# Stabilizing two-qubit entanglement with dynamically decoupled active
feedback

Sacha Greenfield1,2,3 Leigh Martin4,5 Felix Motzoi4,6,7 K. Birgitta Whaley4,6
Justin Dressel3,8 Eli M. Levenson-Falk1,2,9<EMAIL_ADDRESS>1Department of
Physics and Astronomy, University of Southern California, Los Angeles,
California 90089  2Center for Quantum Information Science and Technology,
University of Southern California, Los Angeles, California 90089  3Institute
for Quantum Studies, Chapman University, Orange, California 92866  4Berkeley
Center for Quantum Information and Computation, Berkeley, California 94720 USA
5Department of Physics, University of California, Berkeley, California 94720
USA  6Department of Chemistry, University of California, Berkeley, California
94720 USA  7Peter Grünberg Institute of Quantum Control (PGI-8),
Forschungszentrum Jülich GmbH, Wilhelm-Johnen-Straße, 52428 Jülich, Germany
8Schmid College of Science and Technology, Chapman University, Orange,
California 92866  9Department of Electrical and Computer Engineering,
University of Southern California, Los Angeles, California 90089

###### Abstract

We propose and analyze a protocol for stabilizing a maximally entangled state
of two noninteracting qubits using active state-dependent feedback from a
continuous two-qubit half-parity measurement in coordination with a
concurrent, non-commuting dynamical decoupling drive. We demonstrate that such
a drive can be simultaneous with the measurement and feedback, while also
playing a key part in the feedback protocol itself. We show that robust
stabilization with near-unit fidelity can be achieved even in the presence of
realistic nonidealities, such as time delay in the feedback loop, imperfect
state-tracking, inefficient measurements, dephasing from $1/f$-distributed
qubit-frequency noise, and relaxation. We mitigate feedback-delay error by
introducing a forward-state-estimation strategy in the feedback controller
that tracks the effects of control signals already in transit. More generally,
the steady state is globally attractive without the need for ancillas,
regardless of the error state, in contrast to most known feedback and error
correction schemes.

##  I Introduction

Maximally entangled multi-qubit states are important resources for quantum
information processing tasks. However, entangled states decohere more rapidly
than single-qubit states, and so must either be generated on demand when
needed or stored using some form of stabilization that prevents degradation.
Circuit-based quantum error correction protocols are examples of such active
entanglement stabilization [1], but are resource-intensive since they require
additional ancillary qubits, periodic entangling unitary gates, and single-
qubit measurements. Bath engineering protocols can stabilize entanglement [2,
3, 4, 5, 6, 7], but typically require custom device designs to select the
state.

A natural non-unitary tool to consider for such an entanglement stabilization
task is the direct measurement of a two-qubit observable, such as parity [8,
9]. Periodic or continuous measurements of such an observable can prepare [10]
and then stabilize [11, 12, 13] the entangled eigenspaces of the measurement
and can perform multi-qubit operations [14, 15], even in cases where the
qubits do not directly interact [16, 17, 18, 19, 20, 21, 22, 23]. Stabilizing
a specific state, however, generally requires bath engineering or additional
feedback control [24, 25, 26, 27, 28, 29] to break the symmetry of the
measurement toward the desired outcome.

Implementing robust entanglement stabilization using measurement feedback is
challenging in practice. The stabilized state should not only be a steady
state under ideal conditions for the measurement and other engineered
decoupling drives [30, 31, 32, 33, 34, 35, 36], but remain stabilized in the
presence of realistic nonidealities [37]. Some dynamical states such as
single-qubit Rabi oscillations [38] are less sensitive to environmental noise
and so can be more easily stabilized, but entangled states are more
susceptible to decoherence, which can drastically lower the achievable steady-
state entanglement [16, 17, 18]. Dynamical decoupling can reduce dephasing and
prolong coherence [39, 40, 41], but does not produce true stabilization [42]
and is typically separated from the measurement and feedback. Time delays in
the feedback loop [43, 44] and measurement inefficiency are also unavoidable
in the laboratory [45], and can be disastrous to the stabilization fidelity if
not accounted for in the protocol [46, 33, 47].

In this work, we show that it is possible to combine measurement, a feedback
drive, and a dynamical decoupling drive simultaneously to achieve robust
entanglement generation and stabilization. As a case study we revisit the
simple measurement-feedback protocol introduced in Ref. [33], which stabilizes
a maximally entangled two-qubit Bell state with feedback from a half-parity
measurement. This protocol was vulnerable to dephasing which would mix the
target state with a degenerate measurement eigenstate, thus eliminating
stability. We expand the protocol to include a concurrent drive that provides
both dynamical decoupling [48, 49, 50, 51] and acts to remove an undesired
fixed point of the dynamics. We analyze both the optimum settings and
resulting performance in the presence of experimental nonidealities including
the time delay of the feedback loop, measurement inefficiency, and
environmentally-induced relaxation and dephasing. Notably, we model
environmental dephasing as more realistic non-Markovian qubit-frequency noise
from environmental fluctuators, with either white or $1/f$ noise spectra [52],
which we compare to the standard (Lindblad) Markovian techniques for modeling
dephasing.

Our modified stabilization protocol benefits from two key innovations. First,
our dynamical decoupling drive notably differs from standard approaches since
it neither acts only during idle times nor acts only on auxiliary subspaces.
Instead, the decoupling drive is always on as a second drive that can
optionally be adjusted by the feedback controller. This drive both reduces
dephasing and provides an active pathway for stabilization out of erroneous
states. Second, we develop a forward-state-estimation procedure for the
feedback controller that accounts for the time delay in the feedback loop by
anticipating the evolution occurring from signals already in transit. We find
through numerical simulations that the inclusion of these features
significantly improves the performance of the stabilization protocol. For
dephasing from $1/f$ fluctuator noise, in particular, the modified protocol
can robustly achieve near-unit asymptotic fidelity to the target state in a
wide range of conditions.

##  II State Stabilization Protocol

As in Refs. [16, 33], our system is comprised of two qubits that are not
interacting, described by a joint (system) density operator
$\rho_{\text{sys}}(t)$. Each qubit experiences environmental dephasing at an
average rate $\Gamma_{2}$, detailed later. We assume a rotating frame,
independent control drives, and coupling to a common half-parity measuring
device (e.g., [10, 16, 13]).

The half-parity observable
$\hat{N}=(\hat{\sigma}_{z}^{(1)}+\hat{\sigma}_{z}^{(2)})/2$ has three
eigenvalues: -1, 0, and 1. The eigenvalues -1 and 1 have the even-parity
eigenvectors $\ket{11}$ and $\ket{00}$, respectively, while the eigenvalue 0
has a degenerate odd-parity subspace spanned by $\ket{01}$ and $\ket{10}$. The
two odd-parity entangled Bell states,
$\ket{\psi_{\pm}}\equiv\big{(}{\ket{01}\pm\ket{10}}\big{)}/\sqrt{2}$ are thus
preserved by the measurement, while the even-parity Bell states
$\ket{\phi_{\pm}}\equiv\big{(}{\ket{00}\pm\ket{11}}\big{)}/\sqrt{2}$ get
disturbed.

_The goal of the protocol is to use the two single-qubit control drives and
the joint half-parity measurement to stabilize one maximally entangled Bell
state: $\ket{\psi_{+}}$._

To use feedback to accomplish this goal, we apply control drives that depend
upon the collected measurement record $r(t)$, which has the approximate form,

$r(t)dt=\text{Tr}(\rho_{\text{sys}}(t)\hat{N})dt+\sqrt{\tau}~{}dW,$ (1)

where $dW^{2}=dt$ is a stochastic Wiener increment [24] and
$\tau=1/2\eta\Gamma$ is a timescale set by the measurement rate $\eta\Gamma$,
given efficiency $\eta\in[0,1]$ and measurement-dephasing rate $\Gamma$. This
stochastic process models, e.g., a homodyne measurement of $\hat{N}$ after
phase-sensitive amplification of the informational quadrature, then shifting
and rescaling to normalize the mean signal to the eigenvalues of $\hat{N}$
[53].

We synchronize the applied single-qubit drives to be either corotating or
counterrotating,
$\hat{\sigma}_{y}^{(1)}\pm\hat{\sigma}_{y}^{(2)}\equiv\hat{Y}_{\pm}$, yielding
effective two-qubit drive Hamiltonians,

$\displaystyle\hat{H}_{\Omega}(t)$
$\displaystyle=\hbar\Omega~{}\hat{Y}_{+}/2,$
$\displaystyle\hat{H}_{\Delta}(t)$
$\displaystyle=\hbar\Delta~{}\hat{Y}_{-}/2,$ (2)

with amplitudes $\Omega$ and $\Delta$ that are generally functions of time
$t$, the collected record $r(t-\tau_{d})$ after a delay $\tau_{d}$ from the
feedback loop, and/or an estimated state $\rho_{\text{est}}(t)$. Importantly,
both $\hat{N}$ and $\hat{H}_{\Delta}$ act trivially on the stabilized
$\ket{\psi_{+}}$, meeting the condition for global convergence [25, 37]. The
corotating drive $\hat{H}_{\Omega}$ with matched drive phases generates
rotations in the $\big{\\{}{\ket{\psi_{+}},\ket{\phi_{-}}}\big{\\}}$ subspace
that contains the target state $\ket{\psi_{+}}$, making it the _feedback
drive_ [33]. The counterrotating drive $\hat{H}_{\Delta}$ with relative
$\pi$-phase shift generates rotations in the
$\big{\\{}{\ket{\phi_{+}},\ket{\psi_{-}}}\big{\\}}$ subspace orthogonal to
$\ket{\psi_{+}}$; these rotations also suppress slowly-varying dephasing noise
on $\ket{\psi_{+}}$, making this the _dynamical decoupling drive_.

Figure 1: Schematic for stabilizing the entangled Bell state $\ket{\psi_{+}}$
(red). Two noninteracting qubits $\rho_{\text{sys}}(t)$ undergo a half-parity
measurement with dephasing rate $\Gamma$ (orange), yielding a noisy record $r$
and stochastic transfer $\hat{M}_{r}$ between
$\ket{\phi_{+}}\leftrightarrow\ket{\phi_{-}}$ while stabilizing
$\ket{\psi_{\pm}}$. Environmental dephasing at rate $\Gamma_{2}$ (pink) also
causes stochastic transfer $\hat{H}_{\Gamma_{2}}$ between
$\ket{\phi_{+}}\leftrightarrow\ket{\phi_{-}}$ and
$\ket{\psi_{+}}\leftrightarrow\ket{\psi_{-}}$. A controller uses the
measurement record to internally track an estimated state
$\rho_{\text{est}}(t)$, then applies a feedback drive $\hat{H}_{\Omega}$
(blue) after a delay $\tau_{d}$ at rate $\Omega[\rho_{\text{est}}(t)]$ that
induces a biased transfer $\ket{\phi_{-}}\rightarrow\ket{\psi_{+}}$, as well
as a dynamical decoupling drive $\hat{H}_{\Delta}$ (purple) at rate
$\Delta[\rho_{\text{est}}(t)]$ that stabilizes $\ket{\psi_{+}}$ and
$\ket{\phi_{-}}$ and also transfers
$\ket{\psi_{-}}\leftrightarrow\ket{\phi_{+}}$.

Fig. 1 shows a conceptual schematic of how the two drives and the measurement
both prepare and stabilize the entangled state $\ket{\psi_{+}}$. After
preparing the product state
$|+,+\rangle=\big{(}{\ket{\psi_{+}}+\ket{\phi_{+}}}\big{)}/\sqrt{2}$, the
measurement stabilizes $\ket{\psi_{+}}$ at dephasing-rate $\Gamma$ but
stochastically transfers within the subspace
$\ket{\phi_{+}}\leftrightarrow\ket{\phi_{-}}$. The feedback drive unitarily
rotates between $\ket{\phi_{-}}\leftrightarrow\ket{\psi_{+}}$ at rate
$\Omega$, but the measurement feedback law biases this rotation to accumulate
population in $\ket{\psi_{+}}$. Meanwhile, dephasing induces transfers between
$\ket{\psi_{+}}\leftrightarrow\ket{\psi_{-}}$ and
$\ket{\phi_{+}}\leftrightarrow\ket{\phi_{-}}$ at rate $\Gamma_{2}$, with
$\ket{\psi_{-}}$ invisible to both $\Omega$ and the measurement. Thus,
$\ket{\psi_{-}}$ is an undesired fixed point of the protocol so far. However,
the decoupling drive at rate $\Delta$ unitarily rotates between
$\ket{\psi_{-}}\leftrightarrow\ket{\phi_{+}}$, providing a pathway through
$\Gamma$ and $\Omega$ back to $\ket{\psi_{+}}$. Neither the dephasing
$\Gamma_{2}$ nor the decoupling $\Delta$ were considered in Ref. [33], nor in
similar ancilla-free protocols in [10, 22, 23, 31].

###  II.1 Modeling the Feedback Controller

Simulations of real-time measurement feedback must distinguish between the
true (but inaccessible) state of the system $\rho_{\text{sys}}(t)$ and the
estimated state $\rho_{\text{est}}(t)$ being internally tracked by the
controller, both indicated in Fig. 1. Simulations must evolve both
$\rho_{\text{sys}}(t)$ to determine the measurement record in Eq. (1) and
$\rho_{\text{est}}(t)$ to determine the feedback drives in Eqs. (33) [47, 54].

The evolution of the internal state $\rho_{\text{est}}(t)$ has three key
differences. First, the controller does not know the specific environmental
fluctuations causing ensemble decoherence, so can only use an average
decoherence model. Second, the controller only has the collected record $r(t)$
up to a delay $\tau_{d}$ before the output feedback signal will reach the
system state $\rho_{\text{sys}}(t+\tau_{d})$. Third, whereas the system state
evolves with simulation timestep $dt$, the controller estimate
$\rho_{\text{est}}(t)$ evolves with timestep $\delta t=n~{}dt$, where $n$ is
some integer. This timestep is called a “control cycle” and represents the
time interval in which the controller performs the Bayesian update, estimates
the state, and updates the drive tones. Differentiating the two timesteps is
important because, whereas $dt$ must be much smaller than any other simulation
timescale to maintain accuracy of the discrete state-update, $\delta t$ will
be experimentally constrained by FPGA controller latency and may be comparable
to other timescales (such as a Rabi period).

###  II.2 Evolution With Decoherence and Measurement

We model dephasing as stochastic dispersive energy shifts from weakly coupling
to a noisy environment [52],

$\hat{H}_{\omega}(t)=\hbar\omega\big{(}{\chi_{1}(t)\,\hat{\sigma}_{z}^{(1)}+\chi_{2}(t)\,\hat{\sigma}_{z}^{(2)}}\big{)}/2.$
(3)

Here $\chi_{1,2}(t)$ are independent zero-mean stochastic processes, which we
generate with either white or $1/f$ power spectral densities to encompass the
two extremes of commonly observed frequency noise. From here on we refer to
these as “white noise” and “fluctuator noise”. We calibrate the amplitude
$\hbar\omega$ so that ensemble-averaging the fluctuations yields the dephasing
rate $\Gamma_{fl}$ (Appx. A).

After measuring $\hat{N}$, the observed record $r(t)$ in Eq. (1) integrated
over a small timestep $dt$ is Gaussian-distributed with mean
$\text{Tr}(\rho_{\text{sys}}(t)\hat{N})$ and variance $1/2\eta\Gamma dt$. The
induced partial collapse is thus encapsulated by the Kraus operator update
[53, 47],

$\displaystyle\hat{M}_{r(t)dt}$
$\displaystyle=\exp\left[\eta\Gamma\,dt\left(r(t)\hat{N}-\hat{N}^{2}/2\right)\right],$
(4) $\displaystyle\mathcal{M}_{r(t)}[\rho(t)]$
$\displaystyle=\hat{M}_{r(t)}\rho(t)\hat{M}^{\dagger}_{r(t)}/\text{Tr}\left(\hat{M}^{\dagger}_{r(t)}\hat{M}_{r(t)}\rho(t)\right).$
(5)

For inefficient measurements, $\eta<1$, the uncollected part of the signal
must be averaged over as unknown, causing residual measurement-dephasing. This
dephasing is described with the additional jump/no-jump Lindblad map. We use
the same procedure to model relaxation ($T_{1}$) processes, and group them
together into the single map $\mathcal{L}_{dt}[\rho(t)]$ [55, 56]:

$\displaystyle\mathcal{L}_{dt}[\rho(t)]$
$\displaystyle=\hat{J}_{0}\rho(t)\hat{J}_{0}^{\dagger}+\sum_{k=1}^{3}(\gamma_{k}dt)\hat{J}_{k}\rho(t)\hat{J}_{k}^{\dagger},$
(6a) $\displaystyle\hat{J}_{1,2}$
$\displaystyle=\hat{\sigma}_{-}^{(1,2)}/\sqrt{2},\qquad\qquad\gamma_{1,2}=1/T_{1}$
(6b) $\displaystyle\hat{J}_{3}$
$\displaystyle=\hat{N}/\sqrt{2},\qquad\qquad\gamma_{3}=(1-\eta)\Gamma,$ (6c)
$\displaystyle\hat{J}_{0}$
$\displaystyle=\sqrt{\mathbb{1}-\sum_{k=1}^{3}(\gamma_{k}dt)\hat{J}_{k}^{\dagger}\hat{J}_{k}}.$
(6d)

The composite update map for the system state is thus,

$\displaystyle\rho_{\text{sys}}(t+dt)$
$\displaystyle=[\mathcal{U}_{dt}\circ\mathcal{L}_{dt}\circ\mathcal{M}_{r(t)dt}][\rho_{\text{sys}}(t)],$
(7)

with the unitary update, $\mathcal{U}_{dt}[\rho(t)]=\hat{U}_{dt,\rm
tot}(t)\rho(t)\hat{U}^{\dagger}_{dt,\rm tot}$,

$\displaystyle\hat{U}_{dt,\rm tot}$
$\displaystyle=\exp(dt[\hat{H}_{\Delta}(t)+\hat{H}_{\Omega}(t)+\hat{H}_{\omega}(t)]/i\hbar),$
(8)

recalling that $\Omega$ and $\Delta$ generally depend upon
$\rho_{\text{est}}(t)$. This completely positive map maintains state integrity
on iteration and is accurate to order $dt$, which is chosen to be much smaller
than any other period in the system evolution.

The controller update omits the noisy $\hat{H}_{\omega}$ as unknown and either
replaces it with an average Lindblad dephasing map (Appx. AA.5) or omits it as
negligible over $\delta t$. It likewise uses a Lindblad map to model the
effects of imperfect measurement efficiency and relaxation. It also performs a
Bayesian update based on the measurement record averaged over the control
cycle duration $\delta t$, i.e.
$\tilde{r}(t)\equiv(1/n)\sum_{m=0}^{n-1}r(t-mdt)$, where the full update is
given by Eqs. 7-8 with $\rho_{\rm sys}(t+dt)\rightarrow\rho_{\rm est}(t+\delta
t)$, $r(t)\rightarrow\tilde{r}(t)$, and $dt\rightarrow\delta t$.

###  II.3 Forward Estimation

Due to the feedback loop delay, the measurement record does not reflect the
effects of drives output over the past $\tau_{d}$. This would cause inaccurate
control if $\tau_{d}>\delta t$, as the system would over-correct by repeatedly
trying to rotate the state before the effects of those rotations could be
measured. We therefore implement a novel _foward estimation_ protocol to
correct for the delay. After the measurement update described above, the
controller then makes a separate forward estimation by the delay time
$\tau_{d}$. This forward estimation predicts the effects of the control
signals already output over the preceding $\tau_{d}$, but of course does not
include any measurement information that the controller would not have access
to. For details of the forward estimation procedure, see Appendix D. This
$\rho_{\text{est}}(t+\tau_{d})$ is used to determine the new control signals
$\Omega(t+\delta t)$ and $\Delta(t+\delta t)$ to output, which are then added
to the buffer of emitted control signals used for forward estimation in future
control cycles. We show the effectiveness of this forward-estimation procedure
in Sec. III.3.

###  II.4 Optimized feedback

The choice of feedback drives $\Omega(t)$ and $\Delta(t)$ in Eq. (33) affect
the stabilization effectiveness, so we choose them to extremize the fidelity
to $\ket{\psi_{+}}$,

$\displaystyle\mathcal{F}[\rho_{\text{est}}(t)]$
$\displaystyle=\bra{\psi_{+}}\\!\rho_{\text{est}}(t)\\!\ket{\psi_{+}},$ (9)

after one time step $\delta t$. Imposing the conditions
$\partial\mathcal{F}[\rho_{\text{est}}(t+\delta t)]/\partial\Delta=0$ and
$\partial\mathcal{F}[\rho_{\text{est}}(t+\delta t)]/\partial\Omega=0$ yields

$\displaystyle 2\Delta_{\rm opt}[\rho]~{}\delta t$
$\displaystyle=\tan^{-1}[2\text{Re}(\rho_{\phi_{+},\psi_{-}})/(\rho_{\phi+}-\rho_{\psi-})]$
(10) $\displaystyle 2\Omega_{\rm opt}[\rho]~{}\delta t$
$\displaystyle=\tan^{-1}[2\text{Re}(\rho_{\psi_{+},\phi_{-}})/(\rho_{\psi+}-\rho_{\phi-})]$
(11)

using the notation $\rho_{\psi,\phi}=\bra{\psi}\\!\rho\\!\ket{\phi}$ and
$\rho_{\psi}=\rho_{\psi,\psi}$. Intuitively, feedback coefficients are
weighted _coherences_ for the subspaces in which the drives and measurement
act. In the limit of measurement strength much larger than other noise scales,
this strategy produces a steady state that is self-healing for arbitrary error
operators.

We also consider a dynamical decoupling strategy with an always-on constant
drive $\Delta_{\rm opt}\to\Delta_{\rm d.d.}$. This simplification gives less-
than-optimal fidelity in the next time step, but keeps the drive agnostic to
state estimation errors while increasing decoupling from dephasing
(particularly for fluctuators noise) and still enabling the crucial transfer
$\ket{\psi_{-}}\rightarrow\ket{\phi_{+}}$. The feedback drive $\hat{Y}_{+}$,
however, disturbs the target state $\ket{\psi_{+}}$, so the optimal drive
$\Omega_{\rm opt}$ is necessarily state-dependent even in this simplified
protocol. As shown below, this tradeoff of local optimality for enhanced
decoupling is beneficial in some circumstances.

Figure 2: Fidelity $\mathcal{F}(t)$ of the ensemble-averaged state ($N=1000$
trajectories) to target $\ket{\psi_{+}}$, compared across different feedback
protocols (a-c) and apparatus parameters (d-f). The ensemble is initialized in
state $\ket{+,+}$ and evolves under optimal corotating drive $\Omega_{\rm
opt,0}(\rho)$ and specified counterrotating drive $\Delta$. Except where
specified, simulations use measurement-dephasing rate $\Gamma=1/\mu$s,
environmental-dephasing rate $\Gamma_{2}=1/50\mu$s, feedback loop delay
$\tau_{d}=0.5~{}\mu$s, and measurement efficiency $\eta=0.5$. (a-c) Comparison
of three decoupling strategies, where $\Delta$ is either (1) off, (2) a
constant $\Delta_{\rm d.d.}/2\pi=25$ MHz simultaneous with
$\Omega_{\text{opt,0}}$, or (3) an optimized state-dependent drive
$\Delta_{\text{opt}}(\rho)$ simultaneous with $\Omega_{\text{opt,0}}$. For
dephasing from pure $1/f$ fluctuator noise in (a), simultaneous feedback and
constant counterrotating (decoupling) drive achieves near-unit fidelity
asymptotically. For fluctuators and $T_{1}$ in (b), the state-dependent
counterrotating drive is most effective but saturates to imperfect
stabilization. Pure white noise (c) mirrors this result with lower fidelities.
(d-f) Comparison with different measurement efficiencies $\eta$ and feedback
loop delays $\tau_{d}$, using the best control strategy for each noise type.
For $1/f$ noise in (d), strategy (2) ( constant $\Delta_{\rm d.d.}$) is
applied, and nonidealities are observed to slow down stabilization without
degrading the asymptotic near-unit fidelity. For fluctuators and $T_{1}$ in
(e) and white noise in (f), asymptotic fidelities depend on $\eta$ and
$\tau_{d}$.

##  III Numerical simulations

We first compare protocols with different combinations of feedback and
counterrotating drives across three different noise environments: pure
fluctuators noise, fluctuators noise combined with $T_{1}$ decay, and pure
white noise. To compare in a standard way across noise environments, we define
the dephasing rate $\Gamma_{2}$ as follows: In the case of pure fluctuators or
pure white noise, the total dephasing rate is simply $\Gamma_{2}=\Gamma_{fl}$,
i.e. all non-measurement-induced dephasing is due to averaged fluctuations,
with $\Gamma_{fl}$ calibrated separately (Appx. AA.4). We assume
$T_{1}\rightarrow\infty$ for both. In the fluctuators noise combined with
$T_{1}$ decay case, the dephasing is split evenly between the fluctuation-
induced and $T_{1}$-induced dephasing, such that
$\Gamma_{2}=\Gamma_{fl}+1/2T_{1}$ and $\Gamma_{fl}=1/2T_{1}=\Gamma_{2}/2$,
with $T_{1}$ decay described by Eq. 6.

###  III.1 Optimal Feedback Strategy

Fig. 2 shows the time-dependent target-state fidelity from Eq. (9) after
ensemble-averaging $N=1000$ simulated trajectories per curve, with time
$t\in[0,150]\,\mu$s discretized into bins $dt=1$ ns, and parameters
$\Gamma_{2}=1/50\mu$s, $\Gamma=1/\mu$s, $\eta=0.5$, and $\tau_{d}=500$ ns. We
note that these values are well within the state-of-the-art in the
superconducting qubit community, with recently achieved efficiencies of
$\eta>0.7$ [57, 58] and dephasing rates of $1/300$ $\mu$s [59]. Feedback delay
time is constrained by the travel time for signals to enter and leave the
cryostat and by processing time, leading to typical delay times of $200$-$400$
ns [38]. The drives $\Omega$ and $\Delta$ are updated every $\delta t=10$ ns
control cycle. In Fig. 2a-c, we compare three strategies using $\Omega_{\rm
opt}$ in Eq. (11):

  1. 1.

$\Omega_{\text{opt}}(\rho)$ and $\Delta=0$, a protocol originally presented in
[33];

  2. 2.

$\Omega_{\text{opt}}(\rho)$ and $\Delta_{\rm d.d.}$, where a constant
counterrotating drive acts as simultaneous dynamical decoupling and feedback;

  3. 3.

$\Omega_{\text{opt}}(\rho)$ and $\Delta_{\text{opt}}(\rho)$, where the
counterrotating drive choice is locally optimal (Eq. 10).

We comment on these in turn:

(1) No counterrotating drive.— In the presence of environmental noise, the
corotating feedback drive $\Omega$ is inadequate for robust stabilization,
regardless of noise spectrum ($1/f$ vs. white) or type (pure dephasing or
dephasing and $T_{1}$). Decoherence populates $\ket{\psi_{-}}$, which is
indistinguishable from $\ket{\psi_{+}}$ under the measurement and inaccessible
to the corotating drive.

(2) Simultaneous d.d. and feedback.— For $1/f$ fluctuators noise (Fig. 2a),
which cause a non-exponential short-time dephasing (Appx. A), a state-agnostic
constant decoupling drive simultaneous with the corotating feedback is
asymptotically the best strategy, achieving near-unit fidelity. The same
approach leads to sub-optimal saturating fidelities for fluctuators and
$T_{1}$ (Fig. 2b) or pure white noise dephasing (Fig. 2c). These results make
sense in light of known decoupling properties of $1/f$ noise and white noise
[60] (Appx. A). With white noise dephasing or relaxation, the d.d. drive makes
the $\ket{\psi_{-}}$ population accessible to the rest of the feedback loop
(enabling some stabilization), but true dynamical decoupling does not occur.

(3) Locally optimal feedback.— For noise environments with a white component
(Fig. 2b, c), the locally optimal feedback on both corotating and
counterrotating drives is asymptotically the best strategy. Furthermore, the
qualitative similarities between fidelity timeseries for the two noise
environments implies that the $T_{1}$ decay is mainly disruptive via its
contribution of $1/2T_{1}$ to the overall dephasing rate $\Gamma_{2}$, which
makes sense given that decay of individual qubit populations from
$\ket{1}\mapsto\ket{0}$ is a non-excitation-conserving map and therefore can
be detected by the measurement and subsequently corrected by the feedback.
Furthermore, since we model $T_{1}$ using Lindblad maps, the spectrum of the
effective dephasing contribution is white, which further explains the
similarities with pure white noise. The protocol performs well, particularly
at short times, for pure $1/f$ fluctuators, but is asymptotically sub-optimal
if there is no broadband noise.

Fig. 2d-f show how decreasing measurement efficiency $\eta$ and increasing
time delay $\tau_{d}$ affects stabilization in the three noise environments,
using the best feedback strategy for each environment as determined above.
Notably, for fluctuators noise, decreasing $\eta$ only slows down the
stabilization (due to the resulting decrease in measurement rate $\eta\Gamma$)
but does not decrease the asymptotic fidelity. For white noise, as well as for
fluctuators noise plus $T_{1}$ decay, the asymptotic fidelity is a steady
state that balances the measurement rate $\eta\Gamma$ and dephasing rate
$\Gamma_{2}$, so it directly depends on $\eta$. These results are in agreement
with other work showing d.d. is most effective against non-white (time-
correlated) noise [48].

We note that the locally optimal protocol attempts to optimize fidelity in the
next time step, but assumes that the decoherence rate is constant. However, a
strong constant d.d. drive greatly reduces the effective dephasing rate in the
case of slow ($1/f$) noise. We include this effect in our estimate of the
state’s dephasing (see Appx. A.5 for details). When $1/f$ dephasing is
dominant, this d.d. approach is globally optimal; when white noise or
relaxation are significant, the d.d. is less effective and so the locally
optimal protocol is also globally optimal. Future work could explore whether
hybrid strategies could be optimal for different ratios of slow and fast
decoherence; for instance, a locally optimal protocol with periodic d.d.

Figure 3: Asymptotic fidelity $\mathcal{F}$ of the ensemble-averaged state
($N=1000$ trajectories) to target $\ket{\psi_{+}}$ at $t_{f}=150\,\mu$s, with
color displayed on a log scale. The optimal protocol for each noise type is
used. Except where specified as a sweep variable, parameters default to
$\eta=0.5$, $\tau_{d}=0.5~{}\mu$s, $\Gamma=1/\mu$s, $\Delta_{0}/2\pi=25$ MHz,
$\Gamma_{2}=1/50.0~{}\mu$s. (a) Fluctuators noise is mitigated with
$\Delta_{\text{d.d.}}$, sweeping the counterrotating drive strength
$\Delta_{0}$, for different dephasing rates $\Gamma_{2}\in[0.01,0.1]/\mu$s.
Near-unit fidelity $(>0.99)$ is achieved for any dephasing rate with a
sufficiently fast decoupling drive $\Delta_{0}$. (b) Fluctuators and $T_{1}$
and (c) white noise are mitigated using $\Delta_{\text{opt}}$, sweeping the
measurement rate $\Gamma$ for different $\Gamma_{2}$. Fidelities are improved
with stronger measurements (larger $\Gamma$), but lag behind the fidelities
observed for fluctuators and $\Delta_{\text{d.d.}}$. (d-f) Measurement rate
$\Gamma$ is swept for different feedback loop delay times $\tau_{d}$ for (d)
fluctuators, (e) fluctuators and $T_{1}$, and (f) white noise. All demonstrate
stabilization fidelities that are remarkably robust to loop delay, with short
delays showing large optimal $\Gamma$ (up to $10/\mu$s in the range swept) and
long delays necessitating smaller $\Gamma$. For the realistically achievable
$\Gamma_{2}=1/50.0~{}\mu$s and $\tau_{d}=0.52~{}\mu$s, optimal fidelities
achieved are (d) $\mathcal{F}=0.9999$ at $\Gamma=5.76/\mu$s, (e)
$\mathcal{F}=0.95$ at $\Gamma=10.0/\mu$s, and (f) $\mathcal{F}=0.92$ at
$10.0/\mu$s. Figure 4: Ensemble fidelities averaged from $100$ to $150~{}\mu$s
are compared on a log scale for different measurement rates $\Gamma$ (x-axis),
loop delay times $\tau_{d}$ (color) and state estimation strategies (opaque
vs. translucent), for the fluctuators plus $T_{1}$ noise environment with
$\Gamma_{2}=1/50.0$ $\mu$s, $\eta=0.5$, $\delta t=$ 10 ns. When control does
not take time delay into account, fidelity is close to the purely mixed case
$\mathcal{F}=0.25$ (dashed lines). Forward estimation results in higher
fidelities for all feedback loop delays $\tau_{d}$ after optimizing over the
control parameter $\Gamma$, with typical infidelity reduction by $\sim 20\%$,
demonstrating the optimality of the protocol.

###  III.2 Optimizing Control Parameters

Fig. 3 shows asymptotic fidelities at $t=150~{}\mu$s, again applying the best
strategy for each noise environment. The left panels, Fig. 3a-c, sweep the
best strategy’s main control parameter ($\Delta$ for $\Delta_{\text{d.d.}}$
and $\Gamma$ for $\Omega_{\text{opt}}(\rho)$) vs. the overall dephasing rate
$\Gamma_{2}$. As noted previously, for pure fluctuators or white noise (Figs.
3a, c), dephasing is entirely due to fluctuations ($\Gamma_{2}=\Gamma_{fl}$),
whereas for mixed fluctuators and $T_{1}$ (Fig. 3b), dephasing is evenly split
between fluctuations and $T_{1}$ contributions
($1/2T_{1}=\Gamma_{fl}=\Gamma_{2}/2$). Fixed parameters are
$\tau_{d}=0.5~{}\mu$s and $\Gamma=1/\mu$s for fluctuators. The right panels,
Fig. 3d-f, sweep the measurement rate $\Gamma$ vs. the feedback loop delay
$\tau_{d}$ for each noise environment, with fixed $\Gamma_{2}=1/50.0\mu$s,
$\eta=0.5$, and $\Delta_{0}/2\pi=25$ MHz for fluctuators.

Similarly to Fig. 2, we observe that the pure fluctuators noise environment
shows the highest stabilization fidelities, with $\mathcal{F}(t_{f})>0.98$ for
any $\Gamma_{2}\in[1/100.0\mu\text{s},~{}1/10.0\mu\text{s}]$ after optimizing
counterrotating drive strength $\Delta_{0}$. Similarly high fidelities are
possible for all loop delays
$\tau_{d}\in[0.1~{}\mu\text{s},5.0~{}\mu\text{s}]$ after optimizing over
$\Gamma$. In particular, we highlight an optimal fidelity of
$\mathcal{F}=0.9999$ with control parameters $\Gamma=5.76/\mu$s and
$\Delta_{0}/2\pi=25.0$ MHz, operating at loop delay $\tau_{d}=0.52~{}\mu$s and
dephasing rate $\Gamma_{2}=1/50.0\mu$s. While fluctuators plus $T_{1}$ and
white noise environments demonstrate lower fidelities, we note that optimizing
over measurement strength $\Gamma$ still leads to high fidelities for
realistic experimental parameters: using measurement strength
$\Gamma=10.0/\mu$s while operating at $\Gamma_{2}=1/50.0\mu$s and
$\tau_{d}=0.52~{}\mu$s, optimal fidelities achieved are $\mathcal{F}=0.95$
(fluctuators noise plus $T_{1}$) and $\mathcal{F}=0.92$ (white noise). This
feedback delay is achievable with modern FPGA-based controllers, which we have
tested and verified with placeholder calculations on a Quantum Machines OPX
controller; all other parameters are well within the current state of the art.

###  III.3 Optimality of Forward Estimation

We note that accounting for time delay in our state estimate is essential to
the success of these protocols. If the raw time-delayed state is used in Eqs.
(10)-(11), we see only instability as the controller “overcorrects”, driving
the system when previously-emitted tones would already rotate to the target
state. Our forward estimation protocol addresses this issue. However, the
behavior is reminiscent of a classical feedback controller with a feedback
gain that is too large, which suggests a simpler protocol: turn down the
strength of the feedback drives. We tested this deliberate weakening of the
feedback drives such that $\Omega=\Omega_{\text{opt}}(\delta t/\Delta t)$. In
this simpler protocol, the non-forward-estimated (time-delayed) state is used
as the state estimate, and $\Delta t$ is a free parameter whose optimal value
depends on other system timescales.

Three estimation strategies are compared in Fig. 4 for different loop delays
$\tau_{d}$ and measurement rates $\Gamma$: no forward estimation with $\Delta
t=\delta t$ (no attenuation), no forward estimation with the attenuation
$\Delta t/\delta t$ optimized for each $(\tau_{d},~{}\Gamma)$ configuration,
and forward estimation with $\Delta t=\delta t$. The first case leads to
complete mixing of the state, resulting in a time-averaged fidelity of $0.25$.
This shows that some correction for feedback delay is essential. The second
case — no forward estimation, but with attenuated drive —provides reliable
stabilization, but the asymptotic fidelity achievable (averaged over the
interval $[100,150]~{}\mu$s to reduce noise) is generally lower than for the
forward estimation. In particular, we note that tuning over the controllable
parameter $\Gamma$ indicates forward-estimation as optimal for all $\tau_{d}$
tested, reducing infidelity by $\sim 20\%$ compared to the attenuated drive
protocol. Thus, due to the low computational overhead of forward-estimating
the state, we selected forward estimation as the optimal estimation strategy
for the data presented in Figs. 2-3, where for simplicity $\Delta t=\delta t$
was used for all forward-estimation simulations; optimizing over the feedback
attenuation while still using forward estimation could further improve
fidelity.

The attenuation-without-forward-estimation strategy is slightly simpler and so
may be preferred in an experiment where computational overhead is the limiting
factor, at the cost of lower fidelity. In a similar attempt at simplification,
we also explored a version of the protocol with quasi-linear feedback, where
the feedback signal at time $t$ is linearly proportional to the measurement
signal $r(t)$ (see Appendix C.4). The proportionality constant derived depends
on the prior state estimate, so it is not true linear feedback. However, while
the linear feedback is qualitatively similar to the optimal feedback for
stable trajectories, on average it does not produce robust stabilization,
which we attribute again to the finite time delay (see Appendix C.5).

##  IV Conclusion and Discussion

We have presented and simulated a protocol that reliably generates and
stabilizes an entangled state of two noninteracting qubits under realistic
experimental conditions with dephasing from various realistic noise
environments. More generally, in the limit of measurement strength much larger
than other noise scales, our protocol produces a steady state that is self-
healing for arbitrary error operators and without the use of ancillas. The
success hinges on two key innovations of our protocol. First, we use a drive
that both provides dynamical decoupling and transfers population out of an
undesired fixed point of the dynamics, working concurrently with the feedback
drive. Second, we develop a forward-estimation protocol for the feedback
controller that compensates for loop delay, which can be used to improve other
feedback control protocols. Together these provide evidence that continuous
feedback control can achieve capabilities beyond what can be replicated with
discrete (projective) measurement and gates, even in the presence of realistic
nonidealities.

We note that, while our simulations take into account many experimental non-
idealities, there are still approximations we have made. We treat the qubits
as true two-level systems, while superconducting qubits are often weakly
anharmonic. We note that the bandwidth of our drives ($2/\delta t=50$ MHz) is
small compared to typical anharmonicity of a transmon ($\approx 200$ MHz), and
so this assumption appears justified. We ignore stray qubit-qubit coupling,
which can be eliminated in many architectures via multi-path coupling (e.g.,
[61]). We ignore classical control crosstalk, which can be eliminated through
proper calibration. We also ignore the effects of finite readout cavity
bandwidth (i.e., we make the “bad-cavity” approximation of Markovianity) and
the effects of an improperly tuned measurement that can distinguish $\ket{01}$
from $\ket{10}$ (i.e., we assume a perfect half-parity measurement). These
last assumptions are the most difficult to achieve experimentally. Future work
could examine protocol performance when they are relaxed, for example by
modeling cavity evolution directly as a truncated Fock space and including
asymmetric measurement rates for $\ket{01}$ from $\ket{10}$.

Extensions of our work could optimize tradeoffs between locally optimal
feedback and dynamical decoupling based on noise spectra; optimize the forward
estimation protocol; and explore the impact of non-Markovian models of
relaxation. Furthermore, our work can be generalized to protocols stabilizing
other entangled states, including states with higher numbers of qubits. The
generalization requires three components: (1) a joint measurement that has the
target entangled state as an eigenstate; (2) combined single-qubit drives that
rotate toward this target state from an entangled state that is not a
measurement eigenstate (in our case the co-rotating drive); (3) combined
single-qubit drives that have the target state as an eigenstate, but rotate
all other degenerate measurement eigenstates to non-eigenstate entangled
states (in our case the counter-rotating drive). We leave the rigorous general
analysis of this class of protocols to future work, but note that it would be
fruitful to compare protocols obtained in this way to protocols obtained via
more general reinforcement learning or gradient descent methods [62, 63].
Applying machine-learning methods to finding optimal entanglement-generation
protocols could guide the choice of generalized measurements, while
identifying specific features of the protocols (e.g. the three components
previously mentioned) could restrict the optimization space to reduce
computational overhead of machine-learning. We anticipate that our results
will guide future studies of analog continuous measurement feedback protocols
and provide an experimental roadmap for implementing fast digital processing
while compensating for feedback delay.

Acknowledgements.—ELF thanks the members of QNL for facilitating this
collaboration. SG thanks Arjendu Pattanayak for helpful conversations about
the system-estimate distinction in quantum feedback control and Vinay Tripathi
for discussions of simulating $1/f$ noise using fluctuators. SG also
acknowledges funding from the Graduate Fellowships for STEM Diversity. This
work was supported by the ONR under N00014-21-1-2688, by Research Corp under
Cottrell Scholarship 27550, by NSF-BSF under Grant Award No. 1915015, and by
ARO/LPS under Grant Award No. W911NF-22-1-0258.

##  Appendix A Modeling dephasing

We used three different simulation methods to model dephasing:

  1. 1.

Lindblad: We use time-discrete and completely positive Kraus maps to implement
Markovian dephasing or decay for individual trajectories when more detailed
modeling is not needed.

  2. 2.

Fluctuator noise: As a more physically realistic dephasing model, we simulate
a bath of uncorrelated two-level fluctuators that are dispersively coupled to
each qubit. The random switches of the fluctuators make the qubit frequencies
stochastic processes with $1/f$ noise spectra, leading to non-exponential
decoherence when ensemble-averaged.

  3. 3.

Markovian white noise: For comparison to the fluctuator model, we also
simulate a white noise fluctuation model where the qubit frequencies become
Markovian stochastic processes, leading to exponential decoherence of Lindblad
type when ensemble-averaged.

We now summarize the theoretical background and implementation for each model
and show numerically simulated examples.

###  A.1 Lindblad decoherence

We simulate average-sense decoherence using the “jump / no-jump” method for
minimally unraveling continuous Lindblad evolution into discrete-time
completely positive maps (e.g., see Eqs. (9-10) of [55]):

$\displaystyle\rho(t)\mapsto\rho(t+dt)$
$\displaystyle=\hat{J}_{0}\rho(t)\hat{J}_{0}^{\dagger}+\sum_{k=1}^{n}(\gamma_{k}dt)\hat{J}_{k}\rho(t)\hat{J}_{k}^{\dagger},$
(12a) $\displaystyle\hat{J}_{0}$
$\displaystyle=\sqrt{\mathbb{1}-\sum_{k=1}^{n}(\gamma_{k}dt)\hat{J}_{k}^{\dagger}\hat{J}_{k}}.$
(12b)

This unraveling models rare events that occur through one of $n$ available
channels, modeled by Kraus operators $\\{{\hat{J}_{k}}\\}_{k=1}^{n}$ with
rates $\gamma_{k}$, such that the probability of one event in a short duration
$dt$ approximates $\gamma_{k}dt$. The corresponding non-event Kraus operator
$\hat{J}_{0}$ follows from the POVM normalization condition
$\mathbb{1}=\hat{J}_{0}^{2}+(\gamma_{k}dt)\hat{J}_{k}^{\dagger}\hat{J}_{k}$.
The total update then averages over all $n+1$ event possibilities assuming a
lack of knowledge regarding which event occurred. Using these time-discrete
Kraus maps guarantees that the evolution of each time step is completely
positive, so is less error-prone than naïvely applying generic numerical
integration methods to the Lindblad differential equation. The usual
continuous Lindblad equation can be recovered by taking the limit $dt\to 0$ of
this discrete update, with the anti-commutator term arising from non-events.

While we emphasize fluctuator and white noise models of dephasing as the more
physically realistic models for each sampled evolution trajectory, this pre-
averaged Lindblad decoherence model is an efficient way to model the expected
ensemble-averaged dephasing evolution by applying it to each individual
trajectory instead, acknowledging the absence of more detailed knowledge of
the fluctuation mechanism for that trajectory. In particular, we use this
crude ensemble-averaged model of the dephasing in projected forward-time-
estimations of the system state to compensate for signal propagation delay and
maximize real-time feedback performance. For this purpose, single-qubit
environmental dephasing using Eq. (12) uses the phase-jump operator
$\hat{J}=\hat{\sigma}_{z}/\sqrt{2}$ with rate $\Gamma_{2}$. Similarly,
residual ensemble-dephasing due to inefficient measurement (see Section B.2
below) uses the jump operator $\hat{J}=\hat{N}/\sqrt{2}$ with observable
$\hat{N}=(\hat{\sigma}_{z}^{(1)}+\hat{\sigma}_{z}^{(2)})/2$ and rate
$(1-\eta)\Gamma$, where $\Gamma$ is the ensemble-averaged measurement-
dephasing rate, $\eta\in[0,1]$ is the measurement efficiency, and $\hat{N}$ is
the half-parity operator being measured.

###  A.2 Non-Markovian fluctuator noise

The fluctuators approach to simulating dephasing postulates a set of $N$ two-
level systems, each of which dispersively interacts with the qubit to create
classical telegraph noise. The two-level systems randomly switch between their
two states in a Poisson process with characteristic rates
$\\{\gamma_{i}\\}_{i=1}^{N}$. The rates are distributed log-uniformly between
minimum and maximum frequencies ($\gamma_{1},\gamma_{N}$), i.e.,

$\displaystyle\log\frac{\gamma_{i}}{\gamma_{1}}=\frac{i-1}{N-1}\log\frac{\gamma_{N}}{\gamma_{1}}\qquad$
(13a)
$\displaystyle\implies\qquad\gamma_{i}=\gamma_{1}\,\left(\frac{\gamma_{N}}{\gamma_{1}}\right)^{(i-1)/(N-1)}.$
(13b)

Because this distribution is exponentially weighted towards low frequencies,
the resulting fluctuations have nontrivial temporal correlations, leading to
non-Markovian frequency noise that goes beyond simple Markovian (e.g.,
Lindblad) models of dephasing. This behavior better represents the
experimentally observed correlations caused by two-level systems on-chip,
which are a common source of dephasing in superconducting qubit systems.
Simulating dephasing with fluctuator noise in this way generates a
$(1/f)^{\alpha}$ power spectral density (PSD) [64, 65, 66, 67], which is in
qualitative agreement with experimentally measured PSD, particularly in cases
where the device is charge- or flux-noise sensitive [64, 68, 69, 70]. To
implement the entanglement-generation experiment we consider in the main text,
a flux-tunable qubit design may be necessary to engineer the desired joint
measurement—as in [71]—in which case we should expect $1/f$ flux noise to be
relevant.

For the simulations in the main text, we generate each stochastic telegraph
noise realization with the following procedure:

  1. 1.

We define $N=20$ frequencies $\\{\gamma_{i}=2\pi\,f_{i}\\}$ distributed log-
uniformly between $(f_{1},f_{N})=(5\text{ kHz},50\text{ MHz})$ according to
Eq. (13a).

  2. 2.

For each $\gamma_{i}$, we define a fluctuator to be a classical stochastic
process $s_{i}(t)$ that takes the binary values $-1$ or $1$ for all
$t\in[0,t_{f}]$ up to the simulation duration $t_{f}$. Each fluctuator
randomly switches between its two values as a Poisson process with
characteristic switching rate $\gamma_{i}$, starting from an equiprobably
random initial state $s_{i}(0)=\pm 1$.

For efficient simulation, we use the fact that the durations between random
switches of a Poisson process are exponentially distributed,
$T_{j}^{(i)}\sim\text{Exp}(\gamma_{i})$, i.e., with probability density
function $p(T_{j}^{(i)})=\gamma_{i}\exp(-\gamma_{i}T^{(i)}_{j})$. Thus, we
first sample the random durations between switches until the total duration,
$\sum_{j=1}T_{j}^{(i)}$, exceeds the desired simulation duration $t_{f}$. We
then construct the stochastic process $s_{i}(t)$ as a piecewise-constant
function that flips sign after each duration $T^{(i)}_{j}$ in the sampled
sequence, starting from $s_{i}(0)$.

  3. 3.

Finally, we construct the the total stochastic process that affect each qubit
frequency by averaging the effect of all $N=20$ fluctuators. The resulting
stochastic process $\chi(t)=(1/N)\sum_{i=1}^{N}s_{i}(t)$ has zero mean and a
PSD with ${}\sim 1/f$ dependence, as verified in Fig. 5.

(a) Time series comparison

(b) Power spectral density comparison

Figure 5: Comparison of the temporal and spectral characteristics of white
noise and fluctuator noise. (a) The white noise process (top) is stationary
and self-similar at different timescales, as shown in the inset comparison,
while the fluctuator noise process (bottom) is temporally correlated and thus
more structured. (b) The PSD of white noise is frequency-independent, while
the PSD of our sampled fluctuator noise has approximately $1/f$ frequency-
dependence (blue fit curve) between 0.1 MHz and 25 MHz, with roll-off into a
flat spectrum at both high and low frequencies. Note that the power spectra
have been smoothed with a moving window-average of width $50\Delta f=1$ MHz
for visual clarity.

The simulated fluctuator envelopes modulate the qubit frequencies in the
effective Hamiltonian,

$\hat{H}_{\rm
fluct}(t)=\frac{\omega}{2}\chi_{1}(t)\,\hat{\sigma}_{z}^{1}+\frac{\omega}{2}\chi_{2}(t)\,\hat{\sigma}_{z}^{2},$
(14)

where $\chi_{1,2}(t)$ are independent fluctuator processes generated with the
above procedure, while $\omega$ is a (constant) characteristic frequency that
sets the strength of the dispersive frequency fluctuations.

Notably, after ensemble averaging, these frequency fluctuations produce a
characteristic decay envelope that is distinct from the purely exponential
decay of Markovian (Lindblad) dephasing models, as can be seen in Fig. 5, with
quadratic decay at small times rather than linear decay. However, at longer
times the fluctuator coherence decay more closely approximates exponential
decay, with the fluctuation strength $\omega$ indirectly setting a
characteristic decay rate $\Gamma_{2}$ that can be compared to the exponential
decay rate of Markovian dephasing models. We describe how to calibrate the
scaling frequency $\omega$ in to achieve a desired $\Gamma_{2}$ in Sec. A.4.

###  A.3 Markovian white noise

For comparison with the non-Markovian fluctuator model, we also simulate
explicitly Markovian (white noise) fluctuations that produce purely
exponential decay of the ensemble-averaged coherence. Using the same
Hamiltonian as Eq. (14), each stochastic envelope $\chi_{k}$ is instead
sampled at discrete times $t_{\ell}=t_{0}+\ell\,dt$ from a zero mean Gaussian
distribution, $\chi_{k}(t_{\ell})\sim\text{Norm}(0,\tau/dt)$ with variance
$\tau/dt$ characterized by a timescale $\tau$ and the time discretization
$dt$. The resulting envelope functions $\chi_{k}(t)$ then are Markovian random
processes with a piecewise constant structure that limit to white noise
processes satisfying
$\langle\chi_{k}(t)\chi_{k}(t^{\prime})\rangle=\tau\,\delta(t-t^{\prime})$ in
the time-continuum limit $dt\to 0$. The exponential decay rate of the ensemble
averaged Bloch radius in the $x$-$y$ plane is then,

$\displaystyle\Gamma_{2}$
$\displaystyle=\int_{t_{0}}^{t}\left\langle[\omega\chi_{k}(t)][\omega\chi_{k}(t^{\prime})]\right\rangle
dt^{\prime}=\frac{1}{2}\omega^{2}\tau,$ (15a)
$\displaystyle\implies\qquad\omega=\sqrt{\frac{2\Gamma_{2}}{\tau}}.$ (15b)

We numerically confirm this scaling of the decay rate with $\omega$ in Fig. 7b
of the next section. For simplicity in the simulations, we choose $\tau=dt$ to
be the same as the time step duration, making the sampled $\chi_{k}$ have unit
variance.

###  A.4 Calibrating the fluctuation strength for a target dephasing rate

To fairly compare dephasing timescales produced by the different noise models,
we numerically calibrate the dependence between the noise strength parameter
$\omega$ in the Hamiltonian in Eq. (14) and a rate $\Gamma_{2}$ that
characterizes the dephasing envelope. Importantly, the strength $\omega$ is
not directly measurable experimentally, while the ensemble-averaged decay
profile of the coherence can be observed in a Ramsey experiment. Calibrating
$\omega$ as a function of the characteristic rate $\Gamma_{2}$ thus better
connects our simulated noise models to experimental observation. Moreover,
matching effective decay rates in the absence of feedback is critically
important to meaningfully compare and assess the performance of each feedback
protocol in the main text.

A subtle complication with establishing a meaningful rate comparison is that
while exponential decay is memory-less with a constant decay rate
$\Gamma_{2}$, fluctuator noise has a temporally structured decay that is is
quadratic for short times and exponential at late times, so any fit to purely
exponential decay is problematic. This non-Markovian behavior arises from the
exponential memory kernel of the fluctuator noise time correlations and is
precisely what makes it possible to dynamically decouple from short-time low-
frequency fluctuator noise experimentally (e.g., using a spin echo mid-gate).
Nevertheless, we can still identify a characteristic rate $\Gamma_{2}$ that
sets the time scale of the total decay profile and reduces to the exponential
decay rate in the Markovian limit, which is sufficiently close for the
purposes of the main text. Most importantly, we confirm that the feedback
protocol fidelity results in the main text are not artifacts of any
miscalibration of our fluctuator models, but are instead fundamental to the
choice of time-dependent feedback.

Since we can crudely think of the fluctuator noise spectrum as a mixture of
$1/f$ and white noise, we can postulate a mean decay profile along a
“stretched exponential” that interpolates between the two. To calibrate
$\omega$, we simulate Ramsey experiments as follows: we initialize ensembles
of single-qubit $\ket{+x}$ states undergoing fluctuator or white noise, sweep
the parameter $\omega$ over the range $[0.25,\,1]$ /$\mu$s for fluctuators
noise and $[1,\,5]$ /$\mu$s for white noise, extract the ensemble-averaged
Bloch coordinate $x$ as a function of time, and fit the resulting decay curves
to a stretched-exponential function,

$x(t)=\exp\big{(}{-(\Gamma_{2}t)^{\beta}}\big{)},$ (16)

characterized by both a rate $\Gamma_{2}$ and a dimensionless power $\beta$.
The power $\beta=1$ corresponds to purely exponential decay, while the short-
time decay anticipated for $1/f$ noise should be quadratic with $\beta=2$,
suggesting that an average fit for $\beta$ will be somewhere between 1 and 2
for the full fluctuator noise decay profile. Eq. (16) is a common experimental
choice for fitting non-exponential decay with a single rate parameter and fits
the broad structure of our simulated decay curves reasonably well. Moreover,
the extracted rates $\Gamma_{2}$ for non-exponential decay are also very close
to the rates obtained from naïve exponential fits to same data, which supports
the use of the obtained $\Gamma_{2}$ as a reasonable comparison to purely
exponential decay rates.

(a) Fluctuator Ramsey decay, varying $\omega\in(0.25,1.0)$ rad/s

(b) Fluctuator fit parameter $\Gamma_{2}/\gamma_{\text{geom}}$ vs. varying
$\omega/\gamma_{\text{geom}}$.

(c) Fluctuator fit parameter $\beta$ vs. varying $\omega$.

Figure 6: (a) Ramsey decay curves for fluctuators with different choices of
frequency drift amplitude $\omega$, fit to a two-parameter stretched
exponential decay, $x(t;\,\Gamma_{2},\beta)=\exp(-(\Gamma_{2}t)^{\beta})$. (b)
Relation between drift amplitudes $\omega$ and the dephasing rate parameter
$\Gamma_{2}$, which is fit to a power law used to set the target dephasing
rates in the main text. The frequency reference $\gamma_{\text{geom}}=$
$\sqrt{\gamma_{1}\gamma_{N}}=(2\pi)\,500$ kHz is the geometric mean of the
$N=20$ log-distributed fluctuator frequencies between $\gamma_{1}/2\pi=5$ kHz
and $\gamma_{N}/2\pi=50$ MHz. (c) Relation between drift amplitudes $\omega$
and the dephasing power parameter $\beta$.

(a) White noise Ramsey decay, varying $\omega\in(1.0,5.0)$ rad/s.

(b) White noise fit parameter $\Gamma_{2}~{}dt$ vs. varying $\omega~{}dt$.

(c) White noise fit parameter $\beta$ vs. varying $\omega$.

Figure 7: (a) Ramsey decay curves for white noise with different choices of
frequency drift amplitude $\omega$, fit to a two-parameter stretched
exponential decay, $x(t;\,\Gamma_{2},\beta)=\exp(-(\Gamma_{2}t)^{\beta})$. (b)
Relation between drift amplitudes $\omega$ and the dephasing rate parameter
$\Gamma_{2}$, which is fit to a power law used to set the target dephasing
rates in the main text. The white noise characteristic time $\tau=dt=10$ ns is
set to the simulation timestep duration for convenience. (c) Relation between
drift amplitudes $\omega$ and the dephasing power parameter $\beta$.
Exponential decay corresponds to the power $\beta=1$, confirmed by the white
noise fit up to small deviations due to fit imperfections.

For the fluctuator noise model, the stretched exponential fits shown in Fig.
6a determine the relationship between $\omega$ and $\Gamma_{2}$ shown in Fig.
6b, which has the following polynomial fit:

$\displaystyle\frac{\Gamma_{2}}{\gamma_{\text{geom}}}$
$\displaystyle=0.12\Big{(}{\frac{\omega}{\gamma_{\text{geom}}}}\Big{)}^{1.55}$
(fluctuator noise). (17)

For this fit we choose the reference rate to be the geometric mean of the log-
uniformly spaced fluctuator frequencies,
$\gamma_{\text{geom}}=\sqrt{\gamma_{1}\gamma_{N}}=\exp[(1/N)\sum_{i}\log\gamma_{i}]=(2\pi)(500\text{kHz})=\pi/\mu\text{s}$,
yielding the swept parameter range
$\omega/\gamma_{\text{geom}}\in[1/4\pi,\,1/\pi]$. Other choices of frequency
reference will scale the prefactor of the fit, but won’t change the dependence
on $\omega$. The numerical results for the prefactor (0.12) and power (1.55)
generally depend on other parameters that we keep constant in our simulations,
like the number of fluctuators $N$ and the number of frequency decades
$d=\log_{10}(\gamma_{\text{max}}/\gamma_{\text{min}})$ spanned about the mean
$\gamma_{\text{geom}}$, but these additional dependencies are not important
for the results of the main text. The relationship between $\omega$ and the
power $\beta$ is also shown in Fig. (6c) and is approximately linear,
$\beta(\omega)=1.07+0.47(\omega/\gamma_{\text{geom}})$.

For the white noise fluctuation model, the stretched exponential fits shown in
Fig. 7a yield the relationship between $\omega$ and $\Gamma_{2}$ shown in Fig.
7b, with a polynomial fit,

$\displaystyle\Gamma_{2}dt$ $\displaystyle=0.49\big{(}{\omega
dt}\big{)}^{1.99},$ (white noise) (18)

which confirms the dependence expected from Eq. (15) with the noise timescale
choice of $\tau=dt$. Similarly, the power $\beta$ is consistent with the
exponential-decay power of 1, as shown in Eq. (7c).

These fit equations can then be inverted to obtain the conversion from
$\Gamma_{2}$ to $\omega$ used for the simulations in the main text:

$\displaystyle\omega(\Gamma_{2})$
$\displaystyle=3.9\,\gamma_{\text{geom}}\Big{(}{\frac{\Gamma_{2}}{\gamma_{\text{geom}}}}\Big{)}^{0.65}$
(fluctuators), (19a) $\displaystyle\omega(\Gamma_{2})$
$\displaystyle=\sqrt{\frac{2\Gamma_{2}}{dt}}$ $\displaystyle\text{(white
noise)}.$ (19b)

###  A.5 Effective dephasing rate with constant dynamical decoupling Rabi
drive

In addition to calibrating the relationship between the fluctuation amplitude
$\omega$ in the Hamiltonian in Eq. (14) and the resulting characteristic
dephasing rate $\Gamma_{2}$, we also calibrate the effective dephasing rate
$\Gamma_{\text{eff}}$ observed in the presence of a dynamical decoupling
drive. This is applicable to the “simultaneous d.d. and feedback” protocol in
the main text, where $\Delta(\rho)=\Delta_{\rm d.d.}$ is constant and acts to
decouple the system from slow fluctuations. In the case of a single qubit
undergoing dephasing with no feedback, a constant Rabi drive
$\Delta\,\hat{\sigma}_{y}/2$ around the $y$-axis orthogonal to the frequency
fluctuation $z$-axis $\hat{\sigma}_{z}$ will mitigate the ensemble dephasing
by reducing the net time that the qubit coherence is affected by frequency
fluctuations. The effective dephasing rate $0<\Gamma_{\text{eff}}<\Gamma_{2}$
will be reduced in proportion to the ratio $\Delta/\Gamma_{2}$ of decoupling
and bare dephasing timescales. The effectiveness of such a decoupling
technique depends on the short-time behavior of the ensemble dephasing, which
is linear for exponential decay but has slower quadratic dependence for
fluctuator-based dephasing. Thus, this simple dynamical decoupling technique
is significantly more effective at mitigating coherence decay for fluctuator
noise than for white noise.

Knowing the expected effective dephasing $\Gamma_{\text{eff}}$ in the presence
of a decoupling Rabi drive is important for our feedback protocol in the main
text. The forward state estimation used to compensate for feedback delay uses
an effective dephasing Lindblad term in the filter (see Sec. D), which must
use an accurate effective dephasing rate that accounts for the drive.
Including this effective ensemble dephasing in the forward estimation improves
the state-tracking fidelity and, as a result, the feedback stabilization
performance.

Fig. 8 shows the effective dephasing rate $\Gamma_{\text{eff}}$ for a sweep of
decoupling Rabi amplitudes $\Delta$ and bare dephasing rates $\Gamma_{2}$ for
a single qubit. Because the qubit state vector is rotating in the $x$-$z$
plane on average, the purity (i.e., $P=\text{Tr}\rho^{2}=(1+r^{2})/2$, with
$r^{2}=x^{2}+y^{2}+z^{2}$) is used to extract a time constant. Since the drive
generally decouples from the slow noise components contributing to the non-
exponential decay, a single parameter fit is sufficient,

$P(t)=\frac{1+\exp(-2\Gamma_{\text{eff}}t)}{2}.$ (20)

For $\Delta=0$, $\Gamma_{\text{eff}}=\Gamma_{2}$ as expected.

For white noise, $\Delta/\Gamma_{2}>1/2$ yields the asymptotic limit
$\Gamma_{\text{eff}}/\Gamma_{2}\to 1/2$, as shown in Fig. 8 and predicted from
the white noise correlator
$\langle\chi_{k}(t)\chi_{k}(t^{\prime})\rangle=\tau\delta(t-t^{\prime})$ [72].
The intuition behind this limit is that under a constant Rabi drive the qubit
spends about half its time near the $z$-axis, where it is immune to
$\hat{\sigma}_{z}$ rotations, so dephases at half the rate.

In contrast, for fluctuators there are much larger reductions in
$\Gamma_{\text{eff}}$ as $\Delta$ is increased. As shown in Fig. 8, for larger
$\Delta/\Gamma_{2}\gg 1$ the rate $\Gamma_{\text{eff}}$ decays as roughly
$\Gamma_{2}/\Delta$:

$\Gamma_{\text{eff}}/\Gamma_{2}=0.09~{}(\gamma_{\text{geom}}/\Gamma_{2})^{0.71}(\Gamma_{2}/\Delta)^{1.1},$
(21)

with the $(\gamma_{\text{geom}}/\Gamma_{2})$ prefactor accounting for the
$\Gamma_{2}$-dependent intercepts on the log-log plot.

This $1/\Delta$ behavior allows $\Gamma_{\text{eff}}$ to become arbitrarily
small for a large enough decoupling drive. For simplicity and accuracy, we use
the numerical results shown in Fig. 8 to calibrate $\Gamma_{\text{eff}}$ in
the state-tracking filter used in the main text.

Figure 8: Ratio of effective qubit dephasing rate $\Gamma_{\text{eff}}$ to
bare rate $\Gamma_{2}$ in the presence of a constant decoupling Rabi drive
$\Delta\,\hat{\sigma}_{y}/2$. The qubit frequency fluctuations
$\omega\chi(t)\,\hat{\sigma}_{z}/2$ have the amplitude $\omega$ calibrated to
produce the ensemble-dephasing rate $\Gamma_{2}$ in the absence of the
decoupling drive, based on Fig. 6b and 7b. The small-time quadratic dependence
of the fluctuator-based dephasing dramatically increases the effectiveness of
the decoupling drive, with the effective dephasing rate decreasing as roughly
$1/\Delta$. In contrast, for white noise the effective dephasing drops to
$\Gamma_{\text{eff}}=\Gamma_{2}/2$ for any nonzero $\Delta$ due to the system
state spending approximately half the time close to the dephasing-immune
$z$-axis.

##  Appendix B Measurement model

We use a Bayesian update model of time-stepped sequential weak measurements to
describe the backaction of the joint continuous measurement on the two-qubit
system. For completeness, we briefly review the essentials of this method.

In the time-continuum limit, the monitoring of an observable $\hat{N}$ with a
homodyne readout will produce a normalized signal $r(t)$ that approximates a
stochastic Wiener process

$r(t)dt=\text{Tr}(\rho_{\text{sys}}(t)\hat{N})dt+\sqrt{\tau}~{}dW,$ (22)

with Wiener increment $dW^{2}=dt$ (assuming the Itô picture) and a
characteristic measurement timescale $\tau=1/2\Gamma\eta$ related to the
ensemble measurement-dephasing rate $\Gamma$ through the measurement
efficiency $\eta\in[0,1]$ (see Section B.2 below). This approximation is the
foundation of stochastic Schrödinger equation or master equation methods for
describing continuous measurement. Averaging this readout for a duration $dt$
yields a Gaussian random variate,

$r_{i}=\int_{t_{i}}^{t_{i}+dt}r(t^{\prime})\frac{dt^{\prime}}{dt}=\text{Tr}(\rho_{i}\hat{N})+\sqrt{\frac{\tau}{dt}}~{}\xi_{i},$
(23)

with variance $\tau/dt$, written here in terms of its mean with additive zero-
mean Gaussian noise $\xi_{i}$ with unit variance, i.e.
$\xi_{i}\sim\mathcal{N}(0,1)$, which can be efficiently sampled numerically.
The observable expectation value with the state
$\rho_{i}=\rho_{\text{sys}}(t_{i})$ approximately determines the mean value of
the signal over the time bin of width $dt$. This relation also makes it clear
that the measurement timescale $\tau$ represents the integration time required
to achieve a unit signal-to-noise ratio.

In the laboratory, the finite bandwidth of the signal analyzer discretizes the
time axis into bins $dt$ in this way, making the observed readout signal more
accurately a discrete readout timeseries
$\vec{r}=\\{r_{1},...,r_{i},...,r_{n}\\}$ with approximately Gaussian random
variates. This random time series will determine the measurement backaction on
the system. In other words, for a simulation with time vector
$\vec{t}=\\{t_{1},t_{2},...,t_{n}\\}$, the state evolution will be completely
determined by the randomly sampled Gaussian noise
$\vec{\xi}=\\{\xi_{1},\xi_{2},...,\xi_{n}\\}$ combined with the initial
condition $\rho_{0}$ and full knowledge of the unitary, dissipative, and
measurement evolution.

To obtain the state trajectory
$\vec{\rho}=\\{\rho_{1},\rho_{2},...,\rho_{n}\\}$ from $\vec{\xi}$, we use the
discrete Bayesian filter [53] presented in the main text,

$\rho_{i+1}=\mathcal{L}_{i}\frac{\hat{U}_{i}\hat{M}_{r_{i}}\rho_{i}\hat{M}_{r_{i}}^{\dagger}\hat{U}_{i}^{\dagger}}{\text{Tr}(\hat{M}_{r_{i}}^{\dagger}\hat{M}_{r_{i}}\rho_{i})}.$
(24)

Here, $\mathcal{L}_{i}$ is a Kraus map representing total average dephasing
(due to, e.g., the imperfect measurement efficiency discussed in Section B.2),
$\hat{U}_{i}$ is a unitary representing the coherent evolution between times
$t_{i}$ and $t_{i}+dt$, and $\hat{M}_{r_{i}}$ is the measurement Kraus
operator that implements the measurement backaction and depends on the sampled
$r_{i}$ as described below. The set of $\hat{M}_{r_{i}}$ over all possible
$r_{i}$ construct a Gaussian POVM with completeness condition,

$\int_{-\infty}^{\infty}dr~{}|c(r)|^{2}\hat{M}_{r}^{\dagger}\hat{M}_{r}=\sqrt{\frac{dt}{2\pi\tau}}\int_{-\infty}^{\infty}dr~{}e^{-dt(r-\hat{N})^{2}/2\tau}=\hat{I},$
(25)

with an $r$-dependent normalization $|c(r)|^{2}$ factor that cancels in the
state update of Eq. (24).

###  B.1 Derivation of the measurement operator

To derive the measurement operator specific to our setup directly from the
correspondence principle, we start from the standard form of a measurement
Kraus operator for $\hat{N}=(\hat{\sigma}_{z}^{(1)}+\hat{\sigma}_{z}^{(2)})/2$
that associates the eigenspace projection operators for its three eigenvalues
of $-1,0,1$ with their corresponding conditional readout likelihoods [53, 47],

$\displaystyle c(r)\hat{M}_{r}=$
$\displaystyle\sqrt{P(r|00)}~{}\ket{00}\\!\bra{00}+\sqrt{P(r|11)}~{}\ket{11}\\!\bra{11}$
(26)
$\displaystyle+\sqrt{P(r|~{}01,10)}~{}\big{(}{\ket{01}\\!\bra{01}+\ket{10}\\!\bra{10}}\big{)},$

anticipating a state-independent factor $c(r)$ that will cancel. By
construction, the POVM condition $\int
dr~{}|c(r)|^{2}\,\hat{M}_{r}^{\dagger}\hat{M}_{r}=\hat{I}$ then follows from
the normalization of each likelihood distribution.

For a calibrated readout $r$ averaged over a time bin $dt$, the readout
likelihood distributions,

$\displaystyle P(r|00)$
$\displaystyle=\sqrt{\frac{dt}{2\pi\tau}}\exp\Big{(}{-\frac{dt}{2\tau}(r-1)^{2}}\Big{)}$
(27a) $\displaystyle P(r|11)$
$\displaystyle=\sqrt{\frac{dt}{2\pi\tau}}\exp\Big{(}{-\frac{dt}{2\tau}(r+1)^{2}}\Big{)}$
(27b) $\displaystyle P(r|01,10)$
$\displaystyle=\sqrt{\frac{dt}{2\pi\tau}}\exp\Big{(}{-\frac{dt}{2\tau}r^{2}}\Big{)},$
(27c)

are each Gaussian with variance $\tau/dt$ centered at an eigenvalue of
$\hat{N}$ for each eigenspace, assuming the quantum computing convention
$\hat{\sigma}_{z}=\ket{0}\\!\bra{0}-\ket{1}\\!\bra{1}$. Thus,

$\displaystyle\hat{M}_{r}$
$\displaystyle=e^{(dt/4\tau)(2r-1)}\ket{00}\\!\bra{00}$ (28a)
$\displaystyle~{}+e^{-(dt/4\tau)(2r+1)}\ket{11}\\!\bra{11}+\big{(}{\ket{01}\\!\bra{01}+\ket{10}\\!\bra{10}}\big{)},$
$\displaystyle c(r)$
$\displaystyle=\left(\frac{dt}{2\pi\tau}\right)^{1/4}~{}e^{-(dt/4\tau)r^{2}},$
(28b)

are the forms of the state-dependent Kraus operator $\hat{M}_{r}$ and the
state-independent normalization coefficient $c(r)$ that will cancel in the
state update. Written in terms of the Bell basis, this measurement operator
has the useful closed form,

$\displaystyle\hat{M}_{r}$
$\displaystyle=\ket{\psi_{+}}\\!\bra{\psi_{+}}+\ket{\psi_{-}}\\!\bra{\psi_{-}}$
(29)
$\displaystyle+e^{-dt/4\tau}\Bigg{[}\cosh\left(\frac{rdt}{2\tau}\right)\left(\ket{\phi_{+}}\\!\bra{\phi_{+}}+\ket{\phi_{-}}\\!\bra{\phi_{-}}\right)$
$\displaystyle+\sinh\left(\frac{rdt}{2\tau}\right)\left(\ket{\phi_{+}}\\!\bra{\phi_{-}}+\ket{\phi_{-}}\\!\bra{\phi_{+}}\right)\Bigg{]},$

that depends directly on the measured operator and its square,

$\displaystyle\hat{N}$
$\displaystyle=\frac{\hat{\sigma}_{z}^{(1)}+\hat{\sigma}_{z}^{(2)}}{2}=\ket{\phi_{+}}\\!\bra{\phi_{-}}+\ket{\phi_{-}}\\!\bra{\phi_{+}},$
(30a) $\displaystyle\hat{N}^{2}$
$\displaystyle=\ket{\phi_{+}}\\!\bra{\phi_{+}}+\ket{\phi_{-}}\\!\bra{\phi_{-}},$
(30b) $\displaystyle\hat{N}^{3}$ $\displaystyle=\hat{N}.$ (30c)

Note that this state-dependent part can also be written compactly as

$\displaystyle\hat{M}_{r}=\exp[(dt/2\tau)(r\hat{N}-\hat{N}^{2}/2)],$ (31)

in agreement with the expected form of the Gaussian POVM in Eq. (25). Unlike
for single-qubit observable monitoring (e.g., [47]), the $r$-independent
quadratic term with $\hat{N}^{2}$ is nontrivial for the state update since it
picks out the relevant projection for the subspace in which the measurement
acts.

###  B.2 Inefficient measurement

In a perfectly efficient measurement, the digitized measurement record is
informationally complete, so after conditioning the state on the information
in the collected record no information is lost and the state remains pure.
However, a real measurement apparatus is not perfectly efficient due to signal
loss during transit, as well as a degradation of the remaining information in
the signal from additional noise due to thermal interactions, electronics
noise, or the amplification chain.

The overall measurement efficiency can be characterized with a parameter
$\eta\in[0,1]$, such that $\eta=1$ corresponds to perfectly efficient
measurement with no information loss, $\eta=0$ corresponds to total
information loss, and $0<\eta<1$ is a partially efficient measurement. Any
loss of information increases the measurement time $\tau=1/2\Gamma\eta$ needed
to integrate the signal to achieve a unit signal-to-noise ratio. The
measurement operator in Eq. (31) thus already accounts for this loss through
the parameter $\tau$.

However, any lost information will be effectively averaged over in trajectory
validation tomography, which will produce additional ensemble-average
dephasing. In the limit of $\eta=0$ with no collected signal, the best
estimate one can claim is the ensemble-averaged evolution, which for Markovian
noise should be in accordance with a Lindblad master equation. Thus, we
include the additional ensemble-dephasing from information loss in each
trajectory via the Lindblad map in Eq. (12), using $\hat{J}=\hat{N}/\sqrt{2}$
as the measured operator with a residual ensemble measurement-dephasing rate
$(1-\eta)\Gamma$,

$\displaystyle\mathcal{L}[\rho]$
$\displaystyle=(1-\eta)(dt\,\Gamma/2)\big{(}{\hat{N}\rho\hat{N}}\big{)}+\hat{J}_{0}\rho\hat{J}_{0}^{\dagger}$
(32a) $\displaystyle\hat{J}_{0}$
$\displaystyle=\sqrt{\mathbb{1}-(1-\eta)(dt\,\Gamma/2)\hat{N}^{2}}.$ (32b)

##  Appendix C Derivation of feedback protocols

The feedback task is to apply a control drive that maximizes the fidelity to
the target state $\ket{\psi_{+}}$ while minimizing the fidelity with the
invisible state $\ket{\psi_{-}}$. Such a control drive will be generated by
the co- and counter-rotating drive Hamiltonians defined in the main text,

$\displaystyle\hat{H}_{\Omega}(t)$
$\displaystyle=\Omega\,\frac{\hat{Y}_{+}}{2},$
$\displaystyle\hat{H}_{\Delta}(t)$
$\displaystyle=\Delta\,\frac{\hat{Y}_{-}}{2},$ (33)

where $\hat{Y}_{\pm}\equiv\hat{\sigma}_{y}^{1}\pm\hat{\sigma}_{y}^{2}$ and the
frequencies $\Omega$ and $\Delta$ can generally depend upon the time $t$, the
delayed and normalized measurement record $r(t-\tau_{d})$, and/or the quantum
state estimate $\rho_{\text{est}}(t)$.

We use the standard (Uhlmann) definition for the state fidelity,

$\mathcal{F}(\sigma,\rho)\equiv\Big{(}{\text{Tr}\sqrt{\sqrt{\sigma}~{}\rho\sqrt{\sigma}}}\Big{)}^{2},$
(34)

which for a pure target state $\sigma=\ket{\psi}\\!\bra{\psi}$ and estimated
state $\rho=\rho_{\text{est}}$ simplifies to,

$\mathcal{F}_{\psi}\equiv\mathcal{F}(\ket{\psi}\\!\bra{\psi},~{}\rho_{\text{est}})=\braket{\psi}{\rho_{\text{est}}}{\psi},$
(35)

and can be interpreted as the population of $\rho_{\text{est}}$ in the target
state $\ket{\psi}$.

For constant $\Omega$ and $\Delta$, the unitary evolution generated by the
control Hamiltonian $\hat{H}_{\Omega}+\hat{H}_{\Delta}$ in isolation over a
time interval $dt$ is,

$\displaystyle\hat{U}_{\Omega,\Delta,dt}=~{}e^{-i(\Omega\,\hat{Y}_{+}+\Delta\,\hat{Y}_{-})dt/2}$
(36)
$\displaystyle~{}~{}~{}=e^{-i(\Omega+\Delta)\hat{\sigma}_{y}\,dt/2}\otimes
e^{-i(\Omega-\Delta)\hat{\sigma}_{y}\,dt/2}$
$\displaystyle~{}~{}~{}=~{}(|\psi_{+}\rangle\langle\psi_{+}|+|\phi_{-}\rangle\langle\phi_{-}|)\cos(\Omega
dt)$ $\displaystyle\hskip
34.1433pt+(|\psi_{+}\rangle\langle\phi_{-}|-|\phi_{-}\rangle\langle\psi_{+}|)\sin(\Omega
dt)$ $\displaystyle\hskip
34.1433pt+~{}(|\phi_{+}\rangle\langle\phi_{+}|+|\psi_{-}\rangle\langle\psi_{-}|)\cos(\Delta
dt)$ $\displaystyle\hskip
34.1433pt+(|\phi_{+}\rangle\langle\psi_{-}|-|\psi_{-}\rangle\langle\phi_{+}|)\sin(\Delta
dt),$

where we note the following useful Bell state identities,

$\displaystyle\frac{\hat{Y}_{+}}{2i}~{}~{}$
$\displaystyle=\ket{\psi_{+}}\\!\bra{\phi_{-}}-\ket{\phi_{-}}\\!\bra{\psi_{+}},$
(37a) $\displaystyle\left(\frac{\hat{Y}_{+}}{2}\right)^{2}$
$\displaystyle=|\psi_{+}\rangle\langle\psi_{+}|+|\phi_{-}\rangle\langle\phi_{-}|,$
(37b) $\displaystyle\left(\frac{\hat{Y}_{+}}{2}\right)^{3}$
$\displaystyle=\frac{\hat{Y}_{+}}{2},$ (37c)
$\displaystyle\frac{\hat{Y}_{-}}{2i}~{}~{}$
$\displaystyle=\ket{\phi_{+}}\\!\bra{\psi_{-}}-\ket{\psi_{-}}\\!\bra{\phi_{+}},$
(37d) $\displaystyle\left(\frac{\hat{Y}_{-}}{2}\right)^{2}$
$\displaystyle=|\phi_{+}\rangle\langle\phi_{+}|+|\psi_{-}\rangle\langle\psi_{-}|,$
(37e) $\displaystyle\left(\frac{\hat{Y}_{-}}{2}\right)^{3}$
$\displaystyle=\frac{\hat{Y}_{-}}{2},$ (37f)

which imply the partition of unity,

$\displaystyle\hat{I}$
$\displaystyle=\left(\frac{\hat{Y}_{+}}{2}\right)^{2}+\left(\frac{\hat{Y}_{-}}{2}\right)^{2}$
(38)
$\displaystyle=\ket{\phi_{+}}\\!\bra{\phi_{+}}+\ket{\phi_{-}}\\!\bra{\phi_{-}}+\ket{\psi_{+}}\\!\bra{\psi_{+}}+\ket{\psi_{-}}\\!\bra{\psi_{-}}.$

For completeness, we also note that these operators are part of two natural
sets of Pauli-like two-qubit operators. The other two in the
$\\{\ket{\psi_{+}},\ket{\phi_{-}}\\}$ subspace are,

$\displaystyle\frac{\hat{X}_{+}}{2}=\frac{\hat{\sigma}_{z}^{(1)}\hat{\sigma}_{x}^{(2)}+\hat{\sigma}_{x}^{(1)}\hat{\sigma}_{z}^{(2)}}{2}=\ket{\psi_{+}}\\!\bra{\phi_{-}}+\ket{\phi_{-}}\\!\bra{\psi_{+}},$
(39a)
$\displaystyle\frac{\hat{Z}_{+}}{2}=\frac{\hat{\sigma}_{z}^{(1)}\hat{\sigma}_{z}^{(2)}-\hat{\sigma}_{x}^{(1)}\hat{\sigma}_{x}^{(2)}}{2}=\ket{\phi_{-}}\\!\bra{\phi_{-}}-\ket{\psi_{+}}\\!\bra{\psi_{+}},$
(39b)
$\displaystyle\left[\frac{\hat{X}_{+}}{2},\,\frac{\hat{Y}_{+}}{2}\right]=2i\frac{\hat{Z}_{+}}{2}\quad\text{(and
cyclic perm.)},$ (39c)

while the other two in the orthogonal $\\{\ket{\psi_{-}},\ket{\phi_{+}}\\}$
subspace are,

$\displaystyle\frac{\hat{X}_{-}}{2}=\frac{\hat{\sigma}_{z}^{(1)}\hat{\sigma}_{x}^{(2)}-\hat{\sigma}_{x}^{(1)}\hat{\sigma}_{z}^{(2)}}{2}=\ket{\psi_{-}}\\!\bra{\phi_{+}}+\ket{\phi_{+}}\\!\bra{\psi_{-}},$
(40a)
$\displaystyle\frac{\hat{Z}_{-}}{2}=-\frac{\hat{\sigma}_{z}^{(1)}\hat{\sigma}_{z}^{(2)}+\hat{\sigma}_{x}^{(1)}\hat{\sigma}_{x}^{(2)}}{2}=\ket{\psi_{-}}\\!\bra{\psi_{-}}-\ket{\phi_{+}}\\!\bra{\phi_{+}},$
(40b)
$\displaystyle\left[\frac{\hat{X}_{-}}{2},\,\frac{\hat{Y}_{-}}{2}\right]=2i\frac{\hat{Z}_{-}}{2}\quad\text{(and
cyclic perm.)}.$ (40c)

We now briefly derive two strategies for setting the control frequencies
$\Omega$ and $\Delta$ so this control unitary maximizes the target fidelity.

###  C.1 Optimal feedback without measurement or dephasing

We can derive a feedback protocol if we assume a known initial estimated state
$\rho_{\text{est}}$ from which an optimal drive can be inferred. The updated
state estimate after applying a control drive is
$\rho^{\prime}_{\text{est}}=\hat{U}_{\Omega,\Delta,\delta
t}\,\rho_{\text{est}}\,\hat{U}^{\dagger}_{\Omega,\Delta,\delta t}$, which
should be compared to the states $\ket{\psi_{\pm}}$. For a pure target state
$\sigma=\ket{\psi}\\!\bra{\psi}$ and the post-control estimated state
$\rho=\rho^{\prime}_{\text{est}}$, the state fidelity has the form,

$\displaystyle\mathcal{F}_{\psi}$
$\displaystyle\equiv\mathcal{F}(\ket{\psi}\\!\bra{\psi},~{}\rho^{\prime}_{\text{est}})$
(41) $\displaystyle=\braket{\psi}{\rho^{\prime}_{\text{est}}}{\psi}$
$\displaystyle=\braket{\psi}{\hat{U}_{\Omega,\Delta,\delta
t}\,\rho_{\text{est}}\,\hat{U}^{\dagger}_{\Omega,\Delta,\delta t}}{\psi}.$

The state fidelities to $\ket{\psi_{\pm}}$ following a control drive
$\hat{U}_{\Omega,\Delta,\delta t}$ are thus,

$\displaystyle\mathcal{F}_{\psi+}(\Omega)$
$\displaystyle=\frac{1}{2}\Big{(}\rho_{\psi+}+\rho_{\phi-}+(\rho_{\psi+}-\rho_{\phi-})\cos(2\Omega
dt)$ $\displaystyle+2\text{Re}(\rho_{\psi+,\phi-})\sin(2\Omega~{}\delta
t)\Big{)},$ (42a) $\displaystyle\mathcal{F}_{\psi-}(\Delta)$
$\displaystyle=\frac{1}{2}\Big{(}\rho_{\psi-}+\rho_{\phi+}+(\rho_{\psi-}-\rho_{\phi+})\cos(2\Delta~{}\delta
t)$ $\displaystyle-2\text{Re}(\rho_{\phi+,\psi-})\sin(2\Delta~{}\delta
t)\Big{)},$ (42b)

where $\rho_{\psi\pm}=\braket{\psi_{\pm}}{\rho_{\text{est}}}{\psi_{\pm}}$,
$\rho_{\phi\pm}=\braket{\phi_{\pm}}{\rho_{\text{est}}}{\phi_{\pm}}$,
$\rho_{\psi+,\phi-}=\braket{\psi_{+}}{\rho_{\text{est}}}{\phi_{-}}$, and
$\rho_{\phi+,\psi-}=\braket{\phi_{+}}{\rho_{\text{est}}}{\psi_{-}}$ are the
relevant populations and coherences of the pre-control estimated state
$\rho_{\text{est}}$. Because $\ket{\psi_{+}}$ is invariant under
$\hat{H}_{\Delta}$, and $\ket{\psi_{-}}$ under $\hat{H}_{\Omega}$, their
respective fidelities are only affected by one of the two drive parameters,
$\Omega$ or $\Delta$.

Extremizing these fidelities using the conditions
$d\mathcal{F}_{\psi+}/d\Omega=0=d\mathcal{F}_{\psi-}/d\Delta$ then determines
the optimal state-estimate-dependent constant drives to apply over a single
control cycle $\delta t$,

$\displaystyle\tan\left[2\Omega_{\text{opt,0}}(\rho_{\text{est}})\delta
t\right]$ $\displaystyle=2\Omega_{\text{opt,0}}(\rho_{\text{est}})\delta
t+\mathcal{O}(\delta t^{3})$
$\displaystyle=\frac{2\text{Re}\rho_{\psi+,\phi-}}{\rho_{\psi+}-\rho_{\phi-}},$
(43a) $\displaystyle\tan\left[2\Delta_{\text{opt}}(\rho_{\text{est}})\delta
t\right]$ $\displaystyle=2\Delta_{\text{opt}}(\rho_{\text{est}})\delta
t+\mathcal{O}(\delta t^{3})$
$\displaystyle=\frac{2\text{Re}\rho_{\phi+,\psi-}}{\rho_{\phi+}-\rho_{\psi-}}.$
(43b)

Because the right-hand-side of both expressions are $\mathcal{O}(1)$ in
$\delta t$, which itself is an integer multiple of $dt$, the arctangent is not
expanded in calculations of $\Omega_{\rm opt,0},\Delta_{\rm opt}$ used in
simulations. Note that, unlike state-estimate-independent linear feedback
schemes (e.g., Ref. [47]), the drive $\Omega\,\hat{Y}_{+}$ inherently disrupts
the target state in our protocol, which forces the dependence on a state
estimate to achieve optimal feedback control. This optimal feedback drive
applies when the measurement backaction is ignored or negligible. In
particular, the basic rate $\Omega_{\rm opt,0}$ will acquire corrections when
accounting for the measurement properly, as detailed in the next section.

For completeness, we also note that these optima can be written in terms of
the observables in Eqs. (39) and (40),

$\displaystyle\Omega_{\text{opt,0}}(\rho_{\text{est}})\delta t$
$\displaystyle=\frac{1}{2}\tan^{-1}(C_{+}(\rho_{\rm est})),$ (44a)
$\displaystyle\Delta_{\text{opt}}(\rho_{\text{est}})\delta t$
$\displaystyle=\frac{1}{2}\tan^{-1}(C_{-}(\rho_{\rm est})),$ (44b)

where

$\displaystyle C_{+}(\rho)$
$\displaystyle=\frac{2\text{Re}\rho_{\psi+,\phi-}}{\rho_{\psi+}-\rho_{\phi-}}=-\frac{\langle\hat{X}_{+}\rangle}{\langle\hat{Z}_{+}\rangle},$
(45a) $\displaystyle C_{-}(\rho)$
$\displaystyle=\frac{2\text{Re}\rho_{\phi+,\psi-}}{\rho_{\phi+}-\rho_{\psi-}}=-\frac{\langle\hat{X}_{-}\rangle}{\langle\hat{Z}_{-}\rangle}.$
(45b)

###  C.2 Optimal feedback with measurement but no dephasing

When concurrently monitoring an observable
$\hat{N}=(\hat{\sigma}_{z}^{(1)}+\hat{\sigma}_{z}^{(2)})/2$, the feedback
control will generally depend upon the delayed and normalized measurement
record $r(t-\tau_{d})$. As a minimal feedback protocol, the co-rotating drive
should depend on the record and a state estimate $\rho_{\text{est}}$,

$\displaystyle\Omega(t)$
$\displaystyle=\Omega[r(t-\tau_{d}),\,\rho_{\text{est}}(t)],$ (46)

while the counter-rotating drive can be kept constant or independently
optimized to the $\Delta_{\rm opt}(\rho_{\text{e}st})$ in Eq. (44b). Combining
the control drive in Eq. (LABEL:eq:UGeneral) with the measurement operator
associated with $r(t-\tau_{d})$ yields a composite update operator for the
measurement interaction and evolution with delay time $\tau_{d}$ plus a
following control time step $\delta t$ of $\hat{U}_{\Omega,\Delta,\delta
t,r}\hat{U}_{\tau_{d}+\delta t}\hat{M}_{r}$.

The time delay $\tau_{d}$ complicates the analysis, since the state generally
evolves between when the measurement is performed and when the feedback
control is applied. Distinguishing the control cycle timestep $\delta t$ from
the evolution timestep $dt$ results in a similar complication, since even a
$\tau_{d}=0$ system will undergo evolution that goes uncorrected for an
interval $\delta t$. However, as an initial approximation, we can take the
zero-delay limit $\tau_{d}\to 0$ by ignoring the intermediate evolution
$\hat{U}_{\tau_{d}+\delta t}$ before the control, and take the controller
latency to be negligible by setting $\delta t=dt$. In this zero-delay case,
the fidelity to the target state $\ket{\psi_{+}}$ after a measurement-and-
control timestep $dt$ has the form,

$\displaystyle\mathcal{F}_{\psi+}$
$\displaystyle=\frac{\bra{\psi_{+}}\hat{U}_{\Omega,\Delta,dt,r}\hat{M}_{r}\,\rho_{\text{est}}\,\hat{M}_{r}^{\dagger}\hat{U}^{\dagger}_{\Omega,\Delta,dt,r}\ket{\psi_{+}}}{\text{Tr}[\hat{M}^{\dagger}_{r}\hat{M}_{r}\,\rho_{\text{est}}]}.$
(47)

Treating the parameter $\Omega$ as tunable and using shorthand notation
$\hat{M}=\hat{U}_{\Omega,\Delta,dt,r}\hat{M}_{r}$ for the composite operator,
the extremization condition for this fidelity becomes,

$\displaystyle\frac{\partial\mathcal{F}_{\psi+}}{\partial\Omega}$
$\displaystyle=0=2\text{Re}\bra{\phi_{-}}\hat{M}\rho_{\text{est}}\hat{M}^{\dagger}\ket{\psi_{+}}.$
(48)

Using the Bell state identities,

$\displaystyle\hat{M}^{\dagger}\ket{\psi_{+}}$ $\displaystyle=\cos(\Omega
dt)\ket{\psi_{+}}\sin(\Omega
dt)e^{-dt/4\tau}\Bigg{[}\cosh\left(\frac{rdt}{2\tau}\right)\ket{\phi_{-}}+\sinh\left(\frac{rdt}{2\tau}\right)\ket{\phi_{+}}\Bigg{]},$
(49a) $\displaystyle\hat{M}^{\dagger}\ket{\phi_{-}}$
$\displaystyle=-\sin(\Omega dt)\ket{\psi_{+}}+\cos(\Omega
dt)e^{-dt/4\tau}\Bigg{[}\cosh\left(\frac{rdt}{2\tau}\right)\ket{\phi_{-}}+\sinh\left(\frac{rdt}{2\tau}\right)\ket{\phi_{+}}\Bigg{]},$
(49b)

the fidelity condition evaluates to a combination of components of
$\rho_{\text{est}}$,

$\displaystyle
2\text{Re}\bra{\phi_{-}}\hat{M}\rho_{\text{est}}\hat{M}^{\dagger}\ket{\psi_{+}}$
$\displaystyle=-\sin(2\Omega dt)\,\rho_{\psi+}+2\cos(2\Omega
dt)e^{-dt/4\tau}\Bigg{[}\cosh\left(\frac{rdt}{2\tau}\right)\text{Re}\rho_{\psi+,\phi-}+\sinh\left(\frac{rdt}{2\tau}\right)\text{Re}\rho_{\phi+,\psi+}\Bigg{]}$
(50) $\displaystyle\qquad+\sin(2\Omega
dt)e^{-dt/2\tau}\Bigg{[}\cosh^{2}\left(\frac{rdt}{2\tau}\right)\rho_{\phi-}+\sinh^{2}\left(\frac{rdt}{2\tau}\right)\rho_{\phi+}+\sinh\left(\frac{rdt}{\tau}\right)\text{Re}\rho_{\phi+,\phi-}\Bigg{]}.$

Setting this to zero, we can then solve directly for the optimal drive,

$\displaystyle\tan(2\Omega_{\rm opt}dt)$
$\displaystyle=\frac{2e^{-dt/4\tau}\left[\cosh\left(\frac{rdt}{2\tau}\right)\text{Re}\rho_{\psi+,\phi-}+\sinh\left(\frac{rdt}{2\tau}\right)\text{Re}\rho_{\phi+,\psi+}\right]}{\rho_{\psi+}-e^{-dt/2\tau}\left[\cosh^{2}\left(\frac{rdt}{2\tau}\right)\rho_{\phi-}+\sinh^{2}\left(\frac{rdt}{2\tau}\right)\rho_{\phi+}+\sinh\left(\frac{rdt}{\tau}\right)\text{Re}\rho_{\phi+,\phi-}\right]},$
(51)
$\displaystyle=\frac{2\text{Re}\rho_{\psi+,\phi-}}{\rho_{\psi+}-\rho_{\phi-}}+\frac{rdt}{2\tau}\left[\frac{2\text{Re}\rho_{\psi+,\phi+}}{\rho_{\psi+}-\rho_{\phi-}}+\frac{4\text{Re}\rho_{\psi+,\phi-}\text{Re}\rho_{\phi+,\phi-}}{(\rho_{\psi+}-\rho_{\phi-})^{2}}\right]$
$\displaystyle\quad-\frac{dt}{4\tau}\left[\frac{\text{Re}\rho_{\psi+,\phi-}}{\rho_{\psi+}-\rho_{\phi-}}-\frac{4\text{Re}\rho_{\psi+,\phi+}\text{Re}\rho_{\phi+,\phi-}+2\text{Re}\rho_{\psi+,\phi-}(\rho_{\phi+}-\rho_{\phi-})}{(\rho_{\psi+}-\rho_{\phi_{-}})^{2}}-\frac{2\text{Re}\rho_{\psi+,\phi-}(2\text{Re}\rho_{\phi+,\phi-})^{2}}{(\rho_{\psi+}-\rho_{\phi-})^{3}}\right]$
$\displaystyle\quad+\mathcal{O}(dt^{3/2}),$

where the second equality uses $(rdt)^{2}=\tau dt$.

Identifying the first term as the rescaled coherence $C_{+}(\rho)$ determining
the measurement-free rate $\Omega_{\rm opt,0}$ in Eq. 44a and recalling that
$\tau=1/2\Gamma\eta$, this solution can be rewritten in the form,

$\displaystyle\Omega_{\rm opt}dt$
$\displaystyle=\frac{1}{2}\tan^{-1}\Big{(}C_{+}(\rho_{\rm est})+2\eta\Gamma
P(\rho_{\text{est}})\,rdt+\eta\Gamma
Q(\rho_{\text{est}})\,\frac{dt}{2}\Big{)},$ (52)

with the characteristic state-estimate-dependent factors,

$\displaystyle P(\rho_{\text{est}})$
$\displaystyle=\frac{\text{Re}\rho_{\psi+,\phi+}}{\rho_{\psi+}-\rho_{\phi-}}+C_{+}(\rho_{\rm
est})\frac{\text{Re}\rho_{\phi+,\phi-}}{\rho_{\psi+}-\rho_{\phi-}},$ (53a)
$\displaystyle Q(\rho_{\text{est}})$ $\displaystyle=-\frac{C_{+}(\rho_{\rm
est})}{2}+\frac{4\text{Re}\rho_{\psi+,\phi+}\text{Re}\rho_{\phi+,\phi-}+2\text{Re}\rho_{\psi+,\phi-}(\rho_{\phi+}-\rho_{\phi-})}{(\rho_{\psi+}-\rho_{\phi_{-}})^{2}}+\frac{2\text{Re}\rho_{\psi+,\phi-}(2\text{Re}\rho_{\phi+,\phi-})^{2}}{(\rho_{\psi+}-\rho_{\phi-})^{3}}.$
(53b)

The dominant terms are the measurement-free rate $C_{+}(\rho)$, of order 1,
followed by the white noise term of order $dt^{1/2}$ in
$rdt=2\text{Re}\rho_{\phi+,\phi-}dt+dW/\sqrt{2\Gamma\eta}$. Expanding the
arctangent about the $\mathcal{O}(1)$ term $C_{+}(\rho_{\rm est})$ and keeping
terms of order $dt^{3/2}$ gives

$\displaystyle\Omega_{\rm opt}dt$
$\displaystyle\approx\frac{1}{2}\tan^{-1}(C_{+}(\rho_{\rm
est}))+\frac{\eta\Gamma P(\rho_{\text{est}})\,rdt}{1+C_{+}(\rho_{\rm
est})^{2}}$ (54a) $\displaystyle=\Omega_{\rm opt,0}(\rho_{\rm
est})+\eta\Gamma\Big{(}{\frac{P(\rho_{\text{est}})\,}{1+C_{+}(\rho_{\rm
est})^{2}}}\Big{)}rdt.$ (54b)

Thus, incorporating the most recent measurement information $rdt$ leads to a
linear correction, multiplied by a state-dependent factor. In the case where
the coherence $\rho_{\psi+,\phi-}$ is being consistently and successfully
rotated into the $\rho_{\psi_{+}}$ population, $C_{+}(\rho_{\rm est})\approx
0$ and the feedback is linear up to the state-estimation for $P(\rho_{\rm
est})$:

$\Omega_{\rm opt}dt\approx\eta\Gamma P(\rho_{\text{est}})\,rdt.$ (55)

Note that $P(\rho_{\rm est})$ can also be written in terms of the operators in
Eqs. (39), (40), and (30a) as,

$P(\rho_{\text{est}})=-\frac{\langle\hat{X}_{+}\hat{N}+\hat{N}\hat{X}_{+}\rangle}{4\langle\hat{Z}_{+}\rangle}+\frac{2\langle\hat{X}_{+}\rangle\langle\hat{N}\rangle}{\langle\hat{Z}_{+}\rangle^{2}}.$
(56)

This means that the two dominant terms depend mainly on the pre-measurement
coherence $\braket{\hat{X}_{+}}$ and the measurement-disturbed coherence
$\langle\hat{X}_{+}\hat{N}+\hat{N}\hat{X}_{+}\rangle/2$, both weighted by
population bias $\langle\hat{Z}_{+}\rangle$ in the subspace. This further
motivates thinking of Eq. 55 as the optimal drive when the feedback is on
track: given an initial condition $\rho_{0}$ with $\braket{\hat{X}_{+}}=0$,
the drive following a measurement will just be
$P(\rho_{0})=-\big{(}{\langle\hat{X}_{+}\hat{N}+\hat{N}\hat{X}_{+}\rangle}\big{)}/{4\langle\hat{Z}_{+}\rangle}$.
Since this drive is optimal, it should rotate all measurement-disturbed
coherence into $\ket{\psi_{+}}$, thus restoring $\braket{\hat{X}_{+}}=0$ until
the next measurement. However, the introduction of nonidealities (time-delay
and fluctuator / white noise) may cause the trajectory to deviate from this
optimal feedback, making the $C_{+}(\rho_{\rm est})$ terms critical to the
feedback success. This is explored further in Sec. C.5.

###  C.3 Optimal feedback with measurement and dephasing

The preceding section neglected the average dephasing occuring during the
timestep $dt$ prior to the control being applied. However, this dephasing can
be readily included in principle. Since the specifics of the dephasing are
unknown to the feedback controller, the fidelity optimization condition of Eq.
(48) should be modified to,

$\displaystyle\frac{\partial\mathcal{F}_{\psi+}}{\partial\Omega}$
$\displaystyle=0=2\text{Re}\bra{\phi_{-}}\mathcal{L}_{dt}(\hat{M}\rho_{\text{est}}\hat{M}^{\dagger})\ket{\psi_{+}},$
(57)

with the average dephasing modeled by a Lindblad state update,

$\displaystyle\mathcal{L}_{dt}(\rho)$
$\displaystyle=\frac{dt\Gamma_{2}}{2}\left(\hat{\sigma}_{z}^{(1)}\rho\hat{\sigma}_{z}^{(1)}+\hat{\sigma}_{z}^{(2)}\rho\hat{\sigma}_{z}^{(2)}\right)$
$\displaystyle\quad+\frac{(1-\eta)dt\Gamma}{2}\hat{N}\rho\hat{N}+\hat{J}_{0}\rho\hat{J}_{0}^{\dagger},$
(58a) $\displaystyle\hat{J}_{0}^{2}$
$\displaystyle=\mathbb{1}-\frac{dt\Gamma_{2}}{2}(2)-\frac{(1-\eta)dt\Gamma}{2}\hat{N}^{2},$
(58b)

that includes both the effective single-qubit dephasing and the residual
measurement dephasing due to imperfect efficiency $\eta<1$. However, including
this dephasing over the single timestep $dt$ will only add corrections of
order $dt$ to the deterministic feedback function $Q(\rho_{\text{est}})$ in
Eq. (52), which is already subdominant, so the dephasing corrections can also
be safely neglected.

###  C.4 Linear feedback stochastic Schrödinger equation

Assuming ideally efficient linear feedback in the form of Eq. (52), we can
also derive an effective stochastic Schrödinger equation that describes the
feedback dynamics. For simplicity, consider feedback drives of the minimal
form,

$\displaystyle\Omega(t)$ $\displaystyle=\Gamma P(\rho_{\text{est}})r(t),$
$\displaystyle\Delta$ $\displaystyle=\Gamma c,$ (59)

with a common scaling constant $\Gamma$ set to match the ensemble-average
measurement-dephasing rate. For the co-rotating drive, the tunable state-
estimate-dependent function $P(\rho_{\text{est}})$ scales the measurement
record, again assuming zero time delay for convenience, while the strength of
the counter-rotating drive can be relatively tuned with the dimensionless
constant $c$.

Following the form of the update in Eq. (47) the renormalized state after both
measurement and the unitary feedback drive is,

$\ket{\psi(t+dt)}=\frac{\hat{U}_{\Omega,\Delta,dt,r}\hat{M}_{r}}{\sqrt{\braket{\psi}{\hat{M}_{r}^{\dagger}\hat{M}_{r}}{\psi}}}\ket{\psi}.$
(60)

In this zero-delay case, using the white noise approximation for the readout
in Eq. (22), the form of the measurement operator in Eq. (31), and the form of
the control unitary in Eq. (LABEL:eq:UGeneral) yields the following composite
update map to order $dt$, recalling that
$(r\,dt)^{2}=\tau\,dt+\mathcal{O}(dt^{3/2})$,

$\displaystyle\frac{\hat{U}_{\Omega,\Delta,dt,r}\hat{M}_{r}}{\sqrt{\langle\hat{M}_{r}^{\dagger}\hat{M}_{r}\rangle}}\approx$
(61)
$\displaystyle\quad\approx\hat{I}+(r\,dt)\left[\frac{\hat{N}-\langle\hat{N}\rangle}{2\tau}+\Gamma
P\,\frac{\hat{Y}_{+}}{2i}\right]$ $\displaystyle\qquad+dt\left[\Gamma
c\,\frac{\hat{Y}_{-}}{2i}+\frac{\Gamma
P}{2}\frac{\hat{Y}_{+}}{2i}(\hat{N}-\langle\hat{N}\rangle)\right]$
$\displaystyle\qquad-\frac{dt}{2}\left[\frac{\hat{N}^{2}+2\hat{N}\langle\hat{N}\rangle-3\langle\hat{N}\rangle^{2}}{4\tau}-\tau(\Gamma
P)^{2}\left(\frac{\hat{Y}_{+}}{2i}\right)^{2}\right].$

After substituting the white noise relation $r\,dt=\langle\hat{N}\rangle
dt+\sqrt{\tau}\,dW$ and writing the measurement time $\tau=1/2\Gamma$ in terms
of the dephasing rate $\Gamma$ assuming perfect efficiency $\eta=1$, this
yields a stochastic Schrödinger equation that includes (instantaneous and
perfectly efficient) feedback,

$\displaystyle d\ket{\psi}$
$\displaystyle=\sqrt{\frac{\Gamma}{2}}\,dW\left[(\hat{N}-\langle\hat{N}\rangle)-i\frac{P}{2}\,\hat{Y}_{+}\right]\ket{\psi}$
(62)
$\displaystyle\quad-i\frac{\Gamma}{2}\,dt\left[c\,\hat{Y}_{-}+\frac{P}{2}\hat{Y}_{+}(\hat{N}+\langle\hat{N}\rangle)\right]\ket{\psi}$
$\displaystyle\quad-\frac{\Gamma}{2}\,dt\left[\frac{\left(\hat{N}-\langle\hat{N}\rangle\right)^{2}}{2}+\frac{P^{2}}{8}\hat{Y}_{+}^{2}\right]\ket{\psi}.$

###  C.5 Correspondence of deterministic and record-dependent feedback

The optimal protocols derived in Secs. C.1 and C.2 are equivalent under a
special choice of state estimate. In particular, Eq. 47 can be rearranged as

$\displaystyle\mathcal{F}_{\psi+}$
$\displaystyle=\bra{\psi_{+}}\hat{U}_{\Omega,\Delta,dt,r}\Bigg{(}{\frac{\hat{M}_{r}\,\rho_{\text{est}}\,\hat{M}_{r}^{\dagger}}{\text{Tr}[\hat{M}^{\dagger}_{r}\hat{M}_{r}\,\rho_{\text{est}}]}}\Bigg{)}\hat{U}^{\dagger}_{\Omega,\Delta,dt,r}\ket{\psi_{+}}$
(63a) $\displaystyle=\bra{\psi_{+}}\hat{U}_{\Omega,\Delta,dt,r}\rho_{\rm
est}^{\prime}\hat{U}^{\dagger}_{\Omega,\Delta,dt,r}\ket{\psi_{+}},$ (63b)

which is equivalent to using the Bayesian-updated state $\rho_{\rm
est}^{\prime}$ for the deterministic feedback in Eq. 41. Thus, the optimal
feedback that includes the linear feedback term (Eq. 52) is implicitly
realized by applying deterministic feedback (Eq. 44a) on the Bayesian-updated
state.

This has important practical consequences. The fact that the measurement-
dependent feedback requires state-tracking mitigates its potential advantage
over the deterministic feedback. In a state-independent linear feedback
protocol such as [38], feedback can be implemented in hardware, eliminating
computational overhead. However, since state-estimation is required with both
record-dependent and record-independent expressions we derived, it is optimal
to compute the computationally simpler deterministic expression Eq. 41 after
performing the Bayesian update. In this case, the record-dependence enters the
feedback implicitly through the Bayesian update.

Figure 9: Fidelity $\mathcal{F}(t)$ to target $\ket{\psi_{+}}$, comparing
deterministic feedback $\Omega_{\rm opt,0}(\rho)$ (left) to record-
proportional feedback $\Omega_{\rm opt,1}(\rho)=\eta\Gamma\,P(\rho)\,r$
(right), with $1/f$-noise $\Gamma_{2}=1/52\mu$s, $\Delta_{\rm d.d.}/2\pi=10$
MHz, $\eta=1/2$, $\tau_{d}=500$ ns, and $\Gamma=1/\mu$s. $N=100$ trajectories
(a, b) Overlaid ensemble of $100$ trajectories (translucent purple)
highlighting one trajectory (dark purple) and the ensemble-average (black).
Deterministic feedback in (a) stabilizes the ensemble-mean over time, while
proportional feedback in (b) with a delayed record $r(t-\tau_{d})$ does not
reliably stabilize all trajectories. (c, d) Comparison of the deterministic
drive $\Omega_{\rm opt,0}[\rho(t)]/\eta\Gamma r(t-\tau_{d})$ scaled by the
delayed record (black) and the envelope $P[\rho(t-\tau_{d})]$ used in
proportional feedback with no forward-estimation (aqua). For deterministic
feedback in (c), the drive $\Omega_{\text{opt,0}}$ is computed and used to get
the highlighted trajectory in (a) first, then $P$ is computed for that
observed trajectory and shows close agreement to the deterministic envelope
(inset). For proportional feedback in (d), the $P$ is computed and used to get
the highlighted and stabilized trajectory in (b) first, then
$\Omega_{\text{opt,0}}$ is computed but shows poor agreement.

For completeness, we compare simulations of two approaches in Fig. 9. The left
column depicts deterministic feedback, which uses only the
$\Omega_{\text{opt,0}}$ term from Eq. 44a. The right column depicts record-
proportional feedback, which uses only the first term in Eq. 53a, i.e.

$\displaystyle P(\rho_{\text{est}})$
$\displaystyle=\frac{\text{Re}\rho_{\psi+,\phi+}}{\rho_{\psi+}-\rho_{\phi-}}+\frac{2\text{Re}\rho_{\psi+,\phi-}\text{Re}\rho_{\phi+,\phi-}}{(\rho_{\psi+}-\rho_{\phi-})^{2}}$
(64)
$\displaystyle\approx\frac{\text{Re}\rho_{\psi+,\phi+}}{\rho_{\psi+}-\rho_{\phi-}}.$

This is chosen on the basis that the coherence
$\text{Re}\rho_{\psi_{+},\phi-}$ and the measured $\langle\hat{N}\rangle$ are
proportional to $\sqrt{\rho_{\phi-}}$ so become small near $\ket{\psi_{+}}$.
Including $r$, the noise will dominate over deterministic terms in Eq. 52, so
$\Omega_{\rm opt}dt\approx\eta\Gamma\,P[\rho]\,rdt$ so long as the feedback
remains close to the optimal trajectory.

The deterministic feedback in 9a stabilizes more reliably than the
proportional feedback in 9b. This makes sense given that the deterministic
feedback includes the zero-order term while also including record information
implicitly, whereas the proportional feedback excludes the zero-order term.
Interestingly, for the highlighted trajectory stabilized with deterministic
feedback, the forward-estimated drive $\Omega_{\rm
opt,0}[\rho_{\text{est}}(t)]/\eta\Gamma r(t-\tau_{d})$ scaled by the delayed
record in 9c (black) is in close agreement with the
$P[\rho_{\text{est}}(t-\tau_{d})]$ that would have been calculated for that
trajectory _without forward-estimation_ (aqua). This supports the intuition
that the deterministic feedback implicitly includes the record information,
and also confirms that the proportional term is the dominant effect most of
the time. However, for a trajectory stabilized with proportional feedback
only, the used $P$ in 9d (aqua) shows poor agreement with the drive
$\Omega_{\rm opt,0}/\eta\Gamma r$ (black) that would have been calculated.
This suggests that the infrequent, large discrepancies between
$\Omega_{\text{opt,0}}$ and $P$ in Fig. 9c represent nonlinear corrections
that are critical to the feedback success. Without these large corrections in
the proportional feedback case, the deterministic protocol always predicts a
large correction needed to return to the more stable regime where proportional
feedback becomes reliable.

##  Appendix D Improving feedback control with forward state-estimation

The feedback control derived in the previous section ignored the time delay
$\tau_{d}$ between when the measurement record is collected and the control is
applied. However, in practice this time lag is necessary for information to
travel between the system and measurement apparatus, so concurrent system
evolution will take place during this delay that should not be neglected in
the state estimate $\rho_{\rm est}$ being used to predict the feedback control
that should be applied. Thus, the applied feedback control should anticipate
what the _future state_ will be, accounting for the evolution between when the
measurement interaction occurred in the past to produce the collected record
and when the feedback signal reaches the system.

While unitary control drives are known and may be readily accounted in such a
_forward estimation_ of the future state, other types of evolution like
measurement and fluctuations are not known during the delay. As such, the best
estimate one can make for the evolution during the delay is the _mean
evolution_ that averages over the unknowns and uses the information about what
drives have already been output by the controller but have not yet arrived at
the system.

The feedback loop time delay $\tau_{d}$ between when the measurement microwave
field interacts with the system and when the corresponding feedback control
field reaches the system can be broken into three conceptual parts,

$\displaystyle\tau_{d}$ $\displaystyle=\tau_{s\to c}+\tau_{c}+\tau_{c\to s},$
(65)

where $\tau_{s\to c}$ is the time of flight for the readout to make it through
the measurement chain to the controller, $\tau_{c}$ is the time it takes for
the controller to compute the feedback to apply, and $\tau_{c\to s}$ is the
time of flight for the control signal to reach the system. Thus, the
controller should understand the readout received at time $t$ as
$r(t-\tau_{s\to c})$, meaning it reflects information in the system state at a
time $\tau_{s\to c}$ in the past, but it should issue a control signal that is
appropriate for the future system state $\rho(t-\tau_{s\to
c}+\tau_{d})=\rho(t+\tau_{c}+\tau_{c\to s})$. However, this distinction is
unimportant for the controller, which can operate with a shifted internal
clock time $t\mapsto t+\tau_{s\to c}$ so that its internal $t$ is matched to
the time of the measurement interaction with the past system state, in which
case it can assume that it receives a signal $r(t)$ and must output a feedback
signal for the future state $\rho(t+\tau_{d})$ a full delay time $\tau_{d}$ in
the future.

Other than this full delay time $\tau_{d}$, the only important timescale for
the feedback is the processing time of the controller $\tau_{c}$. This time
sets the minimum duration required to change the output feedback control
signal. Whereas the readout of the system is modeled as a discrete process
with timestep $dt$ according to Eq. 23, the readout processed by the
controller is buffered and averaged in larger bins of duration $\tau_{c}$ to
produce the values $\tilde{r}(t)$. In this way, no signal is lost during the
processing time, but the controller is nevertheless restricted to outputting a
signal that is time-stepped into bins no finer than its processing time
$\tau_{c}$. For the purposes of this work, we assume that this processing time
is what sets the overall control timestep $\delta t=\tau_{c}$ in the analysis.

With the shifted clock, the feedback controller should track an internal state
estimate $\rho_{\text{est}}(t)$ corresponding to the actual system state
$\rho_{\text{sys}}(t)$ that produced the received readout $\tilde{r}(t)$. It
should then assume that the feedback signal will be applied to the appropriate
_forward estimate_ of the system state,

$\displaystyle\rho_{\text{est}}(t+\tau_{d})$
$\displaystyle=\prod_{k=0}^{\lfloor\tau_{d}/\delta
t\rfloor}\mathcal{E}_{\delta t}(t+k\delta t)[\rho_{\text{est}}(t)],$ (66a)
$\displaystyle\mathcal{E}_{\delta t}(t_{k})$
$\displaystyle=\mathcal{L}_{\delta t}\circ\mathcal{U}_{\delta t}(t_{k}),$
(66b)

that has been evolved forward by $\lfloor\tau_{d}/\delta t\rfloor$ timesteps
$\delta t$, including both known unitary dynamics $\mathcal{U}_{\delta
t}(t_{k})[\cdot]=\hat{U}_{\delta t}(t_{k})(\cdot)\hat{U}^{\dagger}_{\delta
t}(t_{k})$ for the drives that were already sent by the controller during the
preceding $\tau_{d}$ (which have not yet arrived at the system) and the
average deocherence $\mathcal{L}_{\delta t}$ that is anticipated during each
timestep $\delta t$ over the delay time $\tau_{d}$.

In practice, the average decoherence is relatively small if the delay
$\tau_{d}$ is sufficiently short, so can be neglected as an initial
approximation. In this case, the forward estimation is entirely determined by
the sequence of control unitaries for the drives that the controller has
already sent. Since the unitary generators $\hat{Y}_{\pm}$ commute, the
controller only needs to save a running buffer sum of the emitted drives,
$(\Omega(t),\Omega(t-\delta t),\ldots,\Omega(t-\tau_{d}))$ and
$(\Delta(t),\Delta(t-\delta t),\ldots,\Delta(t-\tau_{d}))$, from which it can
keep running sums $\Omega_{\tau_{d}}$ and $\Delta_{\tau_{d}}$ over each buffer
by subtracting the oldest and adding the newest when the buffer is updated at
each time step. The forward-estimation unitary operator used for simulations
in the main text is then simply,

$\displaystyle\hat{U}_{\tau_{d}}$
$\displaystyle=\exp\left(-i(\Omega_{\tau_{d}}\hat{Y}_{+}+\Delta_{\tau_{d}}\hat{Y}_{-})\delta
t/2\right),$ (67)

with the corresponding closed form in Eq. (LABEL:eq:UGeneral). This sort of
buffering and calculation is straightforward and fast on an FPGA controller.

## References

  * Lidar and Brun [2013] Daniel A Lidar and Todd A Brun.  _Quantum error correction_.  Cambridge university press, 2013.
  * Shankar et al. [2013] Shyam Shankar, Michael Hatridge, Zaki Leghtas, KM Sliwa, Aniruth Narla, Uri Vool, Steven M Girvin, Luigi Frunzio, Mazyar Mirrahimi, and Michel H Devoret.  Autonomously stabilized entanglement between two superconducting quantum bits.  _Nature_ , 504(7480):419–422, 2013.
  * Kimchi-Schwartz et al. [2016] ME Kimchi-Schwartz, L Martin, E Flurin, C Aron, M Kulkarni, HE Tureci, and I Siddiqi.  Stabilizing entanglement via symmetry-selective bath engineering in superconducting qubits.  _Physical review letters_ , 116(24):240503, 2016.
  * Tacchino et al. [2018] F. Tacchino, A. Auffèves, M. F. Santos, and D. Gerace.  Steady State Entanglement beyond Thermal Limits.  _Physical Review Letters_ , 120(6):063604, February 2018.  doi: 10.1103/PhysRevLett.120.063604.
  * Ma et al. [2019] Sheng-li Ma, Xin-ke Li, Xin-yu Liu, Ji-kun Xie, and Fu-li Li.  Stabilizing Bell states of two separated superconducting qubits via quantum reservoir engineering.  _Physical Review A_ , 99(4):042336, April 2019.  doi: 10.1103/PhysRevA.99.042336.
  * Ma et al. [2021] Sheng-li Ma, Jing Zhang, Xin-ke Li, Ya-long Ren, Ji-kun Xie, Ming-tao Cao, and Fu-li Li.  Coupling-modulation–mediated generation of stable entanglement of superconducting qubits via dissipation.  _Europhysics Letters_ , 135(6):63001, November 2021.  ISSN 0295-5075.  doi: 10.1209/0295-5075/ac2b5c.
  * Santos and Bachelard [2023] Alan C. Santos and R. Bachelard.  Generation of Maximally Entangled Long-Lived States with Giant Atoms in a Waveguide.  _Physical Review Letters_ , 130(5):053601, February 2023.  doi: 10.1103/PhysRevLett.130.053601.
  * Ruskov and Korotkov [2003] Rusko Ruskov and Alexander N. Korotkov.  Entanglement of solid-state qubits by measurement.  _Phys. Rev. B_ , 67(24):241305, June 2003.  ISSN 0163-1829, 1095-3795.  doi: 10.1103/PhysRevB.67.241305.  URL http://arxiv.org/abs/cond-mat/0206396.  arXiv:cond-mat/0206396.
  * Williams and Jordan [2008] Nathan S. Williams and Andrew N. Jordan.  Entanglement genesis under continuous parity measurement.  _Phys. Rev. A_ , 78(6):062322, December 2008.  ISSN 1050-2947, 1094-1622.  doi: 10.1103/PhysRevA.78.062322.  URL http://arxiv.org/abs/0809.3248.  arXiv:0809.3248 [cond-mat, physics:quant-ph].
  * Riste et al. [2013] D. Riste, M. Dukalski, C. A. Watson, G. de Lange, M. J. Tiggelman, Ya M. Blanter, K. W. Lehnert, R. N. Schouten, and L. DiCarlo.  Deterministic entanglement of superconducting qubits by parity measurement and feedback.  _Nature_ , 502(7471):350–354, October 2013.  ISSN 0028-0836, 1476-4687.  doi: 10.1038/nature12513.  URL http://arxiv.org/abs/1306.4002.  arXiv:1306.4002 [cond-mat, physics:quant-ph].
  * Kerckhoff et al. [2010] Joseph Kerckhoff, Hendra I. Nurdin, Dmitri S. Pavlichin, and Hideo Mabuchi.  Designing Quantum Memories with Embedded Control: Photonic Circuits for Autonomous Quantum Error Correction.  _Phys. Rev. Lett._ , 105(4):040502, July 2010.  ISSN 0031-9007, 1079-7114.  doi: 10.1103/PhysRevLett.105.040502.  URL https://link.aps.org/doi/10.1103/PhysRevLett.105.040502.
  * Mohseninia et al. [2020] Razieh Mohseninia, Jing Yang, Irfan Siddiqi, Andrew N. Jordan, and Justin Dressel.  Always-On Quantum Error Tracking with Continuous Parity Measurements.  _Quantum_ , 4:358, November 2020.  ISSN 2521-327X.  doi: 10.22331/q-2020-11-04-358.  URL http://arxiv.org/abs/1907.08882.  arXiv:1907.08882 [quant-ph].
  * Livingston et al. [2022a] William P. Livingston, Machiel S. Blok, Emmanuel Flurin, Justin Dressel, Andrew N. Jordan, and Irfan Siddiqi.  Experimental demonstration of continuous quantum error correction.  _Nat Commun_ , 13(1):2307, April 2022a.  ISSN 2041-1723.  doi: 10.1038/s41467-022-29906-0.  URL https://www.nature.com/articles/s41467-022-29906-0.
  * Hacohen-Gourgy et al. [2018] S. Hacohen-Gourgy, L. P. García-Pintos, L. S. Martin, J. Dressel, and I. Siddiqi.  Incoherent Qubit Control Using the Quantum Zeno Effect.  _Physical Review Letters_ , 120(2):020505, January 2018.  doi: 10.1103/PhysRevLett.120.020505.
  * Blumenthal et al. [2022] E. Blumenthal, C. Mor, A. A. Diringer, L. S. Martin, P. Lewalle, D. Burgarth, K. B. Whaley, and S. Hacohen-Gourgy.  Demonstration of universal control between non-interacting qubits using the Quantum Zeno effect.  _npj Quantum Information_ , 8(1):1–6, July 2022.  ISSN 2056-6387.  doi: 10.1038/s41534-022-00594-4.
  * Roch et al. [2014] Nicolas Roch, Mollie E. Schwartz, Felix Motzoi, Christopher Macklin, Rajamani Vijay, Andrew W. Eddins, Alexander N. Korotkov, K. Birgitta Whaley, Mohan Sarovar, and Irfan Siddiqi.  Observation of measurement-induced entanglement and quantum trajectories of remote superconducting qubits.  _Phys. Rev. Lett._ , 112(17):170501, April 2014.  ISSN 0031-9007, 1079-7114.  doi: 10.1103/PhysRevLett.112.170501.  URL http://arxiv.org/abs/1402.1868.  arXiv:1402.1868 [cond-mat, physics:quant-ph].
  * Motzoi et al. [2015] Felix Motzoi, K Birgitta Whaley, and Mohan Sarovar.  Continuous joint measurement and entanglement of qubits in remote cavities.  _Physical Review A_ , 92(3):032308, 2015.
  * Dickel et al. [2018] Christian Dickel, JJ Wesdorp, NK Langford, S Peiter, Ramiro Sagastizabal, Alessandro Bruno, Ben Criger, F Motzoi, and L DiCarlo.  Chip-to-chip entanglement of transmon qubits using engineered measurement fields.  _Physical Review B_ , 97(6):064508, 2018.
  * Kerckhoff et al. [2009] Joseph Kerckhoff, Luc Bouten, Andrew Silberfarb, and Hideo Mabuchi.  Physical model of continuous two-qubit parity measurement in a cavity-qed network.  _Physical Review A_ , 79(2):024305, 2009.
  * Chantasri et al. [2016] Areeya Chantasri, Mollie E. Kimchi-Schwartz, Nicolas Roch, Irfan Siddiqi, and Andrew N. Jordan.  Quantum Trajectories and Their Statistics for Remotely Entangled Quantum Bits.  _Phys. Rev. X_ , 6(4):041052, December 2016.  ISSN 2160-3308.  doi: 10.1103/PhysRevX.6.041052.  URL https://link.aps.org/doi/10.1103/PhysRevX.6.041052.
  * Hacohen-Gourgy and Martin [2020] S. Hacohen-Gourgy and L. S. Martin.  Continuous measurements for control of superconducting quantum circuits.  _Advances in Physics: X_ , 5(1):1813626, January 2020.  ISSN null.  doi: 10.1080/23746149.2020.1813626.
  * Martin and Whaley [2019] Leigh S. Martin and K. Birgitta Whaley.  Single-shot deterministic entanglement between non-interacting systems with linear optics, November 2019.  URL http://arxiv.org/abs/1912.00067.  arXiv:1912.00067 [quant-ph].
  * Lewalle et al. [2020] Philippe Lewalle, Cyril Elouard, and Andrew N. Jordan.  Entanglement-preserving limit cycles from sequential quantum measurements and feedback.  _Physical Review A_ , 102(6):062219, December 2020.  doi: 10.1103/PhysRevA.102.062219.
  * Wiseman and Milburn [2010] HM Wiseman and GJ Milburn.  Quantum measurement and control cambridge university press: Cambridge.  _NY, USA_ , 2010.
  * Schirmer and Wang [2010] S. G. Schirmer and Xiaoting Wang.  Stabilizing Open Quantum Systems by Markovian Reservoir Engineering.  _Phys. Rev. A_ , 81(6):062306, June 2010.  ISSN 1050-2947, 1094-1622.  doi: 10.1103/PhysRevA.81.062306.  URL http://arxiv.org/abs/0909.1596.  arXiv:0909.1596 [quant-ph].
  * Brif et al. [2010] Constantin Brif, Raj Chakrabarti, and Herschel Rabitz.  Control of quantum phenomena: past, present and future.  _New Journal of Physics_ , 12(7):075008, 2010.
  * Zhang et al. [2017] Jing Zhang, Yu-xi Liu, Re-Bing Wu, Kurt Jacobs, and Franco Nori.  Quantum feedback: theory, experiments, and applications.  _Physics Reports_ , 679:1–60, 2017.
  * de Lange et al. [2014] G. de Lange, D. Riste, M. J. Tiggelman, C. Eichler, L. Tornberg, G. Johansson, A. Wallraff, R. N. Schouten, and L. DiCarlo.  Reversing Quantum Trajectories with Analog Feedback.  _Phys. Rev. Lett._ , 112(8):080501, February 2014.  ISSN 0031-9007, 1079-7114.  doi: 10.1103/PhysRevLett.112.080501.  URL https://link.aps.org/doi/10.1103/PhysRevLett.112.080501.
  * Sayrin et al. [2011] Clément Sayrin, Igor Dotsenko, Xingxing Zhou, Bruno Peaudecerf, Théo Rybarczyk, Sébastien Gleyzes, Pierre Rouchon, Mazyar Mirrahimi, Hadis Amini, Michel Brune, Jean-Michel Raimond, and Serge Haroche.  Real-time quantum feedback prepares and stabilizes photon number states.  _Nature_ , 477(7362):73–77, September 2011.  ISSN 0028-0836, 1476-4687.  doi: 10.1038/nature10376.  URL http://www.nature.com/articles/nature10376.
  * Yamamoto et al. [2007] Naoki Yamamoto, Koji Tsumura, and Shinji Hara.  Feedback control of quantum entanglement in a two-spin system.  _Automatica_ , 43(6):981–992, 2007.
  * Mirrahimi and Van Handel [2007] Mazyar Mirrahimi and Ramon Van Handel.  Stabilizing Feedback Controls for Quantum Systems.  _SIAM J. Control Optim._ , 46(2):445–467, January 2007.  ISSN 0363-0129, 1095-7138.  doi: 10.1137/050644793.  URL http://epubs.siam.org/doi/10.1137/050644793.
  * Yamamoto et al. [2008] Naoki Yamamoto, Hendra I Nurdin, Matthew R James, and Ian R Petersen.  Avoiding entanglement sudden death via measurement feedback control in a quantum network.  _Physical Review A_ , 78(4):042339, 2008.
  * Martin et al. [2015] Leigh Martin, Felix Motzoi, Hanhan Li, Mohan Sarovar, and K. Birgitta Whaley.  Deterministic generation of remote entanglement with active quantum feedback.  _Phys. Rev. A_ , 92(6):062321, December 2015.  ISSN 1050-2947, 1094-1622.  doi: 10.1103/PhysRevA.92.062321.  URL https://link.aps.org/doi/10.1103/PhysRevA.92.062321.
  * Govia et al. [2022] LCG Govia, A Lingenfelter, and AA Clerk.  Stabilizing two-qubit entanglement by mimicking a squeezed environment.  _Physical Review Research_ , 4(2):023010, 2022.
  * Brown et al. [2022] Tristan Brown, Emery Doucet, Diego Ristè, Guilhem Ribeill, Katarina Cicak, Joe Aumentado, Ray Simmonds, Luke Govia, Archana Kamal, and Leonardo Ranzani.  Trade off-free entanglement stabilization in a superconducting qutrit-qubit system.  _Nature communications_ , 13(1):3994, 2022.
  * Doucet et al. [2023] E Doucet, LCG Govia, and A Kamal.  Scalable entanglement stabilization with modular reservoir engineering.  _arXiv preprint arXiv:2301.05725_ , 2023.
  * Motzoi et al. [2016] Felix Motzoi, Eli Halperin, Xiaoting Wang, K. Birgitta Whaley, and Sophie Schirmer.  Backaction-driven, robust, steady-state long-distance qubit entanglement over lossy channels.  _Phys. Rev. A_ , 94(3):032313, September 2016.  ISSN 2469-9926, 2469-9934.  doi: 10.1103/PhysRevA.94.032313.  URL https://link.aps.org/doi/10.1103/PhysRevA.94.032313.
  * Vijay et al. [2012] R. Vijay, C. Macklin, D. H. Slichter, S. J. Weber, K. W. Murch, R. Naik, A. N. Korotkov, and I. Siddiqi.  Stabilizing Rabi oscillations in a superconducting qubit using quantum feedback.  _Nature_ , 490(7418):77–80, October 2012.  ISSN 0028-0836, 1476-4687.  doi: 10.1038/nature11505.  URL http://www.nature.com/articles/nature11505.
  * Pokharel et al. [2018] Bibek Pokharel, Namit Anand, Benjamin Fortman, and Daniel A Lidar.  Demonstration of fidelity improvement using dynamical decoupling with superconducting qubits.  _Physical review letters_ , 121(22):220502, 2018.
  * Tripathi et al. [2022] Vinay Tripathi, Huo Chen, Mostafa Khezri, Ka-Wa Yip, EM Levenson-Falk, and Daniel A Lidar.  Suppression of crosstalk in superconducting qubits using dynamical decoupling.  _Physical Review Applied_ , 18(2):024068, 2022.
  * Pokharel and Lidar [2023] Bibek Pokharel and Daniel A Lidar.  Demonstration of algorithmic quantum speedup.  _Physical Review Letters_ , 130(21):210602, 2023.
  * Khodjasteh et al. [2011] Kaveh Khodjasteh, Támas Erdélyi, and Lorenza Viola.  Limits on preserving quantum coherence using multipulse control.  _Physical Review A_ , 83(2):020305, 2011.
  * Kashima and Yamamoto [2009] Kenji Kashima and Naoki Yamamoto.  Control of quantum systems despite feedback delay.  _IEEE Transactions on Automatic Control_ , 54(4):876–881, 2009.
  * Liu et al. [2021] Yanan Liu, Daoyi Dong, Sen Kuang, Ian R Petersen, and Hidehiro Yonezawa.  Two-step feedback preparation of entanglement for qubit systems with time delay.  _Automatica_ , 125:109174, 2021.
  * Amini et al. [2013] Hadis Amini, Ram A Somaraju, Igor Dotsenko, Clément Sayrin, Mazyar Mirrahimi, and Pierre Rouchon.  Feedback stabilization of discrete-time quantum systems subject to non-demolition measurements with imperfections and delays.  _Automatica_ , 49(9):2683–2692, 2013.
  * Ahn et al. [2002] Charlene Ahn, Andrew C. Doherty, and Andrew J. Landahl.  Continuous quantum error correction via quantum feedback control.  _Phys. Rev. A_ , 65(4):042301, March 2002.  ISSN 1050-2947, 1094-1622.  doi: 10.1103/PhysRevA.65.042301.  URL http://arxiv.org/abs/quant-ph/0110111.  arXiv:quant-ph/0110111.
  * Patti et al. [2017] Taylor Lee Patti, Areeya Chantasri, Luis Pedro García-Pintos, Andrew N. Jordan, and Justin Dressel.  Linear feedback stabilization of a dispersively monitored qubit.  _Phys. Rev. A_ , 96(2):022311, August 2017.  ISSN 2469-9926, 2469-9934.  doi: 10.1103/PhysRevA.96.022311.  URL https://link.aps.org/doi/10.1103/PhysRevA.96.022311.
  * Lidar [2014] Daniel A Lidar.  Review of decoherence-free subspaces, noiseless subsystems, and dynamical decoupling.  _Quantum information and computation for chemistry_ , pages 295–354, 2014.
  * Roy et al. [2011] Soumya Singha Roy, TS Mahesh, and GS Agarwal.  Storing entanglement of nuclear spins via uhrig dynamical decoupling.  _Physical Review A_ , 83(6):062326, 2011.
  * Franco et al. [2014] R Lo Franco, Antonio D’Arrigo, Giuseppe Falci, Giuseppe Compagno, and Elisabetta Paladino.  Preserving entanglement and nonlocality in solid-state qubits by dynamical decoupling.  _Physical Review B_ , 90(5):054304, 2014.
  * McCourt et al. [2023] Trevor McCourt, Charles Neill, Kenny Lee, Chris Quintana, Yu Chen, Julian Kelly, Jeffrey Marshall, VN Smelyanskiy, MI Dykman, Alexander Korotkov, et al.  Learning noise via dynamical decoupling of entangled qubits.  _Physical Review A_ , 107(5):052610, 2023.
  * Paladino et al. [2014a] E. Paladino, Y. M. Galperin, G. Falci, and B. L. Altshuler.  1/f noise: implications for solid-state quantum information.  _Rev. Mod. Phys._ , 86(2):361–418, April 2014a.  ISSN 0034-6861, 1539-0756.  doi: 10.1103/RevModPhys.86.361.  URL http://arxiv.org/abs/1304.7925.
  * Korotkov [2011] Alexander N. Korotkov.  Quantum Bayesian approach to circuit QED measurement.  _arXiv:1111.4016 [cond-mat, physics:quant-ph]_ , November 2011.  URL http://arxiv.org/abs/1111.4016.  arXiv: 1111.4016.
  * Szigeti et al. [2013] S. S. Szigeti, S. J. Adlong, M. R. Hush, A. R. R. Carvalho, and J. J. Hope.  Robustness of system-filter separation for the feedback control of a quantum harmonic oscillator undergoing continuous position measurement.  _Phys. Rev. A_ , 87(1):013626, January 2013.  ISSN 1050-2947, 1094-1622.  doi: 10.1103/PhysRevA.87.013626.  URL https://link.aps.org/doi/10.1103/PhysRevA.87.013626.
  * Khezri et al. [2015] Mostafa Khezri, Justin Dressel, and Alexander N. Korotkov.  Qubit measurement error from coupling with a detuned neighbor in circuit QED.  _Phys. Rev. A_ , 92(5):052306, November 2015.  ISSN 1050-2947, 1094-1622.  doi: 10.1103/PhysRevA.92.052306.  URL http://arxiv.org/abs/1506.06321.  arXiv: 1506.06321.
  * [56] See supplementary material.
  * Eddins et al. [2019] A. Eddins, J.M. Kreikebaum, D.M. Toyli, E.M. Levenson-Falk, A. Dove, W.P. Livingston, B.A. Levitan, L.C.G. Govia, A.A. Clerk, and I. Siddiqi.  High-Efficiency Measurement of an Artificial Atom Embedded in a Parametric Amplifier.  _Physical Review X_ , 9(1), 2019.  ISSN 21603308.  doi: 10.1103/PhysRevX.9.011004.
  * Lecocq et al. [2021] F. Lecocq, L. Ranzani, G. A. Peterson, K. Cicak, X. Y. Jin, R. W. Simmonds, J. D. Teufel, and J. Aumentado.  Efficient Qubit Measurement with a Nonreciprocal Microwave Amplifier.  _Physical Review Letters_ , 126(2):020502, January 2021.  doi: 10.1103/PhysRevLett.126.020502.
  * Place et al. [2021] Alexander P. M. Place, Lila V. H. Rodgers, Pranav Mundada, Basil M. Smitham, Mattias Fitzpatrick, Zhaoqi Leng, Anjali Premkumar, Jacob Bryon, Andrei Vrajitoarea, Sara Sussman, Guangming Cheng, Trisha Madhavan, Harshvardhan K. Babla, Xuan Hoang Le, Youqi Gang, Berthold Jäck, András Gyenis, Nan Yao, Robert J. Cava, Nathalie P. de Leon, and Andrew A. Houck.  New material platform for superconducting transmon qubits with coherence times exceeding 0.3 milliseconds.  _Nature Communications_ , 12(1):1779, March 2021.  ISSN 2041-1723.  doi: 10.1038/s41467-021-22030-5.
  * Gordon et al. [2008] Goren Gordon, Gershon Kurizki, and Daniel A. Lidar.  Optimal Dynamical Decoherence Control of a Qubit.  _Phys. Rev. Lett._ , 101(1):010403, July 2008.  ISSN 0031-9007, 1079-7114.  doi: 10.1103/PhysRevLett.101.010403.  URL http://arxiv.org/abs/0804.2691.  arXiv:0804.2691 [quant-ph].
  * Yan et al. [2018] Fei Yan, Philip Krantz, Youngkyu Sung, Morten Kjaergaard, Daniel L. Campbell, Terry P. Orlando, Simon Gustavsson, and William D. Oliver.  Tunable Coupling Scheme for Implementing High-Fidelity Two-Qubit Gates.  _Physical Review Applied_ , 10(5):054062, November 2018.  doi: 10.1103/PhysRevApplied.10.054062.
  * Porotti et al. [2022] Riccardo Porotti, Antoine Essig, Benjamin Huard, and Florian Marquardt.  Deep Reinforcement Learning for Quantum State Preparation with Weak Nonlinear Measurements.  _Quantum_ , 6:747, June 2022.  doi: 10.22331/q-2022-06-28-747.
  * Porotti et al. [2023] Riccardo Porotti, Vittorio Peano, and Florian Marquardt.  Gradient-Ascent Pulse Engineering with Feedback.  _PRX Quantum_ , 4(3):030305, July 2023.  doi: 10.1103/PRXQuantum.4.030305.
  * Paladino et al. [2014b] E. Paladino, Y. M. Galperin, G. Falci, and B. L. Altshuler.  1 / f noise: Implications for solid-state quantum information.  _Rev. Mod. Phys._ , 86(2):361–418, April 2014b.  ISSN 0034-6861, 1539-0756.  doi: 10.1103/RevModPhys.86.361.  URL https://link.aps.org/doi/10.1103/RevModPhys.86.361.
  * Chen and Lidar [2022] Huo Chen and Daniel A. Lidar.  HOQST: Hamiltonian Open Quantum System Toolkit.  _Commun Phys_ , 5(1):112, May 2022.  ISSN 2399-3650.  doi: 10.1038/s42005-022-00887-2.  URL http://arxiv.org/abs/2011.14046.  arXiv:2011.14046 [quant-ph].
  * Tripathi et al. [2023] Vinay Tripathi, Huo Chen, Eli Levenson-Falk, and Daniel A. Lidar.  Modeling low- and high-frequency noise in transmon qubits with resource-efficient measurement, February 2023.  URL http://arxiv.org/abs/2303.00095.  arXiv:2303.00095 [quant-ph].
  * Yip [2021] Ka Wa Yip.  Open-system modeling of quantum annealing: theory and applications, July 2021.  URL http://arxiv.org/abs/2107.07231.  Number: arXiv:2107.07231 arXiv:2107.07231 [quant-ph].
  * Bylander et al. [2011] Jonas Bylander, Simon Gustavsson, Fei Yan, Fumiki Yoshihara, Khalil Harrabi, George Fitch, David G. Cory, Yasunobu Nakamura, Jaw-Shen Tsai, and William D. Oliver.  Noise spectroscopy through dynamical decoupling with a superconducting flux qubit.  _Nature Phys_ , 7(7):565–570, July 2011.  ISSN 1745-2481.  doi: 10.1038/nphys1994.  URL https://www.nature.com/articles/nphys1994.  Number: 7 Publisher: Nature Publishing Group.
  * Trappen et al. [2023] R. Trappen, X. Dai, M. A. Yurtalan, D. Melanson, D. M. Tennant, A. J. Martinez, Y. Tang, J. Gibson, J. A. Grover, S. M. Disseler, J. I. Basham, R. Das, D. K. Kim, A. J. Melville, B. M. Niedzielski, C. F. Hirjibehedin, K. Serniak, S. J. Weber, J. L. Yoder, W. D. Oliver, D. A. Lidar, and A. Lupascu.  Decoherence of a tunable capacitively shunted flux qubit, July 2023.  URL http://arxiv.org/abs/2307.13961.  arXiv:2307.13961 [quant-ph].
  * Yoshihara et al. [2006] F. Yoshihara, K. Harrabi, A. O. Niskanen, Y. Nakamura, and J. S. Tsai.  Decoherence of Flux Qubits due to $1/f$ Flux Noise.  _Phys. Rev. Lett._ , 97(16):167001, October 2006.  doi: 10.1103/PhysRevLett.97.167001.  URL https://link.aps.org/doi/10.1103/PhysRevLett.97.167001.  Publisher: American Physical Society.
  * Livingston et al. [2022b] William P. Livingston, Machiel S. Blok, Emmanuel Flurin, Justin Dressel, Andrew N. Jordan, and Irfan Siddiqi.  Experimental demonstration of continuous quantum error correction.  _Nat Commun_ , 13(1):2307, April 2022b.  ISSN 2041-1723.  doi: 10.1038/s41467-022-29906-0.  URL https://www.nature.com/articles/s41467-022-29906-0.
  * [72] Daniel Lidar.  _Open Quantum Systems (unpublished book)._