# Attention Based Semantic Segmentation on UAV Dataset For Natural Disaster
Damage Assessment

###### Abstract

The detrimental impacts of climate change include stronger and more
destructive hurricanes happening all over the world. Identifying different
damaged structures of an area including buildings and roads are vital since it
helps the rescue team to plan their efforts to minimize the damage caused by a
natural disaster. Semantic segmentation helps to identify different parts of
an image. We implement a novel self-attention based semantic segmentation
model on a high resolution UAV dataset and attain Mean IoU score of around
$88\%$ on the test set. The result inspires to use self-attention schemes in
natural disaster damage assessment which will save human lives and reduce
economic losses.

Index Terms—  Semantic segmentation, natural disaster damage assessment, self-
attention method, UAV

##  1 Introduction

In recent time, the world has seen numerous natural disasters which have
brought both personal injury and economic loss to several countries all over
the world. A major step to both save valuable human lives and reduce financial
loss is an accurate assessment of the damage inflicted by these events. A
correct estimation of the damage helps to plan efficiently and allows the
rescue team to allocate their efforts and aid properly.

With the advent of new sophisticated technologies capturing events of natural
disaster has improved. Currently satellite imageries [1, 2] have been used for
proper assessment of these disaster events. The rescue team can use UAV
(Unmanned Aerial Vehicle) to capture images of damaged properties and the
whole affected area. Compared to satellite imagery, UAV imagery [3, 4, 5]
provides higher resolution which helps to understand the detailed damage level
of the captured area.

Semantic segmentation is a core part of image understanding. Semantic
segmentation is the task of assigning semantic labels to every pixels of an
image. Besides traditional approach to tackle computer vision problems like
semantic segmentation, deep convolutional neural networks [6] have achieved
several revolutionary achievements in answering complex image understanding
issues. Although many advanced segmentation works have been proposed in the
recent years for popular urban datasets like Cityscapes [7] and PASCAL VOC [8]
dataset, very few methods [1, 2, 9, 10] have been proposed and applied on
natural disaster datasets.

Attention based models are able to capture long range dependencies and have
been applied to several applications including computer vision field [11, 12].
Although self attention based models is gaining popularity in semantic
segmentation of popular public non-disaster datasets like Cityscapes and
PASCAL VOC due their higher performances, to the best of our knowledge no
attention based model has been proposed for the semantic segmentation of
disaster datasets.

Fig. 1: Illustration of after disaster scenes of HRUD dataset. First row shows
original image and the second row shows the corresponding annotations.

We propose a self attention method named ReDNet, a ResNet [13] based Dual
attention method, which aims to capture both spatial and semantic contribution
of the features. We evaluate this model on High Resolution UAV Dataset (HRUD)
[3, 4]. Figure 1 shows sample images and respective colored annotated masks
from HRUD. Most of the already proposed methods [2, 10] have performed
segmentation on flood water, damaged road, and buildings. But we
comprehensively perform semantic segmentation on all of the components present
in the images including debris, water, building, vehicle, road, tree, and
sand. We also compare our method with three other state-of-art semantic
segmentation models named ENet [14], DeepLabv3+ [15], and PSPNet [16].

##  2 Related Works

###  2.1 Semantic Segmentation

Fully Convolutional Networks (FCNs) based methods have achieved excellent
performance in semantic segmentation task. Popular state-of-art semantic
segmentation methods can be classified into two sections: encoder-decoder
based architecture [14] and pooling based architecture [16, 15]. Encoder-
decoder methods use middle and lower level features and generate local
contextual map which creates sharp object boundaries. On the other hand,
pooling based methods like PSPNet [16] and DeepLabv3+ [15] use pyramid pooling
operations to create feature maps which are rich in global contextual
information. Recently self-attention based methods [12, 11] have demonstrated
remarkable performance in collecting global contextual dependencies and thus
superior performance in segmentation.

###  2.2 Natural Disaster Damage Assessment

Recently several research works have been proposed addressing the damage
assessment from natural disaster datasets. Rahnemoonfar et al. implement a
densely connected recurrent neural network in [10] on UAV imageries for river
segmentation. Rudner et al. present a novel approach named Multi3Net in [1]
for segmentation of flooded buildings using satellite imageries. RescueNet is
proposed by Gupta et al. in [2] for joint building segmentation. Zhu et al.
propose an instance segmentation named MSNet in [9] on aerial videos to assess
building damage.

In this paper, inspired by the recent performance of the self-attention based
methods we propose a novel self-attention based network ReDNet and evaluate
its performance on HRUD dataset [3, 4]. To the best of our knowledge this is
the first self-attention based method applied on any natural disaster dataset.

##  3 Dataset

Fig. 2: The details of proposed position attention module.

We use High Resolution UAV Dataset (HRUD) [3, 4] where resolution of the
captured images are 3000 $\times$ 4000\. The dataset is taken from the Center
for Robot-Assisted Search and Rescue open data repository (hrail.crasar.org)
for small UAV imagery for disasters, specifically the Hurricane Michael event.
The dataset is unique for two reasons. One is fidelity: it contains imagery
from sUAV taken during the response phase by emergency responders, thus the
data reflects what is the state of the practice and can be reasonable expected
to be collected during a disaster. Second: it is the only known database of
sUAV imagery for disasters. Note that there are other existing databases of
imagery from unmanned and manned aerial assets collected during disasters,
such as National Guard Predators or Civil Air Patrol, but those are larger,
fixed-wing assets that operate above the 400 feet AGL limitation of sUAV. The
HRUD dataset consists of imagery taken from 80 flights conducted between
October 11-14, 2018, on behalf of the Florida State Emergency Response Team at
Mexico Bearch and other directly impacted areas. All flights were flown at 200
feet AGL, as compared to manned assets which normally fly at 500 feet AGL or
higher. The majority of daylight images were taken with DJI Mavic Pro
quadcopters, though two sets of video were taken with Parrot Disco fixed-wing
sUAV and one set at night with a DJI Inspire and thermal camera.

Fig. 3: An overview of our ReDNet architecture.

The images include debris, water, building, vehicle, road, tree, and sand. The
images are collected after hurricane Michael. In total 1973 images have been
annotated for semantic segmentation. The buildings are annotated into four
classes based on their damage levels. The classes are no damage, medium
damage, major damage, and total destruction. For the experiments in this
paper, we merge no damage, medium damage, and major damage into “Building Not
Totally Destroyed” and total destruction is considered as “Building Totally
Destroyed”.

Table 1: Per-class results on HRUD testing set.

Method Debris Water Building Not Totally Destroyed Building Totally Destroyed
Vehicle Road Tree Sand mIoU ENet[14] 45.97 75.84 66.16 39.52 36.74 61.19 71.64
61.77 54.15 DeepLabv3+[15] 65.8 85.8 84.5 57.3 51.3 73.3 75.9 77.4 69.67
PSPNet[16] 88.76 67.98 85.75 80.51 65.83 82.81 94.53 76.04 80.27 ReDNet(Our
method) 92.66 95.80 92.16 83.01 57.3 89.78 96.41 96.46 87.95

Fig. 4: Visual comparison on HRUD test set.

##  4 Method

###  4.1 PAM: Position Attention Module

Convolution operations lead to local receptive field. Lack of global context
results in intra-class inconsistency and hurts the recognition accuracy [11,
12]. We use ResNet-101 as base recognition model. Different stages of the
ResNet-101 has different recognition capability. Lower stages show encoding of
fine spatial information but poor semantic consistency due to smaller
receptive field view. On other hand, higher stages shows good semantic
consistency due to larger receptive field but poor encoding of spatial
information. Keeping this in mind, we propose a novel self-attention
segmentation network ReDNet which combines two position attention modules by
extracting features from both lower and higher stages of ResNet-101.

Position attention module is shown in Figure 2. Given a local feature map $A$
with dimension $C\times H\times W$ we generate three new feature maps using a
convolution layer with similar shape to given feature map $A$. Matrix
multiplications is performed on transpose of $B$ and $C$. Then using a softmax
layer spatial attention map $S$ is calculated using the following formula.

$s_{ji}=\frac{exp(B_{i}\cdot C_{j})}{\sum_{i=1}^{N}exp(B_{i}\cdot C_{j})}$ (1)

Then matrix multiplication is performed between transpose of $S$ and $D$. The
output from the matrix multiplication is multiplied with $\alpha$. Finally
element-wise sum is performed with feature $A$ to generate final output with
shape $C\times H\times W$.

###  4.2 ReDNet

ReDNet architecture consists of two position attention modules (PAMs) which
are placed in parallel. One PAM takes output features from layer 2 of
ResNet-101 as input while the other one takes output of layer 3 of ResNet-101
as input. Their attention maps are added together using element-wise sum
operation. ReDNet architecture is shown in Figure 3.

##  5 Experiments

###  5.1 Implementation Details

PyTorch has been used for implementation of segmentation networks. As hardware
we use NVIDIA GeForce RTX 2080 Ti GPU and Intel Core i9 CPU. We use “poly”
learning rate with base learning rate 0.0001. Momentum, weight decay, and
power are set to 0.9, 0.0001, and 0.9 respectively. For augmentation we use
random shuffling, scaling, flipping, and random rotation which help models to
avoid overfitting. During training, we resize the images to 713 $\times$ 713
since large crop size is useful for the high resolution images. For semantic
segmentation evaluation metric we use mean IoU.

##  6 Results and Discussion

HRUD is a very challenging dataset due to its variable sized classes along
with similar textures among different classes. Debris makes a great impact on
segmentation performances of the evaluated network models. Similar textures of
debris, sand, and building with total destruction damage are very difficult to
differentiate from segmentation performances’ point of view.

Evaluations of our method ReDNet on HRUD test set is presented in Table 1 and
compared with three other state-of-art segmentation models, ENet [14],
DeepLabv3+ [15], and PSPNet [16]. ReDNet achieves best results in all classes
with mean IoU of $87.95\%$. Superior performance of ReDNet proves that global
feature map generated using proposed self-attention method improves the
segmentation compared to extracted local context map by ENet, and global
feature maps by PSPNet and DeepLabv3+. The visualization of the segmentation
can be seen in Figure 4.

##  7 Conclusion

In this paper, we propose and evaluate a novel self-attention segmentation
model ReDNet on a new high resolution natural disaster dataset named HRUD. We
discuss the challenges of semantic segmentation on this dataset along with the
excellent performance of the proposed model on it. We hope this paper will
facilitate future research on natural disaster damage assessment and will help
achieving less human and economic losses with efficient natural disaster
management.

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# Inferring physical properties of symmetric states from the fewest copies

Da-Jian Zhang<EMAIL_ADDRESS>Department of Physics, Shandong University,
Jinan 250100, China  D. M. Tong<EMAIL_ADDRESS>Department of Physics,
Shandong University, Jinan 250100, China

###### Abstract

Learning physical properties of high-dimensional states is crucial for
developing quantum technologies but usually consumes an exceedingly large
number of samples which are difficult to afford in practice. In this Letter,
we use the methodology of quantum metrology to tackle this difficulty,
proposing a strategy built upon entangling measurements for dramatically
reducing sample complexity. The strategy, whose characteristic feature is
symmetrization of observables, is powered by the exploration of symmetric
structures of states which are ubiquitous in physics. It is provably optimal
under some natural assumption, efficiently implementable in a variety of
contexts, and capable of being incorporated into existing methods as a basic
building block. We apply the strategy to different scenarios motivated by
experiments, demonstrating exponential reductions in sample complexity.

Learning properties of states like expectation values of observables is of
increasing importance in quantum physics, underpinning vast applications in
emerging fields, such as quantum machine learning Biamonte  _et al._ (2017),
quantum computational chemistry McArdle  _et al._ (2020), and variational
quantum computing Cerezo  _et al._ (2021); Tilly  _et al._ (2022). A
fundamental difficulty in this task is that any full reconstruction of a
generic unknown state inevitably consumes exponentially many samples due to
the growth of the state space dimension with the system size. This makes
traditional learning methods like full state tomography hopelessly inefficient
and poses a serious challenge in the Noisy Intermediate-Scale Quantum (NISQ)
era Preskill (2018).

Fortunately, although there are a large number of states in the state space,
most of the physically relevant states are the ones that admit some nontrivial
structures, such as those of low rank Gross  _et al._ (2010); Liu  _et al._
(2012), matrix product states Cramer  _et al._ (2010); Baumgratz  _et al._
(2013), and those with quasi-local structures Anshu  _et al._ (2021); Rouzé
and França (2021). Furthermore, for many purposes, one is only interested in
some specific properties of states, e.g., mean energies of many-body systems,
which makes it unnecessary to fully reconstruct states for obtaining all their
information. These well-motivated physical considerations have led to a series
of proposals for more efficient alternatives to full state tomography, with
excellent examples including compressed sensing Gross  _et al._ (2010); Liu
_et al._ (2012), adaptive tomography Mahler  _et al._ (2013); Qi  _et al._
(2017), self-guided tomography Ferrie (2014); Rambach  _et al._ (2021), and
classical shadow Aaronson (2018); Huang  _et al._ (2020).

While existing methods generally rely on local measurements, it has been
recognized that entangling measurements are typically far more efficient than
local measurements for extracting information from unknown states Peres and
Wootters (1991); Gisin and Popescu (1999); et al. (2022a); Aharonov  _et al._
(2022); Miguel-Ramiro  _et al._ (2022); Larocca  _et al._ (2022). Moreover,
the ongoing development of large-scale quantum computers opens up exciting
possibilities of leveraging quantum computational resources to realize
entangling measurements et al. (2022a); Aharonov  _et al._ (2022). In
particular, motivated by the availability of NISQ computers et al. (2022b), a
number of experiments Tang  _et al._ (2020); Huang  _et al._ (2022); Conlon
_et al._ (2023) have been carried out for realizing entangling measurements as
well as demonstrating their superiority over local measurements. An important
issue following is therefore to explore the usefulness of entangling
measurements in learning properties of states et al. (2022a); Aharonov  _et
al._ (2022).

Here we propose a strategy built upon entangling measurements, for
dramatically reducing sample complexity beyond what can be achieved with local
measurements. The basic idea is to explore symmetric structures of states
which are ubiquitous in physics (see also Ref. Tóth  _et al._ (2010)). Using
information-theoretic tools from quantum metrology Helstrom (1976); Holevo
(2011), we figure out the measurement that can make best use of these
symmetric structures for learning expectation values of observables. This
enables our strategy to operate at the optimal sample efficiency in a variety
of physical contexts, thereby achieving a sought-after goal in quantum
metrology Braunstein and Caves (1994); Zhang and Gong (2020); Zhang and Tong
(2022), which is unlikely, if not impossible, to reach with local
measurements. Taking translational and permutational symmetries as two
examples, we demonstrate that our strategy, when incorporating known efficient
quantum circuits, allows for saving exponentially many samples while merely
consuming polynomial amounts of quantum computational resources. The findings
of this Letter uncover an intriguing route to reducing sample complexity via
taking advantage of symmetric structures of states, which opens opportunities
for leveraging recent breakthroughs on large-scale quantum computers to
accomplish a plethora of learning tasks.

We start with a simple example. Let $\rho$ be an unknown state of a qubit
whose symmetric structures are described by the group
${G}=\\{\openone,\sigma_{z}\\}$, i.e., $\sigma_{z}\rho\sigma_{z}=\rho$, where
$\sigma_{\alpha}$, $\alpha=x,y,z$, denote the Pauli matrices. Any observable
$X$ of the qubit can be written as $X=a\openone+\bm{b}\cdot\bm{\sigma}$, with
$a\in\mathbb{R}$, $\bm{b}=(b_{x},b_{y},b_{z})\in\mathbb{R}^{3}$, and
$\bm{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$. To obtain the expectation
value $\expectationvalue{X}_{\rho}$ of $X$ in $\rho$, the commonly used
approach is to perform the projective measurement of $X$. The quantum
uncertainty in this measurement is $(\Delta
X)^{2}:=\expectationvalue{X^{2}}_{\rho}-\expectationvalue{X}_{\rho}^{2}$.
Resorting to the symmetric structures of $\rho$, we can alternatively measure
$Y=\frac{1}{2}X+\frac{1}{2}\sigma_{z}X\sigma_{z}=a\openone+\bm{b}^{\prime}\cdot\bm{\sigma}$
for obtaining $\expectationvalue{X}_{\rho}$, where
$\bm{b}^{\prime}=\left(0,0,b_{z}\right)$. The subtle difference between $X$
and $Y$ is that, whereas
$\expectationvalue{X}_{\rho}=\expectationvalue{Y}_{\rho}$, $(\Delta
X)^{2}=\norm{\bm{b}}^{2}-(a-\expectationvalue{X}_{\rho})^{2}\geq\norm{\bm{b}^{\prime}}^{2}-(a-\expectationvalue{Y}_{\rho})^{2}=(\Delta
Y)^{2}$, where $\norm{\cdot}$ denotes the Euclidean norm. So, for obtaining
$\expectationvalue{X}_{\rho}$ up to a certain desired precision, the
projective measurement of $Y$ generally consumes less copies of $\rho$ than
that of $X$. We are thus led to the observation that the expectation value of
a given observable may be obtained more efficiently through measuring another
observable than the given one when the state in question admits some symmetric
structures. Below, we systematically analyze how to optimally take advantage
of symmetric structures of states for efficiently learning expectation values
of observables.

We consider the general setting that $\rho$ is an unknown state of a (possibly
many-body) system with the symmetric structures described by a finite or
compact Lie group $G$, i,e, $U_{g}\rho U_{g}^{\dagger}=\rho$ for $g\in G$,
where $U_{g}$ denotes a unitary representation of ${G}$. It is worth noting
that the majority of symmetric structures of interest in quantum physics can
be described in this way. Our aim is to consume as few copies of $\rho$ as
possible to learn the expectation value $\expectationvalue{X}_{\rho}$ of any
given observable $X$ up to a certain desired precision.

To reach this aim, we resort to the methodology of quantum metrology Helstrom
(1976); Holevo (2011) which provides powerful tools to deal with the so-called
parameter estimation problems SM . We first set the stage of our analysis.
Throughout, we assume that we know nothing about $\rho$ except its symmetric
structures, leaving the discussion on this assumption to the end of this
Letter. We treat $\expectationvalue{X}_{\rho}$ as the parameter to be
estimated and use $\beta$ to represent $\expectationvalue{X}_{\rho}$ for later
convenience. Without loss of generality, we can describe the task of learning
$\expectationvalue{X}_{\rho}$ from multiple copies of $\rho$ as first
performing a measurement on $\rho^{\otimes M}$ and then inferring the value of
$\beta=\expectationvalue{X}_{\rho}$ from the measurement outcome et al.
(2022a). Here, $M$ denotes the number of samples consumed, which is to be
determined shortly. Any measurement can be described by a positive operator-
valued measure (POVM) $\\{\Pi_{\bm{y}}\\}_{\bm{y}}$ satisfying
$\sum_{\bm{y}}\Pi_{\bm{y}}=\openone$, where $\bm{y}$ labels the measurement
outcome and could be multivariate in general. Any inference rule amounts to
finding an estimator $\hat{\beta}(\bm{y})$, which is a map from the set of
measurement outcomes to the set of possible values of $\beta$. $\hat{\beta}$
is said to be unbiased if its expected value equals to $\beta$, that is,
$\sum_{\bm{y}}p_{\bm{y}}\hat{\beta}(\bm{y})=\beta$, where
$p_{\bm{y}}=\tr(\Pi_{\bm{y}}\rho^{\otimes M})$ denotes the probability of
getting outcome $\bm{y}$. Associated with a POVM $\\{\Pi_{\bm{y}}\\}_{\bm{y}}$
and an (unbiased) estimator $\hat{\beta}$, the error in estimating $\beta$ can
be quantified by the variance $\textrm{Var}[\hat{\beta}]$ of the estimator
$\hat{\beta}$ Helstrom (1976); Holevo (2011),
$\textrm{Var}[\hat{\beta}]=\sum_{\bm{y}}p_{\bm{y}}[\hat{\beta}(\bm{y})-\beta]^{2}$.
We require that $\textrm{Var}[\hat{\beta}]\leq\epsilon$, where $\epsilon$
characterizes the desired precision.

We next introduce fundamental bounds on $\textrm{Var}[\hat{\beta}]$ and $M$.
Using representation theory of groups, we can characterize $\rho$ in terms of
some unknown parameters, $\rho=\rho(\bm{\theta})$, with
$\bm{\theta}=(\theta_{1},\cdots,\theta_{\mathfrak{p}})$ [see Supplemental
Material (SM) SM  for details]. This enables us to introduce a bound on
$\textrm{Var}[\hat{\beta}]$,

$\displaystyle\textrm{Var}[\hat{\beta}]\geq\frac{\partial\beta[H(\bm{\theta})]^{-1}\partial\beta^{T}}{M},$
(1)

known as the quantum Cramér-Rao bound (QCRB) Braunstein and Caves (1994);
Zhang and Gong (2020); Zhang and Tong (2022). Here,
$\partial\beta=(\partial\beta/\partial\theta_{1},\cdots,\partial\beta/\partial\theta_{\mathfrak{p}})$
and $H(\bm{\theta})$ is a $\mathfrak{p}\times\mathfrak{p}$ symmetric matrix
known as the quantum Fisher information matrix. The $kl$ element of
$H(\bm{\theta})$ is given by $H_{kl}=\tr[\rho(\bm{\theta})(L_{k}\circ
L_{l})]$, where $L_{k}\circ L_{l}=(L_{k}L_{l}+L_{l}L_{k})/2$ denotes the
Jordan product, and $L_{k}$ is the symmetric logarithmic derivative defined as
the Hermitian operator satisfying
$\partial\rho(\bm{\theta})/\partial\theta_{k}=L_{k}\circ\rho(\bm{\theta})$.
The meaning of Eq. (1) is that the precision attainable in any task of
learning $\beta$ from the $M$ samples is fundamentally constrained by the QCRB
Braunstein and Caves (1994); Zhang and Gong (2020); Zhang and Tong (2022),
irrespective of the choices of a POVM and an estimator. Inserting
$\textrm{Var}[\hat{\beta}]\leq\epsilon$ into Eq. (1), we have

$\displaystyle
M\geq\frac{\partial\beta[H(\bm{\theta})]^{-1}\partial\beta^{T}}{\epsilon},$
(2)

implying that quantum mechanics does not allow for consuming less than
$M_{\textrm{min}}:=\lceil\frac{\partial\beta[H(\bm{\theta})]^{-1}\partial\beta^{T}}{\epsilon}\rceil$
samples for reaching the desired precision. Here $\lceil\cdot\rceil$ denotes
the ceiling function.

We then analyze the saturation conditions of the bounds. We derive in SM SM
the following crucial equality

$\displaystyle\partial\beta[H(\bm{\theta})]^{-1}\partial\beta^{T}=(\Delta
Y)^{2},$ (3)

which connects the QCRB to the quantum uncertainty of the observable

$\displaystyle Y=\mathcal{T}(X).$ (4)

Here, $\mathcal{T}$ is the $G$-twirling operation defined as
$\mathcal{T}(X)=\int_{G}d\nu(g)U_{g}XU_{g}^{\dagger}$ with $\nu(g)$ denoting
the normalized Haar measure Halmos (1950). In particular, when $G$ is a finite
group, $\mathcal{T}(X)={\absolutevalue{G}}^{-1}\sum_{g\in
G}U_{g}XU_{g}^{\dagger}$, where $\absolutevalue{G}$ is the cardinality of $G$.
It is interesting to note that $Y$ satisfies $[Y,U_{g}]=0$ for $g\in G$ and is
therefore the symmetrized counterpart of $X$, which is different from $X$ in
most cases ($Y$ equals to $X$ in the special case that $[X,U_{g}]=0$ for all
$g\in G$). On the other hand, using the normalization of the Haar measure
$\int_{G}d\nu(g)=1$ and $U_{g}\rho U_{g}^{\dagger}=\rho$ for all $g\in G$, we
have

$\displaystyle\expectationvalue{Y}_{\rho}=\expectationvalue{X}_{\rho}.$ (5)

Hence, by performing the projective measurement of $Y$ on each of the $M$
samples, we can obtain an unbiased estimate of $\beta$ via the sample mean
estimator $\hat{\beta}(\bm{y})=\sum_{i=1}^{M}y_{i}/M$, where $y_{i}$ denotes a
random variable taking on a value in the spectrum of $Y$. The error thus
produced is $\textrm{Var}[\hat{\beta}]=(\Delta Y)^{2}/M$, which saturates the
QCRB because of Eq. (3). This further implies that we are allowed to only
consume $M=M_{\textrm{min}}$ samples for obtaining $\beta$ up to the desired
precision, as Eq. (2) is a direct consequence of Eq. (1).

Figure 1: Schematic of our strategy. To learn the expectation value
$\expectationvalue{X}_{\rho}$ of an observable $X$ in $\rho$ which is possibly
a many-body state, we perform the projective measurement of the observable $Y$
on each sample. This measurement can be implemented by utilizing the quantum
circuit $V$ to transform the eigenbasis of $Y$ into the computational basis
and then performing the standard measurement in the computational basis.
Repeating this procedure $M_{\textrm{min}}=\lceil(\Delta
Y)^{2}/\epsilon\rceil$ times, we can obtain an estimate of
$\expectationvalue{X}_{\rho}$ up to the desired precision $\epsilon$ by post-
processing the outcomes $y_{i}$ via the sample mean estimator.

With the above analysis, we are ready to specify our strategy in what follows.
To obtain $\expectationvalue{X}_{\rho}$, our strategy is to perform the
projective measurement of $Y=\mathcal{T}(X)$ on each sample, which, after
post-processing the outcomes via the sample mean estimator, can produce the
same expectation value with $X$ but requires $M_{\textrm{min}}=\lceil(\Delta
Y)^{2}/\epsilon\rceil$ samples for reaching the desired precision $\epsilon$.
Note that the projective measurement of $Y$ typically belongs to the class of
entangling measurements, because (some of) the $U_{g}$’s act on multiple
subsystems for most of the symmetric structures of interest in physics. To
implement this measurement, we can first apply the quantum circuit $V$
transforming the eigenbasis of $Y$ into the computational basis and then
perform the standard measurement in the computational basis (see Fig. 1). So,
the working principle of our strategy is to leverage quantum computational
resources for reducing the number of samples required to its minimal value.
Notably, as the projective measurement of $Y$ saturates the QCRB, our
strategy, theoretically speaking, operates at the optimal sample efficiency
allowed by quantum mechanics and therefore outperforms other methods from the
perspective of sample complexity.

Illustrative application 1: translational symmetries. Let us now apply our
strategy to the scenario that $\rho$ is an unknown state of $n$ qubits with
the symmetric structures described by the translation group
$G=\\{T^{i},i=0,\cdots,2^{n}-1\\}$. Here, $T$ is defined as
$T\ket{j}=\ket{j+1}$ with the periodic boundary condition
$\ket{2^{n}}=\ket{0}$, where $\ket{j}$, $j=0,\cdots,2^{n}-1$, denote the
computational basis. Note that translational symmetries are ubiquitous in
condensed-matter physics. The above scenario could arise, e.g., in quantum
simulations of electrons in crystalline solids Bloch  _et al._ (2012);
Georgescu  _et al._ (2014), for which Bloch’s theorem states that solutions to
the Schrödinger equations in periodic potentials are Bloch states and hence
respect translational symmetries.

To show the usefulness of our strategy, we calculate $(\Delta X)^{2}$ and
$(\Delta Y)^{2}$, which are respectively proportional to the numbers of
samples required in the commonly used approach and our strategy for obtaining
$\expectationvalue{X}_{\rho}$ up to $\epsilon$. To this end, we write $\rho$
as $\rho=\sum_{j}p_{j}\ket{f_{j}}\bra{f_{j}}$, which follows from
$[\rho,T]=0$. Here, $p_{j}\geq 0$ satisfies $\sum_{j}p_{j}=1$, and
$\ket{f_{j}}=\sum_{k}e^{2\pi ijk/2^{n}}\ket{k}/{\sqrt{2^{n}}}$ denotes the
Fourier basis which is the eigenbasis of $T$. Then, expressing $X$ as
$X=\sum_{jk}X_{jk}\ket{f_{j}}\bra{f_{k}}$ with
$X_{jk}=\bra{f_{j}}X\ket{f_{k}}$, we have $(\Delta
X)^{2}=(\sum_{jk}p_{j}\absolutevalue{X_{jk}}^{2})-(\sum_{j}p_{j}X_{jj})^{2}$.
Besides, using Eq. (4) and noting that $T\ket{f_{j}}=e^{-2\pi
ij/2^{n}}\ket{f_{j}}$, we have $Y=\sum_{j}X_{jj}\ket{f_{j}}\bra{f_{j}}$, which
further leads to $(\Delta
Y)^{2}=(\sum_{j}p_{j}{X}_{jj}^{2})-(\sum_{j}p_{j}X_{jj})^{2}$. Hence, $(\Delta
X)^{2}-(\Delta Y)^{2}=\sum_{j,k\neq j}p_{j}\absolutevalue{X_{jk}}^{2}$
contains exponentially many nonnegative terms. Roughly speaking, this
indicates that our strategy allows for dramatically reducing sample complexity
for countless choices of $X$. To clearly see this point, we take
$X=\bigotimes_{l=1}^{n}(\sigma_{x}^{l}+\sigma_{z}^{l})$ as an example, where
$\sigma_{\alpha}^{l}$, $\alpha=x,y,z$, are the Pauli matrices acting on the
$l$-th qubit. The corresponding $Y$ can be found by exploiting this expression
to calculate $X_{jj}$ SM . We have $2^{n}-1\leq(\Delta X)^{2}\leq 2^{n}$ but
$0\leq(\Delta Y)^{2}\leq 1$ SM , demonstrating that the reduction allowed by
our strategy can be exponential in $n$.

We point out that our strategy is efficiently implementable on a quantum
computer in the scenario under consideration. Indeed, the eigenbasis of $Y$ is
the Fourier basis, which implies that the quantum circuit $V$ is just the
inverse quantum Fourier transform. That our strategy is efficiently
implementable follows from the known result that the inverse quantum Fourier
transform can be realized as a quantum circuit consisting of only
$\mathcal{O}(n^{2})$ Hadamard gates and controlled phase shift gates Nielsen
and Chuang (2010) (see also Ref. Griffiths and Niu (1996) for a semiclassical
realization without using two-bit gates).

Illustrative application 2: permutational symmetries. Let us consider again
$n$ qubits but in the state whose symmetric structures are described by the
permutation group $G=\\{P_{s},s\in S_{n}\\}$. Here, $s$ labels a permutation
in the symmetric group $S_{n}$ and $P_{s}$ is defined by
$P_{s}\ket{\psi_{1}}\otimes\cdots\otimes\ket{\psi_{n}}=\ket{\psi_{s(1)}}\otimes\cdots\otimes\ket{\psi_{s(n)}}$.
This scenario arises frequently in multipartite experiments Kiesel  _et al._
(2007); Wieczorek  _et al._ (2009); Prevedel  _et al._ (2009), in which the
states involved are typically invariant under permutations Tóth  _et al._
(2010). For example, three well-known states of this type are Werner states
Werner (1989), Dicke states Dicke (1954), and Greenberger–Horne–Zeilinger
(GHZ) states Greenberger  _et al._ (1989), which are key resources in quantum
information processing Horodecki  _et al._ (2009); Gühne and Tóth (2009).
Below, motivated by the fact that Pauli measurements are widely used in
multipartite experiments, we explore our strategy to reduce sample complexity
in Pauli measurements.

We can express a generic Pauli observable as
$X_{\bm{kl}}=\sigma_{x}^{k_{1}}\sigma_{z}^{l_{1}}\otimes\cdots\otimes\sigma_{x}^{k_{n}}\sigma_{z}^{l_{n}}(i)^{\bm{k}\cdot\bm{l}}$,
where $\bm{k}=(k_{1},\cdots,k_{n})$ and $\bm{l}=(l_{1},\cdots,l_{n})$ are two
vectors of binary numbers and $\bm{k}\cdot\bm{l}=\sum_{i=1}^{n}k_{i}l_{i}$
denotes the usual dot product. Note that, associated to each $X_{\bm{kl}}$,
there is a symmetrized counterpart $Y_{\bm{kl}}=\mathcal{T}(X_{\bm{kl}})$. To
illustrate the superiority of the projective measurement of $Y_{\bm{kl}}$ over
the Pauli measurement of $X_{\bm{kl}}$, we evaluate $(\Delta X_{\bm{kl}})^{2}$
and $(\Delta Y_{\bm{kl}})^{2}$ on the GHZ state of $n$ qubits. Hereafter we
assume for simplicity that $n$ is odd. It can be shown that $(\Delta
X_{\bm{kl}})^{2}=1$ for any Pauli observable with $\absolutevalue{\bm{k}}\neq
0$ and $n$ SM . Here $\absolutevalue{\bm{k}}=\sum_{i=1}^{n}k_{i}$. By
contrast, $(\Delta Y_{\bm{kl}})^{2}=1/\binom{n}{\absolutevalue{\bm{k}}}$ for
the same Pauli observable SM . To clearly see the difference between $(\Delta
X_{\bm{kl}})^{2}$ and $(\Delta Y_{\bm{kl}})^{2}$, we consider the Pauli
measurements with
$(1-\delta)\frac{n}{2}<\absolutevalue{\bm{k}}<(1+\delta)\frac{n}{2}$, referred
to as the typical Pauli measurements for ease of language. Here, $0<\delta<1$
is fixed. We show that $(\Delta X)^{2}/(\Delta
Y)^{2}\geq\sqrt{\frac{2}{n\pi(1-\delta^{2})}}\left[\frac{4}{(1-\delta)^{1-\delta}(1+\delta)^{1+\delta}}\right]^{\frac{n}{2}}$
for any typical Pauli measurement SM . Noting that
$(1-\delta)^{1-\delta}(1+\delta)^{1+\delta}<4$ for $0<\delta<1$, we deduce
that $(\Delta X)^{2}/(\Delta Y)^{2}$ is exponential in $n$. Besides, we show
that the number of the typical Pauli measurements is $\geq
4^{n}(1-\frac{1}{n\delta^{2}})$ SM . Since the total number of Pauli
measurements is $4^{n}$, this means that most of Pauli measurements are
typical for a large $n$. Therefore, our strategy allows for exponentially
reducing sample complexity for most of Pauli measurements when $n$ is large.

Notably, our strategy can be efficiently implemented on a quantum computer in
the considered scenario, too. Indeed, as detailed in SM SM , the eigenbasis of
$Y$ can be mapped into the computational basis via the quantum Schur transform
followed by at most $\lceil\frac{n}{2}\rceil$ controlled gates. That our
strategy is efficiently implementable follows from the known result that the
quantum Schur transform can be realized as a quantum circuit of polynomial
size Bacon  _et al._ (2006).

Before concluding, we present a few remarks. We point out that the relations
$\expectationvalue{Y}_{\rho}=\expectationvalue{X}_{\rho}$ and $(\Delta
Y)^{2}\leq(\Delta X)^{2}$ hold as long as $\rho$ admits some symmetric
structures described by a finite or compact Lie group. That is, our strategy
is applicable to any such $\rho$, regardless of whether the foregoing
assumption, i.e., nothing about $\rho$ except its symmetric structures is
known, is satisfied or not. When this assumption is satisfied, our strategy
allows for consuming the fewest samples to learn $\expectationvalue{X}_{\rho}$
up to the desired precision $\epsilon$. On the other hand, while the
assumption is satisfied in numerous scenarios, there are also many scenarios
in which we know other structures of $\rho$ besides its symmetric structures.
For example, apart from translational symmetries, the state of a many-body
system usually admits quasi-local structures, based on which some learning
methods have been proposed Anshu  _et al._ (2021); Rouzé and França (2021). As
such, our strategy could be incorporated into existing methods as a basic
building block for further reducing sample complexity via making best use of
symmetric structures of states.

In conclusion, targeting at learning expectation values of observables, we
have proposed an entangling-measurement-based strategy that can leverage
quantum computational resources to dramatically reduce sample complexity. Our
strategy, which is powered by the exploration of symmetric structures of
states, is to infer the expectation value of an observable $X$ from the
projective measurement of its symmetrized counterpart $Y=\mathcal{T}(X)$
rather than $X$ itself. This enables our strategy to operate at the optimal
sample efficiency in a variety of contexts and thereby go beyond what can be
achieved with local measurements.

To illustrate the significance of our strategy, we have applied it to two
scenarios involving different kinds of symmetric structures of states, i.e.,
those described respectively by the translation and permutation groups, which
are ubiquitous in condensed-matter physics and quantum many-body physics. In
all the scenarios, we have shown that our strategy allows for yielding
exponential reductions in sample complexity while merely consuming polynomial
amounts of quantum computational resources.

The present Letter opens many interesting topics for future work, e.g., how to
optimally take advantage of symmetric structures of states to simultaneously
learn expectation values of multiple observables and further to extend the
scope of discussions from expectation values of observables to other
properties like various resource measures Zhang  _et al._ (2018).

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
through Grant Nos. 12275155 and 12174224.

Supplemental Material

##  I Fundamentals of quantum metrology

Here we recall some fundamentals of quantum metrology, whose mathematical
foundation is the quantum parameter estimation theory Helstrom (1976); Holevo
(2011). The theory deals with the quantum parameter estimation problems, in
which both the state in question and the quantity of interest are
characterized by some unknown parameters. Specifically, the state in question
can be a density matrix $\rho(\bm{\theta})$ characterized by $\mathfrak{p}$
unknown parameters $\bm{\theta}=(\theta_{1},\cdots,\theta_{\mathfrak{p}})$,
and the quantity of interest could be a function of $\bm{\theta}$, denoted as
$\beta(\bm{\theta})$. Given $M$ copies of $\rho(\bm{\theta})$, the task is to
estimate the value of $\beta$ from the outcome of a measurement performed on
$\rho(\bm{\theta})^{\otimes M}$. Any measurement can be described by a
positive operator-valued measure $\Pi_{x}$ satisfying
$\sum_{x}\Pi_{x}=\openone$, where $x$ labels the outcome of the measurement.
Here, albeit written as a single discrete variable, $x$ could be continuous or
multivariate. Any inference rule amounts to an estimator $\hat{\beta}(x)$,
i.e., a map from the set of measurement outcomes to the set of possible values
of $\beta$. The deviation of $\hat{\beta}(x)$ from $\beta$ is quantified by
Braunstein and Caves (1994)

$\displaystyle\delta\beta(x):=\frac{\hat{\beta}(x)}{\absolutevalue{d\expectationvalue{\hat{\beta}}/d\beta}}-\beta.$
(I.0S.1)

Here, $\expectationvalue{\hat{\beta}}$ is the statistical average of
$\hat{\beta}(x)$ over potential outcomes $x$,

$\displaystyle\expectationvalue{\hat{\beta}}=\sum_{x}p_{x}({\bm{\theta}})\hat{\beta}(x),$
(I.0S.2)

where $p_{x}({\bm{\theta}})=\tr[\Pi_{x}\rho(\bm{\theta})^{\otimes M}]$.
$d\expectationvalue{\hat{\beta}}/d\beta$ in Eq. (I.0S.1) is used to remove the
local difference in the “units” of the estimate $\hat{\beta}$ and the value
$\beta$ Braunstein and Caves (1994). The estimation error is defined as
Braunstein and Caves (1994)

$\displaystyle{\expectationvalue{(\delta\beta)^{2}}}=\sum_{x}p_{x}(\bm{\theta})[\delta\beta(x)]^{2},$
(I.0S.3)

i.e., the statistical average of $[\delta\beta(x)]^{2}$ over potential
outcomes $x$. A central result in the quantum parameter estimation theory is
that $\expectationvalue{(\delta\beta)^{2}}$ is bounded from below as Helstrom
(1976); Holevo (2011); Braunstein and Caves (1994),

$\displaystyle\expectationvalue{(\delta\beta)^{2}}\geq\frac{\partial\beta[H(\bm{\theta})]^{-1}\partial\beta^{T}}{M},$
(I.0S.4)

known as the quantum Cramér-Rao bound (QCRB). Here, as specified in the main
text, $H(\theta)$ is the quantum Fisher information (QFI) matrix. The meaning
of the QCRB is that quantum mechanics does not allow for estimating $\beta$
with arbitrarily high precision. Instead, the precision attainable in any
strategy for estimating $\beta$ with the $M$ copies of $\rho(\bm{\theta})$ is
fundamentally constrained by the QCRB, which represents the ultimate precision
allowed by quantum mechanics Braunstein and Caves (1994); Zhang and Gong
(2020); Zhang and Tong (2022). In particular, when $\hat{\beta}(x)$ is
unbiased, i.e., $\expectationvalue{\hat{\beta}}=\beta$, there is
$\delta\beta(x)=\hat{\beta}(x)-\expectationvalue{\hat{\beta}}$. Hence,

$\displaystyle\expectationvalue{(\delta\beta)^{2}}=\textrm{Var}[\hat{\beta}],$
(I.0S.5)

i.e., $\expectationvalue{(\delta\beta)^{2}}$ is simply the variance for an
unbiased estimator. Inserting Eq. (I.0S.5) into Eq. (I.0S.4) gives Eq. (1) in
the main text.

##  II Parameterization of symmetric states

Here we show how to characterize $\rho$ in terms of unknown parameters. To do
this, we introduce a $\mathbb{C}^{*}$-algebra Curtis and Reiner (1962),

$\displaystyle\mathcal{A}=\\{A|[A,U_{g}]=0,\forall g\in G\\},$ (II.0S.6)

which is comprised of all the complex matrices commuting with $U_{g}$.
Standard structure theorems for $\mathbb{C}^{*}$-algebras Curtis and Reiner
(1962) imply that $\mathcal{A}$ has a unique representation of the form

$\displaystyle\mathcal{A}\cong\bigoplus_{\alpha=1}^{s}\textrm{L}(\mathbb{C}^{n_{\alpha}})\otimes\openone_{d_{\alpha}},$
(II.0S.7)

up to unitary equivalence. Here, $\alpha$ labels the $\alpha$-th irreducible
representation of $G$ with dimension $d_{\alpha}$ and multiplicity
$n_{\alpha}$, $\textrm{L}(\mathbb{C}^{n_{\alpha}})$ is the matrix algebra of
all $n_{\alpha}\times n_{\alpha}$ complex matrices, and
$\openone_{d_{\alpha}}$ denotes the $d_{\alpha}\times d_{\alpha}$ identity
matrix. Using Eq. (II.0S.7) and noting that $\rho\in\mathcal{A}$, we can
express $\rho$ as

$\displaystyle\rho\cong\bigoplus_{\alpha=1}^{s}q_{\alpha}\rho_{\alpha}\otimes\frac{\openone_{d_{\alpha}}}{d_{\alpha}},$
(II.0S.8)

where $q_{\alpha}\geq 0$ satisfies $\sum_{\alpha=1}^{s}q_{\alpha}=1$ and
$\rho_{\alpha}$ denotes a density matrix in
$\textrm{L}(\mathbb{C}^{n_{\alpha}})$. We can further expand $\rho_{\alpha}$
by the generators
$\bm{\lambda}_{\alpha}=(\lambda_{\alpha,1},\cdots,\lambda_{\alpha,n_{\alpha}^{2}-1})$
of Lie algebra $\mathfrak{su}(n_{\alpha})$,

$\displaystyle\rho_{\alpha}=\frac{\openone_{n_{\alpha}}}{n_{\alpha}}+\frac{1}{2}\bm{r}_{\alpha}\cdot\bm{\lambda}_{\alpha},$
(II.0S.9)

where $\bm{r}_{\alpha}\in\mathbb{R}^{n_{\alpha}^{2}-1}$ is the generalized
Bloch vector Kimura (2003). The generators satisfy

$\displaystyle\lambda_{\alpha,i}^{\dagger}=\lambda_{\alpha,i},~{}~{}\tr\lambda_{\alpha,i}=0,~{}~{}\tr(\lambda_{\alpha,i}\lambda_{\alpha,j})=2\delta_{ij},$
(II.0S.10)

and they are characterized by structure constants $f_{\alpha,ijk}$ (completely
antisymmetric tensor) and $g_{\alpha,ijk}$ (completely symmetric tensor) as

$\displaystyle[\lambda_{\alpha,i},\lambda_{\alpha,j}]=2i\sum_{k}f_{\alpha,ijk}\lambda_{\alpha,k},$
(II.0S.11)

and

$\displaystyle\\{\lambda_{\alpha,i},\lambda_{\alpha,j}\\}=4\delta_{ij}\frac{\openone_{n_{\alpha}}}{n_{\alpha}}+2\sum_{k}g_{\alpha,ijk}\lambda_{\alpha,k},$
(II.0S.12)

where $\\{\cdot,\cdot\\}$ denotes the anticommutator. From Eqs. (II.0S.8) and
(II.0S.9), it follows that $\rho$ can be characterized by the parameters
$\bm{\theta}:=({\bm{q}},\bm{r}_{1},\cdots,\bm{r}_{s})$. Here, in view of the
constraint that $\sum_{\alpha=1}^{s}q_{\alpha}=1$, we have chosen
${\bm{q}}:=(q_{1},\cdots,q_{s-1})$ as independent parameters without loss of
generality.

##  III Proof of Eq. (3)

Here we present a proof of Eq. (3) in the main text. To derive $\partial\beta$
where $\beta=\expectationvalue{X}_{\rho}$, we make use of the equality

$\displaystyle\expectationvalue{X}_{\rho}=\expectationvalue{Y}_{\rho},$
(III.0S.13)

which follows from the normalization of the Haar measure $\int_{G}d\nu(g)=1$
not  and $U_{g}\rho U_{g}^{\dagger}=\rho$. Besides, the translation-invariant
property of the Haar measure implies that $Y\in\mathcal{A}$, which allows us
to express $Y$ in the form

$\displaystyle
Y\cong\bigoplus_{\alpha=1}^{s}Y_{\alpha}\otimes\openone_{d_{\alpha}},$
(III.0S.14)

with

$\displaystyle
Y_{\alpha}=a_{\alpha}\openone_{n_{\alpha}}+\bm{b}_{\alpha}\cdot\bm{\lambda}_{\alpha}.$
(III.0S.15)

Here, $a_{\alpha}\in\mathbb{R}$ and
$\bm{b}_{\alpha}\in\mathbb{R}^{n_{\alpha}^{2}-1}$ are constants, as $Y$ is
uniquely determined by $X$ through Eq. (4) in the main text. Inserting Eqs.
(II.0S.8), (II.0S.9), (III.0S.14), and (III.0S.15) into Eq. (III.0S.13) to get

$\displaystyle\expectationvalue{X}_{\rho}=\sum_{\alpha=1}^{s}q_{\alpha}l_{\alpha}$
(III.0S.16)

and then differentiating $\expectationvalue{X}_{\rho}$ with respect to
$\bm{\theta}$, we have

$\displaystyle\partial\beta=\left(\bm{l}-l_{s}\bm{e},q_{1}\bm{b}_{1},\cdots,q_{s}\bm{b}_{s}\right),$
(III.0S.17)

where

$\displaystyle
l_{\alpha}=a_{\alpha}+\bm{b}_{\alpha}\bm{r}_{\alpha}^{T},~{}~{}\bm{l}=(l_{1},\cdots,l_{s-1}),$
(III.0S.18)

and $\bm{e}$ is a $(s-1)$-dimensional vector with all components identical to
one. In Sec. IV, we figure out the QFI matrix for $\rho(\bm{\theta})$,

$\displaystyle
H(\bm{\theta})=H({\bm{q}})\bigoplus\left[\bigoplus_{\alpha=1}^{s}H(\bm{r}_{\alpha})\right],$
(III.0S.19)

with

$\displaystyle
H({\bm{q}})=\textrm{diag}(\frac{1}{q_{1}},\cdots,\frac{1}{q_{s-1}})+\frac{1}{q_{s}}\bm{e}^{T}\bm{e}$
(III.0S.20)

and

$\displaystyle
H(\bm{r}_{\alpha})=q_{\alpha}[R_{\alpha}-\bm{r}_{\alpha}^{T}\bm{r}_{\alpha}+\frac{2}{n_{\alpha}}\openone_{n_{\alpha}^{2}-1}]^{-1}.$
(III.0S.21)

Here, all $q_{\alpha}$’s are assumed temporarily to be strictly larger than
zero, and $R_{\alpha}$ is a $(n_{\alpha}^{2}-1)\times(n_{\alpha}^{2}-1)$
symmetric matrix with its $jk$ element defined as

$\displaystyle R_{\alpha,jk}=\sum_{i}r_{\alpha,i}g_{\alpha,ijk},$ (III.0S.22)

where $r_{\alpha,i}$ denotes the $i$-th component of $\bm{r}_{\alpha}$. Using
Eqs. (III.0S.17) and (III.0S.19), we figure out the left-hand side (LHS) of
Eq. (3),

$\displaystyle\partial\beta
H(\bm{\theta})^{-1}\partial\beta^{T}=\left(\sum_{\alpha=1}^{s}q_{\alpha}l_{\alpha}^{2}\right)-\left(\sum_{\alpha=1}^{s}q_{\alpha}l_{\alpha}\right)^{2}$
$\displaystyle+\sum_{\alpha=1}^{s}q_{\alpha}\bm{b}_{\alpha}\left(R_{\alpha}-\bm{r}_{\alpha}^{T}\bm{r}_{\alpha}+\frac{2}{n_{\alpha}}\openone_{n_{\alpha}^{2}-1}\right)\bm{b}_{\alpha}^{T},$
(III.0S.23)

where we have used the equality

$\displaystyle[H({\bm{q}})]^{-1}=\textrm{diag}(q_{1},\cdots,q_{s-1})-{\bm{q}}^{T}{\bm{q}}.$
(III.0S.24)

To figure out the right-hand side (RHS) of Eq. (3), we deduce from Eqs.
(II.0S.8) and (III.0S.14) that

$\displaystyle\expectationvalue{Y^{2}}_{\rho}=\sum_{\alpha=1}^{s}q_{\alpha}\tr(\rho_{\alpha}Y_{\alpha}^{2}).$
(III.0S.25)

Then, using Eqs. (II.0S.9) and (III.0S.15) and noting that

$\displaystyle\tr(\bm{b}_{\alpha}\cdot\bm{\lambda}_{\alpha}\bm{r}_{\alpha}\cdot\bm{\lambda}_{\alpha})=2\bm{b}_{\alpha}\bm{r}_{\alpha}^{T},$
(III.0S.26)
$\displaystyle\tr(\bm{b}_{\alpha}\cdot\bm{\lambda}_{\alpha}\bm{b}_{\alpha}\cdot\bm{\lambda}_{\alpha})=2\bm{b}_{\alpha}\bm{b}_{\alpha}^{T},$
(III.0S.27)
$\displaystyle\tr(\bm{b}_{\alpha}\cdot\bm{\lambda}_{\alpha}\bm{r}_{\alpha}\cdot\bm{\lambda}_{\alpha}\bm{b}_{\alpha}\cdot\bm{\lambda}_{\alpha})=2\bm{b}_{\alpha}R_{\alpha}\bm{b}_{\alpha}^{T},$
(III.0S.28)

we have

$\displaystyle\expectationvalue{Y^{2}}_{\rho}=\sum_{\alpha=1}^{s}q_{\alpha}\left(a_{\alpha}^{2}+2a_{\alpha}\bm{b}_{\alpha}\bm{r}_{\alpha}^{T}+\frac{2}{n_{\alpha}}\bm{b}_{\alpha}\bm{b}_{\alpha}^{T}+\bm{b}_{\alpha}R_{\alpha}\bm{b}_{\alpha}^{T}\right).$

Using this equality and noting that
$\expectationvalue{Y}_{\rho}=\sum_{\alpha=1}^{s}q_{\alpha}l_{\alpha}$, we
obtain

$\displaystyle(\Delta Y)^{2}$ $\displaystyle=$
$\displaystyle\sum_{\alpha=1}^{s}q_{\alpha}(a_{\alpha}^{2}+2a_{\alpha}\bm{b}_{\alpha}\bm{r}_{\alpha}^{T}+\frac{2}{n_{\alpha}}\bm{b}_{\alpha}\bm{b}_{\alpha}^{T}+$
(III.0S.29)
$\displaystyle\bm{b}_{\alpha}R_{\alpha}\bm{b}_{\alpha}^{T})-\left(\sum_{\alpha=1}^{s}q_{\alpha}l_{\alpha}\right)^{2}.$

Rewriting Eq. (III.0S.29) by taking into account

$\displaystyle
a_{\alpha}^{2}+2a_{\alpha}\bm{b}_{\alpha}\bm{r}_{\alpha}^{T}=l_{\alpha}^{2}-\bm{b}_{\alpha}\bm{r}_{\alpha}^{T}\bm{r}_{\alpha}\bm{b}_{\alpha}^{T}$
(III.0S.30)

and further comparing the resultant equation with Eq. (III), we arrive at Eq.
(3),

$\displaystyle\partial\beta H(\bm{\theta})^{-1}\partial\beta^{T}=(\Delta
Y)^{2}.$ (III.0S.31)

As the set of invertible matrices belonging to $\mathcal{A}$ is a dense subset
of $\mathcal{A}$, Eq. (III.0S.31) also holds when one or more $q_{\alpha}$
approach zero.

##  IV Derivation of Eq. (III.0S.19)

Here we derive the QFI matrix for $\rho(\bm{\theta})$. To do this, we need to
figure out the symmetric logarithmic derivatives (SLDs) associated with the
unknown parameters $\bm{\theta}=({\bm{q}},\bm{r}_{1},\cdots,\bm{r}_{s})$. The
SLD associated with $q_{\alpha}$, denoted by $L_{q_{\alpha}}$, is the
Hermitian operator that satisfies

$\displaystyle\partialderivative{\rho}{q_{\alpha}}=\rho\circ L_{q_{\alpha}},$
(IV.0S.32)

where $\alpha\in\\{1,\cdots,s-1\\}$. Noting that the LHS of Eq. (IV.0S.32)
reads

$\displaystyle\partialderivative{\rho}{q_{\alpha}}\cong\rho_{\alpha}\otimes\frac{\openone_{d_{\alpha}}}{d_{\alpha}}-\rho_{s}\otimes\frac{\openone_{d_{s}}}{d_{s}},$
(IV.0S.33)

we have

$\displaystyle
L_{q_{\alpha}}\cong\frac{1}{q_{\alpha}}\openone_{n_{\alpha}}\otimes\openone_{d_{\alpha}}-\frac{1}{q_{s}}\openone_{n_{s}}\otimes\openone_{d_{s}}.$
(IV.0S.34)

Here, $\cong$ means these equalities are up to a unitary transformation. On
the other hand, the SLD associated with $r_{\alpha,i}$, denoted by
$L_{r_{\alpha,i}}$, is the Hermitian operator satisfying

$\displaystyle\partialderivative{\rho}{r_{\alpha,i}}=\rho\circ
L_{r_{\alpha,i}}.$ (IV.0S.35)

To solve Eq. (IV.0S.35), we assume the following ansatz for
$L_{r_{\alpha,i}}$,

$\displaystyle
L_{r_{\alpha,i}}\cong\left(w_{\alpha,i}\openone_{n_{\alpha}}+\bm{v}_{\alpha,i}\cdot\bm{\lambda}_{\alpha}\right)\otimes\openone_{d_{\alpha}},$
(IV.0S.36)

where $w_{\alpha,i}\in\mathbb{R}$ and
$\bm{v}_{\alpha,i}\in\mathbb{R}^{n_{\alpha}^{2}-1}$ are to be determined.
Besides, it is easy to see that the LHS of Eq. (IV.0S.35) reads

$\displaystyle\partialderivative{\rho}{r_{\alpha,i}}\cong\frac{1}{2}q_{\alpha}\lambda_{\alpha,i}\otimes\frac{\openone_{d_{\alpha}}}{d_{\alpha}}.$
(IV.0S.37)

Substituting Eqs. (IV.0S.36) and (IV.0S.37) into Eq. (IV.0S.35), we have

$\displaystyle\left(\frac{\openone_{n_{\alpha}}}{n_{\alpha}}+\frac{1}{2}\bm{r}_{\alpha}\cdot\bm{\lambda}_{\alpha}\right)\circ\left(w_{\alpha,i}\openone_{n_{\alpha}}+\bm{v}_{\alpha,i}\cdot\bm{\lambda}_{\alpha}\right)=\frac{1}{2}\lambda_{\alpha,i},$

which may be viewed as an equation in terms of $w_{\alpha,i}$ and
$\bm{v}_{\alpha,i}$. Solving Eq. (IV) by resorting to the defining properties
of $\bm{\lambda}_{\alpha}$ [specified in Eqs. (II.0S.10), (II.0S.11), and
(II.0S.12)], we have

$\displaystyle w_{\alpha,i}=-\bm{v}_{\alpha,i}\bm{r}_{\alpha}^{T},$ (IV.0S.39)

and

$\displaystyle\bm{v}_{\alpha,i}=\bm{h}_{\alpha,i}\left[R_{\alpha}-\bm{r}_{\alpha}^{T}\bm{r}_{\alpha}+\frac{2}{n_{\alpha}}\openone_{n_{\alpha}^{2}-1}\right]^{-1}.$
(IV.0S.40)

Here, $R_{\alpha}$ is defined in Eq. (III.0S.22), and $\bm{h}_{\alpha,i}$ is a
$(n_{\alpha}^{2}-1)$-dimensional vector with its $i$-th component identical to
one and all others being zero. By the way, we point out that in the case that
$[R_{\alpha}-\bm{r}_{\alpha}^{T}\bm{r}_{\alpha}+\frac{2}{n_{\alpha}}\openone_{n_{\alpha}^{2}-1}]$
is singular,
$[R_{\alpha}-\bm{r}_{\alpha}^{T}\bm{r}_{\alpha}+\frac{2}{n_{\alpha}}\openone_{n_{\alpha}^{2}-1}]^{-1}$is
understood as the Moore–Penrose inverse of
$[R_{\alpha}-\bm{r}_{\alpha}^{T}\bm{r}_{\alpha}+\frac{2}{n_{\alpha}}\openone_{n_{\alpha}^{2}-1}]$.
It is easy to see that for three matrices $A$, $B$, and $C$, there are

$\displaystyle A\circ B=B\circ A,~{}~{}~{}~{}\tr[A(B\circ C)]=\tr[(A\circ
B)C].$ (IV.0S.41)

Using these two equalities and Eqs. (IV.0S.34), (IV.0S.35), and (IV.0S.37), we
have

$\displaystyle\tr\left[\rho\left(L_{q_{\alpha}}\circ
L_{r_{\beta,i}}\right)\right]$ (IV.0S.42) $\displaystyle=$
$\displaystyle\tr\left[\left(\rho\circ
L_{r_{\beta,i}}\right)L_{q_{\alpha}}\right]$ $\displaystyle=$
$\displaystyle\tr(\partialderivative{\rho}{r_{\beta,i}}L_{q_{\alpha}})$
$\displaystyle\cong$
$\displaystyle\tr\left[\frac{1}{2}q_{\beta}\lambda_{\beta,i}\otimes\frac{\openone_{d_{\beta}}}{d_{\beta}}\left(\frac{1}{q_{\alpha}}\openone_{n_{\alpha}}\otimes\openone_{d_{\alpha}}-\frac{1}{q_{s}}\openone_{n_{s}}\otimes\openone_{d_{s}}\right)\right]$
$\displaystyle=$ $\displaystyle 0.$

Here, $\beta$ is a subscript and should be distinguished from the notation
used in the main text. Using Eq. (IV.0S.36), we easily have

$\displaystyle\tr\left[\rho\left(L_{r_{\alpha,i}}\circ
L_{r_{\beta,j}}\right)\right]=0,$ (IV.0S.43)

for $\alpha\neq\beta$. From Eqs. (IV.0S.42) and (IV.0S.43), we deduce that
$H(\bm{\theta})$ can be expressed in the following block-diagonal form

$\displaystyle
H(\bm{\theta})=H({\bm{q}})\bigoplus\left[\bigoplus_{\alpha=1}^{s}H(\bm{r}_{\alpha})\right],$
(IV.0S.44)

where $H({\bm{q}})$ is a $(s-1)\times(s-1)$ symmetric matrix with its
$\alpha\beta$ element defined as

$\displaystyle\left[H({\bm{q}})\right]_{\alpha\beta}=\tr[\rho(L_{q_{\alpha}}\circ
L_{q_{\beta}})],$ (IV.0S.45)

and $H(\bm{r}_{\alpha})$ is a $(n_{\alpha}^{2}-1)\times(n_{\alpha}^{2}-1)$
symmetric matrix with its $ij$ element defined as

$\displaystyle\left[H(\bm{r}_{\alpha})\right]_{ij}=\tr[\rho\left(L_{r_{\alpha,i}}\circ
L_{r_{\alpha,j}}\right)].$ (IV.0S.46)

Inserting Eq. (IV.0S.34) into Eq. (IV.0S.45), we have, after simple algebra,

$\displaystyle\left[H({\bm{q}})\right]_{\alpha\beta}=\delta_{\alpha\beta}\frac{1}{q_{\alpha}}+\frac{1}{q_{s}},$
(IV.0S.47)

that is,

$\displaystyle
H({\bm{q}})=\textrm{diag}(\frac{1}{q_{1}},\cdots,\frac{1}{q_{s-1}})+\frac{1}{q_{s}}\bm{e}^{T}\bm{e}.$
(IV.0S.48)

Substituting Eqs. (IV.0S.36), (IV.0S.37), (IV.0S.39), and (IV.0S.40) into Eq.
(IV.0S.46) gives

$\displaystyle\left[H(\bm{r}_{\alpha})\right]_{ij}$ (IV.0S.49)
$\displaystyle=$ $\displaystyle\tr[\rho\left(L_{r_{\alpha,i}}\circ
L_{r_{\alpha,j}}\right)]$ $\displaystyle=$ $\displaystyle\tr[\left(\rho\circ
L_{r_{\alpha,i}}\right)L_{r_{\alpha,j}}]$ $\displaystyle=$
$\displaystyle\tr[\partialderivative{\rho}{r_{\alpha,i}}L_{r_{\alpha,j}}]$
$\displaystyle=$
$\displaystyle\tr[\left(\frac{1}{2}q_{\alpha}\lambda_{\alpha,i}\otimes\frac{\openone_{d_{\alpha}}}{d_{\alpha}}\right)\left(w_{\alpha,j}\openone_{n_{\alpha}}+\bm{v}_{\alpha,j}\cdot\bm{\lambda}_{\alpha}\right)\otimes\openone_{d_{\alpha}}]$
$\displaystyle=$ $\displaystyle
q_{\alpha}\bm{h}_{\alpha,i}\bm{v}_{\alpha,j}^{T}$ $\displaystyle=$
$\displaystyle
q_{\alpha}\bm{h}_{\alpha,i}\left[R_{\alpha}-\bm{r}_{\alpha}^{T}\bm{r}_{\alpha}+\frac{2}{n_{\alpha}}\openone_{n_{\alpha}^{2}-1}\right]^{-1}\bm{h}_{\alpha,j}^{T}.$

That is,

$\displaystyle
H(\bm{r}_{\alpha})=q_{\alpha}\left[R_{\alpha}-\bm{r}_{\alpha}^{T}\bm{r}_{\alpha}+\frac{2}{n_{\alpha}}\openone_{n_{\alpha}^{2}-1}\right]^{-1}.$
(IV.0S.50)

Now, the QFI matrix for $\rho(\bm{\theta})$ is given by Eqs. (IV.0S.44) with
Eq. (IV.0S.48) and (IV.0S.50).

##  V Details on application 1

Here we prove the two inequalities used in the main text,

$\displaystyle 2^{n}-1\leq(\Delta X)^{2}\leq 2^{n},$ (V.0S.51)

and

$\displaystyle 0\leq(\Delta Y)^{2}\leq 1.$ (V.0S.52)

To do this, we resort to the product representation of the Fourier basis
Nielsen and Chuang (2010),

$\displaystyle\ket{f_{j}}=\frac{\left(\ket{0}+e^{2\pi
i0.j_{n}}\ket{1}\right)\cdots\left(\ket{0}+e^{2\pi i0.j_{1}\cdots
j_{n}}\ket{1}\right)}{2^{n/2}},$

where $0.j_{l}j_{l+1}\cdots j_{m}$ denotes the binary fraction
$j_{l}/2+j_{l+1}/4+\cdots+j_{m}/2^{m-l+1}$. Using Eq. (V) to calculate
$X_{jj}=\bra{f_{j}}X\ket{f_{j}}$ gives

$\displaystyle X_{jj}=\cos(2\pi 0.j_{n})\cdots\cos(2\pi 0.j_{1}j_{2}\cdots
j_{n}).$ (V.0S.54)

Noting that $\absolutevalue{X_{jj}}\leq 1$ and
$\rho=\sum_{j}p_{j}\ket{f_{j}}\bra{f_{j}}$, we have

$\displaystyle\absolutevalue{\expectationvalue{X}_{\rho}}=\absolutevalue{\sum_{j}p_{j}X_{jj}}\leq\sum_{j}p_{j}\absolutevalue{X_{jj}}\leq\sum_{j}p_{j}=1.$
(V.0S.55)

Besides, note that $X^{2}=2^{n}\openone\otimes\cdots\otimes\openone$ and
therefore

$\displaystyle\expectationvalue{X^{2}}_{\rho}=2^{n}.$ (V.0S.56)

Then Eq. (V.0S.51) follows from Eqs. (V.0S.55) and (V.0S.56). On the other
hand, noting that $Y=\sum_{j}X_{jj}\ket{f_{j}}\bra{f_{j}}$, we have

$\displaystyle\absolutevalue{\expectationvalue{Y}_{\rho}}=\absolutevalue{\expectationvalue{X}_{\rho}}\leq
1,$ (V.0S.57)

and

$\displaystyle\expectationvalue{Y^{2}}_{\rho}=\sum_{j}p_{j}X_{jj}^{2}\leq 1.$
(V.0S.58)

It is easy to see that Eq. (V.0S.52) follows from Eqs. (V.0S.57) and
(V.0S.58).

##  VI Details on application 2

###  VI.1 Calculating $(\Delta X_{\bm{kl}})^{2}$ and $(\Delta
Y_{\bm{kl}})^{2}$

Here we figure out the expressions of $(\Delta X_{\bm{kl}})^{2}$ and $(\Delta
Y_{\bm{kl}})^{2}$. Note that

$\displaystyle\sigma_{x}^{k_{1}}\sigma_{z}^{l_{1}}\otimes\cdots\otimes\sigma_{x}^{k_{n}}\sigma_{z}^{l_{n}}\ket{GHZ}=$
$\displaystyle\frac{1}{\sqrt{2}}\left[\ket{k_{1},\cdots,k_{n}}+(-1)^{\absolutevalue{\bm{l}}}\ket{1-k_{1},\cdots,1-k_{n}}\right],$

with the $n$-qubit GHZ state $\ket{GHZ}=\left(\ket{0\cdots 0}+\ket{1\cdots
1}\right)/\sqrt{2}$ and $\absolutevalue{\bm{l}}=\sum_{i=1}^{n}l_{i}$. Using
Eq. (VI.1) and

$\displaystyle
X_{\bm{kl}}=\sigma_{x}^{k_{1}}\sigma_{z}^{l_{1}}\otimes\cdots\otimes\sigma_{x}^{k_{n}}\sigma_{z}^{l_{n}}(i)^{\bm{k}\cdot\bm{l}},$
(VI.1S.60)

we have

$\displaystyle\bra{GHZ}X_{\bm{kl}}\ket{GHZ}=(i)^{\bm{k}\cdot\bm{l}}\frac{1+(-1)^{\absolutevalue{\bm{l}}}}{2}(\delta_{\bm{k},\bm{0}}+\delta_{\bm{k},\bm{1}}),$

where $\delta_{\bm{k},\bm{0}}:=\delta_{k_{1},0}\cdots\delta_{k_{n},0}$ and
$\delta_{\bm{k},\bm{1}}:=\delta_{k_{1},1}\cdots\delta_{k_{n},1}$. Hence,

$\displaystyle\bra{GHZ}X_{\bm{kl}}\ket{GHZ}=$
$\displaystyle\begin{cases}1&\bm{k}=\bm{0}~{}~{}\textrm{and}~{}~{}\absolutevalue{\bm{l}}\in\textrm{even
numbers},\\\
(i)^{\absolutevalue{\bm{l}}}&\bm{k}=\bm{1}~{}~{}\textrm{and}~{}~{}\absolutevalue{\bm{l}}\in\textrm{even
numbers},\\\ 0&\textrm{otherwise}.\end{cases}$ (VI.1S.62)

It follows from Eq. (VI.1) and $X_{\bm{kl}}^{2}=\openone_{2^{n}}$ that

$\displaystyle(\Delta
X_{\bm{kl}})^{2}=\begin{cases}0&\bm{k}=\bm{0},\bm{1}~{}~{}\textrm{and}~{}~{}\absolutevalue{\bm{l}}\in\textrm{even
numbers},\\\ 1&\textrm{otherwise}.\end{cases}$

To calculate $(\Delta Y_{\bm{kl}})^{2}$, we resort to the defining properties
of $P_{s}$,

$\displaystyle
P_{s}\ket{\psi_{1}}\otimes\cdots\otimes\ket{\psi_{n}}=\ket{\psi_{s(1)}}\otimes\cdots\otimes\ket{\psi_{s(n)}}$
(VI.1S.64)

and

$\displaystyle P_{s}A_{1}\otimes\cdots\otimes
A_{n}P_{s}^{\dagger}=A_{s(1)}\otimes\cdots\otimes A_{s(n)},$ (VI.1S.65)

which, in conjunction with the rearrangement theorem, lead to

$\displaystyle
P_{s}\ket{GHZ}=\ket{GHZ},~{}~{}P_{s}Y_{\bm{kl}}=Y_{\bm{kl}}P_{s}.$ (VI.1S.66)

Using Eq. (VI.1S.66) and

$\displaystyle Y_{\bm{kl}}=\frac{1}{n!}\sum_{s\in
S_{n}}\sigma_{x}^{k_{s(1)}}\sigma_{z}^{l_{s(1)}}\otimes\cdots\otimes\sigma_{x}^{k_{s(n)}}\sigma_{z}^{l_{s(n)}}(i)^{\bm{k}\cdot\bm{l}},$

we have

$\displaystyle\bra{GHZ}Y_{\bm{kl}}^{2}\ket{GHZ}$ $\displaystyle=$
$\displaystyle\frac{1}{n!}\sum_{s\in
S_{n}}\bra{GHZ}\sigma_{x}^{k_{s(1)}}\sigma_{z}^{l_{s(1)}}\otimes\cdots\otimes\sigma_{x}^{k_{s(n)}}\sigma_{z}^{l_{s(n)}}(i)^{\bm{k}\cdot\bm{l}}Y_{\bm{kl}}\ket{GHZ}$
(VI.1S.68) $\displaystyle=$ $\displaystyle\frac{1}{n!}\sum_{s\in
S_{n}}\bra{GHZ}P_{s}^{\dagger}\sigma_{x}^{k_{s(1)}}\sigma_{z}^{l_{s(1)}}\otimes\cdots\otimes\sigma_{x}^{k_{s(n)}}\sigma_{z}^{l_{s(n)}}(i)^{\bm{k}\cdot\bm{l}}Y_{\bm{kl}}P_{s}\ket{GHZ}$
$\displaystyle=$ $\displaystyle\frac{1}{n!}\sum_{s\in
S_{n}}\bra{GHZ}P_{s}^{\dagger}\sigma_{x}^{k_{s(1)}}\sigma_{z}^{l_{s(1)}}\otimes\cdots\otimes\sigma_{x}^{k_{s(n)}}\sigma_{z}^{l_{s(n)}}(i)^{\bm{k}\cdot\bm{l}}P_{s}Y_{\bm{kl}}\ket{GHZ}$
$\displaystyle=$ $\displaystyle\frac{1}{n!}\sum_{s\in
S_{n}}\bra{GHZ}\sigma_{x}^{k_{1}}\sigma_{z}^{l_{1}}\otimes\cdots\otimes\sigma_{x}^{k_{n}}\sigma_{z}^{l_{n}}(i)^{\bm{k}\cdot\bm{l}}Y_{\bm{kl}}\ket{GHZ}$
$\displaystyle=$
$\displaystyle\bra{GHZ}\sigma_{x}^{k_{1}}\sigma_{z}^{l_{1}}\otimes\cdots\otimes\sigma_{x}^{k_{n}}\sigma_{z}^{l_{n}}(i)^{\bm{k}\cdot\bm{l}}Y_{\bm{kl}}\ket{GHZ}$
$\displaystyle=$ $\displaystyle\frac{1}{n!}\sum_{s\in
S_{n}}\bra{GHZ}\sigma_{x}^{k_{1}}\sigma_{z}^{l_{1}}\sigma_{x}^{k_{s(1)}}\sigma_{z}^{l_{s(1)}}\otimes\cdots\otimes\sigma_{x}^{k_{n}}\sigma_{z}^{l_{n}}\sigma_{x}^{k_{s(n)}}\sigma_{z}^{l_{s(n)}}(-1)^{\bm{k}\cdot\bm{l}}\ket{GHZ}$
$\displaystyle=$ $\displaystyle\frac{1}{n!}\sum_{s\in
S_{n}}\bra{GHZ}\sigma_{x}^{k_{1}+k_{s(1)}}\sigma_{z}^{l_{1}+l_{s(1)}}\otimes\cdots\otimes\sigma_{x}^{k_{n}+k_{s(n)}}\sigma_{z}^{l_{n}+l_{s(n)}}(-1)^{(\bm{k}+\bm{k}_{s})\cdot\bm{l}}\ket{GHZ}$
$\displaystyle=$ $\displaystyle\frac{1}{n!}\sum_{s\in
S_{n}}(-1)^{(\bm{k}+\bm{k}_{s})\cdot\bm{l}}\frac{1+(-1)^{\absolutevalue{\bm{l}+\bm{l}_{s}}}}{2}(\delta_{\bm{k}\oplus\bm{k}_{s},\bm{0}}+\delta_{\bm{k}\oplus\bm{k}_{s},\bm{1}}),$

where $\bm{k}_{s}=(k_{s(1)},\cdots,k_{s(n)})$,
$\bm{l}_{s}=(l_{s(1)},\cdots,l_{s(n)})$, and $\oplus$ denotes addition modulo
$2$, that is, $\bm{k}\oplus\bm{k}_{s}=(k_{1}+k_{s(1)},\cdots,k_{n}+k_{s(n)})$
(mod $2$). Noting that
$\absolutevalue{\bm{l}+\bm{l}_{s}}=\sum_{i=1}^{n}l_{i}+l_{s(i)}=2\absolutevalue{\bm{l}}$,
we have $[1+(-1)^{\absolutevalue{\bm{l}+\bm{l}_{s}}}]/{2}=1$. Moreover,
assuming that $n$ is odd, we have $\delta_{\bm{k}\oplus\bm{k}_{s},\bm{1}}=0$.
So, the only nonzero terms in the summation of Eq. (VI.1S.68) are those such
that $\bm{k}=\bm{k}_{s}$. The number of such terms is
$\absolutevalue{\bm{k}}!(n-\absolutevalue{\bm{k}})!$. Besides,
$(-1)^{(\bm{k}+\bm{k}_{s})\cdot\bm{l}}=1$ when $\bm{k}=\bm{k}_{s}$. We have

$\displaystyle\bra{GHZ}Y_{\bm{kl}}^{2}\ket{GHZ}=1/\binom{n}{\absolutevalue{\bm{k}}}.$
(VI.1S.69)

Noting that $\bra{GHZ}Y_{\bm{kl}}\ket{GHZ}=\bra{GHZ}X_{\bm{kl}}\ket{GHZ}$ and
using Eqs. (VI.1) and (VI.1S.69), we arrive at the expression of $(\Delta
Y_{\bm{kl}})^{2}$:

$\displaystyle(\Delta
Y_{\bm{kl}})^{2}=\begin{cases}0&\bm{k}=\bm{0},\bm{1}~{}~{}\textrm{and}~{}~{}\absolutevalue{\bm{l}}\in\textrm{even
numbers},\\\
1/\binom{n}{\absolutevalue{\bm{k}}}&\textrm{otherwise}.\end{cases}$

###  VI.2 Lower bound on $(\Delta X)^{2}/(\Delta Y)^{2}$ for typical Pauli
measurements

Here we figure out the lower bound on $(\Delta X)^{2}/(\Delta Y)^{2}$ for
typical Pauli measurements. Using the Stirling’s approximation

$\displaystyle n!\approx\sqrt{2\pi n}\left(\frac{n}{e}\right)^{n}$ (VI.2S.71)

and noting that

$\displaystyle(1-\delta)\frac{n}{2}<\absolutevalue{\bm{k}}<(1+\delta)\frac{n}{2},$
(VI.2S.72)

we have

$\displaystyle{(\Delta X)^{2}}/{(\Delta Y)^{2}}$ $\displaystyle=$
$\displaystyle\binom{n}{\absolutevalue{\bm{k}}}$ (VI.2S.73)
$\displaystyle\geq$ $\displaystyle\binom{n}{(1-\delta)\frac{n}{2}}$
$\displaystyle=$
$\displaystyle\frac{n!}{\left[(1-\delta)\frac{n}{2}\right]!\left[(1+\delta)\frac{n}{2}\right]!}$
$\displaystyle\approx$ $\displaystyle\frac{\sqrt{2\pi
n}\left(\frac{n}{e}\right)^{n}}{\sqrt{2\pi(1-\delta)\frac{n}{2}}\left(\frac{(1-\delta)\frac{n}{2}}{e}\right)^{(1-\delta)\frac{n}{2}}\sqrt{2\pi(1+\delta)\frac{n}{2}}\left(\frac{(1+\delta)\frac{n}{2}}{e}\right)^{(1+\delta)\frac{n}{2}}}$
$\displaystyle=$
$\displaystyle\sqrt{\frac{2}{n\pi(1-\delta^{2})}}\left[\frac{4}{(1-\delta)^{1-\delta}(1+\delta)^{1+\delta}}\right]^{\frac{n}{2}}.$

Here we have assumed for simplicity that $(1-\delta)\frac{n}{2}$ is an
integer. The above proof can be made more rigorous by replacing the Stirling’s
approximation with the inequalities $\sqrt{2\pi
n}\left(\frac{n}{e}\right)^{n}e^{\frac{1}{12n+1}}\leq n!\leq\sqrt{2\pi
n}\left(\frac{n}{e}\right)^{n}e^{\frac{1}{12n}}$ which hold for $n\geq 1$.

###  VI.3 Lower bound on the number of typical Pauli measurements

Here we find a lower bound on the number of typical Pauli measurements. To do
this, we interpret all $k_{i}$’s as independent and identically distributed
binary random variables taking on two values $0$ and $1$ with equal
probabilities $1/2$. Then, all the bit strings $\bm{k}=(k_{1},\cdots,k_{n})$
are with equal probabilities $1/2^{n}$. Using the Chebyshev’s inequality
Mitzenmacher and Upfal (2005) and noting that
$\textrm{Var}[\absolutevalue{\bm{k}}]=\textrm{Var}[k_{1}+\cdots+k_{n}]=n/4$,
we can bound from below the number of the bit strings with
$(1-\delta)\frac{n}{2}<\absolutevalue{\bm{k}}<(1+\delta)\frac{n}{2}$ as

$\displaystyle N$ $\displaystyle=$ $\displaystyle
2^{n}\textrm{Pr}\left((1-\delta)\frac{n}{2}<\absolutevalue{\bm{k}}<(1+\delta)\frac{n}{2}\right)$
(VI.3S.74) $\displaystyle=$ $\displaystyle
2^{n}\left[1-\textrm{Pr}\left(\absolutevalue{\absolutevalue{\bm{k}}-\frac{n}{2}}\geq\frac{n\delta}{2}\right)\right]$
$\displaystyle\geq$ $\displaystyle
2^{n}\left[1-\frac{\textrm{Var}[\absolutevalue{\bm{k}}]}{\left(\frac{n\delta}{2}\right)^{2}}\right]$
$\displaystyle=$ $\displaystyle 2^{n}\left(1-\frac{1}{n\delta^{2}}\right),$

where
$\textrm{Pr}\left((1-\delta)\frac{n}{2}<\absolutevalue{\bm{k}}<(1+\delta)\frac{n}{2}\right)$
stands for the probability that a bit string $\bm{k}$ is with
$(1-\delta)\frac{n}{2}<\absolutevalue{\bm{k}}<(1+\delta)\frac{n}{2}$, and
similarly for the notation
$\textrm{Pr}\left(\absolutevalue{\absolutevalue{\bm{k}}-\frac{n}{2}}\geq\frac{n\delta}{2}\right)$.
The number of typical Pauli measurements is

$\displaystyle 2^{n}N\geq 4^{n}\left(1-\frac{1}{n\delta^{2}}\right),$
(VI.3S.75)

which follows from the fact that there are $2^{n}$ different $\bm{l}$ for each
$\bm{k}$.

###  VI.4 Discussion on the implementation of $V$

Here we explain why the quantum circuit $V$ can be efficiently realized as a
quantum circuit. According to the Schur–Weyl duality Weyl (1950), the Hilbert
space $(\mathbb{C}^{2})^{\otimes n}$ of $n$ qubits can be decomposed into a
direct sum of tensor products,

$\displaystyle(\mathbb{C}^{2})^{\otimes
n}\cong\bigoplus_{j=j_{\textrm{min}}}^{j_{\textrm{max}}}\mathbb{C}^{2j+1}\otimes\mathbb{C}^{d_{n,j}},$
(VI.4S.76)

where $\mathbb{C}^{2j+1}$ and $\mathbb{C}^{d_{n,j}}$ are irreducible
representations of $U(2)$ and $S_{n}$, respectively. $j$ labels the eigen-
subspace of the total squared angular momentum with eigenvalue $j(j+1)$, where
$j=j_{\textrm{min}}$, $j_{\textrm{min}}+1$, $\cdots$, $j_{\textrm{max}}$ with
$j_{\textrm{max}}=n/2$ and $j_{\textrm{min}}=0$, $1/2$ for even and odd $n$,
respectively. The decomposition (VI.4S.76) corresponds to a basis for
$(\mathbb{C}^{2})^{\otimes n}$, $\ket{\Psi_{j,q,p}}$, known as the Schur basis
Bacon  _et al._ (2006), where $j$ labels the subspaces
$\mathbb{C}^{2j+1}\otimes\mathbb{C}^{d_{n,j}}$, and $q$ and $p$ label bases
for $\mathbb{C}^{2j+1}$ and $\mathbb{C}^{d_{n,j}}$, respectively. The Schur
transform, denoted as $U_{\textrm{Sch}}$ hereafter, is defined as the unitary
operator mapping the Schur basis into the computational basis,

$\displaystyle U_{\textrm{Sch}}\ket{\Psi_{j,q,p}}=\ket{j,q,p}.$ (VI.4S.77)

Here, $\ket{j,q,p}$ stands for the computational basis, with $j,q,p$ expressed
as bit strings. A central result of Ref. Bacon  _et al._ (2006) is that
$U_{\textrm{Sch}}$ can be realized as a quantum circuit of polynomial size.

Without loss of generality, we can express $X$ in the form

$\displaystyle
X=\sum_{j,j^{\prime}}\sum_{q,q^{\prime}}\sum_{p,p^{\prime}}X_{jqp;j^{\prime}q^{\prime}p^{\prime}}\ket{\Psi_{j,q,p}}\bra{\Psi_{j^{\prime},q^{\prime},p^{\prime}}},$
(VI.4S.78)

where

$\displaystyle
X_{jqp;j^{\prime}q^{\prime}p^{\prime}}=\bra{\Psi_{j,q,p}}X\ket{\Psi_{j^{\prime},q^{\prime},p^{\prime}}}.$
(VI.4S.79)

Moreover, noting that $\rho$ and $Y$ belong to
$\mathcal{A}\cong\bigoplus_{j=j_{\textrm{min}}}^{j_{\textrm{max}}}\textrm{L}(\mathbb{C}^{2j+1})\otimes\openone_{d_{n,j}}$,
we can express $\rho$ and $Y$ in the form

$\displaystyle\rho=\sum_{j}\sum_{q,q^{\prime}}\sum_{p}p_{j}(\rho_{j})_{qq^{\prime}}\ket{\Psi_{j,q,p}}\bra{\Psi_{j,q^{\prime},p}}$
(VI.4S.80)

and

$\displaystyle
Y=\sum_{j}\sum_{q,q^{\prime}}\sum_{p}(Y_{j})_{qq^{\prime}}\ket{\Psi_{j,q,p}}\bra{\Psi_{j,q^{\prime},p}},$
(VI.4S.81)

where $p_{j}\geq 0$ satisfies $\sum_{j}p_{j}=1$, $\rho_{j}$ is a density
matrix, and $Y_{j}$ is a Hermitian matrix. To find the connection between $X$
and $Y$, we make use of the equality
$\expectationvalue{X}_{\rho}=\expectationvalue{Y}_{\rho}$, which leads to

$\displaystyle({Y}_{j})_{qq^{\prime}}=\frac{1}{d_{n,j}}\sum_{p}X_{jqp;jq^{\prime}p}.$
(VI.4S.82)

It follows from Eqs. (VI.4S.77) and (VI.4S.81) that

$\displaystyle
U_{\textrm{Sch}}YU_{\textrm{Sch}}^{\dagger}=\sum_{j}\sum_{q,q^{\prime}}\sum_{p}(Y_{j})_{qq^{\prime}}\ket{{j,q,p}}\bra{{j,q^{\prime},p}}$
$\displaystyle=\sum_{j}\ket{j}\bra{j}\otimes\left(\sum_{q,q^{\prime}}(Y_{j})_{qq^{\prime}}\ket{q}\bra{q^{\prime}}\right)\otimes\openone_{d_{n,j}}.$

Let the eigendecomposition of $Y_{j}$ be

$\displaystyle Y_{j}=U_{j}D_{j}U_{j}^{\dagger},$ (VI.4S.84)

where $U_{j}$ denotes a unitary matrix and $D_{j}$ is a diagonal matrix with
real entries. We then define $V$ to be of the form

$\displaystyle V=V_{j_{\textrm{min}}}\cdots
V_{j_{\textrm{max}}}U_{\textrm{Sch}},$ (VI.4S.85)

where

$\displaystyle V_{j}=\ket{j}\bra{j}\otimes
U_{j}^{\dagger}+\sum_{j^{\prime}\neq
j}\ket{j^{\prime}}\bra{j^{\prime}}\otimes\openone_{2j+1}$ (VI.4S.86)

is a controlled gate. It is easy to see that the $V$ thus defined transforms
the eigenbasis of $Y$ into the computational basis $\ket{j,q,p}$. As each
$V_{j}$ is a unitary matrix of polynomial size, $V$ can be realized as a
quantum circuit of polynomial size.

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some transit parameter values as priors taken from a TESS and CARMENES joint
fit. Then systematic instrumental effects were removed from the data, which
were used for the final joint fit.

##  Appendix D Joint fit results

During our data fitting test phase we compared the results of a joint fit
performed on TESS photometric data and CARMENES RV measurements but using
different approaches to deal with the RVs. In one test we used GPs to model
the red noise of both datasets (TESS and CARMENES), for a second model we used
GPs for TESS and no GPs for CARMENES, and for the third model we used GPs only
for TESS data and no GPs for CARMENES but assuming different zero-point
offsets for each observed night. The zero-point offset method assumes that
each night has a different radial velocity zero-point offset, meaning that we
added five free $\gamma$ parameters corresponding to each night that the
target was observed. Figure 18 shows the posterior distribution of the radial
velocity semi-amplitude $K$ obtained from all the models we tested, from the
distributions we can see that either using or not using GPs, or the zero-point
offset method, we get the same posterior distribution for $K$; meaning the
that determination of the mass of the planet is unaffected by the use of GPs
to model the systematics present in the RV measurements.

Figure 18: Posterior distribution of the radial velocity semi-amplitude ($K$)
for a joint fit using only TESS and CARMENES RV observations. We show the
results for a fit using GPs to model the red noise of both datasets (light
yellow), GPs for TESS data and no GPs for CARMENES measurements (light blue),
and a fit of TESS photometry including GPs and RVs without GPs but using a
different zero-point (ZP) offset for each night (light purple).

Figure 19 presents the posterior distributions for the fitted orbital
parameters of Wolf 327b using all the data available (TESS, ground-based
transit observations, and RV measurements from CARMENES). The fitted
parameters for the eclipsing binary TIC 4918919 and Gaussian processes
components were left out intentionally from the plot for easy viewing.

Figure 19: Correlation plot for the fitted transit and orbital parameter for
Wolf 327b. The fitted orbital parameters of the eclipsing binary TIC 4918919
and the free parameters used to model the systematic effects were
intentionally left out for easy viewing. The blue lines mark the median values
of the distribution. For plotting purposes the distributions for the central
time of the transit and orbital period were offset by T1 = 2459252 d.