††thanks: Authors S. S. A. and V. E. C. contributed equally††thanks: Authors
S. S. A. and V. E. C. contributed equally

# Robust Quantum Sensing with Multiparameter Decorrelation

Shah Saad Alam  0000-0001-6541-662X JILA and Department of Physics, University
of Colorado, 440 UCB, Boulder, CO 80309, USA  Victor E. Colussi
0000-0002-0972-6276 Infleqtion, Inc., 3030 Sterling Circle, Boulder, CO 80301,
USA  John Drew Wilson  0000-0001-6334-2460 JILA and Department of Physics,
University of Colorado, 440 UCB, Boulder, CO 80309, USA  Jarrod T. Reilly
0000-0001-5410-089X JILA and Department of Physics, University of Colorado,
440 UCB, Boulder, CO 80309, USA  Michael A. Perlin  0000-0002-9316-1596
Infleqtion, Inc.,141 West Jackson Blvd Suite 1875 Chicago, IL 60604, USA
Murray J. Holland  0000-0002-3778-1352 JILA and Department of Physics,
University of Colorado, 440 UCB, Boulder, CO 80309, USA

###### Abstract

The performance of a quantum sensor is fundamentally limited by noise. This
noise is particularly damaging when it becomes correlated with the readout of
a target signal, caused by fluctuations of the sensor’s operating parameters.
These uncertainties limit sensitivity in a way that can be understood with
multiparameter estimation theory. We develop a new approach, adaptable to any
quantum platform, for designing robust sensing protocols that leverages
multiparameter estimation theory and machine learning to decorrelate a target
signal from fluctuating off-target (“nuisance”) parameters. Central to our
approach is the identification of information-theoretic goals that guide a
machine learning agent through an otherwise intractably large space of
potential sensing protocols. As an illustrative example, we apply our approach
to a reconfigurable optical lattice to design an accelerometer whose
sensitivity is decorrelated from lattice depth noise. We demonstrate the
effect of decorrelation on outcomes and Bayesian inferencing through
statistical analysis in parameter space, and discuss implications for future
applications in quantum metrology and computing.

##  I Introduction

Quantum sensing is a promising near-term application of quantum devices, with
the potential to leverage the properties of quantum mechanics to revolutionize
physical measurements [1, 2, 3]. This has been demonstrated in extensive
applications such as atomic clocks [4, 5], inertial sensors [6, 7, 8, 9],
gravitational wave detectors [10, 11, 12, 13], and biological tissue imaging
devices [14]. Quantum sensors also enable searches for new fundamental physics
through, for example, precision tests of the equivalence principle and
measurements of variations in fundamental constants [15, 16] to identify
signatures of physics beyond the standard model [17, 18, 19]. Common amongst
these applications is the goal of improving upon the fundamental limitations
posed by susceptibilities to classical and quantum noise (c.f. [20, 21, 22,
23, 24, 25, 26]).

Figure 1: Machine learning framework for generating decorrelating sensing
protocols (DSP’s) using reinforcement learning in the multiparameter setting.
The agent first takes an action which generates a transition in the
environment resulting in a reward. The reward is formulated in terms of the
score functions $\ell$ that determine the classical Fisher information matrix
[see Eq. (4)]. This reward takes into account correlations between target and
nuisance parameters in determining the overall sensitivity and decorrelating
effect of the control function. With each iteration, the reward is used to
update the neural network and inform the decision making of the agent.

Past works have proposed many methods to mitigate the effects of external
noise. These methods range from quantum error correction [27, 28], to
classical compensation in lasers [29, 30], to modifying the system dynamics
through Hamiltonian steering methods [31], and dynamical decoupling [32, 33,
34]. More recently, programmable quantum sensors have been used to develop
quantum metrology protocols to simultaneously resolve several parameters, such
as the vector components of a field [35, 36, 37, 38]. In this work, we design
sensing protocols that minimize the classical correlations between off-target
parameter uncertainties and the readout signal of interest. Consequently, the
adverse effects of both quantum and classical noise are reduced by ensuring
that they become statistically independent of the estimator for the target
signal.

To engineer “decorrelated” quantum sensing protocols (DSP’s) that
statistically isolate a target signal from other parameters, we task a machine
learning agent with suitable information-theoretic goals [39], as illustrated
in Fig. 1. However, these protocols and rewards would work with any similar
optimization protocol. In the single-parameter setting, the study of sensing
protocols from an information-theoretic perspective has already yielded new
insights about how traditional paradigms [20, 40, 41, 1, 42] can fail to reach
the full metrological potential available on a device. In particular, machine
learning was used in Refs. [20, 40] to find schemes that maximized sensitivity
using an end-to-end design that is unconstrained by traditional sensing
paradigms, which involve distinct state-preparation and interrogation steps.

In this work, we unify these paradigms to propose a robust, decorrelating
method that is agnostic to the choice of quantum device and can be applied to
any multiparameter sensing task. As an illustrative example, we consider an
inertial sensor based on cold atom interferometry in an optical lattice, and
use machine learning to make it robust to noise. We design inertial DSPs to
perform accelerometry in a setting of incomplete information about the optical
lattice depth, for example due to online laser power noise, shot-to-shot
variations in laser settings, or other (possibly unknown) effects.

##  II Background

###  II.1 Multiparameter estimation theory

In this section we discuss multiparameter estimation theory, and establish the
formalism used in the remainder of this work. The goal of multiparameter
estimation is to estimate the values of parameters
$\mathbf{b}=(b^{1},b^{2},\cdots)$ that are encoded into a quantum state
(density matrix) $\hat{\rho}(\mathbf{b})$. In this work, we assume that the
encoding occurs through time evolution via the equation of motion
$\partial\hat{\rho}/\partial t=\mathcal{L}_{\mathbf{b}}\hat{\rho}$, which
takes an initial state $\hat{\rho}_{0}\to\hat{\rho}(\mathbf{b})$. The time
evolution operator $\mathcal{L}_{\mathbf{b}}$ and final state
$\hat{\rho}(\mathbf{b})$ implicitly depend on additional control parameters
$\mathbf{c}$ that parameterize a sensing protocol. Designing a sensing
protocol thus entails choosing control parameters $\mathbf{c}$—such as a
(discrete) choice of gates, or a (continuous) choice of a laser phase
profile—that optimize the inference of the estimation parameters $\mathbf{b}$
from measurements of the state $\hat{\rho}(\mathbf{b})$.

Measurements of a quantum system are performed with respect to a basis,
typically represented by a set of projectors
$\mathbb{\Pi}=\\{\hat{\Pi}_{1},\hat{\Pi}_{2},...\\}$ 111In fact, our results
are unchanged if we replace the projector-valued measure (PVM) $\mathbb{\Pi}$
by a more general positive operator-valued measure (POVM).. Each projector is
associated with a single measurement outcome, and the probability of obtaining
outcome $j$ when measuring the state $\hat{\rho}(\textbf{b})$ is given by the
likelihood function
$P(j|\textbf{b})=\operatorname{Tr}(\hat{\rho}(\mathbf{b})\Pi_{j})$. For a
given prior $P(\textbf{b})$ that describes the a priori likelihood
(probability distribution) for different values of $\mathbf{b}$, Bayes’
theorem 222In this work we take a Bayesian approach to parameter estimation.
However, we emphasize that in the case of a fixed parameterization and flat
prior, Bayesian and frequentist approaches are equivalent [77]. prescribes how
measurement outcomes should be used to update the likelihood distribution for
b. For a sequence of statistically independent measurement outcomes
$M=(m_{1},m_{2},...)$, the conditional likelihood distribution for b becomes

$P({\mathbf{b}}|M)=\frac{P(M|{\mathbf{b}})}{P(M)}P(\mathbf{b})=\left[\prod_{m\in
M}\frac{P(m|{\mathbf{b}})}{P(m)}\right]P(\mathbf{b}),$ (1)

where $P(M|\mathbf{b})=\prod_{m\in M}P(m|\mathbf{b})$ is the joint likelihood
function for outcomes $M$, and $P(M)=\prod_{m\in M}P(m)$ with $P(m)=\sum_{\bf
b}P(m|{\bf b})$ is the probability of finding the measurement record $M$ for
any possible value of $\mathbf{b}$. Given a flat (uniform) prior $P({\bf b})$,
the peak of the Bayesian posterior $P({\mathbf{b}}|M)$ coincides with the
maximum likelihood estimator of the parameters:

$\hat{\mathbf{b}}=\mathrm{argmax}_{\mathbf{b}}\log P(M|{\mathbf{b}}).$ (2)

The ultimate sensitivity of a quantum sensor is decided by the sensitivity of
each possible measurement outcome to a small change in a parameter. This
sensitivity is given by the score function

$\ell_{\mu}(j|{\bf b})\equiv\partial_{\mu}\log P(j|\mathbf{b}),$ (3)

where $\partial_{\mu}\equiv\partial/\partial b^{\mu}$. The score function is
used in the definition of the classical Fisher information matrix (CFIM) [see
Fig. 1]:

$\displaystyle\mathcal{I}_{\mu\nu}(\mathbb{\Pi}|\textbf{b})\equiv\mathbb{E}(\ell_{\mu}\,\ell_{\nu}|\textbf{b})$
(4)
$\displaystyle=\sum_{\Pi_{j}\in\mathbb{\Pi}}P(j|\mathbf{b})\left[\partial_{\mu}\log
P(j|{\mathbf{b}})\right]\left[\partial_{\nu}\log P(j|{\mathbf{b}})\right],$

where $\mathbb{E}(A|{\bf
b})=\sum_{\Pi_{j}\in\mathbb{\Pi}}P(j|\mathbf{b})A(j|{\bf b})$ is the classical
expectation value of $A$. The CFIM quantifies the sensitivity of the
probability distribution over measurement outcomes to small variations in the
estimation parameters.

###  II.2 Optical-lattice interferometry

We now consider multiparameter estimation in the context of accelerometry with
an optical lattice interfereometer, which is a reconfigurable inertial sensing
platform recently demonstrated experimentally in Ref. [45]. In this system, an
ultracold atomic gas is loaded into the ground state of a one-dimensional
optical lattice, described by the time-dependent Hamiltonian

$\displaystyle\hat{H}(t)=\hat{H}_{\text{lat}}(t)+\hat{H}_{\text{acc}}(t),$ (5)
$\displaystyle\hat{H}_{\text{lat}}(t)=\frac{\hat{p}^{2}}{2m}-\frac{V_{L}}{2}\cos\left[2k_{L}\hat{x}+\phi(t)\right],$
(6) $\displaystyle\hat{H}_{\text{acc}}(t)=ma\hat{x},$ (7)

where $\hat{p}$ and $\hat{x}$ are the canonical one-body momentum and position
operators, $m$ is the atomic mass, $V_{L}$ is the lattice depth,
$k_{L}=2\pi/\lambda_{L}$ and $\lambda_{L}$ are the wavenumber and wavelength
of the lattice light, respectively, $\phi(t)$ is a time-dependent phase called
the “control function”, and $a$ is the component of acceleration applied along
the lattice axis 333For simplicity, we do not consider the impact transverse
components of the acceleration [45] in this work.. For concreteness, we
consider an atomic gas of ${}^{87}\mathrm{Rb}$ atoms with mass $m\approx 86.9$
amu trapped in an optical lattice with wavelength $\lambda_{L}=1064$ nm and a
lattice depth of $V_{L}=10E_{R}$ where $E_{R}=\hbar^{2}k_{L}^{2}/2m$ is the
lattice recoil energy of the atoms. The control function $\phi(t)$ is
implemented with a time-dependent phase difference between the two counter-
propagating beams that form the lattice.

When the lattice is “shaken” with a time-varying control function $\phi(t)$,
momentum and energy are exchanged via Bragg scattering between the lattice
light and the atoms. This transfer occurs in multiples of $2\hbar k_{L}$,
which can be seen by expanding
$\cos\left[2k_{L}\hat{x}+\phi(t)\right]\ket{p}\propto(\ket{p+2\hbar
k_{L}}e^{i\phi(t)}+\ket{p-2\hbar k_{L}}e^{-i\phi(t)})$ [47, 48]. Because the
shaking is a global (translation-invariant) modulation of the lattice, the
time-dependent control function does not alter the mean quasimomentum of the
atoms (c.f. [49]), and instead drives transitions between Bloch states. Here,
the energetically lowest (“insulating” bands) Bloch states are weakly
dispersive and not effective for bulk transport unlike the higher-lying
(“conduction” bands) states. The flexibility of the control function allows
multiple sensing tasks to be carried out on the same device, making the
platform reconfigurable. At the end of the shaking sequence, the occupations
of different momentum states are recorded and can, neglecting interactions,
taken to be a measurement record $M$ of size $N_{\mathrm{atoms}}$ independent
single atom measurements where $N_{\mathrm{atoms}}$ is the total number of
atoms in the wavepacket. The measurement record, in turn, is used for Bayesian
inferencing of the acceleration $a$ and lattice depth $V_{L}$ 444Data and code
for Sec. IV are available for reference at Ref. [78].

###  II.3 Previous works

Accelerometry using optical lattice interferometers has been performed within
the Mach-Zehnder configuration [51, 45, 52, 53, 54, 55, 56]. Here, machine
learning techniques were used to obtain control functions that produce beam-
splitting and mirroring components. The reconfigurability of the platform
however allows for other, non-traditional protocols and paradigms to be
considered, which may lead to improved sensitivity.

As discussed in Ref. [41], for conventional three-pulse accelerometers and
gravimeters operating in free space [57, 58], the Mach-Zehnder inteferometer
(MZI) result $I_{\mathrm{MZI}}(p_{0})=(2p_{0}T^{2}/\hbar)^{2}$ for splitting
$p_{0}$, free evolution time $T=\mathcal{T}/2$, and total operating time
$\mathcal{T}$, does not produce the optimal sensitivity from the standpoint of
the Fisher information [41]. In that work, it was found that removal of the
mirroring pulse lead instead to Ramsey interferometry (RI), boosting the
sensitivity as $I_{\mathrm{RI}}(p_{0})=4I_{\mathrm{MZI}}(p_{0})$. However,
this configuration requires a momentum rather than (conventional) spin
population measurement, with the latter necessitating a spatial overlap in the
output for a high contrast signal. The relative compactness of optical-lattice
interferometer allows for resolution of momentum-space occupations directly
via time-of-flight measurements or in-trap via Bragg or Doppler spectroscopy
(c.f. [59, 60, 61, 62, 63]) such that optimal sensitivity in the Ramsey
configuration is in principle achievable. The demonstration of in-situ
detection methods on this device remains the subject of future study.

Nevertheless, a roadblock in achieving the optimal sensitivity of the Ramsey
configuration on an optical lattice interferometer is the long component times
compared to total operating time. Additionally, the readout is constrained by
the fixed basis of the experiment, and the management of these competing
constraints is a task well-suited for machine learning techniques. In this
paper, we pursue the problem of achieving the Ramsey bound while decorrelating
fluctuating off-target parameters from spoiling the sensitivity of the device
to inertial signals. In the following section, we describe a novel approach to
decorrelation within the framework of multiparameter estimation theory
described in Sec. II.1.

##  III Decorrelated Sensing Protocols

Having provided the requisite background concepts in Sec. II, we now outline
the main finding of this work: that the CFIM can be cast as an information-
theoretic goal suitable for input into machine learning techniques to produce
decorrelated sensing protocols (DSP’s). In the present case, these protocols
consist of control functions $\phi(t)$ that dictate the dynamical modulation
of the optical lattice for performing accelerometry.

Recall that the CFIM relates the response of the possible measurement outcomes
to small variations in the parameters. Explicitly, the posterior can be
expanded as

$\log P({\bf b}+d{\bf b}|M)-\log P({\bf b}|M)=-\frac{N}{2}\
\mathcal{I}_{\mu\nu}\ db^{\mu}db^{\nu},$ (8)

where $N=|M|$ is the length of the measurement record, we implicitly sum over
repeated indices, and for brevity we henceforth suppress the explicit
dependence of the CFIM $\mathcal{I}$ on measurement basis $\mathbb{\Pi}$ and
estimation parameters $\mathbf{b}$. Eq. (8) illuminates the geometrical role
of the CFIM as a metric on smooth statistical manifolds [64]. Physically, Eq.
(8) reveals that the resolvability of parameter values, and hence the ultimate
sensitivity of the device, is related to the informational distance between
the two posteriors, and hence by the CFIM which determines distances across
the informational landscape.

Correlations between parameters are quantified by the covariance matrix
$\mathrm{cov}(\mathbf{b}|M)$, whose value at row $\mu$ and column $\nu$ is the
covariance between the estimates of parameters $b^{\mu}$ and $b^{\nu}$, found
from Eq. (2). The diagonal entry with $\nu=\mu$ is just the variance of the
estimate of $b^{\mu}$. The asymptotic lower bound for this uncertainty is
given by the (multiparameter) Cramér-Rao bound [65, 66, 67], which can be
expressed by the inequality

$\mathrm{cov}({\mathbf{b}}|M)\geq\frac{1}{N}\mathcal{I}^{-1}\geq\frac{1}{N}\mathcal{F}^{-1},$
(9)

where $\mathcal{I}^{-1}$ is the inverse of the CFIM, and $A\geq B$ means the
matrix $A-B$ is positive semi-definite (has non-negative eigenvalues) 555In
the case of zero diagonal entries in the CFIM/QFIM, these are omitted when
calculating the Cramer-Rao Bound. In practical systems, however, it’ll be rare
to have zero Fisher information. The minimal bound in Eq. 9 is achieved by
$\mathcal{F}$, the quantum Fisher Information matrix (QFIM), which represents
the matrix upper bound of $\mathcal{I}$. The QFIM is constructed by maximizing
each diagonal element $\mathcal{I}_{\mu\mu}$ over all possible sets of
measurements. In practice, this means replacing the score functions for every
element of the matrix in Eq. (4) with parameter derivatives of the quantum
state. For a pure state, this takes the form [65]

$\mathcal{F}_{\mu\nu}=4\mathrm{Re}\left[\langle\partial_{\mu}\psi|\partial_{\nu}\psi\rangle-\langle\partial_{\mu}\psi|\psi\rangle\langle\psi|\partial_{\nu}\psi\rangle\right].$
(10)

The presence of parameter correlations leads to finite off-diagonal values of
the CFIM, producing a “tilt” of the Bayesian posterior, which we quantify by
the dimensionless correlation

$\displaystyle\mathrm{Corr}(\ell_{\mu},\ell_{\nu})\equiv$
$\displaystyle\frac{\mathcal{I}_{\mu\nu}}{\sqrt{I_{\mu}}\sqrt{I_{\nu}}},$ (11)

where $I_{\mu}=\mathcal{I}_{\mu\mu}$, for brevity. This quantity diagnoses how
correlated the parameter score functions for $b^{\mu}$ and $b^{\nu}$ are (as
indicated in Fig. 1), with $+1$ being perfectly correlated, $0$ being
decorrelated (statistically independent), and $-1$ being anti-correlated. We
note that the connection between Eqs. (8) and (11) becomes apparent in the
normalized parameterization $\tilde{\textbf{b}}^{\mu}\equiv
b^{\mu}/\sqrt{I_{\mu}}$, where the dimensionless correlation enters Eq. (8) in
place of $\mathcal{I}_{\mu\nu}$.

###  III.1 Target and Nuisance Parameters

We now develop further language specific to the task of parameter
decorrelation. The operation of a quantum sensor will, in general, depend on
multiple parameters but only some of these will be of interest. The parameters
of interest are called target parameters and are denoted by the ordered list
$\boldsymbol{\alpha}=(\alpha^{1},\alpha^{2},...)$, while the other parameters
are called nuisance parameters and are denoted by the ordered list
$\boldsymbol{\eta}=(\eta^{1},\eta^{2},...)$, so the total ordered list of
parameters is $\mathbf{b}=(\alpha^{1},\alpha^{2},...,\eta^{1},\eta^{2},...)$.
Our ability to measure the target parameters depends on our degree of
knowledge and control over all parameters, some of which are changing or
initially unknown, such as lattice fluctuations in lattice atomic clocks [69,
70].

Even if nuisance parameters are removed via marginalization, the uncertainties
in the nuisance parameters can feed into the uncertainty for the target
parameters via correlations. To see this, consider the general case of
multiple target parameters for which we wish to marginalize out the dependence
on the nuisance parameters. To do so, we consider the block form of the
CFIM[39],

$\mathcal{I}=\begin{pmatrix}\mathcal{A}&\mathcal{C}\\\
\mathcal{C}^{\text{T}}&\mathcal{N}\end{pmatrix},$ (12)

where $\mathcal{A}$ and $\mathcal{N}$ are the CFIM’s for only
$\boldsymbol{\alpha}$ and $\boldsymbol{\eta}$ respectively, and $\mathcal{C}$
is the cross terms, i.e. $\mathcal{C}_{\mu j}$ is the correlation between
$\alpha^{\mu}$ and $\eta^{j}$. By the block matrix inversion theorem, the
inverse Fisher information that appears in the Cramér-Rao bound of Eq. (9) is

$\left(\mathcal{I}^{-1}\right)^{\mu\nu}=\left(\mathcal{A}_{\mu\nu}-\mathcal{C}_{\mu
i}\left(\mathcal{N}^{-1}\right)^{ij}\mathcal{C}_{j\nu}\right)^{-1},$ (13)

where Greek indices $\mu$, $\nu$ run over target parameters and Latin indices
$i$, $j$ are summed over nuisance parameters[39].

As an illustrative example, consider the case of just one target parameter
$\alpha$ and nuisance parameter $\eta$, so $\mathbf{b}=(\alpha,\eta)$. If
$\eta$ were perfectly known, then it is straightforward to saturate the
Cramér-Rao bound–although often difficult in real experimental contexts.
However, it is rarely the case that all other parameters are known. In this
case, even though we don’t necessarily care about these other parameters, we
have [71, 39]

$\mathrm{var}(\alpha)\geq\left[NI_{\alpha}\left(1-\mathrm{Corr}(\ell_{\alpha},\ell_{\eta})^{2}\right)\right]^{-1},$
(14)

where we’ve used the parameters as labels, rather than indices. This
demonstrates the penalizing effect of having incomplete information about
additional parameters in a system, even when one is only concerned about a
single parameter.

Figure 2: Accelerometry using the SPP (a-d) and DSP (e-h), including control
functions (a,e), momentum space dynamics (b,f), Bloch state occupations (c,g),
and position space dynamics (d,h). Position space visualizations in this work
are made by the convolution of $\psi_{q}(p,t)$ with a Gaussian of width
$\sigma_{p}=0.1\hbar k_{L}$. This illustrates the split trajectories,
initialized as a Gaussian wave packet spanning roughly 10 lattice sites. In
practice (c.f. [45]) the initial wave packet is delocalized over many times
this number of sites such that trajectories are not resolvable. Figure 3:
Relevant classical and quantum Fisher information (FI) elements for (a)
acceleration, (b) covariance, and (c) lattice depth in the SPP and DSP cases.
Here, $v_{R}=\hbar k_{L}/m$ is the recoil velocity. Panel (a) is in units of
the Mach-Zehnder interferometer to illustrate advantage over traditional
setups, and the dashed line indicates the optimal result of the Ramsey
configuration discussed in Sec. II.3.

The goal of a DSP is then to optimize sensitivity to a small change in one or
many target parameters in $\boldsymbol{\alpha}$, while also being invariant to
changes of any nuisance parameters in $\boldsymbol{\eta}$. From Eq. (13) we
see that this is equivalent to maximizing some property of $\mathcal{A}$–such
as its smallest eigenvalue, its trace, or its determinant depending on the
specific use case [72], while minimizing some property of
$\mathcal{C}\mathcal{N}^{-1}\mathcal{C}^{\text{T}}$. In this work, we choose
to input elements of Eq. (13) directly into a machine learning method for
developing DSP’s in the case of one target and one nuisance parameter [see
Fig. 1]. Further details on the machine learning method employed can be found
in App. C. Finally, we comment that the formalism developed in this section
for designing DSP’s targets two common forms of nuisance parameter
fluctuations: miscalibration and shot-to-shot parameter drift such as
frequency drift, laser phase noise in an atomic clock, or lattice noise in an
optical lattice system [73].

###  III.2 Accelerometry

In this section, we analyze the quantum dynamics that result from a DSP
designed to perform accelerometry in the presence of fluctuations in the
nuisance parameter $V_{L}$. In this case, the appropriate element of Eq. (13)
to be used as design goal for developing DSP’s for accelerometry using machine
learning (see Fig. 1) is

$\text{Design Goal =
}\underset{\phi(t)}{\mathrm{max}}\left[1/(\mathcal{I}^{-1})_{a,a}=I_{a}-\frac{\mathcal{I}_{a,V_{L}}^{2}}{I_{V_{L}}}\right].$
(15)

As discussed in App. C, we use reinforcement learning to search for control
parameters $\mathbf{c}$ corresponding to a control function $\phi(t)$ that
maximizes this goal 666We note that the formulation in Eq. (15) and Eq. (13)
is written generally for application with a generic optimizer.. We judge the
performance of the DSP based on sensitivity to the target parameter
$\zeta_{a}=1/(\mathcal{I}^{-1}_{a,a}I_{MZI})$ as well as reduction in the
value of the dimensionless correlation $\mathrm{Corr}(\ell_{a},\ell_{V_{L}})$
introduced in Eq. (11). To illustrate the effect of decorrelation, we compare
results where perfect information about nuisance parameters exists, using the
single-parameter protocol (SPP) shown in Fig. 2(a), against results where we
have no information about the nuisance parameters, using the DSP shown in Fig.
2(e). We expect the dimensionless correlation to take random values between
$[-1,1]$ for the SPP, as the elements of Eq. (13) supplied to the machine
learning method contain information only about the target parameter. Instead,
we expect this ratio to instead be close to $0$ for a successful DSP.
Additionally, since we wish to compare against reference sensing protocols as
well as previous works on optical lattice interferometry [45](see Sec. II.3),
we reject control functions that generate occupations in excess of a few
percent for momentum states with magnitudes greater than $p_{0}=4\hbar k_{L}$.

Fig. 2 shows the time-dependent control functions learned by the reinforcement
learning agent, as well as the resultant quantum state dynamics, for both the
SPP [(a)-(d)] and DSP [(e)-(h)]. Despite differences in the control functions,
within 200 $\mu$s the quantum state is excited from valence $(\nu=0,1,2)$ to
conduction bands $(\nu\geq 3)$ where bulk spatial transport becomes possible
as shown by the split spatial trajectories in Fig. 2 (d,h). During this
transport phase, the Bloch band occupation dynamics differ greatly between the
protocols as shown in Fig. 2 (c,g), with significant occupations of the lowest
conduction bands, which are superpositions of the form
$|p_{0},\pm\rangle\equiv(|p_{0}\rangle\pm|-p_{0}\rangle)/\sqrt{2}$. By $t\sim
550$ $\mu$s, the quantum state is prepared for measurement in the fixed basis
of the experiment. Here we see that in the SPP, population is transferred to
the valence bands, whereas in the DSP a large oscillation occurs between
valence and conduction bands, crucial for decorrelation of nuisance and target
parameters.

From Figs. 2(d,h) we see that a large spacetime area is enclosed without an
obvious mirroring component, characteristic of the Ramsey configuration
discussed in Sec. II.3. This leads in both cases to a sensitivity
$\zeta_{a}\approx 2$, twice that of the conventional Mach-Zehnder
configuration but half that of the Ramsey configuration. Analysis of the
Fisher information matrices (quantum and classical) shown in Fig. 3, reveals
in panel (a) that the optimal result $I_{RI}$ is approached in both cases
where the QFIM is evaluated using Eq. 10 However the final sensitivity is
limited by the finite duration of the basis preparation phase. Fig. 3(b,c)
reveals instead the decorrelating effect of the DSP
($\mathrm{Corr}(\ell_{a},\ell_{V_{L}})\approx 0.004$) compared to the SPP
($\mathrm{Corr}(\ell_{a},\ell_{V_{L}})\approx 0.3$).

From Fig. 3, we see that the DSP also yields more information about the
lattice depth. This is an additional strategy to further reduce the
dimensionless correlation, i.e. by executing a sensing protocol that reduces
our uncertainty in target and nuisance parameters alike, thus reducing the
multiparameter estimation problem to that of a single parameter where an SPP
becomes more appropriate.

##  IV Statistical Analysis

In this section, we analyze the utility of DSPs for performing accelerometry
in the presence of fluctuations and uncertainty in the lattice depth. In Sec.
IV.1, we use Bayes’ theorem (see Sec. II.1) to estimate the target parameter
$a$ from the final momentum distributions obtained by running a DSP over a
range of accelerations and lattice depths. In Sec. IV.2, we further quantify
performance and establish the range of validity of the DSP through an analysis
of the Jensen-Shannon divergences (JSD). This analysis extends to the
multiparameter domain approaches developed in Refs. [45, 40, 51] to handle the
complex, multi-modal output of an optical lattice interferometer.

Figure 4: Final momentum probability distributions $P(p|a,V_{L})$, resulting
from the SPP (left column) and DSP (right column) presented in Fig. 2
evaluated over a range of $a$, $V_{L}$, and $p$. Unlike a conventional Bragg
or three-pulse interferometer that outputs a single interference signal, an
optical lattice interferometer returns fringes for each momentum component
obtained via time-of-flight measurements. For clarity we display occupations
up to momenta $\pm 4\hbar k_{L}$ and note that larger momenta can also be
occupied particularly as the acceleration or lattice depth is varied
significantly. Although not shown, these occupations are accounted for in the
analysis of this section.

###  IV.1 Bayesian estimation

We begin by performing Bayesian estimation (Eq. (1)) on the final momentum
distributions $P(p|a,V_{L})$, which are shown in Fig. 4 for the SPP and DSP
evaluated over the regions $-0.1\leq a/g\leq 0.1$ and $9\leq V_{L}/E_{R}\leq
11$. From Eq. (8), the rate of variation of the distributions in parameter
space can be understood in terms of the CFIM, with the fast variation in $a$
indicative of protocols designed for performing accelerometry (rather than
lattice depth estimation). The multi-modal, multi-parameter nature of the
distributions shown in Fig. 4, however, complicates the statistical analysis
and interpretation of the CFIM. The results of multiparameter Bayesian
updating using these distributions are shown in Fig. 5. Here, the posterior
distribution is shown for a measurement record $M$ assembled after taking $N$
stochastic samples of the distribution function $P(p|a,V_{L})$ evaluated at
the point $(a,V_{L})=(0g,10E_{R})$, corresponding to the coordinates used for
optimization of the DSP in Sec. III.2.

Figure 5: Bayesian updating in the multi-parameter estimation problem using
the SPP (left column) and DSP (right column) as a function of measurement
number $N$. In each subplot, the posterior probability distribution (Eq. (1))
is shown, which narrows on the actual parameter values
$(a,V_{L})=(0g,10E_{R})$ after repeated measurement, allowing for parameter
estimation via the maximum likelihood estimate (Eq. (2)). For sufficiently
large $N$, the uncertainty in the posteriors obeys the Cramér-Rao bound (Eq.
(9)) with the visible tilt of the SPP a consequence of the relatively large
value of $\mathcal{I}_{a,V_{L}}$ for this sequence compared to the DSP. As
discussed in Sec. III this tilt is a reflection also of the dimensionless
correlation. We note that each subplot has been normalized to provide visible
contrast of the posteriors.

Again, the resulting posterior distributions resulting after $N$ measurements
can be understood via Eq. (8), where now the differences between the final
CFIM elements for the SPP and DSP (see Fig. 3) are visually evident. Both
protocols were designed to perform accelerometry, with the large sensitivity
$\zeta_{a}$ reflected in the “narrowness” of the posterior over the range of
accelerations considered. Indeed, as additional measurements are performed,
the standard deviation of the distribution approaches the standard quantum
limit $\sim 1/\sqrt{N}$ according to the Cramér-Rao bound (Eq. (9)) which
determines the lower bounds on sensitivity in terms of the CFIM. Here, the
noticeable “tilt” of the SPP posterior is a reflection of the larger values of
$\mathcal{I}_{a,V_{L}}$ and dimensionless correlation relative to the DSP as
discussed in Sec. III.2. We recall from that section that the dimensionless
correlation differs by two orders of magnitude between SPP and DSP. As a
consequence, Fig. 5 provides a visually intuitive confirmation that the
desired decorrelating effect is achieved by the DSP in the maximum likelihood
estimation. Furthermore, we comment that the reduction of uncertainty in
target and nuisance parameters alike by DSP’s leads to a narrower posterior
along the $V_{L}$ axies for the DSP.

A limitation of the Bayesian analysis shown in Fig. 5 is that it only reveals
performance of the accelerometer locally around the parameters of the
simulated measurement record. It is important, however, to also consider
device operation using the DSP over a broader range of parameter space, which
we consider in the following section.

###  IV.2 Jenson-Shannon Divergence

Generally speaking, the precision of a quantum sensor is determined by how
rapidly the measurement outcomes change with small variations $d{\bf b}$ of
parameters. Consequently, if the outcomes $P(p|{\bf b})$ and $P(p|{\bf
b}+d{\bf b})$ are too “close”, the sensor may fail to distinguish different
regions in parameter space during Bayesian estimation. To quantify this
statistical notion of closeness, we analyze the Jenson-Shannon divergence
(JSD) which, for two probability distributions $P$ and $Q$, is given by

$D_{JS}(P||Q)=\frac{1}{2}\left[S(P||M)+S(Q||M)\right],$ (16)

where $M=\frac{1}{2}(P+Q)$ is a mixture of $P$ and $Q$, and
$S(A||B)=-\sum_{p}A(p)\log_{2}\frac{B(p)}{A(p)}$ is the relative entropy (also
known as the Kullback–Leibler divergence) of $A$ with respect to $B$. The JSD
captures the shared information between $P$ and $Q$ given measurements from a
mixture of the two. Intuitively, a small value of the JSD means the two
probability distributions almost perfectly coincide, and therefore would be
difficult to resolve via Bayesian estimation. In this section, we extend the
usage of the JSD for optical lattice interferometry from Ref. [45] to
performing accelerometry in the multiparameter domain, while emphasizing that
the discussion applies more generally to characterizing any DSP where
information about a (potentially) large space of target and nuisance
parameters must be compressed and analyzed.

We first relate the JSD to the CFIM by expanding for small changes in
parameters for the case of accelerometry considered in the present work (see
App. D for general expressions)

$\displaystyle D_{JS}(P(p|a,V_{L})$ $\displaystyle||P(p|a+da,V_{L}+dV_{L}))$
$\displaystyle\approx\frac{1}{8}\left[I_{a}da^{2}+I_{V_{L}}dV_{L}^{2}+2\mathcal{I}_{a,V_{L}}da\
dV_{L}\right].$ (17)

The CFIM determines the local curvature of the JSD, relating small parameter
variations to displacements on the smooth statistical manifold [64], impacting
Bayesian estimation (Eq. (8)), and the decorrelization effect of the protocol.
Expanded instead using the normalized parameterization ${\bf\tilde{b}}$
discussed in Sec. III.2, the matrix element $\mathcal{I}_{a,V_{L}}$ is
replaced by the dimensionless correlation, highlighting the universal
relevance of this quantity.

Figure 6:  Visualization of the Jensen-Shannon divergence for the DSP as pairs
of parameters are varied, including (a) fixed lattice depth and varying
acceleration, (b) fixed acceleration and varying lattice depth, and (c)
varying acceleration and lattice depth. The non-varied parameters are held
fixed at the $a/g=0$ and $V_{L}/E_{R}=10$ for consistency. In panels (a) and
(c), dashed white lines indicate regions where the phase repeats by $2\pi$,
denoting the effective range of the sensor for performing accelerometry. In
(c), we plot the JSD around the training point for DSP (indicated with a
cross) whose local curvature matches the final CFIM elements presented in Fig.
3. In panels (a) and (b), the blue dotted lines indicate the main diagonal
where the JSD is vanishing due to perfect coincidence of the probability
distributions. Panel (d) shows the
$D_{JS}(P(p|a,V_{L}=10E_{R})||P(p,a^{\prime}=0g,V_{L}=10E_{R})$ as a function
of acceleration $a$ in dotted red lines, which is equivalent to a evaluating a
horizontal slice of (a) at $a^{\prime}=0$. The local $I_{a}$ along this slice
is also shown as a black solid line. We see that the CFI envelope decays
signficantly by $a/g=\pm 0.11$ and the JSD envelope increases along the same
range, allowing an estimation of the sensor’s effective range.

Having established the local connection between JSD and CFIM, we now plot the
JSD over finite domains of interest in parameter space for centered around the
coordinates used for optimizing the DSP from Sec. III.2. In principle, this
requires a four-dimensional analysis however, we consider instead variations
of pairs of parameters in order to illustrate our main findings. Figure 4(a)
shows the JSD with lattice depth fixed at the optimized coordinate with
varying accelerations. The sensitivity of the protocol is reflected in the
curvature surrounding the diagonal according to Eq. (IV.2)) which depends
directly on $I_{a}$. Away from the diagonal, regions of nearly-vanishing JSD
are also observed, indicating repetition of likelihood functions at different
locations in parameter space. The low JSD off-diagonals arise due to the
spatial inversion symmetry of the lattice Hamiltonian in the absence of
shaking as noted in Ref. [45]. Shaking breaks this symmetry, leading to
offsets of these regions from the origin, with the sign of the offset
depending on the directionality of the initial shaking actions. We expect such
considerations to apply generally when target parameters obey similar
symmetries of the Hamiltonian under consideration.

In Fig. 4(a) it is clear that there are in fact numerous diagonals and off-
diagonals, leading to a visible gridded pattern, which is a signature of
aliasing. While visually obvious in our JSD plots, aliasing limits the
accuracy of our sensor, since target parameters that fall on different
diagonals or off-diagonals can have similar likelihood functions and thus be
difficult to distinguish via Bayesian estimation. Thus, for accurate and
efficient determination of acceleration by evaluating a maximum likelihood
estimate, one should be careful to restrict Bayesian updating to the parameter
domain within distinct cells of the aliasing grid. We expect that such
aliasing is a generic feature, present regardless of whether a DSP or SPP is
used.

In addition to information about aliasing, Fig. 4(a) reveals additional
information about the (broader) effective range of the DSP’s precision. This
can be seen in the loss of contrast in the aliasing grid that occurs (roughly)
for accelerations exceeding $0.11g$ in magnitude. This loss of contrast can be
quantified by evaluating $I_{a}$ locally along a horizontal slice of Fig. 6(a)
as shown in panel (d). Here the effective range appears as a slowly varying
envelope on top of the local classical Fisher information, implying a loss of
precision as we move away from $a/g=0$. We see this decay in performance
reflected in $D_{JS}(P(p|a,V_{L})||P(p|a^{\prime}=0,V_{L})$ which is also
shown in panel (d). Here, the overall envevlope of the JSD function increases
as the envelope of $I_{a}$ decreases. The decay in $I_{a}$ is partly due to
the DSP being optimized for measuring acceleration’s around $a/g=0$ as a phase
measurement in the reference frame falling with the atoms (see App. A, Eq.
(23)). In particular, the values $a/g=\pm 0.11$ coincide with $2\pi$ phase
difference of the phase of the lattice. Similar considerations will apply to
any DSP where the target parameter relies on a phase measurement.

Next, we plot the JSD for fixed acceleration and varying lattice depth in Fig.
6(b). We see that the curvature of the JSD away from the diagonal blue line
results in a slow variation in the JSD as a consequence of low $I_{V_{L}}$,
indicating that the likelihood functions in Fig. 4 vary slowly over the
considered values of lattice depths. Furthermore, this curvature is a
reflection of the slow variation of the measurement outcomes found in Fig. 4
for all momentum as the lattice depth is varied at fixed acceleration. Thus,
while the same considerations about aliasing and effective range apply to
variations in the lattice depth, they do not affect estimation of acceleration
via Bayesian updating.

In order to reveal the decorrelation properties of the DSP, we now evaluate
the JSD for simultaneous variation of the lattice depth and acceleration as
shown in Fig. 6(c). Explicitly, we evaluate the JSD as a function of
$P(a,V_{L})$ and $P(a^{\prime},V_{L}^{\prime})$ for fixed $a/g=0$ and
$V_{L}/E_{R}=10$. This form was chosen because the curvature and corresponding
CFIM elements determine the values of the JSD around the origin according to
Eq. (IV.2) and the results shown in Fig. 3. Qualitatively, as acceleration is
varied at fixed lattice depth, we find many distinct horizontal regions of
low-valued JSD. The narrowness of these regions as a function of $a$ reflects
the high sensitivity (large $I_{a}$) of the DSP to the target parameter.
Similarly, these regions are broad as a function of $V_{L}$ due to low
$I_{V_{L}}$. The decorrelating effect of the DSP is visible in an absence of
noticeable tilt in the central region due to the small $\mathcal{I}_{a,V_{L}}$
and dimensionless correlation. The degree to which each region is tilted is
however not uniform as both acceleration or lattice depth is varied away from
the point $a/g=0$, $V_{L}/E_{R}=10$ where the DSP was optimized. Importantly,
this figure provides also additional performance information about the
effective range over which the decorrelating effect of the sensor applies.
This information is encoded in the parameter region beyond which the tilt
reappears, indicating the development of significant correlations between
target and nuisance parameters.

##  V Conclusion

In this work, we have shown how target signals can be decorrelated from
fluctuations of nuisance parameters by desinging sensing protocols using
machine learning and multiparameter estimation theory. This was demonstrated
in the specific case of performing accelerometry on an optical lattice
interferometer, however the formulation of information-theoretical goals in
this work is broadly applicable to decorrelation tasks in quantum metrology
and computing. In addition to generating a concrete example of a decorrelated
sensing protocol (DSP), we also perform statistical analysis on the
measurement outcomes. Here visual signatures of the decorrelating effect of
the sensing protocol were found as a consequence of the connections between
the Fisher information matrix, measures of informational differences between
measurements, and curvatures on statistical manifolds. Additionally, we study
the Jensen-Shannon divergence and establish the effective ranges for a DSP for
operating as an accelerometer with ensured decorrelation. A simple scheme for
lattice depth estimation is also analyzed in App. E for completeness.

Although DSP’s were demonstrated in this work for a specific platform, the
method of providing machine learning with knowledge of the information
geometry of a quantum system in order to arrive at states which are
metrologically advantageous as well as robust is quite general. In particular,
the search for protocols that yield robust gates (c.f. Refs. [75, 76, 42]) and
optimal use of quantum resources [35, 36, 37] is a highly active topic where
these methods may find fruitful applicability. Additionally, the single target
and nuisance parameter example studied in this work can be extended
straightforwardly to the many-parameter setting using the general formalism of
Sec. III. This setting describes metrological applications in both high-
precision metrology as well as the operation of quantum devices in noisy,
real-world environments where intense efforts are underway to transition
quantum technologies out of the lab and into the field.

###### Acknowledgements.

The authors acknowledge Catie LeDesma, Kendall Mehling, John Cooper, Chris
Wilson, Pranav Gohkale, Maximillian Seifert, Dana Anderson, Zack Pagel, and
Noah Fitch for fruitful discussions. This research was supported in part by
NASA under grant number 80NSSC23K1343, NSF OMA 2016244, NSF OSI 2231377, and
by a Sponsored Research Agreement between the University of Colorado Boulder
and Infleqtion.

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##  Appendix A Augmented states

In order to evaluate the Fisher information matrices, one must compute inner
products involving the derivatives of the quantum state
$|\partial_{b^{\mu}}\psi\rangle$ with respect to the parameters of the model.
This can be done in a variety of ways. In the absence of a lattice, it can be
done analytically as discussed in App. B. In the presence of a lattice, one
must resort to numerical methods for instance by running the simulation at
parameter values ${\mathbf{b}}$, ${\mathbf{b}}-d{\mathbf{b}}$, and
${\mathbf{b}}+d{\mathbf{b}}$ in order to evaluate the state derivatives via
finite differencing. This is what was done in Ref. [40]. Alternatively, one
can form an augmented state composed of the quantum state and the state
derivatives
$(|\psi\rangle,|\partial_{b^{1}}\psi\rangle,|\partial_{b^{2}}\psi\rangle,\dots)$.
Utilizing this method of augmented states (c.f. [79, 75]), one obtains the
following equations of motion by taking partial derivatives of the Hamiltonian
(Eq. (5))

$\displaystyle i\hbar\frac{d}{dt}|\psi\rangle=\hat{H}(t)|\psi\rangle,$ (18)
$\displaystyle
i\hbar\frac{d}{dt}|\partial_{b^{\mu}}\psi\rangle=\hat{H}(t)|\partial_{b^{\mu}}\psi\rangle+(\partial_{b^{\mu}}\hat{H}(t))|\psi\rangle.$
(19)

The partial derivative of the Hamiltonian with respect to the lattice depth is
given by

$\displaystyle\partial_{V_{L}}\hat{H}(t)=-\frac{1}{2}\cos(2k_{L}\hat{x}+\phi(t)).$
(20)

To evaluate the partial derivative with respect to the acceleration, we
perform the frame transformation discussed in Ref. [45]. Here, application of
the Galilean boost generator $\hat{\mathcal{K}}(t)=\exp(-imat\hat{x}/\hbar)$
yields the transformed Hamiltonian [80]

$\displaystyle\hat{H}_{\mathrm{acc}}(t)$
$\displaystyle=\hat{\mathcal{K}}(t)\hat{H}(t)\hat{\mathcal{K}}(t)^{\dagger}-i\hbar\left[\partial\hat{\mathcal{K}}(t)/\partial
t\right]\hat{\mathcal{K}}(t)^{\dagger}$
$\displaystyle=\frac{\hat{\pi}^{2}(t)}{2m}-\frac{V_{L}}{2}\cos\left[2k_{L}\hat{x}+\phi(t)\right].$
(21)

where the inertial force $ma\hat{x}$ is now absent and the kinematic momentum
becomes $\hat{\pi}(t)\equiv\hat{p}-mat$. In this frame, one finds

$\partial_{a}\hat{H}^{(1)}(t)=-\hat{p}t$ (22)

where we ignore a term contributing a constant to the transformed Hamiltonian
and note that Eq. (20) is unchanged
($\partial_{V_{L}}\hat{H}(t)=\partial_{V_{L}}\hat{H}_{\mathrm{acc}}(t)$) in
the boosted frame. Finally, we note that an additional frame transformation
was discussed in Ref. [45] where hte acceleration appears instead as an
additional phase shift. Here, the transformation is to the reference frame
falling with the atoms, yielding

$\hat{H}^{(2)}(t)=\frac{\hat{p}^{2}}{2m}-\frac{V_{L}}{2}\cos\left[2k_{L}+\phi_{a}(t)\right].$
(23)

where $\phi_{a}(t)=\phi(t)-k_{L}at^{2}$. Thus, measuring acceleration amounts
to a phase measurement $\phi_{a}$ around $a=0g$. This also implies that the
$\phi_{a}$ phase wraps evey $a_{0}=\frac{2\pi}{k_{L}t^{2}}$, treating
acceleration values that differ by an $a_{0}$ shift equally.

While the state vector has unit norm, the norm of the state derivatives, which
is initially set to zero, is not fixed due to the source term in Eq. (19),
which is responsible for generating quantum Fisher information, related to
parameter $b^{\mu}$ along the quantum trajectory. Generally, the dynamics of
the augmented quantum state vector has a geometrical interpretation as the
system moves along a trajectory in Hilbert space and acquires sensitivity to
the parameters of the system. In practice, the augmented state vector can be
propagated in time using an RK4 algorithm.

##  Appendix B Free-space Fisher information bounds

In this section, we derive the result
$I_{\mathrm{RI}}(p_{0})=4I_{\mathrm{MZI}}(p_{0})$ for the Fisher information
in the Ramsey configuration discussed in Sec. II.3 in connection with Ref.
[41]. This result can be analytically derived in free space using the
augmented state method outlined in App. A Eqs. (18) and (19) can be calculated
analytically as we now discuss.

We begin by considering an arbitrary initial state $|\psi(t=0)\rangle$, and
then analytically obtain the evolution of the associated augmented state
vector by solving Eqs. (18) and (19). The Fisher information at the terminal
time $\mathcal{T}$ can be then be straightforwardly extracted. The equation of
motion for the quantum state is given by Eq. (18), which can be solved as
$|\psi(t)\rangle=\hat{U}(t)|\psi(t=0)\rangle$, where
$\hat{U}(t)=\exp(-i\int^{t}ds\hat{H}(s))$ is the evolution operator

$\displaystyle\hat{U}(t)=$
$\displaystyle\exp\left[-\frac{i}{2m\hbar}\int^{\mathcal{T}}_{0}ds\
\hat{\pi}(t)^{2}\right],$ (24) $\displaystyle=$
$\displaystyle\exp{\left[-\frac{i}{2m\hbar}\left(\hat{p}^{2}\mathcal{T}-ma\mathcal{T}^{2}\hat{p}+\frac{m^{2}a^{2}\mathcal{T}^{3}}{3}\right)\right].}$
(25)

Likewise, the equation of motion (Eq. (19)) for the state derivative
$|\partial_{a}\psi(t)\rangle$ can be solved as

$\displaystyle|\partial_{a}\psi(t)\rangle=$ $\displaystyle\
\partial_{a}\hat{U}(t)|\psi(0)\rangle,$ $\displaystyle=$
$\displaystyle-i\hat{\mathcal{G}}_{a}(t)\hat{U}(t)|\psi(0)\rangle$ (26)

where

$\displaystyle\hat{\mathcal{G}}_{a}(t)=$ $\displaystyle\int_{0}^{t}ds\
\partial_{a}\hat{H}(s),$ $\displaystyle=$
$\displaystyle-\frac{t^{2}}{2\hbar}\left(\hat{p}-\frac{2mat}{3}\right)$ (27)

is the generator associated with the acceleration [41]. Note that this
generator is the same in origin as the source term in the App. A.

At time $t=\mathcal{T}$, one obtains

$\ket{\partial_{a}\psi(\mathcal{T})}=\left[-\frac{i}{2m\hbar}\left(\frac{2am^{2}\mathcal{T}^{3}}{3}-m\mathcal{T}^{2}\hat{p}\right)\right]\ket{\psi(\mathcal{T})},$
(28)

We see from the generator that acceleration encodes a phase difference between
two momentum states based on their momentum difference. Thus, a maximal phase
difference can be generated by choosing an equal superposition of two momentum
states with a high momentum splitting.

To obtain the Fisher information, we specify an initial condition
$\ket{\psi(t=0)}=\ket{\pm p_{0}}$ for the instantaneous application of an
ideal beam-splitter to the the quantum state at $t=0$. In this state,
$\langle\hat{p}\rangle=0$ and $\langle\hat{p}^{2}\rangle=p_{0}^{2}$, and Eq.
(10) gives
$F_{\mathrm{RI}}(p_{0})=\left(\frac{p_{0}\mathcal{T}^{2}}{\hbar}\right)^{2}$.
The instantaneous application of an ideal recombiner at $t=\mathcal{T}$
achieves $I_{\mathrm{RI}}(p_{0})=F_{\mathrm{RI}}(p_{0})$ [41].

##  Appendix C Reinforcement learning

Reinforcement learning is a particularly well suited for method for generating
control functions for quantum problems, such as the end-to-end construction of
DSP’s considered in this work, where the dimensionality of the problem can
become very high and the memory footprint of quantum optimal control methods
becomes prohibitive [40, 51, 45, 52, 53, 54, 55, 56]. We employ a specific,
model-free kind of reinforcement learning using the double-deep-Q approach
[81, 82], which has been applied previously to design control functions for
optical lattice interferometers [40, 51, 45]. Briefly, the reinforcement
learning agent selects an action $\mathbf{a}$ when exposed to the state
$\mathbf{s}$ of an environment. The environment subsequently carries out that
action and returns to the agent a reward $\mathbf{r(s^{\prime})}$ based on the
“quality” of the resulting state $\mathbf{s^{\prime}}$. We use Q-learning for
the decision making process of the agent which involves the evaluation of
quality factors, called Q-values $\mathbf{Q(s,a)}\in\mathbb{R}$, using neural
networks, as depicted Fig. 1, where each element represents the desirability
of a possible action $\mathbf{a}$ given a state $\mathbf{s}$. For action
selection, the agent uses a policy $\pi$ based on the Q-value to decide the
next action - for our case, we employ a greedy policy selects the action
corresponding to the largest Q-value for the current state. To strike a
balance between learning and exploration, we employ epsilon greedy
exploration, where the agent decides the next action either randomly or based
on $\pi$ using a probability $\epsilon$. The probability of exploration
$\epsilon$ is steadily decreased with subsequent training episodes.

As discussed in Sec. III, the goal of a DSP is to optimize sensitivity to a
target parameter while minimizing the effect of nuisance parameters.
Therefore, we formulate rewards as $\mathbf{r}[\mathcal{I}]$. We comment that
Ref. [40], considered the reward $\mathbf{r(s^{\prime})}=I_{b^{1}}$,
appropriate for single-parameter estimation with perfect information about all
other parameters of the system, producing control functions maximizing single-
parameter sensitivity (SPP’s). For DSP’s, the agent requires access to more
information in order to perform the dual tasks of sensitivity optimization and
decorrelation. Here, the natural candidate is to construct reward functions
for DSP’s from elements of Eq. (13), which ultimately determine the
theoretical sensitivity of the device in the multiparameter setting (see Eq.
(9)). Explicitly, the results of Secs. III.2 and IV were produced using
$\mathbf{r}[\mathcal{I}]=f[\mathcal{I}]/(1-f[\mathcal{I}])$ with
$f[\mathcal{I}]=(2I_{\mathrm{MZI}}(\mathcal{I})^{-1}_{a,a})^{-1}=\zeta_{a}/2$.
We note that this can be translated to a reward function for SPP’s as the
assuption of perfect information about untargeted parameters produces the
mapping $\mathcal{I}^{-1}_{a,a}\to 1/I_{a}$ Additionally, DSP-appropriate cost
functions for quantum control methods can be constructed analogously although
we do not pursue that avenue in this work. In Q-learning, the necessary
condition that must be satisfied in order to describe optimality is given by
the Bellman equation [81]

$\mathbf{Q_{\pi}(s,a)}=\mathbf{r(s^{\prime})}+\mathbf{\gamma\
\underset{a^{\prime}}{\mathrm{max}}\ Q_{\pi}(s^{\prime},a^{\prime})}.$ (29)

This equation states that optimality is achieved when the Q-value
($\mathbf{Q(s,a)}$) of taking action $\mathbf{a}$ in state $\mathbf{s}$ under
policy $\mathbf{\pi}$ is equal to the Q-value
($\mathbf{Q(s^{\prime},a^{\prime})}$) in the expected next state $s^{\prime}$
discounted by the factor $\mathbf{\gamma}$ and maximized over all possible
actions $\mathbf{a^{\prime}}$ under $\mathbf{\pi}$ plus the reward received
“along the way” $\mathbf{r(s^{\prime})}$. Essentially, the value of taking a
particular action in a particular state is determined by the reward received
in the following state plus the value of making optimal decisions informed by
the Q-values thereafter.

The Bellman equation is a recursive relation between Q-values and in low-
dimensional problems, a tabular approach can be taken to record and iterate
the Q-values [81]. However due to the enormous dimensionality of the present
problem, we employ neural networks in the double-deep-Q-approach and train
using repeated runs of a numerical simulation or experiment. In practice, two
neural nets are used in order to evaluate Eq. (29) and provide additional
stability [82, 83, 51], the Q-network and a target network. The Q-network is
used to evaluate $\mathbf{Q(s,a)}$ while $\mathbf{Q(s^{\prime},a^{\prime})}$
is evaluated using the target network, and then the difference between the
left and right hand sides of the Bellman Equation (Eq. (29)) is evaluated and
backpropagated through to update the Q-network. The target network is then
updated via a weighted average, with weighting $\mathbf{\tau}$. Additional
details of the double-deep-Q algorithm applied to optical lattice
interferometry can be found in Refs. [51, 40].

As indicated in Table 1, each neural net has a single hidden layer with 64
nodes all fully connected. The set of actions corresponds to 16 possible
discrete amplitude values

$\mathbf{\mathcal{A}}=\\{n\pi/12|n=0,1,\dots,15\\}.$ (30)

These actions determine the control parameters $\mathbf{c}$ which translate
into shaking over a half period oscillation of the form
$\phi(t)=\mathrm{Amp}(t)\times\sin(\omega_{s}t)$ where
$\omega_{s}=11.5\omega_{R}$ with $\omega_{R}=E_{R}/\hbar$. The time interval
between transitions is then $\pi/11.5\approx 0.27\omega_{R}$. This is a
variant of the action set considered in Ref. [51] where it was found to work
well in driving transitions between low-lying Bloch states.

The selection of features $s$ to input in the neural net requires physical
intuition. In Ref. [51], the momentum-space occupations were used. In Ref.
[40], position-space expectation values to provide the agent additional
information about angular momentum for rotational sensing. From Eq. (5), we
see then that the expectation value $\langle\hat{x}\rangle$ should be added to
the feature set. Physically, this expectation value provides information to
the agent about the free-fall of the non-inertial frame as appropriate for
accerometry. In App. A, it was shown how the acceleration enters the
Hamiltonian proportional to $t$ (see Eq. (A)). The features consist then of
the dimensionless flag $t/\mathcal{T}$ along with the parity-sensitive
populations of states $|p,\pm\rangle$, which provides additional information
allowing the agent to distinguish, for instance, between nearly-degenerate,
opposite-parity Bloch states in the valence band. The flag $t/\mathcal{T}$
also allows the agent to distinguish between mirrored matter-wave
trajectories, for instance in the Mach-Zehnder configuration.

Table 1: Choice of hyperparameters for the design of DSP’s. The exploration probability decays exponentially for each episode between 1.0 and 0.1 at the rate represented by $\mathbf{\epsilon}$ decay. The “Adam” optimizer is a commonly-used method for stochastic gradient descent [84]. Hyperparameters | Values
---|---
$\mathbf{\gamma}$ | 0.99
$\mathbf{\tau}$ | 0.8
$\mathbf{\alpha}$ | $10^{-3}$
episodes | 5000
$\mathbf{\epsilon}$ decay | $2.5\times 10^{-5}$
Hidden size | 64
Batch size | 100
Optimizer | Adam

##  Appendix D Expansions of the Jensen-Shannon divergence

We provide further details on the analytic derivation of the expansions of the
Jensen-Shannon divergence and connections with the CFIM given in Sec. IV. We
begin by expanding $D_{JS}(P({\bf b})||P({\bf b}+d{\bf b}))\equiv f({\bf
b}+d{\bf b})$ in a Taylor series to second order in the differential parameter
vector $d{\bf b}=(db^{1},db^{2},...)$, yielding

$f({\bf b}+d{\bf b})\approx f({\bf b})+[\nabla_{\mu}f({\bf
b})]db^{\mu}+\frac{1}{2}H_{f}({\bf b})_{\mu\nu}\ db^{\mu}db^{\nu},$ (31)

where $H_{f}({\bf b})$ is the (symmetric) Hessian matrix whose elements are
defined as $(H_{f}({\bf b}))_{ij}\equiv\partial^{2}f/\partial b^{i}\partial
b^{j}$.

We discuss the evaluation of each term in the expansion in Eq. (31). The
contribution $f({\bf b})$ is vanishing by virtue of the definition of the JSD
(Eq. (16)). Likewise, the contribution of the gradient also vanishes because
the minimum of the JSD is absolute. The Hessian matrix is however nonzero,
making the contribution

$H_{f}({\bf b})_{\mu\nu}=\frac{\mathcal{I}_{\mu\nu}}{4}.$ (32)

which establishes the connection between the CFIM and the local curvature of
the JSD, quantified by the Fisher information metric (Eq. (4))

$D_{JS}(P({\bf b})||P({\bf b}+d{\bf b}))\
\approx\frac{1}{8}\mathcal{I}_{\mu\nu}\ db^{\mu}db^{\nu}\Big{|}_{\bf{b}}.$
(33)

This metric can be used to calculate information distances on smooth
statistical manifolds [64]. In the case considered in this work, only two
components of the differential element $d{\bf b}=(da,dV_{L})$ are varied and
one finds Eq. (IV.2).

##  Appendix E Decorrelated sensing protocol for lattice depth estimation

Figure 7: Lattice depth estimation using a DSP, including the control function
(a) momentum space dynamics (b), Bloch state occupations (c), and position
space dynamics (d). The position space visualizations use the same convolution
method described in the caption of Fig. 2. In practice the wave packet in an
optical lattice interferometer is delocalized over many lattice sites. Figure
8: Relevant classical and quantum Fisher information (FI) elements for (a)
acceleration, (b) covariance, and (c) lattice depth for the DSP designed for
lattice depth estimation. In (c) a $F_{V_{L}}\propto t^{3}$ power is found
once the valence band superposition is prepared (see Fig. 7(c)). Although not
aimed at accelerometry, we provide in panel (a) the same rescaling in units of
the Mach-Zehnder configuration result with dashed line indicating the optimal
result in the Ramsey configuration discussed in Sec. II.3.

Sections III.2 and IV presented and analyzed a DSP for accelerometry with
lattice depth treated as a nuisance parameter. It is however valuable to have
a precise estimate of $V_{L}$ prior to performing quantum sensing or
simulation tasks [85, 86, 87, 88]. Here, the strength $a$ of the inertial
force can be viewed instead as a nuisance parameter as the device may be
operating in an un-characterized or fluctuating inertial environment. In this
section, we present and analyze a DSP for this scenario, placing the result in
the context of existing methods, which typically provide uncertainties in the
$10-20\%$ range [89, 90, 91, 92].

The same set of actions (see App. C) are available to the agent for lattice
depth estimation as for accelerometry, and the reward function
$\mathbf{r}[\mathcal{I}]$ is again a function of the CFIM. However, we use an
alternative non-linear function to boost large rewards and speedup learning
$\mathbf{r}[\mathcal{I}]=\sinh(4/(\mathcal{I})^{-1}_{V_{L},V_{L}})$. We see in
Fig. 7(a) that the found DSP is largely zero after an initial repeated
pattern. We emphasize that a succession of repeated values found by the agent
is a highly-nontrivial pattern as the Q-values that the agent uses for
decision making are randomly chosen from the $16^{32}$ possible distinct
control functions. The initial forays of the agent in a reinforcement learning
algorithm are also exploratory in nature before becoming less random as the
training iterates over many episodes (see App. C and [81]).

In Fig. 7(b-d), we see that the DSP yields a periodic evolution in momentum
space after the quantum state is placed in a superposition of valence bands.
From Fig. 7(d) it is clear that transport of the wave packet between sites is
suppressed, which results from the gapped nature and weak dispersion of the
valence bands. This periodic oscillation between momentum components is
heavily reminiscent of the dipole oscillations studied in Ref. [93], and we
note that center of mass displacements can be seen even in Fig. 7(d).
Consequently, decorrelation is added to the list of robust features of this
method.

We now perform a Fisher analysis to understand why this method is favored as
well as how it achieves decorrelation. Figure 8 shows the Fisher information
matrices where it is clear that this DSP performs poorly as an accelerometer.
On the other hand, we find monotonic growth in the lattice-depth sensitivity
$F_{V_{L}}\propto t^{3}$ once the state is prepared in the valence band
superposition. Here, the actions serve to time a maximum of $I_{V_{L}}$ with
measurement, aided by the two nonzero actions following the initial pattern.

In terms of decorrelation, the DSP minimizes the penalty due to the nuisance
parameter $a$, with $\mathrm{Corr}(\ell_{a},\ell_{V_{L}})\approx 0.081$. How
does the sensitivity of this DSP compare to traditional methods?
Quantitatively, the result $I_{V_{L}}E_{R}^{2}=7.31$ yields an uncertainty in
the lattice depth of $0.37E_{R}$, providing a lower bound on the single-shot
error of $\sim 3\%$ for lattice depth estimation. This is comparable to the
experimental results of Ref. [93] using the sudden phase shift method, however
in that case many shots were required to achieve the fringe visibility
required for lattice depth estimation.

Figure 9: Final momentum probability distributions $P(p|a,V_{L})$, resulting
from the DSP for lattice-depth estimation, and presented over a range of $a$,
$V_{L}$, and $p$. We display occupations up to momenta $\pm 2\hbar k_{L}$ and
note that for larger momenta are not significantly occupied in this DSP as the
lattice depth or acceleration are varied significantly.

We now provide a brief statistical analysis of these results, extending the
discussion of Sec. IV to lattice depth estimation. The momentum probability
distributions $P(p|a,V_{L})$ are given in Fig. 9. Comparing to Fig. 4, we note
the reduced contrast and periodicity as acceleration is varied at fixed
lattice depth. This is a reflection of the poor performance of the lattice-
depth DSP as an accelerometer–as expected. However, The variation of the
distributions with lattice depth at fixed acceleration is more dramatic–again
due to the different design goals. Consequently, care should be taken when
Bayesian inferencing for lattice depth estimation to restrict to smaller
regions of $V_{L}$.

Figure 10: Visualization of the Jensen-Shannon divergence for the lattice-
depth focused DSP as pairs of parameters are varied, including (a) fixed
lattice depth and varying acceleration, (b) fixed acceleration and varying
lattice depth, and (c) varying acceleration and lattice depth. As in Fig. 6,
the non-varied parameters are held fixed at $a/g=0$ and $V_{L}/E_{R}=10$
although a different DSP is used in this case. In panels (a) and (c), dashed
white lines indicate the effective range of acceleration for the DSP. In (b) a
larger portion of the aliasing grid is shown for the lattice depth however the
effective range for the DSP is not reached over the parameter range
considered. In (c) the cross indicates the origin whose local curvature
matches the CFIM elements presented in Fig. 8. The green dotted lines in (a)
and (b) indicate the main diagonals where the JSD vanishes. (d) Evaluation of
the JSD (dotted red line) as $D_{JS}(V_{L}||V_{L}^{\prime})$ with
$V^{\prime}_{L}/E_{R}=10$ and fixed acceleration, along with the local
$I_{V_{L}}$ evaluated along the $V_{L}=10E_{R}$ horizontal of panel (b). The
effective range of the DSP for performing lattice depth estimation is not
exceeded over the range of $V_{L}$ shown, such that an envelope analogous to
the one found in Fig. 6(d) is not yet visible.

Furthermore, we compress the multi-fringed probability distributions into
entropy measures using the Jenson-Shannon divergence with results shown in
Fig. 10. First, we note that in panel (a) the aliasing grid pattern is harder
to observe compared to the accelerometry DSP. In particular, diagonal lines
where the JSD achieves low values are significantly broader now as a
consequence of the lower values of $I_{a}$ compared to Fig. 6(a). The
periodicity of the aliasing pattern found in Fig. 6 for accelerometry is here
greatly enlarged with only four unit cells visible over the same range
presented in Fig. 6(a). Instead, the aliasing pattern in lattice depth is much
more visible as shown in panel (b). The JSD curvature is also shown in Fig.
(c), reflecting the reduced correlation produced by the DSP, larger lattice-
depth curvature due to enhanced $I_{V_{L}}$, and lower acceleration curvature
due to reduced $I_{a}$–as expected. In panel (d) we show the local Fisher
information $I_{V_{L}}$ as a function of lattice depth, along with
$D_{JS}(P(p|a,V_{L}||P(p|a,V^{\prime}_{L})$ with $V^{\prime}_{L}/E_{R}=10$ and
$a=0g$. Here, greater lattice-depth sensitivites are found away from the point
$(0g,10E_{R})$ at which the machine learning agent was trained. However,
decorrelation of the two parameters is still only guaranteed near the training
point. Because we do not consider a wide enough range of $V_{L}$, we do not
reach values at which the effective operating range of the sensor in terms of
the lattice depth is reached (unlike the remarks made surrounding Fig. 6(d)).
Here we caution that for larger lattice depths, the competition of
interactions and kinetic energy can drive a cold-atom system across an
insulating phase transition, which is beyond the scope of the present work
[94].