# Unified theory of classical and quantum signal sensing with a qubit

Wen-Long Ma<EMAIL_ADDRESS>State Key Laboratory of Superlattices and
Microstructures, Institute of Semiconductors, Chinese Academy of Sciences,
Beijing 100083, China  College of Materials Science and Opto-Electronic
Technology, University of Chinese Academy of Sciences, Beijing 100049, China

###### Abstract

Quantum sensing protocols typically uses a quantum sensor to sense classical
signals with the standard Ramsey inteferometry measurements. The classical
signals are often real numbers determining the sensor Hamiltonian. However,
for a senor embedded in a quantum environment, the signal to detect may be a
quantum operator on a target quantum system. There is still no systematic
method to detect such a quantum signal. Here we provide a general framework to
sense static quantum signals with a qubit sensor by the Ramsey interferometry
measurements, with the static classical signal sensing incorporated as a
special case. This framework is based on a novel approach to simultaneously
estimating the eigenvalues of the quantum signal operator with sequential
projective measurements of the sensor, which can extract useful information
about the target quantum system. The scheme can also be extended to sense ac
quantum signals with dynamical decoupling control of the sensor. As an
example, we show that a qubit sensor can simultaneously detect the individual
coupling strengths with multiple target qubits in a spin-star model.

Quantum sensing concerns the task of measuring physical quantities with a
quantum sensor Degen2017 ; Giovannetti2004 ; Giovannetti2011 ; Pirandola2018 .
Quantum properties, such as quantum coherence or entanglement, can make the
quantum sensor have better measurement sensitivity or precision than classical
sensors Bollinger1996 ; Giovannetti2006 ; Balasubramanian2008 ; Maze2008 ;
Rondin2014 ; Barry2020 . Quantum sensing protocols often follow the generic
steps including sensor initialization, interaction with the signal, sensor
measurement and signal estimation. A canonical sensing protocol is the Ramsey
interferometry measurement (RIM) Ramsey1950 ; Lee2002 ; Taylor2008 ; Liu2019 ,
where a classical signal acts as a static parameter in the qubit sensor
Hamiltonian [Eq. (1)] (this signal is classical in the sense that it is only a
real number). With the RIM sequence, the classical signal influences the
probabilities of the measurement results in a single RIM. The measurement
outcomes of different RIMs are assumed to be independent and identically
distributed. So by repeating the RIMs sequentially and averaging over the
measurement results, the classical signal can be accurately estimated within
standard quantum limit. Various other schemes based on sequential or
continuous measurements have been proposed to estimate unknown parameters in
open quantum systems Mabuchi1996 ; Gambetta2001 ; Catana2012 ; Guta2011 ;
Gammelmark2013 ; Gammelmark2014 ; Horssen2015 ; Kiilerich2015 ; Burgarth2015 ;
Demkowicz2017 ; Zhou2018 .

Figure 1:  Quantum circuits and measurement statistics of classical and
quantum signal sensing with sequential RIMs. (a) Quantum circuit for sensing a
classical signal $v$ [$R_{z}(vt)=e^{-ivt\sigma_{z}}$] with a qubit sensor.
Each measurement has two outcomes $\\{0,1\\}$. The probability to obtain
outcome 0 is $p_{0}=[1-\cos(2vt+\phi)]/2$, and the number of outcome 0 in $m$
measurements is denoted as $m_{0}$. (b) The probability of the frequency
$f_{0}=m_{0}/m$ of outcome 0 for classical signal sensing, which obeys a
binomial distribution around $p_{0}$. (c) Quantum circuit for sensing a
quantum signal $V$ ($U_{0}=e^{-iVt}$ and $U_{1}=U_{0}^{\dagger}$) with the set
of eigenvalues $\\{v_{k}\\}_{k=1}^{s}$. (d) The probability distribution of
$f_{0}$ for quantum signal sensing, which can display multiple binomial
distributions around $\\{p_{10},\cdots,p_{s0}\\}$ with
$p_{k0}=[1-\cos(2v_{k}t+\phi)]/2$.

In practice, a quantum sensor is embedded in a quantum environment, with the
target system as part of this quantum environment, such as a nitrogen-vacancy
(NV) electron spin to detect some nuclear spins in a spin bath Zhao2011 ;
Zhao2012 ; Kolkowitz2012 ; Taminiau2012 ; Shi2014 ; Abobeih2019 . So generally
the signal to detect should be a quantum signal, that is, a Hermitian operator
on the target system [Eq. (6)]. The quantum signal operator can be either
static or time-varying. For a time-varying quantum signal operator, its
influence on the sensor can sometimes be approximated by a classical
stochastic noise Abobeih2019 ; Cywinski2008 ; Witzel2014 ; Ma2015 , but
exploiting the quantum nature of the signal operator enables the detection of
high-order quantum correlations and quantum non-linear spectroscopy Wang2019 ;
Wu2022 ; Wang2021 ; Meinel2022 ; Shen2023 . A more basic problem is to fully
sense a static quantum signal operator (such as its eigenvalues), which
corresponds exactly to the static classical signal sensing with RIMs. This
poses several interesting questions: can we design a scheme to sense such
static quantum signals? What information about the signals can we detect? What
is the relationship between the classical and quantum signal sensing in this
sense?

In this paper, we answer these questions by developing a unified framework to
sense both classical and quantum signals with a qubit senor. The important
finding is that the sequential RIM protocol generally works for sensing the
eigenvalues of static quantum signals, with the static classical signal
sensing as a special case. But in contrast to classical signal sensing, the
measurement statistics of sequential RIMs are not independent and identically
distributed. Each RIM of the sensor induces a quantum channel (or generalized
quantum measurement) on the target quantum system. For sequential RIMs of the
sensor, the target system is gradually steered to the fixed points of the
quantum channel, corresponding to the states in different eigenspaces of
quantum signal operator. Conversely, the gradual state change of the target
system will influence the sensor measurement statistics. Surprisingly, the
sensor measurement statistics can display multiple binomial distributions [Eq.
(19)], with each distribution concentrating around some value determined by
the eigenvalues of the quantum signal operator. When the target system
initially start from an eigenstate of the quantum signal Hamiltonian, the
measurement statistics shows a single binomial distribution and is thus
equivalent to the case of classical signal sensing [Eq. (5)]. With dynamical
decoupling (DD) control of the sensor, the sequential RIM scheme can also
detect ac quantum signals. As an example, we show sequential RIMs can be
utilized to estimate the individual coupling strengths in a spin-star model.

RIMs for sensing classical signals. We start by reviewing the standard RIM for
sensing a classical signal [Fig. 1(a)]. A sensor qubit is directly coupled to
a static classical signal with a sensing Hamiltonian

$H_{c}=v\sigma_{z}=v(|0\rangle_{a}\langle 0|-|1\rangle_{a}\langle 1|),$ (1)

where $v$ is the classical signal amplitude and $\sigma_{i}$ is the Pauli-$i$
operator of the ancilla qubit ($i=x,y,z$). Hereafter we denote the sensor
rotation along the axis in equatorial plane as
$R_{\phi}(\theta)=e^{-i(\cos\phi\sigma_{x}+\sin\phi\sigma_{y})\theta/2}$ with
$\phi$ denoting the rotation axis and $\theta$ being the rotation angle. In a
single RIM sequence, the qubit initialized to state $|0\rangle_{a}$, first
undergoes a rotation $R_{\phi_{1}}(\pi/2)$, then evolves with the sensor
Hamiltonian $H$ for time $t$, undergoes another rotation $R_{\phi_{2}}(\pi/2)$
with the resultant state

$\displaystyle R_{\phi_{2}}(\pi/2)R_{z}(vt)R_{\phi_{1}}(\pi/2)|0\rangle_{a}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left[{\begin{array}[]{*{20}{c}}{{e^{-ivt}-e^{i(\Delta\phi+vt)}}}\\\
{-ie^{i\phi_{2}}\left({e^{-ivt}+e^{i(\Delta\phi+vt)}}\right)}\end{array}}\right]_{a},$
(4)

where $R_{z}(vt)=e^{-ivt\sigma_{z}}$, and $\Delta\phi=\phi_{1}-\phi_{2}$.
Finally a projective measurement (PM) is performed on the sensor with the
readout basis $\\{|0\rangle,|1\rangle\\}$. The probability distribution for
the outcomes $\\{0,1\\}$ is $F^{\prime}=(p_{0},p_{1})$ with
$p_{0}=[1-\cos(2vt+\Delta\phi)]/2$ and $p_{1}=1-p_{0}$.

The RIM are repeated $m$ times, and the frequencies of measurement outcomes
are recorded as $F=(f_{0},f_{1})$. Then $F$ obeys a binomial distribution
Bengtsson2017

$p_{c}(F)=\frac{m!}{(mf_{0})!(mf_{1})!}p_{0}^{mf_{0}}p_{1}^{mf_{1}}\approx
e^{-mS(F\|F^{\prime})},$ (5)

where $S(F\|F^{\prime})=\sum_{i=0,1}f_{i}\ln(f_{i}/p_{i})$ is the relative
entropy between $F$ and $F^{\prime}$. For large $m$, the above probability
distribution is concentrated around $F^{\prime}$ as a Gaussian distribution
since
$S(F\|F^{\prime})\approx\sum_{i=0,1}(f_{i}-p_{i})^{2}/(2p_{i})=(f_{0}-p_{0})^{2}/(2p_{0}p_{1})$
[Fig. 1(b)]. Therefore, $F^{\prime}$ can be approximated by the most probable
$F$ with the standard deviation $\sqrt{p_{0}p_{1}/m}$, and the classical
signal $v$ can be inferred from $F$ with a known free evolution time $t$.

RIMs for sensing quantum signals. We provide a basic model to sense a static
quantum signal [Fig. 1(c)]. Suppose the sensor is coupled to a $d$-dimensional
target quantum system with the coupling Hamiltonian Degen2017

$H_{q}=\sigma_{z}\otimes V=\sum_{k=1}^{s}v_{k}\sigma_{z}\otimes P_{k},$ (6)

where $V$ is a Hermitian operator on the target system,
$\\{v_{k}\\}_{k=1}^{s}$ is the set of eigenvalues of $V$, and
$\\{P_{k}\\}_{k=1}^{s}$ is the set of projection operators satisfying
$\sum_{k=1}^{s}P_{k}=\mathbb{I}$ with $\mathbb{I}$ being the identity operator
on the target system. Our objective is to characterize the quantum signal $V$
by estimating the set of eigenvalues $\\{v_{k}\\}_{k=1}^{s}$.

With the target system prepared in an eigenstate with eigenvalue $v_{k}$, the
quantum signal Hamiltonian [Eq. (6)] can be replaced by a classical one [Eq.
(1)]. So we can estimate $v_{k}$ by RIMs, with the same steps as those in
sensing classical signals. This is quite analogous to the quantum phase
estimation algorithm Nielsen2010 ; Cleve1998 , which is also designed to
estimate eigenvalues of a unitary.

For the target system in an arbitrary state $\rho$, a single RIM on the sensor
induces a quantum channel Wolf2010 ; Caruso2014 ; Watrous2018  on the target
system, which can be represented in the Stinespring representation as
Stine1955

$\Phi(\rho)={\rm Tr}_{a}[U(t)(\rho_{a}\otimes\rho)U^{\dagger}(t)],$ (7)

where $U(t)=e^{-iH_{q}t}=|0\rangle_{a}\langle 0|\otimes
U_{0}+|1\rangle_{a}\langle 1|\otimes U_{1}$ with
$U_{0}=\sum_{k=1}^{s}e^{-iv_{k}t}P_{k}$ and $U_{1}=U_{0}^{\dagger}$,
$\rho_{a}=|\psi\rangle_{a}\langle\psi|$ with
$|\psi\rangle_{a}=R_{\phi_{1}}(\pi/2)|0\rangle_{a}$, and ${\rm Tr}_{a}$
denotes the partial trace over the sensor. This quantum channel can also be
transformed to the Kraus representation as Kraus1983

$\Phi(\rho)=M_{0}\rho M_{0}^{\dagger}+M_{1}\rho M_{1}^{\dagger},$ (8)

with the Kraus operators $M_{0}=(U_{0}-e^{i\Delta\phi}U_{1})/2$ and
$M_{1}=(U_{0}+e^{i\Delta\phi}U_{1})/2$. Note that $\Phi$ is independent of the
second rotation $R_{\phi_{2}}(\pi/2)$, while $M_{0}$ and $M_{1}$ depend on the
phase difference $\Delta\phi$ between $R_{\phi_{1}}(\pi/2)$ and
$R_{\phi_{2}}(\pi/2)$. The probability to obtain result 0 is $p_{0}={\rm
Tr}(M_{0}\rho M_{0}^{\dagger})=[1-\sum_{k=1}^{s}{\rm
Tr}(P_{k}\rho)\cos(2v_{k}t+\Delta\phi)]/2$.

If the probability distributions of each RIM are assumed to be independent and
identical, we can still repeat the RIMs, estimate $\\{v_{k}\\}_{k=1}^{s}$ by
recording $p_{0}(t)$ as a function of $t$ and then perform Fourier analysis to
find the frequency components. This is the method to perform spectral analysis
of a unitary in the model of deterministic quantum computation with one
quantum bit (DQC1) Knill1998 ; Datta2005 ; Datta2008 ; Cable2016 . However,
this method neglects the backaction of sensor measurements on the target
system state in the sequential RIM scheme. Actually the target system
undergoes different state jumps with different measurement outcomes of the
sensor in a single RIM, which will influence the statistics of the next RIM.
Therefore, the statistics of sequential RIMs will show some non-classical
features. The main finding in this paper is that such non-classical features
of measurement statistics can be utilized to efficiently estimate
$\\{v_{k}\\}_{k=1}^{s}$. Below we exactly solve the backaction and measurement
statistics of sequential RIMs for sensing quantum signals.

Backaction of sequential RIMs. To analyse the backaction of sequential RIMs on
the target system, it is illuminating to study the behaviors of sequential
applications of the quantum channel $\Phi$. Previous works have studied the
asymptotic behaviors of sequential quantum channels Albert2019 ; Burgarth2013
; Novotny2018 ; Blume2010 . Our recent work shows that sequential quantum
channels with normal and commuting Kraus operators can simulate a PM in the
asymptotic limit Ma2023 . Our derivations below make use of this recent
theoretical finding.

First we introduce the natural representation of a quantum channel on the
Hilbert-Schmidt (HS) space of the target system Bengtsson2017 ; Watrous2018 .
The space of operators on the Hilbert space of the target system form a linear
vector space called the HS space, as can be seen by the matrix reshaping,
$X=\sum_{i,j=1}^{d}x_{ij}|i\rangle\langle
j|\leftrightarrow|X\rangle\rangle=\sum_{i,j=1}^{d}x_{ij}|ij\rangle\rangle$.
The inner product in the HS space is defined as $\langle\langle
Y|X\rangle\rangle={\rm Tr}(Y^{\dagger}X)$. Then the superoperator $X(\cdot)Y$
on the Hilbert space is equivalent to a linear operator $X\otimes Y^{T}$ on
the HS space, so the channel $\Phi$ can be naturally represented as
$\hat{\Phi}=\hat{\mathcal{M}}_{0}+\hat{\mathcal{M}}_{1}$ with
$\hat{\mathcal{M}}_{\alpha}=M_{\alpha}\otimes M_{\alpha}^{*}$ with
$\alpha\in\\{0,1\\}$ (note that we add hats for operators acting on the HS
space). With the HS space, the probability to get the $\alpha$th outcome is
$\langle\langle\mathbb{I}|\hat{\mathcal{M}}_{\alpha}|\rho\rangle\rangle={\rm
Tr}(M_{\alpha}\rho M_{\alpha}^{\dagger})$.

The channel $\hat{\Phi}$ can be recast into a neat form if we rewrite the
Kraus operators as Ma2023

$\displaystyle\left[{\begin{array}[]{*{20}{c}}{{M_{0}}}\\\
{{M_{1}}}\end{array}}\right]=\left[{\begin{array}[]{*{20}{c}}{{\bm{\lambda}_{1}}}&\cdots&{{\bm{\lambda}_{s}}}\end{array}}\right]\left[{\begin{array}[]{*{20}{c}}{{P_{1}}}\\\
\vdots\\\ {{P_{s}}}\end{array}}\right],$ (15)

where
$\bm{\lambda}_{k}=[e^{-iv_{k}t}-e^{i(\Delta\phi+v_{k}t)},-ie^{i\phi_{2}}(e^{-iv_{k}t}+e^{i(\Delta\phi+v_{k}t)})]^{T}/2$
is a unit column vector satisfying
$\langle\bm{\lambda}_{k},\bm{\lambda}_{k}\rangle=1$, with
$\langle\cdot,\cdot\rangle$ denoting the Euclidean inner product in the
complex vector space. Then $\\{\bm{\lambda}_{k}\\}_{k=1}^{s}$ is a set of such
unit vectors. Intriguingly, $\bm{\lambda}_{k}$ is similar to the sensor
wavefunction in Eq. (Unified theory of classical and quantum signal sensing
with a qubit). Then $\hat{\Phi}$ becomes a diagonal operator in the HS space
as

$\hat{\Phi}=\sum_{k,l=1}^{s}\langle\bm{\lambda}_{l},\bm{\lambda}_{k}\rangle
P_{k}\otimes P_{l},$ (16)

with the eigenvalues
$\\{\langle\bm{\lambda}_{l},\bm{\lambda}_{k}\rangle\\}_{k,l=1}^{s}$. Since
$|\langle\bm{\lambda}_{l},\bm{\lambda}_{k}\rangle|\leq\langle\bm{\lambda}_{l},\bm{\lambda}_{l}\rangle\langle\bm{\lambda}_{k},\bm{\lambda}_{k}\rangle=1$
due to the Cauthy-Schwarz inequality Garcia2017 , all the eigenvalues of
$\hat{\Phi}$ lie within the unit disk of the complex plane. The eigenvectors
of $\hat{\Phi}$ with eigenvalue 1 are called fixed points Arias2002 , and and
those with eigenvalues $e^{i\varphi}$ with $\varphi\neq 0$ are rotating points
Albert2019 . The HS subspace spanned by the fixed points and rotating points
are called asymptotic subspace (also known as peripheral or attractor
subspace). If any two unit vectors in $\\{\bm{\lambda}_{k}\\}_{k=1}^{s}$ are
not parallel, then the asymptotic subspace contains only the fixed points,
which are all eigenstates of $V$. With sequential applications of
$\hat{\Phi}$, the projections to the asymptotic subspace remain unchanged,
while the projections to the other eigenspaces gradually vanish Albert2019 ;
Burgarth2013 ; Novotny2018 ; Blume2010 . So for large $m$,
$\hat{\Phi}^{m}\approx\sum_{k=1}^{s}\hat{\mathcal{P}}_{k}$ approximates a PM
on the target system Ma2023 ; Ma2018 ; Wang2023 ; Liu2017 ; Rao2019 ;
Dasari2022 , with $\hat{\mathcal{P}}_{k}=P_{k}\otimes P_{k}$ corresponding to
the projection superoperator $P_{k}(\cdot)P_{k}$.

Measurement statistics of sequential RIMs. The normal and commuting Kraus
operators [Eq. (15)] also allow an exact solution of the measurement
statistics for sequential RIMs. Since
$[\hat{\mathcal{M}}_{0},\hat{\mathcal{M}}_{1}]=0$, we can expand
$\hat{\Phi}^{m}$ according to the binomial theorem,
$\hat{\Phi}^{m}=\sum_{F}\hat{\Phi}^{m}(F)$ with

$\hat{\Phi}^{m}(F)=\frac{m!}{(mf_{0})!(mf_{1})!}\hat{\mathcal{M}}_{0}^{mf_{0}}\hat{\mathcal{M}}_{1}^{mf_{1}}.$
(17)

With a reasoning similar to that in Eq. (5), we get

$\hat{\Phi}^{m}(F)\approx\sum_{k=1}^{s}e^{-mS(F\|F_{k})}\hat{\mathcal{P}}_{k},$
(18)

where $F_{k}=(p_{k0},p_{k1})$ with $p_{k0}=[1-\cos(2v_{k}t+\Delta\phi)]/2$. If
$\bm{\lambda}_{k}$ is regarded as a wavefunction [Eq. (Unified theory of
classical and quantum signal sensing with a qubit)], $F_{k}$ is just the
probability amplitude distribution. Then the probability distribution for the
measurement result $F$ is

$p_{q}(F)=\langle\langle\mathbb{I}|\hat{\Phi}^{m}(F)|\rho\rangle\rangle\approx\sum_{k=1}^{s}e^{-mS(F\|F_{k})}{\rm
Tr}(P_{k}\rho),$ (19)

where we have used
$\langle\langle\mathbb{I}|\hat{\mathcal{P}}_{k}|\rho\rangle\rangle={\rm
Tr}(P_{k}\rho)$.

Eq. (19) represents a summation of at most $s$ different binomial
distributions around $F_{1},\cdots,F_{s}$ [Fig. 1(d)]. The weight of the $k$th
binomial distribution is ${\rm Tr}(P_{k}\rho)$, that is, the projection of the
initial target system $\rho$ on the $k$th eigenspace of the quantum signal
operator $V$. So if the target system starts from an eigenstate of $V$, the
single binomial distribution [Eq. (5)] in classical signal sensing is
recovered. Interestingly, the initial maximally mixed state of the target
system is desirable for quantum sensing, since then all the binomial
distributions can appear, and integration of the measurement results for $k$th
distribution also heralds a selective PM $\hat{\mathcal{P}}_{k}$ on the target
system Ma2023 . Any two binomial distributions around $F_{k}$ and $F_{l}$ can
are well separated if the distance between $F_{k}$ and $F_{l}$ is larger than
the sum of the respective half widths of the distribution. This requires
$m>2|\ln\eta|(\sqrt{p_{k0}p_{k1}}+\sqrt{p_{l0}p_{l1}})/|p_{k0}-p_{l0}|$, where
$\eta$ is the ratio of the minimum hight to the maximum hight within the
distribution width. Then for large $m$, all the binomial distributions can be
well distinguished and therefore the eigenvalue set $\\{v_{k}\\}_{k=1}^{s}$ of
the quantum signal operator $V$ can be accurately estimated within the
standard quantum limit.

Extension to ac quantum signal sensing. So far we have assumed that the
quantum signal is static and determinant, but the RIM scheme can also sense a
time-dependent quantum signal operator. First recall that for a classical
time-dependent signal $v(t)$, the RIM scheme can detect the integration of
such a signal $\int_{0}^{t}v(t^{\prime})dt^{\prime}$. For quantum signal
sensing, it is crucial to note that the RIM scheme actually detect the
eigenvalues of the unitary operator $U_{0}^{\dagger}U_{1}$ with the condition
$[U_{0},U_{1}]=0$. For a time-dependent quantum signal operator $V(t)$,
$U_{0}=\mathcal{T}e^{-i\int_{0}^{t}V(t^{\prime})dt^{\prime}}$ and
$U_{1}=\mathcal{T}e^{i\int_{0}^{t}V(t^{\prime})dt^{\prime}}$ with
$\mathcal{T}$ being the time-ordering operator. If $U_{0}$ and $U_{1}$ can be
well approximated by the first-order Magnus expansion, i.e., $U_{0}\approx
e^{-i\int_{0}^{t}V(t^{\prime})dt^{\prime}}\approx U_{1}^{\dagger}$, then
$[U_{0},U_{1}]=0$ and the RIM scheme can approximately detect the eigenvalues
of the operator $\int_{0}^{t}V(t^{\prime})dt^{\prime}$.

In particular, the RIM scheme can perform ac quantum signal sensing with DD
control of the sensor. The $N$-pulse DD control of the sensor consists of a
sequence of $\pi$ flips at times $\\{t_{1},t_{2},\cdots,t_{N}\\}$ for the
sensor evolution from 0 to $t$. If the target system has a free Hamiltonian
and the sensor is under DD control within each RIM cycle, the Hamiltonian
becomes

$H^{\prime}_{q}=f(t)\sigma_{z}\otimes V+H_{0},$ (20)

where $H_{0}=\sum_{i=1}^{d}\varepsilon_{i}|i\rangle\langle i|$ is the free
Hamiltonian, $V=\sum_{i,j=1}^{d}V_{ij}|i\rangle\langle
j|=\sum_{\omega}V(\omega)$ with
$V(\omega)=\sum_{\varepsilon_{i}-\varepsilon_{j}=\omega}V_{ij}|i\rangle\langle
j|$, and $f(t)$ is the DD modulation function jumping between $+1$ and $-1$
every time the sensor is flipped by a DD pulse. If $[V,H_{0}]=0$, the quantum
signal is still static and thus the above RIM scheme works. However, if
$[V,H_{0}]\neq 0$, the quantum signal becomes
$V(t)=e^{iH_{0}t}Ve^{-iH_{0}t}=\sum_{\omega}e^{i\omega t}V(\omega)$ in the
interaction picture, which can be termed as an ac quantum signal. Then DD
control of the sensor can select some specific component $V(\omega)$ and
filter out the other parts in $V$, if $f(t)$ is periodic and its frequency
matches $\omega$. Then $U_{0}^{\dagger}U_{1}\approx{\rm
exp}\\{2i[F(\omega,t)V(\omega)+F^{*}(\omega,t)V^{\dagger}(\omega)])\\}$ with
$F(\omega,t)=\int_{0}^{t}f(t^{\prime})e^{i\omega t}dt^{\prime}$, whose
eigenvalues can be estimated with sequential RIM sequences.

Figure 2:  Monte Carlo simulation of quantum signal sensing in a spin-star
model. (a,b) For the model in Eq. (21) with $\omega\neq 0$, the histogram of
the number of samples as a function of the frequency $f_{0}$ for different
repetition times in sequential RIMs: (a) $m$ = 50, (b) $m$ = 200. The
parameters are $A_{1}t=0.13$ rad, $A_{2}t=0.26$ rad. (c,d) is similar to (a,b)
but corresponds to the model in Eq. (21) with $\omega/A_{1}=2\omega/A_{2}=10$,
$\mathbf{n}_{1}=(0.50,0.50,0.71)$, $\mathbf{n}_{1}=(0.10,0.10,0.99)$ and the
CPMG pulse number $N=4$. The red solid lines represent the exact probability
distribution in Eq. (19), and the initial state of the target qubits are an
equal superposition of all the four eigenstates of the quantum signal
operator. All the Monte Carlo simulations contain $10^{4}$ samples with
$\phi=\pi/2$.

Example: quantum signal sensing in a spin-star model. As an example, we
consider a spin-star model with the Hamiltonian

$H_{\rm
star}=\sigma_{z}\otimes\sum_{j=1}^{J}\frac{A_{j}}{4}(\bm{\sigma}_{j}\cdot\mathbf{n}_{j})+\frac{\omega}{2}\sum_{j=1}^{J}\sigma_{j}^{z},$
(21)

where $\bm{\sigma}_{j}=(\sigma_{j}^{x},\sigma_{j}^{y},\sigma_{j}^{z})$is the
Pauli vector for the $j$th target spin and
$\mathbf{n}_{j}=(n_{j}^{x},n_{j}^{y},n_{j}^{z})$ is a unit vector. Here the
first term denotes the inhomogeneous coupling between the sensor and $N$
target spins, and the second term is an homogeneous Zeeman term of the target
spins induced by an external magnetic field.

We consider two cases of this model for quantum signal sensing: (i) Without
the Zeeman term ($\omega=0$), the quantum signal is a static operator
$V=\sum_{j=1}^{J}A_{j}(\bm{\sigma}_{j}\cdot\mathbf{n}_{j})$ with eigenvalues
$v_{k}=\sum_{j=1}^{J}\xi_{jk}A_{j}/4$ for $\xi_{jk}\in\\{1,-1\\}$, so we can
fully determine $\\{v_{k}\\}_{k=1}^{2^{J}}$ with sequential RIMs and thus the
coupling strengths $\\{A_{j}\\}_{j=1}^{J}$. (ii) With the Zeeman term
($\omega\neq 0$), we can apply the $N$-pulse Carr-Purcell-Meiboom-Gill (CPMG)
control to the sensor with the flip pulses at times
$\\{t_{l}=(2l-1)\tau\\}_{l=1}^{N}$ ($2\tau=t/N$). With the resonant DD
condition ($2\tau=\pi/\omega$),
$U_{0}^{\dagger}U_{1}\approx\prod_{j=1}^{J}{\rm
exp}[iA_{j}^{\perp}N(\cos\kappa_{j}\sigma_{j}^{x}-\sin\kappa_{j}\sigma_{j}^{y})/(2\omega)]$
with $A_{j}^{\perp}=A_{j}[(n_{j}^{x})^{2}+(n_{j}^{y})^{2}]$ and
$\kappa_{j}={\rm arg}(F(\omega,t))+{\rm arctan}(A_{j}^{y}/A_{j}^{x})$, so by
sequential RIMs we can estimate $\\{A_{j}^{\perp}\\}$, which are the
components of the interaction fields $\\{A_{j}\mathbf{n}_{j}\\}$ perpendicular
to the external magnetic field. With several different directions of the
external magnetic field, we can estimate both $\\{A_{j}\\}$ and
$\\{\mathbf{n}_{j}\\}$ Ma2016 .

For both cases of the spin-star model, we perform Monte Carlo calculations to
simulate the measurement statistics of sequential RIMs for two target spins.
The results in Fig. 2 show four distinct binomial distributions as the
repetition number $m$ of RIMs increases, which correspond to the four
different eigenvalues of the quantum signal operators.

Conclusions and outlooks. We develop a unified theoretical framework to sense
both classical and quantum signals with sequential RIMs. For quantum signal
sensing, the measurement statistics can display multiple distribution peaks,
which can be used to precisely estimate the eigenvalues of the quantum signal
operator on a target quantum system. The measurement statistics of classical
signal sensing can be recovered when the target system starts from an
eigenstate of the quantum signal operator. The concept and proposal of quantum
signal sensing greatly broaden the scope of quantum sensing concerning mostly
about classical signals before. The unified framework also raises further
problems to explore, which can be inspired by topics in classical signal
sensing. In particular, it will be interesting to consider a more complex
sensor (such as a sensor with multiple qubits, a qudit or even a harmonic
oscillator) for quantum signal sensing, and explore the role of quantum
coherence and entanglement in improving the sensitivity beyond the standard
quantum limit.

###### Acknowledgements.

W.L.M acknowledges support from Chinese Academy of Sciences (No. E0SEBB11, No.
E27RBB11), National Natural Science Foundation of China (No. 12174379, No.
E31Q02BG), and Innovation Program for Quantum Science and Technology (No.
2021ZD0302300).

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