# Autonomous stabilization with programmable stabilized state

Ziqian Li† James Franck Institute, University of Chicago, Chicago, Illinois
60637, USA  Department of Physics, University of Chicago, Chicago, Illinois
60637, USA  Department of Applied Physics, Stanford University, Stanford,
California 94305, USA  These authors contributed equally  Tanay Roy†,∗ James
Franck Institute, University of Chicago, Chicago, Illinois 60637, USA
Department of Physics, University of Chicago, Chicago, Illinois 60637, USA
These authors contributed equally  Yao Lu  Present address: Superconducting
Quantum Materials and Systems Center, Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA James Franck Institute, University of Chicago,
Chicago, Illinois 60637, USA  Department of Physics, University of Chicago,
Chicago, Illinois 60637, USA  Eliot Kapit  Department of Physics, Colorado
School of Mines, Golden, CO 80401  David I. Schuster  James Franck Institute,
University of Chicago, Chicago, Illinois 60637, USA  Department of Physics,
University of Chicago, Chicago, Illinois 60637, USA  Pritzker School of
Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA
Department of Applied Physics, Stanford University, Stanford, California
94305, USA

###### Abstract

Reservoir engineering is a powerful technique to autonomously stabilize a
quantum state. Traditional schemes involving multi-body states typically
function for discrete entangled states. In this work, we enhance the
stabilization capability to a continuous manifold of states with programmable
stabilized state selection using multiple continuous tuning parameters. We
experimentally achieve $84.6\%$ and $82.5\%$ stabilization fidelity for the
odd and even-parity Bell states as two special points in the manifold. We also
perform fast dissipative switching between these opposite parity states within
$1.8\text{\,}\mathrm{\SIUnitSymbolMicro s}$ and
$0.9\text{\,}\mathrm{\SIUnitSymbolMicro s}$ by sequentially applying different
stabilization drives. Our result is a precursor for new reservoir engineering-
based error correction schemes.

##  I Introduction

Entanglement is one major resource any quantum protocol utilizes to achieve
quantum advantage [1, 2]. Generally, the entanglement is created by unitary
operations, where dissipation is considered detrimental and should be
maximally avoided. Inspired by laser cooling, an alternative approach is to
use tailored dissipation for stabilizing entanglement. By coupling the qubit
system to some cold reservoirs, one can engineer the Hamiltonian such that the
population will flow directionally to the stabilized point in the Hilbert
space, and extra entropy is autonomously dumped into the cold reservoir during
the process. This provides an extra route to state preparation. In a
multiqubit-reservoir coupled system, dissipation engineering can enhance the
capabilities of quantum simulation, as predicting the final state of a driven
dissipative quantum system is more complex than its unitary counterpart [3]
when all local qubit and reservoir interactions are simultaneously turned on.
Dissipation stabilization also inspires autonomous quantum error correction
codes (AQEC) [4, 5, 6, 7, 8, 9] that achieve hardware efficiency in the
experiment.

Stabilization has been theoretically proposed and experimentally realized in
different platforms, such as superconducting qubits [10, 11, 12, 13, 14, 15,
16, 17] and trapped ions [18, 19], focusing on stabilizing a single special
state per device, such as even or odd parity Bell states. Unlike universal
quantum state preparation through unitary gate decomposition, dissipative
stabilization requires individual Hamiltonian engineering for each stabilized
state through different drive combinations or hardware. This makes the tunable
dissipative stabilization a challenging task. A generalized scheme that allows
one to programmatically choose stabilized states from a large class of states
per device will expand the toolbox for state preparation. For instance, the
ability to choose an arbitrary stabilized state can be used for the
implementation of density matrix exponentiation [20, 21] by enabling an
efficient reset of the input density matrix.

In this work, we realize an autonomous stabilization protocol with
superconducting circuits that allows selection from a broad class of states,
including the maximally entangled states. We use microwave-only drives with
tunable parameters such as drive detunings and strengths that allow fast
programmable switching between Bell states of different parities. The system
is based on a two-transmon inductive coupler design [22, 17, 23, 8] that
allows fast parametric interactions between qubits without significantly
compromising their coherence. The readout resonators are also used as cold
reservoirs, eliminating the requirement for extra components. We perform
stabilization spectroscopy and demonstrate a fidelity over $78\%$ for all
stabilized states. For odd and even parity Bell pairs, we measured $84.6\%$
and $82.5\%$ stabilization fidelity and a stabilization time of
$1.8\text{\,}\mathrm{\SIUnitSymbolMicro s}$ and
$0.9\text{\,}\mathrm{\SIUnitSymbolMicro s}$ respectively. The structure of the
paper is as follows. First, we explain the Hamiltonian construction of the
stabilization protocol. Then we discuss the experimental measurement of
individual stabilized state and demonstrate a dissipative switch of Bell state
parity.

##  II Stabilization theory

Figure 1: (a) General stabilization scheme. Two qubits’ eigenstates
$\\{\left|A\right\rangle,\left|B\right\rangle,\left|C\right\rangle,\left|D\right\rangle\\}$
are plotted in the energy level diagram. When the energy relation
$E_{D}+E_{A}=E_{B}+E_{C}$ is satisfied, $\left|A\right\rangle$ is stabilized.
Qubit-resonator interactions and resonator photon decay rate $\kappa$ are
shown in blue and orange arrows. Qubit decay rate $\gamma$ is assumed slowest
and not plotted. (b) Stabilization of entangled states
$\left|\Psi_{\theta}\right\rangle=\sin\left(\theta/2\right)\left|gg\right\rangle-\cos\left(\theta/2\right)\left|ee\right\rangle$
or
$\left|\Phi_{\theta}\right\rangle=\sin\left(\theta/2\right)\left|ge\right\rangle-\cos\left(\theta/2\right)\left|eg\right\rangle$.
(c) A special case of (b) that stabilizes the odd and even parity bell states
$\left|\Phi_{-}\right\rangle$ and $\left|\Psi_{-}\right\rangle$. Circulating
arrows are color-coded to represent red (exchange-like) and blue (two-photon-
pumping) sidebands respectively. The QQ and QR sideband rates are separate
$\Omega$ and $W_{j}$, and the QR sideband is detuned in frequency by
${\Omega}/{2}$.

We consider a system of two coupled qubit-resonator pairs $\\{Q_{1},Q_{2}\\}$
and $\\{R_{1},R_{2}\\}$. The lossy resonators serve as both cold baths and
dispersive readouts for the qubits. We label the ground and the first excited
states of the qubits $Q_{1/2}$ as $\left|g\right\rangle$ and
$\left|e\right\rangle$, and of the resonators $R_{1/2}$ as
$\left|0\right\rangle$ and $\left|1\right\rangle$, with the full system state
being represented as $\left|Q_{1}Q_{2}R_{1}R_{2}\right\rangle$. The system
Hamiltonian $H_{\rm sys}=H_{QQ}+H_{QR1}+H_{QR2}$ includes the dominant two-
qubit interaction $H_{QQ}$ and qubit-resonator interactions
$H_{QRj},j=\\{1,2\\}$ acting as perturbations. We label the four eigenstates
of $H_{QQ}$ as
$\\{\left|A\right\rangle,\left|B\right\rangle,\left|C\right\rangle,\left|D\right\rangle\\}$
with eigenenergies $\\{E_{A}<E_{B}\leq E_{C}\leq E_{D}\\}$ so that
$\left|A\right\rangle$ is the target state to stabilize. Our stabilization
scheme involves engineering a one-way flow of population to
$\left|A\right\rangle$ connecting all intermediate eigenstates of the system.

Figure 2: Experimental demonstration of $\left|\Psi_{-}\right\rangle$ ((a),
(b)) and $\left|\Phi_{-}\right\rangle$ ((c), (d)) stabilization with the
initial state $\left|gg\right\rangle$. Two-qubit state tomography is performed
at each time point, and the reconstructed density matrix is used to calculate
the target state fidelity. The density matrices reconstructed with 5000 single
shot measurements at $49\text{\,}\mathrm{\SIUnitSymbolMicro s}$ are plotted.
Lab frame simulation results are shown in dash lines, which matched well in
both short and long time scales. Parameters used in simulation:
$\\{\Omega,W_{1},W_{2},\Gamma_{1},\Gamma_{2}\\}/2\pi=\\{2.0,0.47,0.47,0.33,0.43\\}$
MHz for $\left|\Psi_{-}\right\rangle$ and $\\{3.0,0.36,0.36,0.33,0.43\\}$ MHz
for $\left|\Phi_{-}\right\rangle$. Qubit coherence time is chosen as
$\\{T_{1}^{q1},T_{1}^{q2},T_{\phi}^{q1},T_{\phi}^{q2}\\}=\\{25,12,25,25\\}\
\mu$s.

We now derive the energy matching requirements for an efficient stabilization
protocol in our two-qubit-two-resonator system depicted in Figure 1(a). We
control the form of the target stabilized state $\left|A\right\rangle$ by
choosing different two-qubit interaction strengths and detunings that control
$H_{QQ}$. We change the resonator photon energy in the rotating frame by
detuning the QR interactions. The dynamics of $H_{\rm sys}$ are captured by
considering the following set of eigenstates:
$\\{\left|A\right\rangle,\left|B\right\rangle,\left|C\right\rangle,\left|D\right\rangle\\}\otimes\\{\left|00\right\rangle,\left|10\right\rangle,\left|01\right\rangle\\}$.
We neglect the resonator state $\left|11\right\rangle$ as the probability of
simultaneous population in both resonators $\\{R_{1},R_{2}\\}$ is extremely
low when resonator decay rate $\kappa$ is much larger than the qubit decay
rate $\gamma$ (assumed identical). The central column in Fig. 1(a) shows the
eigenstates of $H_{QQ}$ with no photons in the resonators. The left column
represents the same states with one photon in the left ($R_{1}$) resonator and
similarly for the right column is associated with the second resonator
($R_{2}$). We engineer the photon energies in $R_{1}$ and $R_{2}$ to be
$E_{B}-E_{A}$ and $E_{C}-E_{A}$ respectively through tuning the QR
interactions $H_{QRj}$. This condition puts two transitions
$\left|A01\right\rangle\leftrightarrow\left|C00\right\rangle$ and
$\left|A10\right\rangle\leftrightarrow\left|B00\right\rangle$ on resonance,
shown in Fig. 1(a). If $\left\langle A01\right|H_{QR1}\left|C00\right\rangle$
and $\left\langle A10\right|H_{QR2}\left|B00\right\rangle$ are non-zero, two
on-resonance oscillations between $\left|C00\right\rangle$,
$\left|A01\right\rangle$ and between $\left|A10\right\rangle$,
$\left|B00\right\rangle$ will be created. Since both resonators are lossy, the
oscillation will quickly damp to $\left|A00\right\rangle$. To complete the
downward stabilization path, we need to also connect $\left|D00\right\rangle$
into the flow. We further require that the following terms are non-zero so
that the transfer path is not blocked: $\left\langle
B01\right|H_{QR1}\left|D00\right\rangle$, $\left\langle
C10\right|H_{QR2}\left|D00\right\rangle$. If all four interaction strengths
(shown in green double-headed arrows in Fig. 1(a)) are dominant over the qubit
decay rate, populations in $\left|B\right\rangle$, $\left|C\right\rangle$, and
$\left|D\right\rangle$ will flow to $\left|A\right\rangle$. From Fermi’s
golden rule, the interaction strength between two states is quadratically
suppressed by their energy gap and maximized when on-resonance [24]. This
imposes a simple energy-matching requirement for efficient stabilization:
$E_{D}+E_{A}=E_{B}+E_{C}$. Energy degeneracy within
$\\{\left|B\right\rangle,\left|C\right\rangle,\left|D\right\rangle\\}$ will
not affect the stabilization scheme, because it will not block the dissipative
flow to $\left|A00\right\rangle$ in Fig. 1(a).

As an explicit demonstration, we first stabilize a continuous set of entangled
states
$\left|\Psi_{\theta}\right\rangle=\sin\left(\theta/2\right)\left|gg\right\rangle-\cos\left(\theta/2\right)\left|ee\right\rangle$,
illustrated in Fig. 1(b). Here, $\theta$ can be regarded as a “blending angle”
between the two even parity states $\left|gg\right\rangle$ and
$\left|ee\right\rangle$. We introduce three sideband transitions into the
system: qubit-qubit (QQ) blue sideband
$\left|gg\right\rangle\leftrightarrow\left|ee\right\rangle$ with rate $\Omega$
and two qubit-resonator (QR) blue sidebands
$\left|g0\right\rangle\leftrightarrow\left|e1\right\rangle$ between $Q_{j}$
and $R_{j}$ with rate $W_{j}$. To ensure that $H_{QRj}$ act as perturbations
over $H_{QQ}$, we adjust the drive strengths to satisfy $\Omega\gg W_{j}$. We
further detune the QQ, QR1, and QR2 blue sideband by $\delta$,
$(\Delta-\delta)/2$, and $(\Delta+\delta)/2$ in frequencies, with
$\Delta=\sqrt{\Omega^{2}+\delta^{2}}$. The detuning $\delta$ determines the
blending angle $\theta=\tan^{-1}\left(\frac{\delta+\Delta}{\Omega}\right)$
with a range of $[0,\frac{\pi}{2})$. In the presence of these three drives,
the rotating frame Hamiltonian $H_{\rm sys}$ is

$\displaystyle{H_{\rm sys}}=$
$\displaystyle\frac{\Omega}{2}\left(a_{q1}a_{q2}+h.c.\right)+\delta
a_{q1}^{{\dagger}}a_{q1}$
$\displaystyle+\frac{W_{1}}{2}\left(a_{q1}a_{r1}+h.c.\right)+\frac{W_{2}}{2}\left(a_{q2}a_{r2}+h.c.\right)$
$\displaystyle+\frac{\Delta+\delta}{2}a_{r1}^{{\dagger}}a_{r1}+\frac{\Delta-\delta}{2}a_{r2}^{{\dagger}}a_{r2}.$
(1)

Here $a_{qj}$ and $a_{rj}$ are separately the $j$-th qubit’s and resonator’s
annihilation operator. Under the combined conditions $\Omega\gg
W_{j}\sim\kappa\gg\gamma$ and $W_{j}=W$, the eigenstates with zero resonator
photons are
$\\{\left|\Psi_{\theta}00\right\rangle,\left|ge00\right\rangle,\left|eg00\right\rangle,\left|\Psi_{\pi-\theta}00\right\rangle\\}$,
with corresponding eigenenergies
$\\{\left(\delta-\Delta\right)/2,0,\delta,\left(\delta+\Delta\right)/2\\}$.
Assuming the lossy resonator has a Lorentzian energy spectrum, the two-step
refilling rate $\Gamma_{t}$ from $\left|eg00\right\rangle$ to
$\left|\Psi_{\theta}00\right\rangle$
($\left|eg00\right\rangle\leftrightarrow\left|\Psi_{\theta}01\right\rangle$,
$\left|\Psi_{\theta}01\right\rangle\rightarrow\left|\Psi_{\theta}00\right\rangle$)
is [24]

$\displaystyle\Gamma_{t}=\frac{W^{2}\cos^{2}\left(\theta/2\right)\kappa}{\kappa^{2}+W^{2}\cos^{2}\left(\theta/2\right)}.$
(2)

The other two-step transitions
$\left|ge00\right\rangle\rightarrow\left|\Psi_{\theta}00\right\rangle$,
$\left|\Psi_{\theta-\pi}00\right\rangle\rightarrow\left|ge00\right\rangle$,
and $\left|\Psi_{\theta-\pi}00\right\rangle\rightarrow\left|eg00\right\rangle$
also have the same rate. Therefore, the steady-state fidelity
$\mathcal{F}_{\infty}$ for $\left|\Psi_{\theta}00\right\rangle$ is (ignoring
all off-resonant transitions, see Appendix. B for detail)

$\displaystyle\mathcal{F}_{\infty}=\left(\frac{\Gamma_{t}+\gamma\sin^{2}\left(\theta/2\right)}{\Gamma_{t}+\gamma}\right)^{2}.$
(3)

Similarly, we can stabilize another set of entangled states with odd parity
$\left|\Phi_{\theta}\right\rangle=\sin\left(\theta/2\right)\left|ge\right\rangle-\cos\left(\theta/2\right)\left|eg\right\rangle$.
We introduce three sideband interactions: QQ red
$\left|eg\right\rangle\leftrightarrow\left|ge\right\rangle$, QR1 red
$\left|e0\right\rangle\leftrightarrow\left|g1\right\rangle$, and QR2 blue
$\left|g0\right\rangle\leftrightarrow\left|e1\right\rangle$ with rates
$\\{\Omega,W_{3},W_{4}\\}$ and frequency detunings
$\\{\delta,(\Delta+\delta)/2,(\Delta-\delta)/2\\}$ respectively. Under this
condition, four resonant interactions will appear:
$\left|gg00\right\rangle\leftrightarrow\left|\Phi_{\theta}01\right\rangle$,
$\left|ee00\right\rangle\leftrightarrow\left|\Phi_{\theta}10\right\rangle$,
$\left|ee01\right\rangle\leftrightarrow\left|\Phi_{\theta-\pi}00\right\rangle$,
and
$\left|gg10\right\rangle\leftrightarrow\left|\Phi_{\theta-\pi}00\right\rangle$.
The detuning similarly sets the blending angle
$\theta=\arctan\left(\frac{\delta+\Delta}{\Omega}\right)$.

With the above construction, we create a stabilization protocol that can
freely tune the blending angles. As a special case, when QQ sideband detuning
$(\delta=0)$, the blending angle for both cases is $\theta=\frac{\pi}{2}$,
which corresponds to the odd and even parity Bell states
$\left|\Phi_{-}\right\rangle=(\left|ge\right\rangle-\left|eg\right\rangle)/\sqrt{2}$
and
$\left|\Psi_{-}\right\rangle=(\left|gg\right\rangle-\left|ee\right\rangle)/\sqrt{2}$,
shown in Fig. 1(c).

In fact, this stabilization protocol can be generalized to stabilize an even
larger group of states, including both entangled and product states, as long
as the energy matching requirement $E_{D}+E_{A}=E_{B}+E_{C}$ is satisfied when
engineering $H_{QQ}$. The following is a list of tunable parameters to
engineer $H_{QQ}$: QQ sideband strength $\Omega$, QQ sideband detunings
$\delta$, single qubit Rabi drive strength, and single qubit Rabi drive
detunings. Corresponding stabilized state $\left|A\right\rangle$ is determined
from $H_{QQ}$. Details about the stabilizable manifold are discussed in
Appendix. D.

##  III Experimental results

We perform the stabilization experiment in a system with two transmons
capacitively coupled to two lossy resonators (See Appendix. A). Two transmons
are inductively coupled through a SQUID loop. All QQ sidebands and QR red
sidebands are realized through modulating the SQUID flux at corresponding
transition frequencies. QR blue sidebands are achieved by sending a charge
drive to the transmon at half the transition frequencies. The experimentally
measured qubit coherence are $T_{1}=$24.3\text{\,}\mathrm{\SIUnitSymbolMicro
s}$\ ($9.1\text{\,}\mathrm{\SIUnitSymbolMicro s}$)$, $T_{\rm
Ram}=$15.2\text{\,}\mathrm{\SIUnitSymbolMicro s}$\
($9.8\text{\,}\mathrm{\SIUnitSymbolMicro s}$)$, $T_{\rm
echo}=$24.6\text{\,}\mathrm{\SIUnitSymbolMicro s}$\
($14.3\text{\,}\mathrm{\SIUnitSymbolMicro s}$)$ for $Q_{1}(Q_{2})$, and the
measured resonator decay rate $\kappa/2\pi$ are $\\{0.33,0.43\\}$ MHz for
$R_{1}$ and $R_{2}$ respectively.

Figure 3: Spectroscopy of universal Bell-state stabilization.
$\left|\Psi_{\theta}\right\rangle$ (top) and
$\left|\Phi_{\theta}\right\rangle$ (bottom) are separately stabilized with a
measured fidelity above $78\%$ among different blending angle $\theta$. The
fidelities are measured after $40\text{\,}\mathrm{\SIUnitSymbolMicro s}$ of
stabilization. For stabilizing $\left|gg\right\rangle$, no external drives are
applied. For $\left|\Phi_{\theta}\right\rangle$ case, the fidelity dropped to
$0$ near $\theta=\pi$. The dotted lines indicate simulated fidelities for the
odd and even parity Bell state stabilization. All parameters used in the
simulation are the same as in Fig. 2.

Figure 2 shows the time evolution of state fidelity for the odd and even
parity Bell state stabilization. To stabilize $\left|\Psi_{-}\right\rangle$, a
$\Omega=2\pi\times 2.0$ MHz QQ blue sideband, $W_{1}=W_{2}=2\pi\times 0.47$
MHz QR blue sidebands are simultaneously applied to the system. Both QR
sidebands are detuned by $\Omega/2=2\pi\times 1.0$ MHz in frequency to
implement the stabilization scheme depicted in Fig. 1(c). For each
stabilization experiment, we reconstruct the system density matrix through
two-qubit state tomography using 5000 repetitions of 9 different pre-
rotations. The stabilization fidelity measured at
$49\text{\,}\mathrm{\SIUnitSymbolMicro s}$ (much longer than single qubit
$T_{1}$ and $T_{\rm Ram}$) is $82.5\%$. To stabilize
$\left|\Phi_{-}\right\rangle$, a $\Omega=2\pi\times 3.0$ MHz QQ red sideband,
$W_{1}=W_{2}=2\pi\times 0.36$ MHz QR1 red and QR2 blue sidebands are
simultaneously applied to the system, with both QR sidebands detuned by
$\Omega/2=2\pi\times 1.5$ MHz. The stabilization fidelity measured at
$49\text{\,}\mathrm{\SIUnitSymbolMicro s}$ is $84.6\%$. The two-qubit state
tomography data at $49\text{\,}\mathrm{\SIUnitSymbolMicro s}$ after ZZ
coupling correction [25] are shown for both stabilization cases.

Next, we introduce QQ sideband detunings $\delta$ and stabilize more general
entangled states $\left|\Psi_{\theta}\right\rangle$ and
$\left|\Phi_{\theta}\right\rangle$. We choose the same sideband strengths
($\\{\Omega,W_{1},W_{2}\\}/2\pi=\\{2.0,0.47,0.47\\}(\\{3.0,0.36,0.36\\})$ MHz
for $\left|\Psi_{\theta}\right\rangle$($\left|\Phi_{\theta}\right\rangle$)
case) and detune QR sideband frequencies accordingly to maximize the
stabilization fidelity measured at $40\text{\,}\mathrm{\SIUnitSymbolMicro s}$.
The experimentally measured state fidelity and state purity as a function of
$\theta$ are shown in Fig. 3. Under the current QR sideband color combination,
$\left|\Phi_{\theta}\right\rangle$ fails to stabilize near
$\theta=180^{\circ}$. This is because the interaction strength $\left\langle
gg00\right|H_{\rm sys}\left|\Phi_{\theta}01\right\rangle$ and $\left\langle
ee00\right|H_{\rm sys}\left|\Phi_{\theta}10\right\rangle$ are close to $0$.
Swapping QR1 and QR2 sidebands’ color and detuning performs a transformation
$\theta\rightarrow\theta-\pi$ in the stabilized state. This ensures a high
stabilization fidelity for arbitrary stabilization angles. Details about
changing sideband colors and detunings to ensure high fidelity are presented
in Appendix. E

The flexibility in our schemes and easy access to different sidebands in our
device allow a further demonstration — fast dissipative switching between
stabilized states. Here, we implement such an operation that can flip the
parity of the stabilized Bell pair by changing sideband combinations, shown in
Fig. 1. To quantify the stabilized parity, we measure the system’s density
matrix $\rho$ and define the parity signature as $2(|\left\langle
ee\left|\rho\right|gg\right\rangle|-|\left\langle
ge\left|\rho\right|eg\right\rangle|)$ describing the difference in relevant
coherence parameters. The results are shown in Fig. 4. The scaling factor is
chosen such that the ideal even and odd Bell pairs have parity signatures of
$\pm 1$. Starting from the ground state
$\left|Q_{1}Q_{2}\right\rangle=\left|gg\right\rangle$, the stabilized state is
set to even parity Bell pair
$(\left|gg\right\rangle-\left|ee\right\rangle)/\sqrt{2}$, and we switch the
parity every $20\text{\,}\mathrm{\SIUnitSymbolMicro s}$. At
$20\text{\,}\mathrm{\SIUnitSymbolMicro s}$, the stabilized state is switched
to odd parity Bell pair
$(\left|ge\right\rangle-\left|eg\right\rangle)/\sqrt{2}$, and stabilization
happens quickly with a time constant
$\tau_{r}=$1.80\text{\,}\mathrm{\SIUnitSymbolMicro s}$$. At
$40\text{\,}\mathrm{\SIUnitSymbolMicro s}$, the switching from odd to even
parity results in a faster stabilization with
$\tau_{b}=$0.91\text{\,}\mathrm{\SIUnitSymbolMicro s}$$. The switching at
$60\text{\,}\mathrm{\SIUnitSymbolMicro s}$ to odd Bell state shows a similar
$\tau_{r}$ of $2.20\text{\,}\mathrm{\SIUnitSymbolMicro s}$. We leave the
stabilization drives turned on for another
$25\text{\,}\mathrm{\SIUnitSymbolMicro s}$ to prove that the performance is
not degraded after a few switching operations.

Figure 4: Dissipative switching of Bell state parity. The initial state is
$\left|gg\right\rangle$, and the switch status is set to even parity between
$\left[$0\text{\,}\mathrm{\SIUnitSymbolMicro
s}$,$20\text{\,}\mathrm{\SIUnitSymbolMicro s}$\right]$ and
$\left[$40\text{\,}\mathrm{\SIUnitSymbolMicro
s}$,$60\text{\,}\mathrm{\SIUnitSymbolMicro s}$\right]$, and to odd parity
between $\left[$20\text{\,}\mathrm{\SIUnitSymbolMicro
s}$,$40\text{\,}\mathrm{\SIUnitSymbolMicro s}$\right]$ and
$\left[$60\text{\,}\mathrm{\SIUnitSymbolMicro
s}$,$85\text{\,}\mathrm{\SIUnitSymbolMicro s}$\right]$. Each experimental
point is measured with the two-qubit state tomography. Stabilization time is
calculated by fitting the parity signature to exponential decay after each
switching event.

Further improvement of the stabilized state’s fidelity is possible by reducing
the transition ratio $\frac{\gamma}{\Gamma_{t}}$ (from Eq. 3) and increasing
QQ sideband rate $\Omega$ for a larger energy gap. Increasing qubit dephasing
time also improves stabilization fidelity (discussed in the Appendix. C). To
speed up the stabilization, i.e., reduce time constants, we need to increase
the refilling rate $\Gamma_{t}$. Since QR sideband rate $W$ is bounded by the
QQ sideband rate $\Omega$ to ensure the validity of the perturbative
approximation, given a fixed $W$, $\Gamma_{t}$ is maximized when the resonator
decay rate $\kappa=W\cos(\theta/2)$. For the even and odd parity Bell states,
further increase in both resonators’ $\kappa$ compared to our current
parameters would thus be beneficial. More details about stabilization
robustness are discussed in Appendix. C. To stabilize a more general set of
states shown in Appendix. D, longer qubit coherence is needed to improve the
experimental resolution between different stabilized states in this manifold
and is a subject of future work.

##  IV Conclusion

In conclusion, we demonstrate a two-qubit programmable stabilization scheme
that can autonomously stabilize a continuous set of entangled states. We
develop an inductively coupled two-qubit device that provides access to both
QQ and QR sideband interactions required. The stabilization fidelity among all
stabilization angles is above $78\%$, specifically, we achieved high Bell pair
stabilization fidelity ($84.6\%$ for the odd parity and $82.5\%$ for the even
parity) as two special points. We further demonstrate a parity switching
capability between the Bell pairs with fast stabilization time constants ($<2\
\mu$s). We believe such freedom in choosing stabilized states will inspire
generalization to autonomous stabilization of larger systems, large-scale
many-body entanglement [3], remote entanglement [26], density matrix
exponentiation [20, 21], and new AQEC logical codewords in the future.

##  V Acknowledgment

This work was supported by AFOSR Grant No. FA9550-19-1-0399 and ARO Grant No.
W911NF-17-S0001. Devices are fabricated in the Pritzker Nanofabrication
Facility at the University of Chicago, which receives support from Soft and
Hybrid Nanotechnology Experimental (SHyNE) Resource (NSF ECCS-1542205), a node
of the National Science Foundation’s National Nanotechnology Coordinated
Infrastructure. This work also made use of the shared facilities at the
University of Chicago Materials Research Science and Engineering Center,
supported by the National Science Foundation under award number DMR-2011854.
EK’s research was additionally supported by NSF Grant No. PHY-1653820.

##  Appendix A Device Hamiltonian and realized sidebands

Figure 5 shows the inductively coupled two-qubit device used in the
stabilization demonstration. Two transmon qubits (red) share a common ground,
which is interrupted by a SQUID (purple). Each transmon is capacitively
coupled to a lossy resonator (green) also serving as the dispersive readout.

Figure 5: The device. (a) False-colored optical image of the device. The inset
shows a zoomed-in image of the inductive coupler. (b) Circuit diagram of the
device. Coupler junctions’ intrinsic capacitance $C_{qc}$ is included in the
quantization analysis.

$\varphi_{\rm dc}=0.0$ |  $T_{1}$$\left(\mu s\right)$ |  $T_{\rm Ram}$$\left(\mu s\right)$ |  $T_{\rm echo}$$\left(\mu s\right)$
---|---|---|---
$Q_{1}$ | $31.6$ | $28.4$ | $26.6$
$Q_{2}$ | $2.8$ | $4.9$ |

$\varphi_{\rm dc}=0.3795\pi$ |  $T_{1}$$\left(\mu s\right)$ |  $T_{\rm Ram}$$\left(\mu s\right)$ |  $T_{\rm echo}$$\left(\mu s\right)$
---|---|---|---
$Q_{1}$ | $24.3$ | $15.2$ | $24.6$
$Q_{2}$ | $9.1$ | $9.8$ | $14.3$
$R_{1}$ | $0.48$ |  |
$R_{2}$ | $0.37$ |  |

Parameter | Symbol | Value/$2\pi$
---|---|---
$Q_{1}$ ge frequency | $\omega_{q1}/2\pi$ |  $3.2046$ (GHz)
$Q_{2}$ ge frequency | $\omega_{q2}/2\pi$ |  $3.6624$ (GHz)
$Q_{1}$ anharmonicity | $\alpha_{1}/2\pi$ |  $-116.3$ (MHz)
$Q_{2}$ anharmonicity | $\alpha_{2}/2\pi$ |  $-159.5$ (MHz)
Readout1 frequency | $\omega_{r1}/2\pi$ |  $4.9946$ (GHz)
Readout2 frequency | $\omega_{r2}/2\pi$ |  $5.4505$ (GHz)
$\left(E_{\left|ee\right\rangle}-E_{\left|ge\right\rangle}\right)-\left(E_{\left|eg\right\rangle}-E_{\left|gg\right\rangle}\right)$ | $ZZ/2\pi$ | -261 (kHz)
Readout1 fidelity |  | $88.87\%$
Readout2 fidelity |  | $81.76\%$

Table 1: Device coherence and frequencies at the coupler biasing point
$\varphi_{\rm dc}=0.3795\pi$ (top and middle), and the coherence at the
coupler sweet spot $\Phi_{\rm dc}=0\pi$ (bottom). $Q_{2}$ experiences higher
loss at the sweet spot from a near two-level system. At the sweet spot,
$Q_{1}$’s decoherence and echo time are similar to those at our experiment’s
biasing point.

The flux line near the SQUID provides a continuous DC bias $\varphi_{\rm
dc}=0.3795\pi$ in our experiment. The measured qubit coherence and frequencies
at this flux point are shown in Table. 1. By sending RF flux drives at
appropriate frequencies through the flux line, the inductive coupler can
provide either QQ red sideband or QQ blue sideband. The Hamiltonian for qubits
and coupler is shown in Eq. 4a. For a detailed description on the adiabatic
approximation of the inductive coupler, we draw the attention of the readers
to the references 12 and 8. Following is a summary. The Hamiltonian of the
circuit containing two qubits and the coupler is described by

$\displaystyle H=$
$\displaystyle\overrightarrow{n}^{\intercal}C_{L}^{-1}\overrightarrow{n}-E_{j1}\cos{\left(\phi_{c}-\phi_{1}\right)}-E_{j2}\cos{\left(\phi_{2}-\phi_{c}\right)},$
$\displaystyle-E_{jc}\cos{\left(\frac{\Phi_{\rm
ext}}{\Phi_{0}}\right)}\cos{\left(\phi_{c}\right)},$ (4a) $\displaystyle
C_{L}=$ $\displaystyle\begin{bmatrix}C_{q1}+C_{q12}&-C_{q12}&0\\\
-C_{q12}&C_{q2}+C_{q12}&0\\\ 0&0&C_{q1}+C_{q2}+C_{qc}\end{bmatrix}$ (4b)
$\displaystyle\overrightarrow{n}^{\intercal}=$
$\displaystyle\left(n_{1},n_{2},n_{c}\right).$

Here, phase variables $\phi_{1,2,c}$, charge variables $n_{1,2,c}$, and
Josephson energy $E_{j1,j2,jc}$ are for qubit 1, qubit 2, and the coupler, and
$C_{qc}$ is the intrinsic capacitance from the coupler junction. After
quantization, one can arrive at the following:

$\displaystyle H_{ab}=$
$\displaystyle\omega_{q1}a_{q1}^{\dagger}a_{q1}+\omega_{q2}a_{q2}^{\dagger}a_{q2}$
$\displaystyle+\frac{\alpha_{1}}{2}a_{q1}^{\dagger}a_{q1}^{\dagger}a_{q1}a_{q1}+\frac{\alpha_{2}}{2}a_{q2}^{\dagger}a_{q2}^{\dagger}a_{q2}a_{q2}$
$\displaystyle+g_{1}\left(t\right)\left(a_{q1}^{\dagger}+a_{q1}\right)\left(a_{q2}^{\dagger}+a_{q2}\right)$
$\displaystyle+g_{2}\left(-a_{q1}^{\dagger}+a_{q1}\right)\left(-a_{q2}^{\dagger}+a_{q2}\right),$
(5a) $\displaystyle g_{1}\left(t\right)=$
$\displaystyle\frac{\sqrt{E_{j1}E_{j2}}}{2E_{jc}\cos{\left(\frac{\Phi_{\rm
ext}\left(t\right)}{\Phi_{0}}\right)}}\sqrt{\omega_{q1}\omega_{q2}},$ (5b)
$\displaystyle g_{2}=$
$\displaystyle\frac{\sqrt{C_{q1}C_{q2}}}{2C_{q12}}\sqrt{\omega_{q1}\omega_{q2}}.$
(5c)

Figure 6: Coupler DC flux sweep of (a) qubits frequencies and (b) ZZ coupling
strength between qubits. Circuit quantization results and experimentally
measured data are separately shown in dash lines and dots.

In the experiment, we have $E_{jc}=$1700\text{\,}\mathrm{GHz}$\gg
E_{j1}(E_{j2})=$12.2\text{\,}\mathrm{GHz}$($12.3\text{\,}\mathrm{GHz}$)$,
making the coupler mode much heavier than the qubit modes. This strong
asymmetry enables the adiabatic removal of the coupler dynamics: We treat the
coupler as a linear inductance and assume the coupler mode is static. By
removing the linear mode through minimizing the system energy, we arrive at
the approximated Hamiltonian in Eq. 5a. When modulating the external RF flux
threaded the coupler, the transverse coupling strength $g_{1}(t)$ will also be
modulated accordingly. By plugging the RF flux modulation $\Phi_{\rm
ext}/\Phi_{0}=\varphi_{\rm dc}+\epsilon\cos\left(\omega_{d}t\right)$ into Eq.
5b and assuming the flux modulation $\epsilon$ is much smaller than $\Phi_{\rm
dc}$, we have:

$\displaystyle g_{1}\left(t\right)=$
$\displaystyle\frac{\sqrt{E_{j1}E_{j2}}}{2E_{jc}}\sqrt{\omega_{q1}\omega_{q2}}\frac{1}{\cos\left(\varphi_{\rm
dc}+\epsilon\cos\left(\omega_{d}t\right)\right)}$ $\displaystyle\approx$
$\displaystyle\frac{\sqrt{E_{j1}E_{j2}}}{2E_{jc}}\sqrt{\omega_{q1}\omega_{q2}}\frac{\left(1+\epsilon\sin\left(\omega_{d}t\right)\tan\left(\varphi_{\rm
dc}\right)\right)}{\cos\left(\varphi_{\rm dc}\right)}$ (6a)

Therefore the QQ sideband rate is roughly
$\epsilon\frac{\sqrt{E_{j1}E_{j2}}}{2E_{jc}}\sqrt{\omega_{q1}\omega_{q2}}\frac{\tan\left(\varphi_{\rm
dc}\right)}{\cos\left(\varphi_{\rm dc}\right)}$, which is proportional to the
flux modulation rate and increases with $\varphi_{\rm dc}$. We choose
$\varphi_{\rm dc}=0.3795\pi$ based on qubit coherence, sideband rate, and
small ZZ coupling between qubits. When the resonator is capacitively coupled
to the qubit at strength $g_{qr}$ and frequency difference $\Delta$, the QR
sideband can also be modulated through the flux line, and the sideband rate is
multiplied by an extra mode dressing coefficient $\frac{g_{qr}}{\Delta}$. By
changing the RF modulation frequency $\omega_{d}$, we can activate different
colors of QQ and QR sidebands.

Both QQ red, QQ blue, and QR red sidebands are generated through flux
modulation at a decent rate. The QR blue sidebands have a different choice to
activate [27]: The direct charge drives at half of the transition frequency
with amplitude $\epsilon_{q}$ can provide an effective rate of
$W={16g_{qr}^{3}\epsilon_{q}^{2}}/{\Delta^{4}}$. This turned out to be easier
to realize with our device, and we achieve at least $0.5\text{\,}\mathrm{MHz}$
QR blue sideband rate.

Figure 7: Experimentally realized QQ and QR sidebands. From top to bottom are
separately (a) QQ red sideband
$\left|ge\right\rangle\leftrightarrow\left|eg\right\rangle$, (b) QQ blue
sideband $\left|gg\right\rangle\leftrightarrow\left|ee\right\rangle$, (c) QR
blue sideband between QR1 and QR2
$\left|g0\right\rangle\leftrightarrow\left|e1\right\rangle$, and (d) QR1 red
sideband $\left|e0\right\rangle\leftrightarrow\left|g1\right\rangle$. Readout
on $Q_{1}$ (red) and $Q_{2}$ (blue) are scaled between $0$
($\left|g\right\rangle$) and $1$ ($\left|e\right\rangle$). Data points are
connected for visual guidance.

Figure 7 demonstrates all realized QQ and QR sidebands needed for the
stabilization experiments. Fast QQ red sidebands at $8.5\text{\,}\mathrm{MHz}$
and modest QQ blue sidebands at $3.9\text{\,}\mathrm{MHz}$ are performed in
the experiment through RF flux modulation of the inductive coupler. Both
readouts for QQ red sideband (with initial state $\left|eg\right\rangle$) and
QQ blue sideband (with initial state $\left|gg\right\rangle$) are shown in
Fig. 7(a) and (b). The QR blue sidebands are generated through the charge
lines that are coupled to the qubit pads, shown in Fig. 7(c) with initial
state $\left|g0\right\rangle$. The QR1 red sidebands are activated through the
coupler flux modulation. The on-resonance readout trace for $Q_{1}$ starting
at $\left|e0\right\rangle$ is plotted in Fig. 7(d).

##  Appendix B Derivation of steady-state fidelity

We take the stabilization of
$\left|\Psi_{\theta}\right\rangle=\sin\left(\theta/2\right)\left|gg\right\rangle-\cos\left(\theta/2\right)\left|ee\right\rangle$
as an example to compute the steady-state fidelity in detail. Suppose the
steady state population at the four basis states
$\\{\left|\Psi_{\theta}\right\rangle,\left|ge\right\rangle,\left|eg\right\rangle,\left|\Psi_{\theta-\pi}\right\rangle\\}$
are separately $\\{w,x,y,z\\}$. We assume the photon population in both
resonators are transitional and ignore their contribution to the steady-state
fidelity. This means $w+x+y+z=1$. The steady-state configuration should
balance the following two processes:

(a) two-step refilling process:
$\left|ge\right\rangle\rightarrow\left|\Psi_{\theta}\right\rangle$,
$\left|eg\right\rangle\rightarrow\left|\Psi_{\theta}\right\rangle$,
$\left|ge\right\rangle\rightarrow\left|\Psi_{\theta}\right\rangle$, and
$\left|ge\right\rangle\rightarrow\left|\Psi_{\theta}\right\rangle$. All the
transition rates are the same

$\displaystyle\Gamma_{t}$
$\displaystyle=\frac{W^{2}\cos^{2}\left(\theta/2\right)\kappa}{\kappa^{2}+W^{2}\cos^{2}\left(\theta/2\right)}$
(7)

(b) Single photon loss in each qubit. The following four transitions have the
same rate $\sin^{2}\left(\theta/2\right)\gamma$:
$\left|ge\right\rangle\rightarrow\left|\Psi_{\theta}\right\rangle$,
$\left|eg\right\rangle\rightarrow\left|\Psi_{\theta}\right\rangle$,
$\left|ge\right\rangle\rightarrow\left|\Psi_{\theta-\pi}\right\rangle$, and
$\left|eg\right\rangle\rightarrow\left|\Psi_{\theta-\pi}\right\rangle$. The
reversed four transitions have the same rate
$\cos^{2}\left(\theta/2\right)\gamma$.

Therefore, the steady-state population should satisfy the following equations

$\begin{cases}\left(\Gamma_{t}+\sin^{2}\left(\theta/2\right)\gamma\right)(x+y)-2\cos^{2}\left(\theta/2\right)\gamma
w&=0,\\\
\left(\Gamma_{t}+\sin^{2}\left(\theta/2\right)\gamma\right)z+\cos^{2}\left(\theta/2\right)\gamma
w-(\Gamma_{t}+\gamma)x&=0,\\\
\left(\Gamma_{t}+\sin^{2}\left(\theta/2\right)\gamma\right)z+\cos^{2}\left(\theta/2\right)\gamma
w-(\Gamma_{t}+\gamma)y&=0,\\\
2\cos^{2}\left(\theta/2\right)\gamma(x+y)-\left(2\Gamma_{t}+2\sin^{2}\left(\theta/2\right)\gamma\right)z&=0,\\\
w+x+y+z&=1.\\\ \end{cases}$ (8)

This gives the following populations

$\begin{cases}w&=\left(\frac{\Gamma_{t}+\gamma\sin^{2}\left(\theta/2\right)}{\Gamma_{t}+\gamma}\right)^{2},\\\
x&=\frac{\cos^{2}\left(\theta/2\right)\gamma}{\Gamma_{t}+\gamma-\cos^{2}\left(\theta/2\right)\gamma}w,\\\
y&=x,\\\
z&=\frac{\cos^{2}\left(\theta/2\right)\gamma}{\Gamma_{t}+\sin^{2}\left(\theta/2\right)\gamma}x.\\\
\end{cases}$ (9)

And $w=\mathcal{F}_{\infty}$ is the steady state fidelity.

##  Appendix C Stabilization robustness

Figure 8: Rotating frame simulation of even and odd parity bell states’
stabilization fidelity: (a) Sweeping qubit dephasing time. The other
simulation parameters are the same as the experiments. (b) Sweeping the ratio
between resonator decay rate $\kappa$ and QR sideband strength $W$ without
qubit dephasing. (c) 2D sweep of $\kappa$ and QQ sideband strength $\Omega$
without qubit dephasing, setting $W=\kappa$. Infidelities are shown on the
contours.

We study the stabilization robustness for $\left|\Psi_{-}\right\rangle$ and
$\left|\Phi_{-}\right\rangle$ in this section. For other stabilization angles,
the discussion is similar. Figure 8 shows the rotating frame simulation of
steady-state fidelity by sweeping different stabilization parameters. For the
$\left|\Psi_{-}\right\rangle$ case, the Hamiltonian used in the simulation is
Eq. 1, and for the $\left|\Phi_{-}\right\rangle$ case the Hamiltonian is
modified accordingly with a different sideband combinations (See Fig. 1(c)).
We study the state fidelity by varying parameters step-by-step towards the
ideal case. First, we show that longer qubit dephasing time helps improve
steady-state fidelity. In Fig. 8 (a), we sweep qubit’s dephasing time
(assuming the same for both qubits) while choosing the following parameters
$\\{\Omega,\frac{W_{1}}{2},\frac{W_{2}}{2},\Gamma_{1},\Gamma_{2}\\}/2\pi$ in
the simulation

$\displaystyle\left|\Psi_{-}\right\rangle\text{case}:$
$\displaystyle\\{1.4,0.35,0.35,0.30,0.33\\}\text{MHz},$
$\displaystyle\left|\Phi_{-}\right\rangle\text{case}:$
$\displaystyle\\{3.0,0.32,0.32,0.30,0.33\\}\text{MHz}.$

We set the following qubit decoherence time
$\\{T_{1}^{q1},T_{1}^{q2}\\}=\\{21,9\\}\mu$s. The steady state fidelities rise
above $80\%$ quickly after $T_{\phi}$ exceeds
$10\text{\,}\mathrm{\SIUnitSymbolMicro s}$. The fidelity for odd and even
parity bell pairs saturate at $87.3\%$ and $85.0\%$ with the parameters used
in the simulation. This demonstrates that steady-state fidelity increases as
qubit dephasing time increases.

In Fig. 8(b), we ignore the qubit dephasing and only sweep resonator decay
rate $\kappa$. For simplicity, we assume QR sideband rates and resonator decay
rate are the same: $W_{1}=W_{2}=W$ and $\kappa_{1}=\kappa_{2}=\kappa$. The
fidelity peak for both parity pairs appears at $W=\kappa$. This can be
understood as the refilling rate $\Gamma_{t}$ (Eq. 2) achieves the maximum at
this point, therefore the steady-state fidelity (Eq. 3) is also maximized at
this point.

Finally, we choose the maximum refilling rate set $W=\kappa$ in Fig. 8(c) and
sweep both the QQ sidebands rate $\Omega$ and resonator decay rate $\kappa$.
The infidelity of the steady states is shown in contours. Larger $\Omega$ and
$\kappa$ suppresses the infidelity efficiently. This indicates that our
steady-state fidelity in the experiment is mainly limited by the sideband
strengths. By increasing the QQ sidebands rate to above
$2\pi\times$10\text{\,}\mathrm{MHz}$$, in simulation, it is possible to
achieve stabilization fidelity above $98\%$.

##  Appendix D Other stabilization combinations

Here we provide a list of states that can be stabilized with our protocol.

Case 1: Any two-qubit product states.

$\displaystyle\left|\psi_{\phi_{1},\phi_{2}}\right\rangle=$
$\displaystyle(\cos(\phi_{1}/2)\left|g\right\rangle+\sin(\phi_{1}/2)\left|e\right\rangle)\otimes$
$\displaystyle(\cos(\phi_{2}/2)\left|g\right\rangle+\sin(\phi_{2}/2)\left|e\right\rangle).$
(11)

This can be achieved by applying two detuned single qubit rabi drives on both
$Q_{1}$ and $Q_{2}$ with rate $\\{A_{1},A_{2}\\}$ and detunings
$\\{\delta_{1},\delta_{2}\\}$. The two-qubit rotating frame Hamiltonian
becomes

$\displaystyle H_{p}=\begin{bmatrix}0&A_{2}/2&A_{1}/2&0\\\
A_{2}/2&\delta_{2}&0&A_{1}/2\\\ A_{1}/2&0&\delta_{1}&A_{2}/2\\\
0&A_{1}/2&A_{2}/2&\delta_{1}+\delta_{2}\end{bmatrix}.$ (12)

It can be easily verified that the four eigenenergies
$\\{E_{A}<E_{B}<E_{C}<E_{D}\\}$ satisfy the requirements
$E_{A}+E_{D}=E_{B}+E_{C}$. Therefore, the lowest energy eigenstate can be
efficiently stabilized by detuning two QR sideband frequencies. This is also a
direct extension of the single-qubit stabilization scheme [12] to the two-
qubit case.

Case 2: Dressed parity Bell states. The stabilized state set can be described
by one continuous variable $\theta_{1}$

$\displaystyle\left|\zeta_{\theta_{1}}\right\rangle=$
$\displaystyle\cos(\theta_{1}/2)\left|\Psi_{-}\right\rangle+\sin(\theta_{1}/2)\left|\Phi_{-}\right\rangle.$
(14)

In this case, we apply on-resonant QQ blue and single qubit rabi drive on
$Q_{1}$ with rate $\Omega$ and $A_{1}$ to dress the stabilized state’s parity.
The two-qubit Hamiltonian $H_{b}$ can be written as

$\displaystyle H_{b}=\begin{bmatrix}0&0&A_{1}/2&\Omega/2\\\ 0&0&0&A_{1}/2\\\
A_{1}/2&0&0&0\\\ \Omega/2&A_{1}/2&0&0\end{bmatrix}.$ (15)

Figure 9: Stabilizing a one-dimensional set of entangled states. Blue and red
lines represent separately using the QQ blue and QQ red sideband in the
rotating frame simulation. Parameter used in the simulation:
$\\{\Omega,W_{1},W_{2},\kappa_{1},\kappa_{2}\\}/2\pi=\\{5.0,0.5,0.5,0.3,0.33\\}$
MHz. Qubit coherence time are $\\{T_{1},T_{\phi}\\}=\\{30,30\\}\mu$s.

One can verify that the four eigenenergies $E_{A}<E_{B},<E_{C}<E_{D}$ of
$H_{b}$ satisfy the requirement $E_{A}+E_{D}=E_{B}+E_{C}$. By choosing QR
sidebands detunings as $E_{B}-E_{A}$ and $E_{C}-E_{A}$, the eigenstate
$\left|A\right\rangle=\left|\zeta_{\theta_{1}}\right\rangle$ is stabilized,
with the dressing angle being

$\displaystyle\theta_{1}=2\arctan\left(\frac{2A_{1}}{\Omega+\sqrt{4A_{1}^{2}+\Omega^{2}}}\right).$
(16)

Similarly, we apply on-resonant QQ red sideband and single qubit rabi drive on
$Q_{1}$ with rate $\Omega$ and $A_{1}$. The two-qubit Hamiltonian $H_{r}$ is

$\displaystyle H_{r}=\begin{bmatrix}0&0&A_{1}/2&0\\\ 0&0&\Omega/2&A_{1}/2\\\
A_{1}/2&\Omega/2&0&0\\\ 0&A_{1}/2&0&0\end{bmatrix}.$ (17)

The dressing angle $\theta_{1}$ under this case is

$\displaystyle\theta_{1}=\pi-2\arctan\left(\frac{2A_{1}}{\Omega+\sqrt{4A_{1}^{2}+\Omega^{2}}}\right).$
(18)

Steady-state fidelity is calculated through Qutip simulation for both cases,
shown in Fig. 9. By combining different QQ sideband colors, all dressing
angles are stabilized with the scheme, except the small band region around
$\theta_{1}=\pi$ where the effective transition rates provided by QR sidebands
are close to $0$.

Figure 10: Stabilizing a 2D set of entangled states. Steady states’ fidelity
(a) and purity (b) are simulated in the rotating frame and plotted.
$\frac{A_{1}}{\Omega}$ and $\frac{\delta}{\Omega}$ are separately two free
variables that are swept to stabilize different states. Parameter used in the
simulation:
$\\{\Omega,W_{1},W_{2},\kappa_{1},\kappa_{2}\\}/2\pi=\\{5.0,0.5,0.5,0.3,0.33\\}$
MHz. Qubit coherence time are$\\{T_{1},T_{\phi}\\}=\\{30,30\\}\mu$s.

Case 3: Rabi-dressed entangled states. This is a more general case where a
two-dimensional set of entangled states is stabilized. We simultaneously apply
a single qubit rabi drive on $Q_{1}$ with the rate $A_{1}$ and detuned QQ blue
sideband with rate $\Omega$ and detuning $\delta$. The rotating frame
Hamiltonian is

$\displaystyle H_{g}=\begin{bmatrix}-\delta/2&0&A_{1}/2&\Omega/2\\\
0&\delta/2&0&A_{1}/2\\\ A_{1}/2&0&-\delta/2&0\\\
\Omega/2&A_{1}/2&0&\delta/2\end{bmatrix}.$ (19)

The four eigenenergies of the Hamiltonian can also be grouped into two pairs
sharing the same sum. By appropriately choosing two QR sideband detunings, the
lowest energy eigenstate is stabilized

$\displaystyle\left|\xi_{\delta,A_{1}}\right\rangle=$ $\displaystyle
E_{00}\left|gg\right\rangle+E_{01}\left|ge\right\rangle+E_{10}\left|eg\right\rangle-\left|ee\right\rangle,$
$\displaystyle x=$
$\displaystyle\sqrt{4\delta^{2}A_{1}^{2}+4A_{1}^{2}\Omega^{2}+\Omega^{4}},$
$\displaystyle y=$
$\displaystyle\sqrt{\delta^{2}+2(2A_{1}^{2}+\Omega^{2}+x)},$ $\displaystyle
E_{00}=$ $\displaystyle\frac{(\delta-y)(\delta^{2}+\Omega^{2}+x+\delta
y)}{2\Omega(2A_{1}^{2}+\Omega^{2}+x)},$ $\displaystyle E_{01}=$
$\displaystyle\frac{A_{1}(\delta-y)}{2A_{1}^{2}+\Omega^{2}+x},$ $\displaystyle
E_{10}=$ $\displaystyle-\frac{A_{1}(\delta^{2}+\Omega^{2}+x+\delta
y)}{\Omega(2A_{1}^{2}+\Omega^{2}+x)}.$ (20)

For brevity, the stabilized state $\left|\xi_{\delta,A_{1}}\right\rangle$ is
not normalized. The form of the state is determined by two independent
variables $\frac{\delta}{\Omega}$ and $\frac{A_{1}}{\Omega}$. We sweep these
two variables and plot the simulated fidelity and purity in Fig. 10. This a
general map covering all stabilized entangled states in the programmable
operation: the vertical cut $\frac{\delta}{\Omega}=0$ represents the blue line
in case 2 Dressed parity Bell states, the horizontal cut
$\frac{A_{1}}{\Omega}=0$ represents the stabilized states shown in Fig. 1(b),
and the bottom left point $(\frac{\delta}{\Omega},\frac{A_{1}}{\Omega})=(0,0)$
is the even parity bell state $\left|\Psi_{-}\right\rangle$. Using a modest
sideband rate combination, most of the states on the plot can be stabilized
with fidelity over $90\%$. Correspondingly, changing the QQ sideband color to
red can stabilize another 2D set of entangled states which are dual to this
case. All possible programmable stabilization operations can be chosen
accordingly through the map provided here. States with both $E_{01}>1$ and
$E_{10}>1$ cannot be stabilized under this case for instance.

##  Appendix E QR sideband colors and detunings

Changing sideband colors and detuning frequency signs stabilize different
states. This is important for stabilizing $\left|\Psi_{\theta}\right\rangle$
and $\left|\Phi_{\theta}\right\rangle$: at certain blending angles the steady
state fidelity is low because of the small effective refilling rate
$\Gamma_{t}$ (See Eq. 2).

Figure 11: (a) Use both QR red sidebands to stabilize
$\left|\Psi_{\theta}\right\rangle$. (b) Use opposite QR blue sideband
detunings to stabilize $\left|\Psi_{\theta-\pi}\right\rangle$. The QQ sideband
rates and detunings are separate $\Omega$ and $\delta$, and the QR sideband is
detuned in frequency by $\frac{\Delta+\delta}{2}$, $\frac{\Delta-\delta}{2}$
in (a) and $-\frac{\Delta+\delta}{2}$, $-\frac{\Delta-\delta}{2}$ in (b). Here
$\Delta=\sqrt{\Omega^{2}+\delta^{2}}$.

As an explicit example, we consider the $\left|\Psi_{\theta}\right\rangle$
stabilization case. Instead of using two QR blue sidebands, we use two QR red
sidebands with the same detuning and sideband rate. The rotating frame
Hamiltonian now becomes

$\displaystyle{H_{\rm color}}=$
$\displaystyle\frac{\Omega}{2}\left(a_{q1}a_{q2}+h.c.\right)+\delta
a_{q1}^{{\dagger}}a_{q1}$
$\displaystyle+\frac{W_{1}}{2}\left(a_{q1}^{{\dagger}}a_{r1}+h.c.\right)+\frac{W_{2}}{2}\left(a_{q2}^{{\dagger}}a_{r2}+h.c.\right)$
$\displaystyle+\frac{\Delta+\delta}{2}a_{r1}^{{\dagger}}a_{r1}+\frac{\Delta-\delta}{2}a_{r2}^{{\dagger}}a_{r2}.$
(21)

Fig. 11 (a) shows the level diagram, where the QR red sidebands now connect
$\left|ge00\right\rangle$ to $\left|\Psi_{\theta}01\right\rangle$, which is
different from the QR blue sidebands case where $\left|ge00\right\rangle$ and
$\left|\Psi_{\theta}10\right\rangle$ are connected. The two-step refilling
rate $\Gamma_{tc}$ for
$\left|ge00\right\rangle\rightarrow\left|\Psi_{\theta}00\right\rangle$ is

$\displaystyle\Gamma_{tc}$
$\displaystyle=\frac{W^{2}\sin^{2}\left(\theta/2\right)\kappa}{\kappa^{2}+W^{2}\sin^{2}\left(\theta/2\right)}.$
(22)

The other three two-step transitions
$\left|eg00\right\rangle\rightarrow\left|\Psi_{\theta}00\right\rangle$,
$\left|\Psi_{\theta-\pi}00\right\rangle\rightarrow\left|ge00\right\rangle$,
and $\left|\Psi_{\theta-\pi}00\right\rangle\rightarrow\left|eg00\right\rangle$
have the same refilling rate. Therefore, the steady-state fidelity for
$\left|\Psi_{\theta}\right\rangle$ is

$\displaystyle\mathcal{F}_{\infty}=\left(\frac{\Gamma_{t}+\gamma\cos^{2}\left(\theta/2\right)}{\Gamma_{t}+\gamma}\right)^{2}.$
(23)

One can thus choosing the QR sideband color for higher $F_{\infty}$. When the
$F_{\infty}$ drops significantly near $\theta=\pi$ for $\Phi_{\theta}$
stabilization case, one can flip the QR sideband color for better performance.

We can also keep the same QR sideband color while choosing opposite QR
sideband detunings. The Hamiltonian becomes

$\displaystyle{H_{\rm opp}}=$
$\displaystyle\frac{\Omega}{2}\left(a_{q1}a_{q2}+h.c.\right)+\delta
a_{q1}^{{\dagger}}a_{q1}$
$\displaystyle+\frac{W_{1}}{2}\left(a_{q1}^{{\dagger}}a_{r1}+h.c.\right)+\frac{W_{2}}{2}\left(a_{q2}^{{\dagger}}a_{r2}+h.c.\right)$
$\displaystyle-\frac{\Delta+\delta}{2}a_{r1}^{{\dagger}}a_{r1}-\frac{\Delta-\delta}{2}a_{r2}^{{\dagger}}a_{r2}.$
(24)

The level diagram is shown in Fig. 11 (b). All population flows to
$\left|\Psi_{\theta-\pi}\right\rangle$, with the same refilling rate (Eq. 2)
and steady-state fidelity (Eq. 3) as the $\left|\Psi_{\theta}\right\rangle$
case.

##  Appendix F Measurement setup

Figure 12 illustrates the measurement setup used in the experiment. Both
single qubit, QR sidebands, and QQ sidebands signals are generated with a
4-channel AWG (Keysight M8195 65 Gsa/s, 16 Gsa/s per channel) to maintain
phase-locking. DC flux bias for the coupler and two Josephson Parametric
Amplifiers (JPAa) are generated with three current sources (Yokogawa GS200).
The bandpass filters on both charge lines are chosen such that the stop band
covers both readout and qubits’ frequencies. The flux drive is delivered using
three separate coaxial cables for DC bias, red sideband and blue sideband
frequencies. The RF lines are merged using a combiner which is then added to
the DC bias using a bias tee. The Stepped impedance Purcell filter (SIPF)
inserted in the flux line is a home-made filter with a stop band between
$2\text{\,}\mathrm{GHz}$ and $5.5\text{\,}\mathrm{GHz}$ blocking any qubit and
resonator signals. Each transmitted signal is first amplified by a JPA with a
$+15$ dB gain, followed by a HEMT amplifier and three room-temperature
amplifiers. The final signal is pre-amplified after demodulation and digitized
with an Alazar ATS 9870 (1GSa/s) card.

Figure 12: Detailed measurement setup.

## References

  * Arute  _et al._ [2019] F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. S. L. Brandao, D. A. Buell, B. Burkett, Y. Chen, Z. Chen, B. Chiaro, R. Collins, W. Courtney, A. Dunsworth, E. Farhi, B. Foxen, A. Fowler, C. Gidney, M. Giustina, R. Graff, K. Guerin, S. Habegger, M. P. Harrigan, M. J. Hartmann, A. Ho, M. Hoffmann, T. Huang, T. S. Humble, S. V. Isakov, E. Jeffrey, Z. Jiang, D. Kafri, K. Kechedzhi, J. Kelly, P. V. Klimov, S. Knysh, A. Korotkov, F. Kostritsa, D. Landhuis, M. Lindmark, E. Lucero, D. Lyakh, S. Mandrà, J. R. McClean, M. McEwen, A. Megrant, X. Mi, K. Michielsen, M. Mohseni, J. Mutus, O. Naaman, M. Neeley, C. Neill, M. Y. Niu, E. Ostby, A. Petukhov, J. C. Platt, C. Quintana, E. G. Rieffel, P. Roushan, N. C. Rubin, D. Sank, K. J. Satzinger, V. Smelyanskiy, K. J. Sung, M. D. Trevithick, A. Vainsencher, B. Villalonga, T. White, Z. J. Yao, P. Yeh, A. Zalcman, H. Neven, and J. M. Martinis, Quantum supremacy using a programmable superconducting processor, Nature 574, 505 (2019).
  * Wu  _et al._ [2021] Y. Wu, W.-S. Bao, S. Cao, F. Chen, M.-C. Chen, X. Chen, T.-H. Chung, H. Deng, Y. Du, D. Fan, M. Gong, C. Guo, C. Guo, S. Guo, L. Han, L. Hong, H.-L. Huang, Y.-H. Huo, L. Li, N. Li, S. Li, Y. Li, F. Liang, C. Lin, J. Lin, H. Qian, D. Qiao, H. Rong, H. Su, L. Sun, L. Wang, S. Wang, D. Wu, Y. Xu, K. Yan, W. Yang, Y. Yang, Y. Ye, J. Yin, C. Ying, J. Yu, C. Zha, C. Zhang, H. Zhang, K. Zhang, Y. Zhang, H. Zhao, Y. Zhao, L. Zhou, Q. Zhu, C.-Y. Lu, C.-Z. Peng, X. Zhu, and J.-W. Pan, Strong quantum computational advantage using a superconducting quantum processor, Phys. Rev. Lett. 127, 180501 (2021).
  * Kapit  _et al._ [2020] E. Kapit, P. Roushan, C. Neill, S. Boixo, and V. Smelyanskiy, Entanglement and complexity of interacting qubits subject to asymmetric noise, Phys. Rev. Res. 2, 043042 (2020).
  * Verstraete  _et al._ [2009] F. Verstraete, M. M. Wolf, and J. Ignacio Cirac, Quantum computation and quantum-state engineering driven by dissipation, Nature Physics 5, 633 (2009).
  * Kapit [2016] E. Kapit, Hardware-efficient and fully autonomous quantum error correction in superconducting circuits, Phys. Rev. Lett. 116, 150501 (2016).
  * Ma  _et al._ [2020] Y. Ma, Y. Xu, X. Mu, W. Cai, L. Hu, W. Wang, X. Pan, H. Wang, Y. P. Song, C.-L. Zou, and L. Sun, Error-transparent operations on a logical qubit protected by quantum error correction, Nature Physics 16, 827 (2020).
  * Gertler  _et al._ [2021] J. M. Gertler, B. Baker, J. Li, S. Shirol, J. Koch, and C. Wang, Protecting a bosonic qubit with autonomous quantum error correction, Nature 590, 243 (2021).
  * Li  _et al._ [2023a] Z. Li, T. Roy, D. R. Perez, K.-H. Lee, E. Kapit, and D. I. Schuster, Autonomous error correction of a single logical qubit using two transmons (2023a), arXiv:2302.06707 [quant-ph] .
  * Li  _et al._ [2023b] Z. Li, T. Roy, D. R. Pérez, D. I. Schuster, and E. Kapit, Hardware efficient autonomous error correction with linear couplers in superconducting circuits (2023b), arXiv:2303.01110 [quant-ph] .
  * Shankar  _et al._ [2013] S. Shankar, M. Hatridge, Z. Leghtas, K. M. Sliwa, A. Narla, U. Vool, S. M. Girvin, L. Frunzio, M. Mirrahimi, and M. H. Devoret, Autonomously stabilized entanglement between two superconducting quantum bits, Nature 504, 419 (2013).
  * Kimchi-Schwartz  _et al._ [2016] M. E. Kimchi-Schwartz, L. Martin, E. Flurin, C. Aron, M. Kulkarni, H. E. Tureci, and I. Siddiqi, Stabilizing entanglement via symmetry-selective bath engineering in superconducting qubits, Phys. Rev. Lett. 116, 240503 (2016).
  * Lu  _et al._ [2017] Y. Lu, S. Chakram, N. Leung, N. Earnest, R. K. Naik, Z. Huang, P. Groszkowski, E. Kapit, J. Koch, and D. I. Schuster, Universal stabilization of a parametrically coupled qubit, Phys. Rev. Lett. 119, 150502 (2017).
  * Huang  _et al._ [2018] Z. Huang, Y. Lu, E. Kapit, D. I. Schuster, and J. Koch, Universal stabilization of single-qubit states using a tunable coupler, Phys. Rev. A 97, 062345 (2018).
  * Andersen  _et al._ [2019] C. K. Andersen, A. Remm, S. Lazar, S. Krinner, J. Heinsoo, J.-C. Besse, M. Gabureac, A. Wallraff, and C. Eichler, Entanglement stabilization using ancilla-based parity detection and real-time feedback in superconducting circuits, npj Quantum Information 5, 69 (2019).
  * Ma  _et al._ [2019] S.-l. Ma, X.-k. Li, X.-y. Liu, J.-k. Xie, and F.-l. Li, Stabilizing bell states of two separated superconducting qubits via quantum reservoir engineering, Phys. Rev. A 99, 042336 (2019).
  * Bultink  _et al._ [2020] C. C. Bultink, T. E. O’Brien, R. Vollmer, N. Muthusubramanian, M. W. Beekman, M. A. Rol, X. Fu, B. Tarasinski, V. Ostroukh, B. Varbanov, A. Bruno, and L. DiCarlo, Protecting quantum entanglement from leakage and qubit errors via repetitive parity measurements, Science Advances 6, eaay3050 (2020), https://www.science.org/doi/pdf/10.1126/sciadv.aay3050 .
  * Brown  _et al._ [2022] T. Brown, E. Doucet, D. Ristè, G. Ribeill, K. Cicak, J. Aumentado, R. Simmonds, L. Govia, A. Kamal, and L. Ranzani, Trade off-free entanglement stabilization in a superconducting qutrit-qubit system, Nature Communications 13, 3994 (2022).
  * Lin  _et al._ [2013] Y. Lin, J. P. Gaebler, F. Reiter, T. R. Tan, R. Bowler, A. S. Sørensen, D. Leibfried, and D. J. Wineland, Dissipative production of a maximally entangled steady state of two quantum bits, Nature 504, 415 (2013).
  * Cole  _et al._ [2022] D. C. Cole, S. D. Erickson, G. Zarantonello, K. P. Horn, P.-Y. Hou, J. J. Wu, D. H. Slichter, F. Reiter, C. P. Koch, and D. Leibfried, Resource-efficient dissipative entanglement of two trapped-ion qubits, Phys. Rev. Lett. 128, 080502 (2022).
  * Lloyd  _et al._ [2014] S. Lloyd, M. Mohseni, and P. Rebentrost, Quantum principal component analysis, Nature Physics 10, 631 (2014).
  * Kjaergaard  _et al._ [2022] M. Kjaergaard, M. E. Schwartz, A. Greene, G. O. Samach, A. Bengtsson, M. O’Keeffe, C. M. McNally, J. Braumüller, D. K. Kim, P. Krantz, M. Marvian, A. Melville, B. M. Niedzielski, Y. Sung, R. Winik, J. Yoder, D. Rosenberg, K. Obenland, S. Lloyd, T. P. Orlando, I. Marvian, S. Gustavsson, and W. D. Oliver, Demonstration of density matrix exponentiation using a superconducting quantum processor, Phys. Rev. X 12, 011005 (2022).
  * Lu [2019] Y. Lu, Parametric control of flux-tunable superconducting circuits (2019).
  * Roy  _et al._ [2023] T. Roy, Z. Li, E. Kapit, and D. Schuster, Two-qutrit quantum algorithms on a programmable superconducting processor, Phys. Rev. Appl. 19, 064024 (2023).
  * Kapit  _et al._ [2014] E. Kapit, M. Hafezi, and S. H. Simon, Induced self-stabilization in fractional quantum hall states of light, Phys. Rev. X 4, 031039 (2014).
  * Roy  _et al._ [2021] T. Roy, Z. Li, E. Kapit, and D. I. Schuster, Tomography in the presence of stray inter-qubit coupling (2021), arXiv:2103.13611 .
  * li Ma  _et al._ [2021] S. li Ma, J. Zhang, X. ke Li, Y. long Ren, J. kun Xie, M. tao Cao, and F. li Li, Coupling-modulation–mediated generation of stable entanglement of superconducting qubits via dissipation, Europhysics Letters 135, 63001 (2021).
  * Wallraff  _et al._ [2007] A. Wallraff, D. I. Schuster, A. Blais, J. M. Gambetta, J. Schreier, L. Frunzio, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Sideband transitions and two-tone spectroscopy of a superconducting qubit strongly coupled to an on-chip cavity, Phys. Rev. Lett. 99, 050501 (2007).