve single qubit gates with coherence limited fidelities at $99.98\%$ in
$25\,$\mathrm{ns}$$, making it competitive with established gate schemes for
both protected and unprotected qubits. However, non-optimized gates suffer
from $T_{1}$ decay during the lowering of the barrier and results in a
$T_{1}$-limited gate fidelity of $99.1\%$. Likewise, we show that by lowering
the barriers of two capacitively coupled DSFQs, that high fidelity two qubit
gates in the fSim-space can be performed. Specifically, we find CPHASE and
$\sqrt{\text{iSWAP}}$ gates with a coherence limited fidelity above $99.9\%$
in $69\,$\mathrm{ns}$$ and $33\,$\mathrm{ns}$$ respectively without residual
$ZZ$-interactions. Again, the two-qubit gates are limited by relaxation and
the $T_{1}$-limited fidelities are $F_{\text{CPHASE}}=91.4\%$ and
$F_{\sqrt{\text{iSWAP}}}=99.7\%$ respectively The fidelities and gate times
can be further improved by using optimized protocols.

We have further shown that readout can be performed efficiently in the
$T_{1}$-protected regime by adjusting the external flux bias away from the
flux frustration point. Near half a flux bias, the dispersive shift is not
only reduced due the the qubit-resonator detuning, but also due to the
approximate symmetry between the two wells. With the order-of-magnitude
variations in dispersive shift and separated double-wells, the DSFQ is robust
againt noise channels arising from the coupling to the resonator.

In addition to the $T_{1}$-protection, we have also proposed a gradiometric
double-loop variation of the DSFQ which is exponentially insensitive to global
flux noise while remaining linearly sensitive to local flux noise. We show
that area variability of the loops can be compensated for by making the
tunable junction slightly asymmetric without being sensitive to the noise in
the tunable junctions.

In total, the DSFQ presents an experimentally available platform for studying
qubits with a variable level of $T_{1}$-protection, where gates can be
performed without involving non-computational states. This contribution may
help pave the way for achieving fast, high fidelity gates on protected qubits
using this novel gate implementation.

##  V Acknowledgments

We acknowledge helpful discussions from András Gyenis and Jonas Vinther and
are grateful to Terry P. Orlando for comments on the manuscript. This research
was supported by the Danish National Research Foundation, the Danish Council
for Independent Research $|$ Natural Sciences. This project has received
funding from the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation programme under Grant Agreement No.
856526. We acknowledge support from the Deutsche Forschungsgemeinschaft (DFG)
– project grant 277101999 – within the CRC network TR 183 (subproject C03). MK
gratefully acknowledges support for this research in part by the U.S. Army
Research Office Grant No. W911NF-22-1-0042 and in part by the Villum
Foundation (grant 37467) through a Villum Young Investigator grant.

Figure 6: (a)-(c) Pulse sequence for the $\sigma_{x}$, $\sigma_{y}$ and
$\sigma_{xy}=(\sigma_{x}-\sigma_{y})/\sqrt{2}$. Parameters for the pulse
envelope and the $\alpha$-profile is identical to those in Fig. 3. The phase
offset and drive frequency are in the three cases: (a)
$\phi_{\text{offset}}=0\pi,\omega_{d}=0.979\omega_{q}$, (b)
$\phi_{\text{offset}}=0.5\pi,\omega_{d}=0.979\omega_{q}$ and (c)
$\phi_{\text{offset}}=0.26\pi,\omega_{d}=0.977\omega_{q}$. (d)-(f)
Corresponding evolution of the states during the gate operation. The
fidelities in the three panels are (d) $F_{x}=99.98\%$, (e) $F_{y}=99.98\%$
and (f) $F_{xy}=99.93\%$. Figure 7: Plot of the $ZZ$-interaction strength due
to the capacitive coupling. When one or none of the barriers are lowered, the
interaction strength is suppressed to the $10\,$\mathrm{kHz}$$ level.

##  Appendix A Gradiometric DSFQ

To better understand the dependence on the global flux, we look at the
potential energy for the double-loop qubit

$\displaystyle H_{J}=$ $\displaystyle-
E_{J}\cos(\phi_{1})-E_{J}\cos(\phi_{2})+H_{J\alpha},$ (12) $\displaystyle
H_{J\alpha}=$
$\displaystyle-\frac{E_{J}}{2}\left[\alpha_{1}\cos(\phi_{\mathrm{ext,1}}+2\phi)+\alpha_{2}\cos(\phi_{\mathrm{ext,2}}+2\phi)\right].$

Here the flux-induced phases are given by

$\phi_{\mathrm{ext}1,2}=\frac{\pm 2\pi A_{1,2}B_{1,2}}{\Phi_{0}},$ (13)

where $A_{1,2}$ are the areas of the two loops and $B_{1,2}=(B\pm b)/2$ are
the field through them.

In the symmetric case when $A_{1}B_{1}=A_{2}B_{2}$, $\alpha_{1}=\alpha_{2}$
the flux-dependent term becomes

$H_{J\alpha}=-\frac{E_{J}}{2}\alpha_{1}\cos(2\phi)\cos(\phi_{\mathrm{ext,1}}),$
(14)

and we see that the potential maintains the symmetry with two degenerate
minima for all values of the global field $B$. However, it is not realistic to
assume that the two areas can fabricated to be identical. Therefore, we
consider the situation where they differ by some (small) amount. To study this
case, we write $H_{J\alpha}$ as

$H_{J\alpha}=-\frac{E_{J}}{2}\left[V_{c}\cos(2\phi)-V_{s}\sin(2\phi)\right],$
(15)

where

$\displaystyle V_{c}$
$\displaystyle=\alpha_{1}\cos(\phi_{\mathrm{ext,1}})+\alpha_{2}\cos(\phi_{\mathrm{ext,2}}),$
(16a) $\displaystyle V_{s}$
$\displaystyle=\alpha_{1}\sin(\phi_{\mathrm{ext,1}})+\alpha_{2}\sin(\phi_{\mathrm{ext,2}}).$
(16b)

The splitting of the degeneracy of the minima of $V_{c}$ is controlled by the
second term $V_{s}$. One could, in principle, choose a set parameters
$(\alpha_{1},\alpha_{1},B_{1},B_{2})$ such that $V_{s}=0$ and regain the
degenerate double-well potential. However, the degeneracy is lifted linearly
in both the global external field $B$ and the tuning of the Josephson
junctions, and the situation is therefore worse than before. Instead, we
search for a point where the qubit is split by the different well depths, but
with at least quadratic protection against deviations from the mentioned set
of parameters. If both junctions in the outer SQUID-loop are tunable
junctions, we have to minimize with respect to both which gives the condition
$\sin(\phi_{\mathrm{ext,1}})=\sin(\phi_{\mathrm{ext,2}})$ at the operating
point. Consequently, the condition for the junctions when minimizing with
respect the global field $B$ is

$A_{1}\alpha_{1}=A_{2}\alpha_{2}.$ (17)

If the tunable junctions are parameterized as
$\alpha_{2}=(1+\delta)\alpha_{1}$, the condition obtaining the sweet spot
where the splitting is quadratic or better in $\delta$ and $B$ is

$\displaystyle\frac{\partial V_{s}}{\partial\delta}\ =0$ $\displaystyle\to\
\sin(\phi_{\mathrm{ext,2}})=0,$ (18a) $\displaystyle\frac{\partial
V_{s}}{\partial B}=0\ $ $\displaystyle\to\
\cos(\phi_{\mathrm{ext,1}})=\frac{(1+\delta)A_{2}}{A_{1}}\cos(\phi_{\mathrm{ext,2}}).$
(18b)

Note that the condition in Eq. (18a) results in a $V_{s}$ which is insensitive
to $\delta$ for all $\delta$. If the flux through the two loops is controlled
by a single global field (i.e., $b=0$), the two equations above can be
combined to give the following condition on $\delta$,

$\delta=-1+\frac{1+r}{1-r}\cos\left(\frac{2\pi r}{1-r}\right)\approx 2r,\quad
r=\frac{A_{1}-A_{2}}{A_{1}+A_{2}},$ (19)

for small $r$.

##  Appendix B IQ-mixing

We show that our single qubit gate scheme is compatible with IQ-mixing in Fig.
6. The pulses are parametrized by
$\varepsilon(t)\cos(\omega_{d}t+\phi_{\text{offset}})$, where $\varepsilon(t)$
is the envelope with cosine ramp up/down and $\phi_{\text{offset}}$ is the
phase offset that determines the $I$ and $Q$ components. We display three flip
gates $\sigma_{x}$ (also found in Fig. 3), $\sigma_{y}$ and
$\sigma_{xy}=(\sigma_{x}-\sigma_{y})/\sqrt{2}$ with similar fidelities
$>99.9\%$ and a $25\,$\mathrm{ns}$$ gate time. The pulse parameters can be
found in the caption of Fig. 6.

##  Appendix C Q-Q coupling

Two coupled DSFQs are shown in Fig. 4(a). The Lagrangian for the total circuit
is

$\displaystyle\mathcal{L}$
$\displaystyle=\frac{C}{2}\dot{\phi}_{1}^{2}+\frac{C}{2}\dot{\phi}_{2}^{2}+\frac{C}{2}\dot{\phi}_{3}^{2}+\frac{C}{2}\dot{\phi}_{4}^{2}$
$\displaystyle+\frac{C_{g}}{2}\quantity(\dot{\phi}_{1}-\dot{\phi}_{3})^{2}$
$\displaystyle+E_{J}\cos\phi_{1}+E_{J}\cos\phi_{2}+\alpha_{1}E_{J}\cos\quantity(\phi_{1}-\phi_{2}+\phi_{\mathrm{ext,1}})$
$\displaystyle+E_{J}\cos\phi_{3}+E_{J}\cos\phi_{4}+\alpha_{2}E_{J}\cos\quantity(\phi_{3}-\phi_{4}+\phi_{\mathrm{ext,2}}).$
(20)

By performing a Legendre transformation, we arrive at the result

$\displaystyle H_{1(2)}$
$\displaystyle=4E_{C}\quantity(\frac{C+C_{g}}{C+2C_{g}})n_{1(3)}^{2}+4E_{C}n_{2(4)}^{2}$
$\displaystyle-E_{J}\cos\phi_{1(3)}+E_{J}\cos\phi_{2(4)}$
$\displaystyle+\alpha_{1(2)}E_{J}\cos\quantity(\phi_{1(3)}-\phi_{2(4)}+\phi_{\mathrm{ext}_{1(2)}}),$
$\displaystyle H_{\mathrm{Q-Q}}$
$\displaystyle=4E_{C}\quantity(\frac{C_{g}}{C+C_{g}})n_{1}n_{3},$ (21)

where 1(2) refers to qubit 1(2) with charge and phase operators
$n_{1(3)},n_{2(4)},\phi_{1(3)},\phi_{2(4)}$. The full Hamiltonian is a sum the
two qubit Hamiltonians and the interaction term,
$H=H_{1}+H_{2}+H_{\mathrm{Q-Q}}$. The qubit Hamiltonians have been
renormalized due to the coupling capacitance between the two circuits. In Fig.
7, we show the $\sigma_{z}^{(1)}\sigma_{z}^{(2)}$ coupling due to the
capacitive coupling defined by
$\zeta_{ZZ}=\omega_{00}-\omega_{01}-\omega_{10}+\omega_{11}$. In Fig. 7, it is
apparent that the $\sigma_{z}^{(1)}\sigma_{z}^{(2)}$ coupling is suppressed
unless both barriers are lowered. Thus, single qubit gates where only one
barrier is lowered do not give rise to unwanted
$\sigma_{z}^{(1)}\sigma_{z}^{(2)}$ interactions. However, we are limited to
only half-grid single qubit gates if we neglect the next nearest neighbor
stray capacitances. As a final remark, we would like to point to the half-
circular suppression of $\zeta_{\mathrm{<<}}$ in Fig. 7. This interesting
feature appears when the sign of the $\sigma_{z}^{(1)}\sigma_{z}^{(2)}$
interaction changes. In colloquial terms, the
$\sigma_{z}^{(1)}\sigma_{z}^{(2)}$ interaction is exactly cancelled when the
“push” or “pull” on the $\ket{11}$ state from states below it is exactly
compensated for by the push/pull from states above it.

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