# Pokemon: Protected Logic Qubit Derived from the 0-$\pi$ Qubit

J. Q. You<EMAIL_ADDRESS>Interdisciplinary Center of Quantum Information,
State Key Laboratory of Modern Optical Instrumentation, and Zhejiang Province
Key Laboratory of Quantum Technology and Device, Department of Physics,
Zhejiang University, Hangzhou 310027, China  Franco Nori  Theoretical Quantum
Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama
351-0198, Japan  Physics Department, University of Michigan, Ann Arbor,
Michigan 48109-1040, USA

###### Abstract

We propose a new protected logic qubit called pokemon, which is derived from
the 0-$\pi$ qubit by harnessing one capacitively shunted inductor and two
capacitively shunted Josephson junctions embedded in a superconducting loop.
Similar to the 0-$\pi$ qubit, the two basis states of the proposed qubit are
separated by a high barrier, but their wave functions are highly localized
along both axis directions of the two-dimensional parameter space, instead of
the highly localized wave functions along only one axis direction in the
0-$\pi$ qubit. This makes the pokemon qubit more protected. For instance, the
relaxation of the pokemon qubit is exponentially reduced by two equally
important factors, while the relaxation of the 0-$\pi$ qubit is exponentially
reduced by only one factor. Moreover, we show that the inductor in the pokemon
can be replaced by a nonlinear inductor using, e.g., a pair or two pairs of
Josephson junctions. This offers an experimentally promising way to implement
next-generation superconducting qubits with even higher quantum coherence.

Introduction.—Quantum computers can outperform their classical counterparts in
simulating many-body quantum systems Nori-RMP-14 ; Aspuru-Guzik-RMP-20  and
implementing important algorithms Nielsen-Chuang , owing to their
exponentially large capacity in storing and processing information. These
quantum advantages can be demonstrated when qubits in quantum computers
achieve sufficiently high quantum coherence. In recent years, superconducting
qubits Clarke-PRL-85 ; Devoret-PScip-98 ; Nakamura-Nature-1999 ; Mooij-Sci-99
; Vion-Science-02 ; Yu-Sci-02 ; Martinis-PRL-02 ; Nori-PhysRep-17  based on
Josephson-junction circuits have indeed been considerably improved in their
quantum coherence, showing quantum advantage using tens of superconducting
qubits Martinis-Nature-19 ; Zhu-PRL-21 .

This quantum-coherence enhancement of superconducting qubits was achieved by
shunting a large capacitance to the small Josephson junction in the circuit to
reduce the sensitivity of the qubits to the charge noise Siddiqi-NatRev-21 ;
Tsai-JAP-21 . This was proposed in the capacitively shunted flux qubit You-
PRB-07  and later implemented experimentally Steffen-PRL-10 ; Yan-NC-16 .
Also, it was proposed for the capacitively shunted Cooper-pair box, called the
transmon Koch-PRA-07 . To be tunable and easily coupled to the neighboring
qubits, the transmon was latter modified as the Xmon Martinis-PRL-13  by both
replacing the small Josephson junction with a superconducting quantum
interference device (SQUID) and connecting a cross-shaped electrode to the
SQUID. In addition, the gatemon Larsen-PRL-15 , which is also a transmon-like
device, and the capacitively shunted fluxonium Earnest-PRL-18  were
implemented. Generally speaking, these high-coherence superconducting qubits
are encoded only on one degree of freedom related to the small Josephson
junction (or SQUID) in the circuit, whereas other circuit elements are to
adjust the anharmonicity of the qubit Tsai-JAP-21 ; Wendin-Review-17 .

However, a practical quantum computer should be fault-tolerant, requiring a
significantly increased system scale Preskill-Quantum-18 . This needs qubits
with even higher quantum coherence. A protected logic qubit called the 0-$\pi$
qubit was proposed Brooks-PRA-13  and implemented very recently Gyenis-PRXQ-21
, which harnesses two Josephson junctions and two inductors embedded in a
superconducting loop and shunted with two intersecting capacitors [see Fig.
1(a)]. This superconducting qubit is encoded on the two degrees of freedom of
the Josephson junctions, with two isolated wave functions in the two-
dimensional (2D) parameter space acting as the basis states of the qubit.
While these two states are separated by a high barrier, they are highly
localized only along one axis direction in the 2D parameter space. Also, the
circuitry of the 0-$\pi$ qubit becomes complex when implementing each of the
two inductors with a large number of Josephson junctions Gyenis-PRXQ-21 .

In this work, we propose a simplified structure to implement a new protected
logic qubit, called pokemon, which is derived from the 0-$\pi$ qubit by using
one capacitively shunted inductor and two capacitively shunted Josephson
junctions embedded in a superconducting loop. The two basis states of the
proposed qubit are also separated by a high barrier. Moreover, they are highly
localized along both axis directions of the 2D parameter space, instead of the
highly localized basis states along only one axis direction in the 0-$\pi$
qubit. These make the pokemon more protected. For instance, the relaxation of
the proposed qubit is exponentially reduced by two equally important factors,
while the relaxation of the 0-$\pi$ qubit is exponentially reduced by only one
factor. Also, we show that the inductor in the pokemon can be replaced by a
nonlinear inductor using, e.g., a pair or two pairs of Josephson junctions,
instead of using a large number of Josephson junctions to achieve a linear
inductor. This provides an experimentally promising way to implement next-
generation superconducting qubits with even higher coherence to carry out more
demanding tasks in quantum computing.

Figure 1: (a) The circuit for the 0-$\pi$ qubit in Ref. Brooks-PRA-13 ,
composed of a superconducting loop with two identical Josephson junctions
(each has a coupling energy $E_{J}$ and capacitance $C_{J}$), two identical
inductors $L$, and two intersecting capacitors of the same capacitance
$C_{s}$. (b) The circuit for the protected logic qubit called pokemon,
consisting of a superconducting loop with two capacitively shunted Josephson
junctions and an inductor $L$ shunted by a capacitance $C_{L}$. Each junction
has a coupling energy $E_{J}$ and capacitance $C_{J}$, and is also shunted
with a large capacitance $C_{s}$. Without the (green) capacitance $C_{L}$, the
circuit is reduced to a bifluxon qubit bifluxon-20 , but each junction is
shunted by a large capacitance. In both (a) and (b), as well as in Fig. 3
below, each (red) arrow indicates the assigned phase-drop direction for the
junction or inductor. The reduced magnetic flux threading the loop is
$f=\Phi_{\rm ext}/\Phi_{0}$.

Pokemon derived from the 0-$\pi$ qubit.—In contrast to the 0-$\pi$ qubit in
Fig. 1(a), the protected logic qubit called pokemon is composed of two
(identical) capacitively shunted Josephson junctions and a capacitively
shunted inductor embedded in a superconducting loop [see Fig. 1(b)]. Also, we
harness a large shunting capacitance for each Josephson junction to reduce the
noise effect on the qubit.

In this pokemon, the phase drops across the two Josephson junctions and the
inductor are constrained by the fluxoid quantization condition:
$\varphi_{1}-\varphi_{2}-\varphi_{3}+2\pi f=0$, with $f\equiv\Phi_{\rm
ext}/\Phi_{0}$, where $\Phi_{\rm ext}$ is the applied magnetic flux threading
the loop and $\Phi_{0}=h/2e$ is the flux quantum. The voltages across the two
Josephson junctions and the inductor are related to the corresponding phase
drops by $V_{i}=(\Phi_{0}/2\pi)\dot{\varphi}_{i}$. The electric energy of the
pokemon is
$T=\frac{1}{2}(C_{s}+C_{J})(V_{1}^{2}+V_{2}^{2})+\frac{1}{2}C_{L}V_{3}^{2}=\frac{1}{4}(C_{s}+C_{J})(V_{1}+V_{2})^{2}+\frac{1}{4}(C_{s}+C_{J}+2C_{L})V_{3}^{2}$.
Using the relationship between the voltage and phase drop, as well as the
canonical coordinates $\theta\equiv\frac{1}{2}(\varphi_{1}+\varphi_{2})$ and
$\varphi\equiv\frac{1}{2}\varphi_{3}$, we can express the electric energy as
$T=(\Phi_{0}/2\pi)^{2}[(C_{s}+C_{J}){\dot{\theta}}^{2}+(C_{s}+C_{J}+2C_{L}){\dot{\varphi}}^{2}]$.

The Lagrangian of the pokemon is ${\cal L}=T-U$, where the potential is
$U=E_{J}(1-\cos\varphi_{1})+E_{J}(1-\cos\varphi_{2})+(\Phi_{0}/2\pi)^{2}\varphi_{3}^{2}/2L=2E_{J}-2E_{J}\cos\theta\cos(\varphi-\pi
f)+2(\Phi_{0}/2\pi)^{2}\varphi^{2}/L$. Conjugate to $\theta$ and $\varphi$,
the canonical momenta are $P_{\theta}=\frac{\partial{\cal
L}}{\partial{\dot{\theta}}}=2(\Phi_{0}/2\pi)^{2}(C_{s}+C_{J}){\dot{\theta}}$,
and $P_{\varphi}=\frac{\partial{\cal
L}}{\partial{\dot{\varphi}}}=2(\Phi_{0}/2\pi)^{2}(C_{s}+C_{J}+2C_{L}){\dot{\varphi}}$.
Then, the Hamiltonian of the pokemon,
$H_{q}=P_{\theta}{\dot{\theta}}^{2}+P_{\varphi}{\dot{\varphi}}^{2}-{\cal L}$,
can be expressed as

$H_{q}=\frac{P_{\theta}^{2}}{4(\Phi_{0}/2\pi)^{2}(C_{s}+C_{J})}+\frac{P_{\varphi}^{2}}{4(\Phi_{0}/2\pi)^{2}(C_{s}+C_{J}+2C_{L})}+U,$
(1)

with the commutation relations $[\theta,P_{\theta}]=i\hbar$, and
$[\varphi,P_{\varphi}]=i\hbar$. When introducing number operators for Cooper
pairs, $\hat{N}_{\theta}=-i\frac{\partial}{\partial\theta}$ and
$\hat{N}_{\varphi}=-i\frac{\partial}{\partial\varphi}$, Hamiltonian (1) is
converted to

$H_{q}=H_{J}+E_{L}\varphi^{2},$ (2)

with the inductive energy $E_{L}=2(\Phi_{0}/2\pi)^{2}/L$, and

$H_{J}=4E_{c\theta}\hat{N}_{\theta}^{2}+4E_{c\varphi}\hat{N}_{\varphi}^{2}+2E_{J}-2E_{J}\cos\theta\cos(\varphi-\pi
f).$ (3)

Here $E_{c\theta}=e^{2}[4(C_{s}+C_{J})]^{-1}\equiv\tilde{E}_{c\theta}$ and
$E_{c\varphi}=e^{2}[4(C_{s}+C_{J}+2C_{L})]^{-1}\equiv\tilde{E}_{c\varphi}$ are
single-electron charging energies relevant to the two junctions.

For the 0-$\pi$ qubit in Fig. 1(a), the phase drops across the two Josephson
junctions and the two inductors are constrained by
$\varphi_{1}-\varphi_{2}-\varphi_{3}+\varphi_{4}+2\pi f=0$. The Hamiltonian
can be written as Dempster-PRA-14  $H_{\rm tot}=H_{q}+H_{\rm osc}$, where
$H_{q}$ has the same form as in Eq. (2), but
$E_{c\theta}=\tilde{E}_{c\theta}$, $E_{c\varphi}=e^{2}[4C_{J}]^{-1}$, and
$E_{L}=(\Phi_{0}/2\pi)^{2}/L$. Here $H_{\rm
osc}=4E_{c\chi}\hat{N}_{\chi}^{2}+E_{L}\chi^{2}$, with
$E_{c\chi}=e^{2}[4C_{s}]^{-1}$ and
$\hat{N}_{\chi}=-i\frac{\partial}{\partial\chi}$, is the Hamiltonian of a
harmonic oscillator. The canonical coordinates conjugate to the number
operators $\hat{N}_{\theta}$, $\hat{N}_{\varphi}$ and $\hat{N}_{\chi}$ are
$\theta\equiv\frac{1}{2}(\varphi_{1}+\varphi_{2})$,
$\varphi\equiv\frac{1}{2}(\varphi_{3}-\varphi_{4})$, and
$\chi\equiv\frac{1}{2}(\varphi_{3}+\varphi_{4})$, respectively. Since $H_{\rm
osc}$ is fully decoupled from $H_{q}$, the Hamiltonian of the 0-$\pi$ qubit
can be reduced to $H_{q}$.

Figure 2: (a) Contour plot of the potential
$U_{J}=2E_{J}-2E_{J}\cos\theta\cos(\varphi-\pi f)$ at $f=0$. (b) and (c): Wave
functions of the two basis states $|0\rangle$ and $|1\rangle$ for (b) the
0-$\pi$ qubit and (c) the pokemon, where the contour plot corresponds to the
potential $U=U_{J}+E_{L}\varphi^{2}$. The parameters are set to be
$E_{c\theta}=0.1$, $E_{J}=10$, and $E_{L}=1$, but $E_{c\varphi}=10$ in (b),
and $E_{c\varphi}=0.09$ in (c).

For both the 0-$\pi$ qubit in Fig. 1(a) and the pokemon in Fig. 1(b), when the
external magnetic flux is absent (i.e., $f=0$), the potential without the
inductive energy, i.e., $U_{J}=2E_{J}-2E_{J}\cos\theta\cos(\varphi-\pi f)$,
has degenerate minima at points $(\theta,\varphi)=(0,0)$ and ($\pi$,$\pi$) in
the 2D parameter space [see Fig. 2(a)]. Nevertheless, the inductive potential
$E_{L}\varphi^{2}$ removes this degeneracy [see the contour plot in Figs. 2(b)
and 2(c)]. When $E_{J}\gg E_{L}$, as in the 0-$\pi$ qubit Brooks-PRA-13 , we
can use the lowest-energy states localized around (0,0) and ($\pi$,$\pi$) to
encode the pokemon qubit. Therefore, both the pokemon and the 0-$\pi$ qubit
are actually logic qubits encoded on two degrees of freedom $\theta$ and
$\varphi$. However, there are distinct differences between them. First, the
pokemon has a simpler circuitry, providing an advantage in sample fabrication.
Second, the pokemon has two small single-electron charging energies
$E_{c\theta}=\tilde{E}_{c\theta}$ and $E_{c\varphi}=\tilde{E}_{c\varphi}$,
because $C_{s}\gg C_{J}$. This corresponds to a heavy particle with
anisotropic masses $M_{\theta}=2(\Phi_{0}/2\pi)^{2}(C_{s}+C_{J})$, and
$M_{\varphi}=2(\Phi_{0}/2\pi)^{2}(C_{s}+C_{J}+2C_{L})$, moving in quantum
wells separated by a barrier of height $E_{J}\gg E_{c\theta},E_{c\varphi}$.
Owing to these large masses in both $\theta$ and $\varphi$ directions as well
as the higher inter-well barrier, the ground and first excited states
$|0\rangle$ and $|1\rangle$ of the pokemon are highly localized in the
vicinity of (0,0) and ($\pi$,$\pi$), respectively, along both $\theta$ and
$\varphi$ directions [Fig. 2(c)]; and the transition frequency of the pokemon
qubit is mainly determined by the inductive-potential difference between these
two points. As for the 0-$\pi$ qubit, the effective mass along the $\theta$
direction is as large as $M_{\theta}$ of the pokemon, but the effective mass
along the $\varphi$ direction, $M_{\varphi}=2(\Phi_{0}/2\pi)^{2}C_{J}$, is
much smaller. The ground and first excited states of the 0-$\pi$ qubit are
hence only highly localized at (0,0) and ($\pi$,$\pi$) along the $\theta$
direction [Fig. 2(b)]. This indicates that the pokemon can become more
protected.

In addition, when the capacitance $C_{L}$ is absent in Fig. 1(b), the pokemon
is reduced to the circuit configuration of the bifluxon bifluxon-20 , but
without the gate, and each Josephson junction there is now shunted by a large
capacitance $C_{s}$. This capacitively shunted bifluxon has the same
Hamiltonian as in Eq. (2), but with
$E_{c\theta}=E_{c\varphi}=\tilde{E}_{c\theta}$, i.e., the effective masses
become isotropic.

Robustness against both charge and flux noises.—Below we reveal the robustness
of the pokemon against both charge and flux noises, which are usually two
major decoherence sources in superconducting circuits. For the 0-$\pi$ qubit,
since the ground and first excited states $|0\rangle$ and $|1\rangle$ are less
localized around (0,0) and ($\pi$,$\pi$) along the $\varphi$ direction, it is
suitable to numerically study its quantum coherence Dempster-PRA-14 . On the
contrary, the ground and first excited states $|0\rangle$ and $|1\rangle$ of
the pokemon are highly localized around (0,0) and ($\pi$,$\pi$) along both
$\theta$ and $\varphi$ directions. These two basis states of the pokemon can
be well approximated by
$|0\rangle=\sqrt{\frac{\alpha_{\theta}\alpha_{\varphi}}{\pi}}e^{-\frac{1}{2}(\alpha_{\theta}^{2}\theta^{2}+\alpha_{\varphi}^{2}\varphi^{2})}$,
and
$|1\rangle=\sqrt{\frac{\alpha_{\theta}\alpha_{\varphi}}{\pi}}e^{-\frac{1}{2}[\alpha_{\theta}^{2}(\theta-\pi)^{2}+\alpha_{\varphi}^{2}(\varphi-\frac{E_{J}}{E_{J}+E_{L}}\pi)^{2}]}$,
with $\alpha_{\theta}^{2}=\frac{1}{2}\sqrt{\frac{E_{J}}{E_{c\theta}}}$, and
$\alpha_{\varphi}^{2}=\frac{1}{2}\sqrt{\frac{E_{J}+E_{L}}{E_{c\varphi}}}$. Due
to the inductive potential, the center of $|1\rangle$ is shifted to
$\frac{E_{J}}{E_{J}+E_{L}}\pi$ in the $\varphi$ direction, and the level
difference between $|1\rangle$ and $|0\rangle$ is
$\varepsilon_{10}=\frac{E_{J}E_{L}\pi^{2}}{E_{J}+E_{L}}\sim E_{L}\pi^{2}$ for
$E_{J}\gg E_{L}$.

When both charge and flux fluctuations are considered, the Hamiltonian (2) is
changed, at $f=0$, to

$\displaystyle H_{t}$ $\displaystyle=$ $\displaystyle
4E_{c\theta}(\hat{N}_{\theta}-\delta
N_{\theta})^{2}+4E_{c\varphi}(\hat{N}_{\varphi}-\delta N_{\varphi})^{2}$ (4)
$\displaystyle+2E_{J}-2E_{J}\cos\theta\cos(\varphi-\pi\delta\\!f)+E_{L}\varphi^{2},$

where $\delta N_{\theta(\varphi)}\equiv\delta Q_{\theta(\varphi)}/2e$ is the
reduced charge fluctuation and $\delta\\!f\equiv\delta\Phi_{\rm ext}/\Phi_{0}$
is the reduced flux fluctuation. Up to second-order perturbations, the
Hamiltonian (4) can be written as
$H_{t}=H_{q}+H^{\prime}_{\theta}+H^{\prime}_{\varphi}+H^{\prime}_{f}$, with
$H_{q}$ given by Eq. (2) and

$\displaystyle H^{\prime}_{\theta(\varphi)}$ $\displaystyle=$ $\displaystyle
X_{\theta(\varphi)}^{(1)}\delta
N_{\theta(\varphi)}+\frac{1}{2}X_{\theta(\varphi)}^{(2)}\delta
N^{2}_{\theta(\varphi)},$ $\displaystyle H^{\prime}_{f}$ $\displaystyle=$
$\displaystyle X_{f}^{(1)}\delta\\!f+\frac{1}{2}X_{f}^{(2)}\delta\\!f^{2},$
(5)

where $X_{\theta(\varphi)}^{(1)}\equiv(\frac{\partial H_{t}}{\partial\,\delta
N_{\theta(\varphi)}})|_{\delta
N_{\theta(\varphi)}=0}=-8E_{c\theta(\varphi)}\hat{N}_{\theta(\varphi)}$,
$X_{\theta(\varphi)}^{(2)}\equiv(\frac{\partial^{2}H_{t}}{\partial\,\delta
N^{2}_{\theta(\varphi)}})|_{\delta
N_{\theta(\varphi)}=0}=8E_{c\theta(\varphi)}$,
$X_{f}^{(1)}\equiv(\frac{\partial
H_{t}}{\partial\,\delta\\!f})|_{\delta\\!f=0}=-2\pi
E_{J}\cos\theta\sin\varphi$, and
$X_{f}^{(2)}\equiv(\frac{\partial^{2}H_{t}}{\partial\,\delta\\!f^{2}})|_{\delta\\!f=0}=2\pi
E_{J}\cos\theta\cos\varphi$.

Note that the perturbation arising from the charge noise terminates at the
second order owing to the specific form of the Hamiltonian (4). We can derive
that $\langle 1|X_{\theta(\varphi)}^{(1)}|1\rangle-\langle
0|X_{\theta(\varphi)}^{(1)}|0\rangle=0$, and $\langle
1|X_{\theta(\varphi)}^{(2)}|1\rangle-\langle
0|X_{\theta(\varphi)}^{(2)}|0\rangle=0$. These imply that the charge
fluctuation does not induce dephasing to the pokemon. However, for the flux
noise, we have nonzero $A_{f}\equiv\frac{1}{2}(\langle
1|X_{f}^{(1)}|1\rangle-\langle 0|X_{f}^{(1)}|0\rangle)$:

$A_{f}=2\pi
E_{J}\sin\left(\frac{E_{L}\pi}{E_{J}+E_{L}}\right)\exp\left\\{-\frac{1}{2}\left(\sqrt{\frac{E_{c\theta}}{E_{J}}}+\sqrt{\frac{E_{c\varphi}}{E_{J}+E_{L}}}\right)\right\\},$
(6)

which yields dephasing to the pokemon. The corresponding dephasing-induced
decay factor is $e^{-\eta(t)}$, with supplemental

$\eta(t)=\frac{1}{\hbar^{2}}|A_{f}|^{2}\\!\\!\int^{+\infty}_{\omega_{c}}\\!\\!d\omega\;S_{\\!f}(\omega)\frac{\sin^{2}(\omega
t/2)}{2\pi(\omega/2)^{2}},$ (7)

where $\omega_{c}$ is a low-frequency cutoff of the flux-noise power spectrum
$S_{\\!f}(\omega)=\int^{+\infty}_{-\infty}dt\langle\delta\\!f(t^{\prime}+t)\delta\\!f(t^{\prime})\rangle
e^{-i\omega t}$. The dephasing rate can be defined as
$\Gamma_{\phi}=1/T_{\phi}$, with the dephasing time $T_{\phi}$ determined by
$\eta(T_{\phi})=1$.

Moreover, we can derive that $\langle 0|X_{f}^{(1)}|1\rangle=0$ and $\langle
0|X_{f}^{(2)}|1\rangle=0$ for the flux noise. Similar to $X_{f}^{(1)}$ and
$X_{f}^{(2)}$, higher-order perturbations arising from flux noise also contain
either $\cos\theta\sin\varphi$ or $\cos\theta\cos\varphi$, so $\langle
0|X_{f}^{(n)}|1\rangle=0$ for $n\geq 3$. Hence the flux noise does not induce
relaxation to the pokemon. Instead, the first-order charge fluctuations yield
relaxation to the pokemon, because

$\displaystyle B_{\theta}\equiv\langle 0|X_{c\theta}^{(1)}|1\rangle$
$\displaystyle=$ $\displaystyle i2\pi\sqrt{E_{c\theta}E_{J}}\;e^{-\gamma},$
$\displaystyle B_{\varphi}\equiv\langle 0|X_{c\varphi}^{(1)}|1\rangle$
$\displaystyle=$ $\displaystyle
i2\pi\sqrt{E_{c\varphi}(E_{J}+E_{L})}\;e^{-\gamma},$ (8)

where the decay rate $\gamma$ is

$\gamma=\frac{\pi^{2}}{8}\left[\sqrt{\frac{E_{J}}{E_{c\theta}}}+\left(\frac{E_{J}}{E_{J}+E_{L}}\right)^{2}\sqrt{\frac{E_{J}+E_{L}}{E_{c\varphi}}}\right].$
(9)

According to the Fermi golden rule Ithier-PRB-05 , the relaxation rate for
each charge noise can be obtained as

$\Gamma_{1,i}=\frac{1}{\hbar^{2}}|B_{i}|^{2}S_{\\!i}(\omega_{10}),$ (10)

where each charge-noise power spectrum is defined as
$S_{\\!i}(\omega)=\int^{+\infty}_{-\infty}dt\langle\delta
N_{i}(t^{\prime}+t)\delta N_{i}(t^{\prime})\rangle e^{-i\omega t}$, with
$i=\theta$, $\varphi$, and $\omega_{10}=\varepsilon_{10}/\hbar$ is the
transition frequency of the pokemon. The total relaxation rate is
$\Gamma_{1}=\Gamma_{1,\theta}+\Gamma_{1,\varphi}$, which gives the relaxation
time $T_{1}=1/\Gamma_{1}$ of the pokemon.

For superconducting circuits, low-frequency noise plays a pivotal role in
decoherence, which can be modeled as $1/f$ noise via
$S_{i}(\omega)=K_{i}/|\omega|$, where $i=\theta$, $\varphi$, and $f$.
Typically, $K_{\theta(\varphi)}\sim 1.7\times 10^{-6}$ for charge noise
Nakamura-PRL-02  and $K_{f}\sim 3\times 10^{-12}$ for flux noise Mooij-PRL-05
. For $E_{c\theta(\varphi)},E_{L}\ll E_{J}$, $A_{f}$ reduces to a small
quantity $A_{f}\sim 2\pi^{2}E_{L}$ [see Eq. (6)], in comparison with $E_{J}$.
Also, because $K_{f}$ is several orders of magnitude smaller than
$K_{\theta(\varphi)}$, a long dephasing time $T_{\phi}$ is given to satisfy
$\eta(T_{\phi})=1$ in Eq. (7), so it gives a small dephasing rate
$\Gamma_{\phi}$. Then, we can ignore this dephasing rate and write the
decoherence rate as
$\Gamma_{2}=\frac{1}{2}\Gamma_{1}+\Gamma_{\phi}\approx\frac{1}{2}\Gamma_{1}$.
Moreover, it follows from Eqs. (Pokemon: Protected Logic Qubit Derived from
the 0-$\pi$ Qubit)-(10) that the relaxation rate is proportional to
$e^{-2\gamma}$, which is exponentially reduced by both
$\sqrt{\frac{E_{J}}{E_{c\theta}}}$ and
$\sqrt{\frac{E_{J}+E_{L}}{E_{c\varphi}}}$. Because
$E_{c\theta(\varphi)},E_{L}\ll E_{J}$, these two quantities are both large,
yielding a small relaxation rate as well. Therefore, the pokemon is a well-
protected logic qubit, owing to its small rates in both dephasing and
relaxation. In contrast, the 0-$\pi$ qubit has $E_{c\theta}\ll E_{J}$ but not
$E_{c\varphi}\ll E_{J}$, so its relaxation rate is only exponentially reduced
by $\sqrt{\frac{E_{J}}{E_{c\theta}}}$, without a further exponential reduction
by
$\sqrt{\frac{E_{J}+E_{L}}{E_{c\varphi}}}\approx\sqrt{\frac{E_{J}}{E_{c\varphi}}}$
in the pokemon.

Figure 3: (a) The pokemon modified by replacing the capacitively shunted
inductor in Fig. 1(b) with two (i.e., a pair of) Josephson junctions, both
having the same coupling energy $E^{\prime}_{J}$ and capacitance
$C^{\prime}_{J}$. (b) The pokemon modified by replacing the capacitively
shunted inductor with two pairs of Josephson junctions, where each junction
has the same coupling energy $E^{\prime}_{J}$ and capacitance
$C^{\prime}_{J}$. Extension of the pokemon applies when modified by replacing
the capacitively shunted inductor with more Josephson junctions.

Pokemon with the loop inductance replaced by a nonlinear inductance.—For both
the 0-$\pi$ qubit in Fig. 1(a) and pokemon in Fig. 1(b), it is required that
$E_{J}\gg E_{L}$, needing a large loop inductance $L$. Here we propose to
implement the pokemon by replacing the loop inductance with a nonlinear one.

We first replace the capacitively shunted inductor in Fig. 1(b) with two
Josephson junctions, both having the same coupling energy $E^{\prime}_{J}$ and
capacitance $C^{\prime}_{J}$ [see Fig. 3(a)]. The total Hamiltonian of the
pokemon can be written as supplemental

$H_{\rm tot}=H_{J}+2E_{J}^{\prime}(1-\cos\varphi)\cos\chi+H_{\chi},$ (11)

where $H_{J}$ has the same form as in Eq. (3), but with
$E_{c\theta}=\tilde{E}_{c\theta}$, and
$E_{c\varphi}=e^{2}[4(C_{s}+C_{J}+C_{J}^{\prime})]^{-1}$;
$H_{\chi}=4E_{c\chi}N_{\chi}^{2}+2E_{J}^{\prime}(1-\cos\chi)$, with
$E_{c\chi}=e^{2}[4C_{J}^{\prime}]^{-1}$. For the subsystem with $H_{\chi}$,
the corresponding harmonic-oscillator frequency is
$\omega_{\chi}=\frac{4}{\hbar}\sqrt{E_{J}^{\prime}E_{c\chi}}$. By suitably
choosing parameters to have a sufficiently large $\omega_{\chi}$, the system
stays in the ground state $|g\rangle$ of $H_{\chi}$, and the Hamiltonian (11)
can be reduced to $H_{q}=H_{J}+E^{(1)}_{J}(1-\cos\varphi)$, where
$E^{(1)}_{J}=2E^{\prime}_{J}\langle g|\cos\chi|g\rangle$. When $E_{c\chi}\ll
E^{\prime}_{J}$, we have $E^{(1)}_{J}=2E^{\prime}_{J}e^{-\eta}$, with
$\eta=\frac{1}{2}\sqrt{\frac{E_{c\chi}}{E^{\prime}_{J}}}$. Comparing with Eq.
(2), the last term in $H_{q}$, i.e., $E^{(1)}_{J}(1-\cos\varphi)$, is the
inductive potential of a nonlinear inductor, which also removes the degeneracy
of the basis states $|0\rangle$ and $|1\rangle$ at $(0,0)$ and $(\pi,\pi)$.
The energy-level difference between these two basis states of the pokemon is
$\varepsilon_{10}=2E^{(1)}_{J}\sim 4E^{\prime}_{J}$. To avoid any unwanted
levels in between $|0\rangle$ and $|1\rangle$, we can take
$\omega_{\chi}>\omega_{q}\equiv\varepsilon_{10}/\hbar$.

When replacing the capacitively shunted inductor with four (i.e., two pairs
of) Josephson junctions, each junction having coupling energy $E^{\prime}_{J}$
and capacitance $C^{\prime}_{J}$ [see Fig. 3(b)], the Hamiltonian of the
pokemon can be reduced to supplemental
$H_{q}=H_{J}+E^{(2)}_{J}[1-\cos(\varphi/2)]$. When $E^{\prime}_{J}\gg
E_{c\xi}\equiv e^{2}[4C^{\prime}_{J}]^{-1}$, we have
$E^{(2)}_{J}=4E^{\prime}_{J}e^{-(\eta+\eta^{\prime})}$, with
$\eta=\frac{1}{2}\sqrt{\frac{E_{c\xi}}{E^{\prime}_{J}}}$, and
$\eta^{\prime}=\frac{1}{2}\eta e^{\eta/2}$. Also, $H_{J}$ has the same form as
in Eq. (3), but with $E_{c\theta}=\tilde{E}_{c\theta}$, and
$E_{c\varphi}=e^{2}[4(C_{s}+C_{J}+C_{J}^{\prime}/2)]^{-1}$. Similarly, the
last term in $H_{q}$, i.e., $E^{(2)}_{J}[1-\cos(\varphi/2)]$, acts as the
inductive potential of a nonlinear inductor, which yields an energy-level
difference between the two basis states $|0\rangle$ and $|1\rangle$ of the
pokemon, $\varepsilon_{10}=E^{(2)}_{J}\sim 4E^{\prime}_{J}$. Here we have
shown that a small number of Josephson junctions can be used to achieve the
inductive potential of a nonlinear inductor to remove the state degeneracy
around (0,0) and ($\pi$,$\pi$) for the two basis states of the pokemon. This
avoids the complex circuitry in the experimentally realized 0-$\pi$ qubit
Gyenis-PRXQ-21  where a large number of Josephson junctions were used to
implement a linear inductor.

Conclusions.—In summary, a new protected logic qubit called the pokemon is
proposed by harnessing two capacitively shunted Josephson junctions and one
capacitively shunted inductor embedded in a superconducting loop. Similar to
the 0-$\pi$ qubit, the two basis states of the proposed qubit are also
separated by a high barrier, but their wave functions can be highly localized
along both axis directions of the 2D parameter space, in sharp contrast to the
highly localized wave functions along only one axis direction in the 0-$\pi$
qubit. This makes the pokemon qubit more protected. Here we find that the flux
noise does not induce relaxation but weak dephasing to the pokemon, while the
charge noise does not cause dephasing and the induced relaxation can be
exponentially reduced by the large capacitors shunted to the two Josephson
junctions. Furthermore, we show how to achieve the pokemon by replacing the
capacitively shunted inductor with a few (one or two) pairs of Josephson
junctions. This offers an experimentally feasible method to implement
protected logic qubits with even higher quantum coherence. In the near future,
an experimental comparison between the properties of a pokemon and a transmon
would be insightful. It would be the superconducting qubit analog of Godzilla
meets King Kong.

## Acknowledgments

We thank Guo-Qiang Zhang for technical assistance. J.Q.Y. is supported by the
National Natural Science Foundation of China (Grants No. 11934010 and No.
U1801661), the National Key Research and Development Program of China (Grant
No. 2016YFA0301200), and the Zhejiang Province Program for Science and
Technology (Grant No. 2020C01019). F.N. is supported in part by Nippon
Telegraph and Telephone Corporation (NTT) Research, the Army Research Office
(ARO) (Grant No. W911NF-18-1-0358), the Japan Science and Technology Agency
(JST) [via the Quantum Leap Flagship Program (Q-LEAP) program and the Centers
of Research Excellence in Science and Technology (CREST) Grant No.
JPMJCR1676], the Japan Society for the Promotion of Science (JSPS) (via the
Grants-in-Aid for Scientific Research (KAKENHI) Grant No. JP20H00134 and the
Japan Society for the Promotion of Science (JSPS)-Russian Foundation for Basic
Research (RFBR) Grant No. JPJSBP120194828), the Asian Office of Aerospace
Research and Development (AOARD) (via Grant No. FA2386-20-1-4069), and the
Foundational Questions Institute Fund (FQXi) via Grant No. FQXi-IAF19-06.

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