# Three-Josephson Junctions Flux Qubit Couplings

María Hita-Pérez<EMAIL_ADDRESS>Gabriel Jaumà  Manuel Pino  Juan
José García-Ripoll  Institute of Fundamental Physics IFF-CSIC, Calle Serrano
113b, Madrid 28006, Spain

###### Abstract

We analyze the coupling of two flux qubits with a general many-body projector
into the low-energy subspace. Specifically, we extract the effective
Hamiltonians that controls the dynamics of two qubits when they are coupled
via a capacitor and/or via a Josephson junction. While the capacitor induces a
static charge coupling tunable by design, the Josephson junction produces a
magnetic-like interaction easily tunable by replacing the junction with a
SQUID. Those two elements allow to engineer qubits Hamiltonians with $XX$,
$YY$ and $ZZ$ interactions, including ultra-strongly coupled ones. We present
an exhaustive numerical study for two three-Josephson junctions flux qubit
that can be directly used in experimental work. The method developed here,
namely the numerical tool to extract qubit effective Hamiltonians at strong
coupling, can be applied to replicate our analysis for general systems of many
qubits and any type of coupling.

Qubit-qubit interactions are a fundamental tool for quantum information
processing, both in existing quantum computers [1] as well as in quantum
simulators [2, 3, 4]. Qubits that interact directly by physical means, usually
develop dipolar-like couplings with a well-defined
orientation—$\sigma^{x}_{1}\sigma^{x}_{2}$ or $\sigma^{z}_{1}\sigma^{z}_{2}$
on the qubit basis—due to the electromagnetic nature of those interactions.
This is the case of transmons and flux qubits, objects which exhibit
capacitive and inductive type couplings, respectively.

In recent years there has been growing interest in enlarging the families of
interactions between superconducting qubits [5, 6, 7, 8] while preserving
their strength. This has obvious advantages in the world of quantum computing
by making possible the implementation of a rich family of gates [9]. It is
also relevant in the development of quantum simulators, enabling non-
stoquastic models that are harder to simulate classically [10, 11, 12, 13, 6],
and which could eventually lead to universal adiabatic quantum computation
[14, 15, 16].In this context, flux qubits [6, 7, 8] gain extra relevance when
compared to transmon qubits [5]. The large anharmonicity of flux qubits
ensures the preservation of a well-defined qubit basis even for ultra-strong
couplings that dominate over the qubit’s local Hamiltonian. In comparison, an
ultra-strong interaction between transmon qubits enables transitions from
qubit states to the weakly anharmonic excitations, so that the effective
dynamics cannot be captured by a spin-$1/2$ model.

In this work we perform an exhaustive study of inductive and capacitive
couplings between three-Josephson-junction flux qubits (3JJQ) [17], as an
alternative to the commonly used rf-SQUID qubits [6], analyzing the origin of
the interactions and the effective Hamiltonians that they produce. The
capacitive interaction is implemented via a capacitor joining the two qubits,
and the inductive-like coupling comes from a shared Josephson junction. One of
our main goals is to determine the best circuit designs and parameter region
that provides large couplings while retaining acceptable qubits properties.
This is relevant in a scenario where there is a non-linear dependence of
physical properties on the circuit parameters.

We are interested in the extraction of the effective low-energy Hamiltonian
for this type of interacting circuits. Our tool to do so is an improved
version of the Schrieffer-Wolff transformation (SWT) introduced in Ref. 7.
This method has allowed us to demonstrate that capacitively and inductively
coupled 3JJQs reproduce a fairly large family of spin-1/2 models, with strong
and non-stoquastic couplings on the qubit basis. If we denote the Hamiltonian
of one flux qubit as $H_{q}=\frac{\Delta}{2}\sigma^{z}$, the Josephson
junction coupling produces a magnetic-like interaction of the
$J_{xx}\sigma^{x}_{1}\sigma^{x}_{2}$ form, while the capacitive coupling
creates interactions along orthogonal directions
$J_{yy}\sigma_{1}^{y}\sigma_{2}^{y}$ and $J_{zz}\sigma_{1}^{z}\sigma_{2}^{z}$
[18, 6, 7, 19]. Our study shows regimes of strong coupling with $J_{xx}\approx
J_{yy}\approx J_{zz}>\Delta$ using realistic qubit designs as those proposed
by Ref. 20.

We first analyze the numerical method that allows us to extract the effective
Hamiltonians $H_{\text{eff}}$ from a general many-body system by means of the
SWT. A unitary mapping between two non-orthogonal subspaces of the same
dimension [21, 22], represented by projectors $P_{0}$ and $P$

$U=\sqrt{(2P_{0}-\openone_{0})(2P-\openone)},$ (1)

satisfying $UPU^{\dagger}=P_{0}$. Here, $P_{0}$ projects onto the low-energy
or qubit subspace of a set of non-interacting circuits $H_{0}=\sum_{i}H_{i}$,
being Hi the independent Hamiltonian of each of them, and $P$ accesses the
low-energy subspace when those circuits interact producing a Hamiltonian
$H=H_{0}+V$. $\openone_{0}$ and $\openone$ are the projectors onto the full
set of eigenstates of the Hamiltonians. For moderate interactions the subspace
$P$ remains gapped from non-qubit states and allows us to interpret the
physical interaction $V$ as a coupling between qubits in the unperturbed qubit
subspace

$H_{\text{eff}}=P_{0}UPHPU^{\dagger}P_{0}.$ (2)

Consani et al. [7] computed $U$ directly from its definition (1), and showed
that $H_{\text{eff}}$ includes qubit-qubit interactions [7] not captured by
perturbative methods. To make the method affordable, Consani et al. [7]
express $H$ in the basis of eigenstates of the uncoupled qubits with up to
$N_{T}$ states—a number determined by converegence—. In this basis, $H$ is
approximately diagonalized to recover the interacting eigenstates and $P$, and
$U$ is computed using equation (1). This step dominates the complexity of the
algorithm, due to working with matrices of size $N_{T}\times N_{T}$.

We propose to compute the rank-$d$ matrices $P_{0}UP$ and $PU^{\dagger}P_{0}$,
using only the $d$ eigenstates of $H_{0}$ and $H$ that span their low-energy
subspaces. Thus, instead of computing $U$ in the full basis, we only need to
estimate $d\times d$ matrices. The cost of the algorithm is now dominated by
the calculation of the $d$ eigenstates, a step also present in Consani’s work,
which is done efficiently using Lanczos techniques. To develop this
simplification, we note that $P_{0}UP$ involves a rank-$d$ transformation $A$

$P_{0}UP=\sum_{i,j=1}^{d}A_{ij}\ket{\psi_{i}^{0}}\\!\bra{\psi_{j}}=P_{0}AP,$
(3)

among the basis $\ket{\psi_{j}^{0}}$ and $\ket{\psi_{j}}$ in which we
represent $P_{0}$ and $P$. Using these basis, $H_{\text{eff}}$ becomes

$H_{\text{eff}}=\sum_{ijk=1\ldots
d}\ket{\psi_{i}^{0}}\\!A_{ij}\braket{\psi_{j}}{H}{\psi_{j}}(A^{\dagger})_{jk}\bra{\psi^{0}_{k}}.$
(4)

We introduce a rank-$d$ operator $B=P_{0}P$, such that
$(P_{0}UP)^{2}=(P_{0}AP)^{2}=P_{0}AB^{\dagger}AP=P_{0}BP$, hence
$AB^{\dagger}A=B$. And its singular value decomposition [23] $B=W\Sigma
V^{\dagger}$, in terms of two unitary transformations
$W,V\in\mathbb{C}^{d\times d}$, and a non-negative diagonal matrix $\Sigma$.
Using these matrices, we can verify that the previous equation is satisfied by
$A=WV^{\dagger}$.

Figure 1:  (a) Two identical c-shunted 3JJQs with grounds in nodes $\phi_{0}$
and $\phi_{0}^{\prime}$ coupled through a capacitor with capacitance $C_{c}$
and a Josephson junction with Josephson energy $E_{J,c}$. (b) Analysis of one
single 3JJQ. At left panel, unit cell for the periodic nonlinear potential at
full frustration, the potential strength is represented in a scale of colors
from the maximum in white to the minimum in black, as a function of the phase
variables $\varphi_{i}=2\pi\phi_{i}/\Phi_{0}$. The two qubit states are
indicated as $\ket{L},\ket{R}$ for the unit cell ($\ket{R^{\prime}}$
correspond to a qubit state at an adjacent cell). The inset shows the barrier
height through the intra and inter-cell tunneling directions. Right panel:
ratio of energy differences between $E_{12}/\Delta$, with $E_{12}$ the energy
difference between second and first exited levels and $\Delta$ the qubit gap,
as a function of $\alpha$ for multiple values of $r=E_{J}/E_{C}$ and
$\beta=0$. Notice that the relative anharmonicity is
$\alpha_{r}=E_{12}/\Delta-1$.

We are interested on using the SWT on pairs of superconducting qubits, with a
bare Hamiltonian $H_{0}=H_{1}+H_{2}$ and an interaction $\gamma V$ controlled
by a parameter $\gamma$. Provided there exists a gapped low-energy subspace
$P$, we can express the circuit model $H=H_{0}+\gamma V$ in the charge basis
and compute $H_{\text{eff}}$, which we expand in the complete basis of Pauli
matrices as

$H_{\text{eff}}=\sum_{i=x,y,z}\left(\frac{h_{1i}}{2}\sigma_{1}^{i}+\frac{h_{2i}}{2}\sigma_{2}^{i}\right)+\sum_{i,j=x,y,x}J_{ij}\sigma_{1}^{i}\sigma_{2}^{j}.$
(5)

We apply this technique to 3JJQs [17, 24], which are composed of a
superconducting loop interrupted by three Josephson junctions, two identical
and one a factor $\alpha$ times smaller [cf. Fig. 1]. At full frustration
$\Phi_{\rm ext}=\frac{1}{2}\Phi_{0}$, the periodic inductive potential has a
unit cell depicted in Fig. 1(b). For $\alpha>0.5$ this unit cell has two
minima that correspond to two persistent current states flowing in opposite
direction. Quantum tunneling couples these current states, creating an
effective qubit subspace $H_{q}=\frac{\Delta}{2}\sigma^{z}$ with a gap
$\Delta$. We choose our qubit parameters in the range $0.6<\alpha<0.9$. The
first inequality guarantees that the qubit subspace $P_{0}$ is gaped from non-
qubit states with a large anharmonicity $E_{21}>2\Delta$. The second
inequality ensures that the intra-cell tunneling along direction $d_{1}$
dominates over the inter-cell tunneling $d_{2}$ [cf. Fig. 1(b)], reducing the
sensitivity to phase slips and charge noise [17]. We also study qubits with a
shunting capacitor in parallel to the small junction because, despite their
reduced anharmonicity, they exhibit lower sensitivity to flux noise and better
reproducibility [25].

Figure 2: Coupling strengths for two 3JJQs with ground in
$\phi_{0}$($\phi_{0}^{\prime}$) coupled through a capacitor connecting nodes
$\phi_{2}-\phi_{1}^{\prime}$. (a) Effective Hamiltonian parameters as a
function of $\gamma$ for $\alpha=0.70$, $r=50$ and $\beta=0$. (b-d) Ratios
between the coupling strengths ($J_{ii}$) and the qubit gap ($\Delta$) for
fixed: (b) $r$ and $\beta$, (c) $\alpha$ and $\beta$, (d) $\alpha$ and $r$.
The legend in plot (a) holds for (b-d). We represent $-J_{yy}$ for the sake of
clarity.

Now we move on the discussion of qubit’s coupling. We consider circuits such
as the one in Fig. 1(a), as well as 11 other topologies that change the places
where the capacitor and inductor are attached and the positions of the
grounds. These circuits are quantized and brought to the form
$H=H_{1}+H_{2}+H_{\text{int}}$, where the $H_{1,2}$ are identified as
renormalized single-qubit Hamiltonians, and the remaining interactions are
grouped into a perturbation $H_{\text{int}}:=\gamma V$ (see Supplementary
Material A). The complete Hamiltonian is then analyzed using the SWT and
brought into a form (5). Working at the symmetry point, $\Phi_{\rm
ext}=\frac{1}{2}\Phi_{0}$, we find that the only nonzero terms are the
renormalized single-qubit gaps $h_{1z}=\Delta_{1}$ and $h_{2z}=\Delta_{2}$,
and the qubit-qubit interactions $\sigma^{x}_{1}\sigma^{x}_{2}$,
$\sigma^{y}_{1}\sigma^{y}_{2}$ and $\sigma^{z}_{1}\sigma^{z}_{2}$, modulated
by the coupling strengths $J_{xx}$, $J_{yy}$ and $J_{zz}$.

Let us discuss first the capacitive coupling of two identical 3JJQs shown in
Fig. 1(a) with $C_{c}=\gamma C$, $E_{J,c}=0$ and grounds at
$\phi_{0}=\phi_{0}^{\prime}=0$. From the analytical perturbative study
performed in Ref. 19, we expect three types of interactions
$\sigma^{y}_{1}\sigma^{y}_{2},\ \sigma^{z}_{1}\sigma^{z}_{2}$ and
$\sigma^{x}_{1}\sigma^{x}_{2}$, corresponding to first, second and third order
coupling in $\gamma$. The $J_{yy}$ terms are explained by the matrix elements
of the charge operator within the qubit subspace [19], while $J_{zz}$ and
$J_{xx}$ are interactions mediated by states outside the qubit space.
Perturbation theory agrees qualitatively with the numerically exact results
shown in Fig. 2(a) for small interaction strengths $\gamma$, but its
prediction fails for moderate interactions where the $J_{yy}$ and $J_{zz}$
couplings reach a maximum similar in magnitude and then slowly start to decay.

Figs. 2(b-d) display the growth of the relative interaction strength
$J/\Delta$ for the design parameters in Fig. 1(a) ($\alpha$, $r$ and $\beta$),
illustrating the crossover from weak $J/\Delta\ll 1$ to strong coupling regime
$J/\Delta\approx 1$. For small $\gamma$, the behavior of the coupling is
dominated by the perturbative tendencies in $J_{ii}$. For larger couplings,
the growth of $J/\Delta$ is dominated by the exponential decrease [17] of the
gap $\log{(\Delta)}=\mathcal{O}(\sqrt{C_{q}/C})$ with the renormalized qubit
capacitance, which grows with $\gamma,\beta$ and $\alpha$. This competition
explains the non-monotonical behaviour found in $J_{zz}/\Delta,J_{yy}/\Delta$
with respect to $\alpha,r$ [cf. Figs. 2(b-c)], as $J$ decreases while
$1/\Delta$ increases with those parameters. Finally, for the limited range of
$\gamma$ where the gap is not negligible, $J_{zz}/\Delta$ and $J_{yy}/\Delta$
always decrease with the shunting $\beta$.

Note that at the same time as the intra-cell tunneling is suppressed, which
produces the exponential decay in the qubit’s gap, the tunneling along the
$d_{2}$ direction may get activated, see Fig. 1. This phenomenon is due to the
renormalization of the capacitances along different directions and is thus
dependent on the qubit’s parameters and the circuit topology. A consequence of
this activation is the fast growth of the $J_{xx}\sigma^{x}_{1}\sigma^{x}_{2}$
interaction. This is, in our opinion, a regime to be avoided. First, because
the $J_{xx}$ can be obtained by other (inductive) means. And second, because
the activation of the inter-cell tunneling is accompanied by a greater
sensitivity to electrostatic field fluctuations.

Different coupling topologies produce qualitatively similar plots, although
the relative coupling strength $J_{ii}/\Delta=1$ may be reached for lower or
higher values of the capacitance $\gamma$, and the relative sign of the
interactions might change. We have also studied different grounding schemes.
Topologically, there are two distinct combinations: we can place the grounds
between the small and big junctions —i.e. $\phi_{1}=0$ or $\phi_{2}=0$—or we
can place them between the big junctions $\phi_{0}=\phi_{0^{\prime}}=0$.
Choosing between $\phi_{1}=0$ or $\phi_{2}=0$ is equivalent to flipping the
flux passing through the qubit, and changes the sign of the $\sigma^{y}$ and
$\sigma^{x}$ operators. If we choose topologically equivalent grounds for both
qubits, we obtain coupling strengths with similar magnitude as the ones seen
before. However, there are somewhat pathological choices—e.g.
$\phi_{0}=\phi_{1^{\prime}}=0$ connecting nodes 0 and $1^{\prime}$—where the
qubits experience different renormalizations and their gaps differ as
interaction grows.

It must be remarked that for all choices of connecting nodes and ground nodes
we always obtain _both_ $J_{yy}\sigma^{y}_{1}\sigma^{y}_{2}$ and
$J_{zz}\sigma^{z}_{1}\sigma^{z}_{2}$ interactions simultaneously, with very
similar magnitude. This means that we can engineer effective qubit-qubit
interactions of the approximate form
$J(\sigma_{1}^{z}\sigma_{2}^{z}\pm\sigma_{1}^{y}\sigma_{2}^{y})$, with
$J\approx\Delta$ and with a sign that depends on the topology. This could
produce a spectral signature that is similar to the one observed in Ref. 6,
but without the flexibility suggested in that experimental work.

We have additionally studied the inductive coupling between two identical
3JJQs with a Josephson junction, using the circuit topology in Fig. 1(a), with
$E_{J,c}=\gamma E_{J}$ and $C_{c}=0$, but with grounds
$\phi_{1}=\phi_{2^{\prime}}=0$ (notice that we neglect the junction’s
capacitance). The interaction is so strong that around $\gamma\approx 0.1$ it
produces a full hybridization of the low and high energy subspaces where we
cannot isolate a qubit subspace.

Before this regime, for $0<\gamma<0.05$, as illustrated by Fig. 3(a),
interactions are dominated by the coupling
$J_{xx}\sigma_{1}^{x}\sigma_{2}^{x}$ between the effective dipolar magnetic
moments of both qubits. In addition to this, we find some residual
$J_{zz}\sigma^{z}_{1}\sigma^{z}_{2}$ and $J_{yy}\sigma^{y}_{1}\sigma^{y}_{2}$
contributions, that are up to three orders of magnitude weaker and can be
neglected.

Figure 3: Coupling strengths for two 3JJQs with ground in
$\phi_{1}$($\phi_{2}^{\prime}$) coupled through a Josephson junction
connecting nodes $\phi_{2}-\phi_{1}^{\prime}$. (a) Effective Hamiltonian
parameters as a function of $\gamma$ for $\alpha=0.70$, $r=50$ and $\beta=0$.
(b) Ratios between the coupling strengths ($J_{ii}$) and the qubit gap
($\Delta$) for fixed $\alpha$ and $r$. We represent $-J_{xx}$ for the sake of
clarity.

The dependency of the coupling strengths on the 3JJQ parameters offers a
simple picture, where the dominant inductive term
$J_{xx}\sigma^{x}_{1}\sigma^{x}_{2}$ grows with $\alpha$ and $r$ (data not
shown). This tendency is accompanied by a reduction of the qubit gap for
increasing $\alpha$ and $r$. Finally, as it can be extracted from Fig. 3(b),
adding a shunting capacitor to the 3JJQs reduces the qubits gap while
strengthening the $\sigma^{x}_{2}\sigma^{x}_{2}$ inductive coupling. This
allows for arbitrarily large ratios between the coupling strength and the gap
of the qubit leading to ultrastrong coupling but also favors the crossing
between levels inside and outside the qubit subspace for increasingly small
values of $\gamma$.

Similar to the capacitive circuit, changing the circuit topology does not
affect the qualitative behavior of the interaction with the coupling strength
$\gamma$. At most, the choice of coupling and ground nodes can speed up or
slow down the growth of interactions with $\gamma$ or change the sign of the
corresponding qubit operator.

We have used dimensionless quantities $(\alpha,\beta,\gamma,r)$ in our
previous analysis. We now present results using dimensional parameters close
to the experimental ones in Ref. 20. Employing the machinery previously
developed, we show that those parameters allow obtaining qubit-qubit strong
couplings that can be implemented in experiments.

Figure 4: Coupling strengths for the reference circuits using real
experimental 3JJQs parameters [20].(a) Effective Hamiltonian parameters as a
function of $\gamma$ for $\alpha=0.70$, $r=35$, $E_{C}=7.4$ (GHz) and
$\beta=0$ for the capacitive coupling. (b) Effective Hamiltonian parameters as
a function of $\gamma$ for $\alpha=0.80$, $r=35$, $E_{C}=7.4$ (GHz) and
$\beta=0$ for the Josephson junction coupling (including the capacitor that
accompanies it). We represent $-J_{yy}$ in (a) and $-J_{xx}$ in (b) for the
sake of clarity.

For the capacitive coupling, the parameters used in Ref. 20 are ideal to
produce a Hamiltonian of the form (5) with a negligible magnetic interaction.
Indeed, the small ratio between the Josephson energy and the capacitive energy
($E_{C}=7.4$), $r=35$, ensures that $J_{xx}\approx 0$ for the circuit
configuration considered above, while $\alpha=0.8$ ensures a good separation
of the two-qubit subspace from higher-energy excitations, even after the
coupling. However, such a large value of $\alpha$ gives rise to a strong
renormalization of the gap, limiting the coupling constants to $J_{ii}<0.2$
GHz, a small value for practical applications.

To overcome this limit, we propose to slightly reduce the qubit’s
anharmonicity, using $\alpha=0.7$. As seen in Fig. 4(a), this modification
produces effective qubit couplings
$J(\sigma_{1}^{y}\sigma_{2}^{y}+\sigma_{1}^{z}\sigma_{2}^{z})$ which are
strong ($J/\Delta\approx 1$) and values of the qubit gap and interactions that
can be measured experimentally, free of thermal fluctuations. Note that models
with $J(\sigma_{1}^{y}\sigma_{2}^{y}-\sigma_{1}^{z}\sigma_{2}^{z})$ couplings
are obtained by changing one of the flux threading one qubit, or the
connection topology.

Fig. 4(b) presents a similar analysis for the inductive coupling, using
Chiorescu et al’s original qubits [20], considering both the junction and its
capacitance. Since the inductive coupling between qubits is extremely strong
and does not allow for large values of $\gamma$, it can be seen that the added
capacitor does not really affect the coupling results. Note the lack of strong
qubit renormalization allows for strong $\sigma^{x}_{1}\sigma^{x}_{2}$
coupling ($J_{xx}/\Delta>2$) without having to modify the original qubit.

Summing up, we have performed an extensive analysis of a system composed of
two 3JJQs coupled via a capacitor and a Josephson junction. This type of
circuits can be implemented experimentally to produce qubits models with
strong coupling in different directions. Indeed, we have obtained arbitrary
interactions of the form
$J^{\text{cap}}(\sigma_{1}^{y}\sigma_{2}^{y}\pm\sigma_{1}^{z}\sigma_{2}^{z})+(J_{xx}^{\text{cap}}+J_{xx}^{\text{JJ}})\sigma_{1}^{x}\sigma_{2}^{x}$.

Our results confirm the idea that flux qubits may be used to simulate strong
non-stoquastic spin Hamiltonians, but also reveal that not all interactions
are independent as found by the simultaneous appearance of
$\sigma^{y}_{1}\sigma^{y}_{2}$ and $\sigma^{z}_{1}\sigma^{z}_{2}$ terms. This
may have consequences for the interpretation of works that argue the classical
simulability of superconducting quantum circuits [26, 27].

Out of the two interactions studied, $J_{xx}^{\text{JJ}}$ admits
straightforward tuneability, replacing the junction with a dc-SQUID. We
believe that this avoids the complex dynamic of the usual rf-SQUID tuneable
couplers, lifting its geometric constrains [7, 6, 28, 29, 30]. We also believe
that the capacitive interactions $J^{\text{cap}}$ can be tuned with the help
of mediating circuits, such as capacitively connected, tuneable frequency
qubits and resonators.

Designing the type of circuits discussed here is a difficult task. A source of
problems is the non-linear dependence of physical quantities on the design
parameters, as the exponential renormalization of the qubit gap with the
coupling capacitance. Our methods could help experimentalists in the design
and optimization of qubits and couplers. As an example of the utility of our
work, we have found acceptable parameters close to earlier experimental
setups, which provide strong capacitively or inductively coupled qubits.

Additionally, we have developed an improved numerical scheme to derive
effective Hamiltonians of coupled quantum systems, reducing the complexity of
the original algorithm in Ref. 7 down to the size of the computational space
under analysis, times the size for the representation of the relevant
eigenstates. This allows us to treat a wider class of models such as qubits
interacting with resonators [19], and can be extended to treat larger systems
if a clever representation—e.g. tensor networks—is used to describe the low-
energy states. This way, we expect to scale our simulations up to $4$ or $5$
qubits, exploring the gap renormalization due to capacitive couplings, and
whether the effective models can still be obtained as a sum of two- or at most
three-body operators in the case of strong qubit interactions.

Finally, our work leaves open questions, such as the application of our
coupling scheme in the context of quantum computation, where the tunability of
the capacitive couplings may become relevant. We expect to analyze this
question in future works, inducing a mediated capacitive coupling via, for
instance, microwave resonators [19] or other qubits [9].

## Supplementary Material

See supplementary material for the full derivation of the system’s Hamiltonian
(Supplementary Material A) and a further study of the multiple circuit
configurations (Supplementary Material B).

###### Acknowledgements.

This work has been supported by European Commission FET-Open project AVaQus GA
899561 and CSIC Quantum Technologies Platform PTI-001. Financial support by
Fundación General CISC (Programa Comfuturo) is acknowledged. The numerical
computations have been performed in the cluster Trueno of the CSIC.

## Data Availability Statement

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

## Authors Declarations

The authors have no conflicts to disclose.

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##  Supplementary Material A: Extraction of the circuit Hamiltonian

Following the procedure presented in Ref. 31 for circuit quantization, we find
that the general Lagrangian for the circuits contemplated in this paper reads:

$\mathcal{L}=\mathcal{L}_{0}+\mathcal{L}_{int}$ (S.1)

where
$\mathcal{L}_{0}=\mathcal{L}_{1}(\boldsymbol{\phi},\dot{\boldsymbol{\phi}})+\mathcal{L}_{2}(\boldsymbol{\phi}^{\prime},\dot{\boldsymbol{\phi}^{\prime}})$
is the sum of the Lagrangians of the single flux qubits (c-shunted or not)
[17]

$\displaystyle\mathcal{L}_{q}(\boldsymbol{\phi},\dot{\boldsymbol{\phi}})=$
$\displaystyle\frac{1}{2}\dot{\boldsymbol{\phi}}\textbf{C}_{q}\dot{\boldsymbol{\phi}}+E_{J}\cos{\left(\frac{\phi_{1}-\phi_{0}}{\varphi_{0}}\right)}+$
(S.2)
$\displaystyle+E_{J}\cos{\left(\frac{\phi_{2}-\phi_{0}}{\varphi_{0}}\right)}+\alpha
E_{J}\cos{\left(\frac{\phi_{2}-\phi_{1}-\Phi}{\varphi_{0}}\right)}$

with $E_{J}$ the characteristic Josephson energy of the junctions, $\Phi$ the
external flux, $\phi_{i}$ the flux variables, and $\textbf{C}_{q}$ the
capacitance matrix of the flux qubit which depends on the characteristic
capacitance of the junctions $C$ and the qubits parameter $\alpha$. And
$\mathcal{L}_{int}$ gives the interaction between qubits and depends on the
specific coupling. For a Josephson junction connecting two nodes, $i$ and
$j^{\prime}$, of different flux qubits we obtain a contribution of the form:

$\mathcal{L}_{int}^{JJ}=\gamma_{ij^{\prime}}^{JJ}E_{J}\cos{\left(\frac{\phi^{\prime}_{j}-\phi_{i}}{\varphi_{0}}\right)}$
(S.3)

While a capacitor coupling gives a contribution:

$\mathcal{L}_{int}^{cap}=\gamma_{ij^{\prime}}^{cap}\frac{C}{2}(\dot{\phi^{\prime}_{j}}-\dot{\phi_{i}})^{2}$
(S.4)

In both cases, $\gamma_{ij^{\prime}}$ represents the proportionality constant,
between coupling Josephson energy (capacitance) and the flux qubits reference
Josephson energy, $E_{J}$ (capacitance, $C$).

Using canonical variables,
$Q_{i}=\frac{\partial\mathcal{L}}{\partial\dot{\phi}_{i}}$, and the Legendre
transformation,
$H(\mathbf{Q},\boldsymbol{\phi})=\mathbf{Q}\dot{\boldsymbol{\phi}}-\mathcal{L}(\boldsymbol{\phi},\dot{\boldsymbol{\phi}})$,
we can conclude from the previous Lagrangian (S.1) that the general
Hamiltonian for two coupled 3JJQ is:

$H=H_{0}+H_{\text{int}}$ (S.5)

Here,
$H_{0}=H_{1}(\mathbf{Q},\boldsymbol{\phi})+H_{2}(\mathbf{Q}^{\prime},\boldsymbol{\phi}^{\prime})$
is the sum of the single flux qubits Hamiltonians (c-shunted or not) whose
inverse capacitance matrix may be modified by the action of the coupling
(renormalization or capacitive loading[7]), $\widetilde{\mathbf{C}}_{q}^{-1}$,

$\displaystyle
H_{q}(\mathbf{Q},\boldsymbol{\phi})=\frac{1}{2}\mathbf{Q}\widetilde{\mathbf{C}}_{q}^{-1}\mathbf{Q}-E_{J}\cos{\left(\frac{\phi_{1}-\phi_{0}}{\varphi_{0}}\right)}$
(S.6) $\displaystyle-
E_{J}\cos{\left(\frac{\phi_{2}-\phi_{0}}{\varphi_{0}}\right)}-\alpha
E_{J}\cos{\left(\frac{\phi_{2}-\phi_{1}-\Phi}{\varphi_{0}}\right)}$

Here ${H}_{int}$ describes the interaction between pairs of different flux
qubits. For the inductive coupling mediated by a Josephson junction, we find
that

$H_{\text{int}}^{JJ}=-\gamma^{JJ}_{ij^{\prime}}E_{J}\cos{\left(\frac{\phi^{\prime}_{j}-\phi_{i}}{\varphi_{0}}\right)}$
(S.7)

where only the two connected nodes, $i$ and $j$, are implicated and no
renormalization for the qubits Hamiltonians has to be considered. However, for
the electrostatic interaction mediated by a capacitor we get an interaction
term of the form

$H_{\text{int}}^{cap}=\mathbf{Q}\mathbf{C}_{c}^{-1}\mathbf{Q}^{\prime}$ (S.8)

which gives connections between all node charges of different qubits and
depends on the inverse mutual capacitance matrix, $\mathbf{C}_{c}^{-1}$. The
single qubits Hamiltonians are renormalized by rescaling the inverse of the
capacitance matrix,
$\mathbf{C}_{q}^{-1}(\alpha)\rightarrow\widetilde{\mathbf{C}}_{q}^{-1}(\alpha,\gamma^{cap}_{ij})$.

To fully understand this procedure it is necessary to define the capacitance
matrices and inverse capacitance matrices that have been mentioned during the
explanation. The full capacitance matrix for our system (including all nodes
in both qubits) is defined as follows

$\mathbf{C}=\begin{pmatrix}\widetilde{\mathbf{C}}_{q}&-\mathbf{C}_{c}\\\
-\mathbf{C}^{T}_{c}&\widetilde{\mathbf{C}}_{q}^{\prime}\end{pmatrix}.$ (S.9)

For example, the full capacitance matrix for the capacitively coupled circuit
in Fig.1(a) is

${\mathbf{C}}=C\begin{pmatrix}1+\alpha+\beta&-(\alpha+\beta)&0&0\\\
-(\alpha+\beta)&1+\alpha+\beta+\gamma&-\gamma&0\\\
0&-\gamma&1+\alpha+\beta+\gamma&-(\alpha+\beta)\\\
0&0&-(\alpha+\beta)&1+\alpha+\beta\end{pmatrix}.$ (S.10)

Thus, $\widetilde{\mathbf{C}}_{q}$ is the renormalized (or not) 3JJQs
capacitance matrix whose elements have the form

$\displaystyle(\widetilde{\mathbf{C}}_{q})_{ii}=(\mathbf{C}_{q})_{ii}+\sum_{j^{\prime}}\gamma_{ij^{\prime}}C$
(S.11) $\displaystyle(\widetilde{\mathbf{C}}_{q})_{ij}=(\mathbf{C}_{q})_{ij}$
$\displaystyle(\widetilde{\mathbf{C}}_{q}^{\prime})_{i^{\prime}i^{\prime}}=(\mathbf{C}_{q})_{i^{\prime}i^{\prime}}+\sum_{j}\gamma_{ji^{\prime}}C$
$\displaystyle(\widetilde{\mathbf{C}}_{q}^{\prime})_{i^{\prime}j^{\prime}}=(\mathbf{C}_{q})_{i^{\prime}j^{\prime}}$

Here $\mathbf{C}_{q}=\mathbf{C}_{q}^{\prime}$ is the non-renormalized single
3JJQ capacitance matrix. The elements $(C_{q})_{ii}$ are given by the sum of
all the capacitances connected to the node $i$ when uncoupled, and the
elements $-(C_{q})_{ij}$ are given by the sum of all capacitances connecting
nodes $i$ and $j$.The $\gamma_{ij^{\prime}}$ are the coupling parameters in
(S.4) which are $0$ if there’s no capacitive coupling involving the
corresponding nodes. And $\mathbf{C}_{c}$ is the mutual capacitance which
accounts for the capacitive couplings between nodes

$(\mathbf{C}_{c})_{ij^{\prime}}=\gamma_{ij^{\prime}}C.$ (S.12)

This way, we find the renormalized inverse capacitance matrix of the qubits,
$\widetilde{\mathbf{C}}_{q}^{-1}$ and
$\widetilde{\mathbf{C}}_{q}^{{}^{\prime}-1}$, and the inverse mutual
capacitance matrix, $\mathbf{C}_{c}^{-1}$, by performing the inversion of the
full capacitance matrix of the system:

$\mathbf{C}^{-1}=\begin{pmatrix}\widetilde{\mathbf{C}}_{q}^{-1}&\mathbf{C}_{c}^{-1}\\\
(\mathbf{C}_{c}^{-1})^{T}&\widetilde{\mathbf{C}}_{q}^{{}^{\prime}-1}\end{pmatrix}.$
(S.13)

##  Supplementary Material B: Complementary graphs for the different
configurations and parameters studies

In this supplementary material, we introduce some complementary graphs and
explanations to provide further support and evidence for the results presented
in the letter. In Fig. S.5, we show the energy spectra obtained from the
reference capacitive coupling configuration (Fig.1(a)) for different values of
the qubit parameters. As shown in Fig. S.5(a) for small values of the
parameter $\alpha$ the qubit’s anharmonicity is not sufficiently large and
hence there is no distinction between the qubit subspace (four lower energy
levels) and the exited subspace. Increasing $\alpha$ results in an increase of
the qubit’s anharmonicity avoiding this problem but significantly reducing the
magnitude of the interaction and qubit’s gap, this is shown in Figs. S.5 (b)
and (c). The effect in the spectrum of increasing $\beta$ and $r$ is really
similar to that of $\alpha$, as noted in the main text: the subspace of the
coupled qubits gains definition while the interactions and the qubit’s gap
decrease (Fig. S.5(d) and (e)). The results commented here motivate the
election of an value of $\alpha$ between $0.6$ and $0.9$ for our study and are
in concordance with the results presented in the letter.

Figure S.5: Low energy subspace as a function of $\gamma$ for the reference
circuit, two 3JJQs with ground in $\phi_{0}$($\phi_{0}^{\prime}$) coupled
through a capacitor connecting nodes $\phi_{2}-\phi_{1}^{\prime}$, and
different qubit parameters: (a) $r=50$, $\alpha=0.6$, $\beta=0$, (b) $r=50$,
$\alpha=0.7$, $\beta=0$, (c) $r=50$, $\alpha=0.8$, $\beta=0$, (d) $r=70$,
$\alpha=0.7$, $\beta=0$, (e) $r=50$, $\alpha=0.7$, $\beta=0.5$.

In Fig. S.6 we present the coupling strengths and qubits gap for the different
configurations of the circuit. First thing to check is that all the different
circuit configurations result on a similar qualitative behavior of the
constants when increasing $\gamma$: the qubits gap diminishes due to the
renormalization, $J_{yy}$ grows until it reaches a maximum and starts to
slowly decay and $J_{zz}$ has a similar behaviour.

Figure S.6: Effective Hamiltonian parameters as a function of $\gamma$ for
different configurations of a circuit composed of two 3JJQs with $\alpha=0.7$
and $r=50$ coupled through a capacitor. (a-e) Ground in
$\phi_{1}$($\phi_{2}^{\prime}$): (a) coupled connecting
$\phi_{2}-\phi_{1}^{\prime}$, (b) coupled connecting coupled connecting
$\phi_{0}-\phi_{0}^{\prime}$, (c) coupled connecting
$\phi_{0}-\phi_{1}^{\prime}$, (d) coupled connecting
$\phi_{2}-\phi_{1}^{\prime}$ and $\phi_{0}-\phi_{0}^{\prime}$, (e) coupled
connecting $\phi_{2}-\phi_{0}^{\prime}$ and $\phi_{0}-\phi_{1}^{\prime}$.(f-i)
Ground in $\phi_{0}$($\phi_{0}^{\prime}$): (f) coupled connecting
$\phi_{1}-\phi_{1}^{\prime}$, (g) coupled connecting
$\phi_{1}-\phi_{2}^{\prime}$, (h) coupled connecting
$\phi_{1}-\phi_{1}^{\prime}$ and $\phi_{2}-\phi_{2}^{\prime}$, (i) coupled
connecting $\phi_{1}-\phi_{2}^{\prime}$ and $\phi_{2}-\phi_{1}^{\prime}$.

We consider two different ground configurations. In Fig. S.6(f-i) we plot the
results for all configurations compound by two 3JJQs with ground in
$\phi_{0}=\phi_{0}^{\prime}=0$ coupled through one or two capacitors
connecting different nodes. Here one must note that, since both qubits are
identical and symmetric with respect to the $\alpha$-junction when their
grounds are in $\phi_{0}$ and $\phi_{0}^{\prime}$, connecting nodes
$\phi_{1}-\phi_{1}^{\prime}$ is equivalent to the connecting nodes
$\phi_{2}-\phi_{2}^{\prime}$, whereas connecting nodes
$\phi_{1}-\phi_{2}^{\prime}$ is equivalent to connecting nodes
$\phi_{2}-\phi_{1}^{\prime}$. Comparing Fig. S.6(f) and (g) it can be
extracted that changing the connection of the coupling capacitor from one
corner of the qubit to the opposite corner does not change the magnitudes of
the interactions, but it does change the sign of $J_{yy}$ and $J_{xx}$. The
main reason for this is that changing the capacitors connections this way is
equal to flipping the sign of the external flux threading one of the qubits,
provoking a change in the sign of its charge and flux operators. Fig. S.6(h)
and (i) show that performing a double coupling of the qubits does not really
increase the magnitude of the interaction but rather accelerates the
renormalization of the gap and the appearance of the maximum coupling strenght
for $J_{y}y$ and $J_{z}z$.

In Fig. S.6(a-e) we plot the results for all configurations composed of two
3JJQs with grounds in $\phi_{1}=\phi_{2}^{\prime}=0$ coupled through one or
two capacitors connecting different nodes. Here it is important to have in
mind that the symmetries that applied in the previous case are no longer valid
since the election of the ground breaks the qubits symmetry and equality,
nevertheless, there are other equivalences that can be considered in this case
given that the qubit’s parameters are identical, and changing the qubit’s
ground from one upper corner to the other equates to changing the external
flux’s sign. For instance, configuration (c) is equivalent to connecting node
$\phi_{2}-\phi_{0}^{\prime}$ and any of the results that can be found by
changing the ground connections from one upper corner to the other can be
extracted from these graphics only by taking into account the change in the
interactions sign. Broadly speaking, looking into Fig. S.6(a-c) we observe
that connecting the coupling capacitor to the node opposite to the
$\alpha$-junction or one of the upper corner nodes of the flux qubit produces
different effects on its gap. This happens due to the way in which the
capacitance matrix is modified when introducing the coupling, resulting in a
quick renormalization of the gap for connections to the upper corner nodes,
(a), and an slow renormalization of the gap for connections to the lower node,
(b). We can clearly see the difference in the circuit including a capacitor
connecting an upper corner node with the lower node, (c). In this case we find
a completely antisymmetric scheme where both qubits are renormalized
differently. Focusing on Fig. S.6(d-e), one encounters again that doubling the
coupling does not increase the magnitude of the coupling but rather
accelerates the evolution of the coupling.

Figure S.7: Low energy subspace as a function of $\gamma$ for the reference
circuit, two 3JJQs with ground in $\phi_{1}$($\phi_{2}^{\prime}$) coupled
through a Josephson junction connecting nodes $\phi_{2}-\phi_{1}^{\prime}$,
and different qubit parameters: (a) $r=50$, $\alpha=0.6$, $\beta=0$, (b)
$r=50$, $\alpha=0.7$, $\beta=0$, (c) $r=50$, $\alpha=0.8$, $\beta=0$, (d)
$r=70$, $\alpha=0.7$, $\beta=0$, (e) $r=50$, $\alpha=0.7$, $\beta=0.5$.

In Fig. S.7 we show the lower energy spectra of the reference Josephson
junction coupling circuit for different values of the qubit parameters. For
the range of $\gamma$ considered none of the values of the qubit parameters
selected result in an extreme reduction of the coupling and qubit gaps or
compromise the application of the SWT (the two subspaces remain separated),
nevertheless, there are two important considerations that we have to make when
interpreting these graphs. On one hand, it is shown in Fig. S.7(a) that, even
though no levels of the qubits subspace cross with levels in the high energy
subspace, the two subspaces are not clearly differentiated in practice. The
reduction of the qubit’s anharmonicity for small values of $\alpha$ makes the
distance between the two subspaces comparable to the qubits gaps. On the other
hand, by looking into Fig. S.7(b-e) we can check that increasing $\alpha$, $r$
or $\beta$ reduces the qubits gap (distances between levels 1-2 and 3-4) but
potentiates the growing of the coupling with $\gamma$ (distance between levels
2 and 3), providing with really large values of the coupling-gap ratio.
However, this fast growth of the coupling may lead to crossing between the
systems subspaces at relatively small values of $\gamma$.

Figure S.8: Effective Hamiltonian parameters as a function of $\gamma$ for
different configurations of a circuit compound by two 3JJQs with $\alpha=0.7$
and $r=50$ coupled through a Josephson junction. (a-e) Ground in
$\phi_{1}$($\phi_{2}^{\prime}$): (a) coupled connecting
$\phi_{2}-\phi_{1}^{\prime}$, (b) coupled connecting coupled connecting
$\phi_{0}-\phi_{0}^{\prime}$, (c) coupled connecting
$\phi_{0}-\phi_{1}^{\prime}$, (d) coupled connecting
$\phi_{2}-\phi_{1}^{\prime}$ and $\phi_{0}-\phi_{0}^{\prime}$, (e) coupled
connecting $\phi_{2}-\phi_{0}^{\prime}$ and $\phi_{0}-\phi_{1}^{\prime}$.(f-i)
Ground in $\phi_{0}$($\phi_{0}^{\prime}$): (f) coupled connecting
$\phi_{1}-\phi_{1}^{\prime}$, (g) coupled connecting
$\phi_{1}-\phi_{2}^{\prime}$, (h) coupled connecting
$\phi_{1}-\phi_{1}^{\prime}$ and $\phi_{2}-\phi_{2}^{\prime}$, (i) coupled
connecting $\phi_{1}-\phi_{2}^{\prime}$ and $\phi_{2}-\phi_{1}^{\prime}$.

In Fig. S.8 we present the coupling strengths and qubit gaps for different
circuit configurations. As it can be extracted from the graphs the qualitative
behavior of the coupling constants coincides for all circuit configurations:
the qubit gaps remain practically unaltered, a predominant and growing
$J_{zz}$ term appears and some residual $J_{zz}$ and $J_{yy}$ are found. The
considerations that have to be made regarding these graphs do not differ from
those made previously for the capacitive coupling, except for one important
remark: in this case connecting two pairs of nodes with two Josephson
junctions truly results in an advantage, since it results in larger coupling
magnitudes (and not only larger $J/\Delta$ as in the double capacitive
coupling) for small values of $\gamma$.