# Universal shot-noise limit for quantum metrology with local Hamiltonians

Hai-Long Shi  Innovation Academy for Precision Measurement Science and
Technology, Chinese Academy of Sciences, Wuhan 430071, China  Xi-Wen Guan
Innovation Academy for Precision Measurement Science and Technology, Chinese
Academy of Sciences, Wuhan 430071, China  Hefei National Laboratory, Hefei
230088, China  Department of Fundamental and Theoretical Physics, Research
School of Physics, Australian National University, Canberra ACT 0200,
Australia  Jing Yang<EMAIL_ADDRESS>Nordita, KTH Royal Institute of
Technology and Stockholm University, Hannes Alfvéns vag 12, 10691 Stockholm,
Sweden

###### Abstract

Quantum many-body interactions can induce quantum entanglement among
particles, rendering them valuable resources for quantum-enhanced sensing. In
this work, we derive a universal and fundamental bound for the growth of the
quantum Fisher information. We apply our bound to the metrological protocol
requiring only separable initial states, which can be readily prepared in
experiments. By establishing a link between our bound and the Lieb-Robinson
bound, which characterizes the operator growth in locally interacting quantum
many-body systems, we prove that the precision cannot surpass the shot noise
limit at all times in locally interacting quantum systems. This conclusion
also holds for an initial state that is the non-degenerate ground state of a
local and gapped Hamiltonian. These findings strongly hint that when one can
only prepare separable initial states, nonlocal and long-range interactions
are essential resources for surpassing the shot noise limit. This observation
is confirmed through numerical analysis on the long-range Ising model. Our
results bridge the field of many-body quantum sensing and operator growth in
many-body quantum systems and open the possibility to investigate the
interplay between quantum sensing and control, many-body physics and
information scrambling.

Introduction.—Quantum entanglement is a valuable resource in quantum
information processing. In quantum metrology, quantum Fisher information (QFI)
[1, 2, 3, 4, 5, 6], quantifying the precision of the sensing parameter, scales
linearly with the number of probes if the probes are uncorrelated, known as
the shot noise limit (SNL), also known as the standard quantum limit, which
also appears in sensing with classical resources. Quantum entanglement can
lead to a quadratic scaling known as the Heisenberg limit (HL) or even beyond
the quadratic scaling, i.e., the so-called super-HL. Entanglement can be
utilized in two ways, either in the stage of state preparation [7, 8, 9, 10,
11, 12] or in the stage of signal sensing via the many-body interactions
between individual sensors [13, 14, 15, 16, 17], which is the main essence of
many-body quantum metrology. Recently, the subject matter has gained renewed
interest. However, the existing protocols require to prepare the initial state
in the highly entangled Greenberger–Horne–Zeilinger (GHZ)-like states, whose
preparation is very challenging and time-consuming. One natural way to address
this issue is to combine the protocols of quantum state preparation and
quantum metrology, see e.g., Ref. [18, 19, 20], where an entangled initial
state is prepared before the sensing process. Nevertheless, to evaluate the
time cost of preparing a highly entangled state from separable states but
taking into account the restrictions of the accessible Hamiltonians can be
very challenging [21, 22, 23].

Figure 1: Comparison between our protocol (a) with the protocol in Ref. [18]
(b). In our protocol (a), the information of the estimation parameter is
encoded into the many-body quantum states through the many-body dynamics
$U_{\lambda}(t)=e^{-\text{i}(\lambda\sum_{i}h_{X_{i}}+H_{1})t}$ while in Ref.
[18], the encoding dynamics given by
$U_{\lambda}=e^{-\text{i}\lambda\sum_{i}h_{X_{i}}}$ with $X_{j}=\\{j\\}$. In
our protocol, the initial state is chosen to be either a separable state or
the non-degenerate ground states of a gapped and local Hamiltonian while in
Ref. [18] the initial state is prepared through the many-body dynamics
$U_{0}(t)$.

As such, as an alternative way to get around the overhead of quantum state
preparation, in this work we propose to prepare the probes or sensors
initially in a separable state, which can be prepared with the current
experimentally feasible technology [24, 25, 26]. In our protocol shown in Fig.
1(a), entanglement is established in the signal sensing process due to the
interactions in the many-body sensing Hamiltonian. It is in sharp contrast
with the protocol in Ref. [18], also shown in Fig. 1(b), where the entangled
initial state is explicitly prepared through the time evolution generated by a
locally interacting preparation Hamiltonian while the sensing Hamiltonian is
non-interacting. As we have alluded to earlier, the time cost to prepare the
entangled initial state in the protocol is difficult to estimate.

It is well known in the literature that when the initial states are restricted
to separable states, for a non-interacting sensing Hamiltonian that is
multiplicative with respective to the estimation parameter, the precision is
limited by SNL [7, 8, 9]. In our protocol, due to the many-body interactions,
the state can become entangled after the sensing process. The central question
we ask is whether many-body interactions can break the SNL? It is also
intimately related to recent studies on operator growth and quantum chaos in
quantum many-body systems [27, 28, 29, 30, 31].

To answer this question, we derive a universal bound for the growth of the
quantum Fisher information in time. Our bound can characterize the role of
quantum entanglement in information scrambling, operator growth, and quantum
chaos. We apply our bound to the quantum sensing protocol with time-
independent many-body Hamiltonians as shown in Fig. 1(a) and estimate our
bound using the celebrated Lieb-Robinson bound [32, 33, 34, 35] for quantum
many-body systems with local interactions. We find that it is impossible to
surpass the SNL with local interactions. Such observation is not only valid
for separable initial states, but also holds when the initial state is the
non-degenerate ground state of a locally gapped Hamiltonian, which can be
experimentally prepared by cooling. Therefore, if only separable states are
accessible in experiments, nonlocal or long-range interactions are essential
to beat the SNL and bring real quantum advantage in many-body quantum
metrology. We exemplify our findings in magnetometry with the short-range
transverse-field Ising (TFI) model, the chaotic Ising (CI) model, and the
long-range Ising (LRI) model.

Universal bound on the growth of the QFI. —We consider the following sensing
Hamiltonian

$H_{\lambda}(t)=H_{0\lambda}(t)+H_{1}(t),$ (1)

where $\lambda$ represents the estimation parameter, and $H_{1}(t)$ involves
interactions among sensors induced by either intrinsic interactions or
external coherent controls. In the formal case, $H_{1}$ is usually time-
independent, while in the later case, $H_{1}(t)$ becomes time-dependent. The
generator for the quantum sensing $\lambda$ [13, 36] is given by

$G(t)=\int_{0}^{t}\left[\partial_{\lambda}H_{\lambda}(\tau)\right]^{(\text{H})}d\tau,$
(2)

where an operator in the Heisenberg picture is defined as
$\mathcal{O}^{(\text{H})}(t)=U^{\dagger}(t)\mathcal{O}^{(\text{S})}(t)U(t)$.
The QFI is determined by the variance of $G(t)$ over the initial state denoted
as $\ket{\psi_{0}}$, i.e.,

$I(t)=\text{4Var}[G(t)]{}_{\ket{\psi_{0}}}.$ (3)

Optimal control theory has been proposed to simultaneously optimize the
initial state $\ket{\psi_{0}}$ and $H_{1}(t)$, resulting in a bound $I(t)\leq
4\left(\int_{0}^{t}\|\partial_{\lambda}H_{\lambda}^{(\text{S})}(\tau)\|d\tau\right)^{2}$
[13, 37, 38, 15]. Here, the semi-norm $\|\cdot\|$ is defined by the spectrum
width of an operator, i.e., the difference between its maximum eigenvalue and
minimum eigenvalue.

Eq. (3) hints that
$\dot{G}(t)=\left[\partial_{\lambda}H_{\lambda}(t)\right]^{(\text{H})}$ can
characterize the growth of quantum Fisher information qualitatively. In this
regard, we derive a universal bound [39]:

$\frac{d\sqrt{I(t)}}{dt}\leq\Gamma(t)\equiv
2\sqrt{\text{Var}\left(\left[\partial_{\lambda}H_{\lambda}(t)\right]^{(\text{H})}\right){}_{\ket{\psi_{0}}}}.$
(4)

The saturation condition can be found in the Supplemental Materials [39].
Alternatively, one can rewrite

$\Gamma(t)=2\sqrt{\text{Var}\left(\partial_{\lambda}H_{\lambda}^{\text{(S)}}(t)\right){}_{\ket{\psi(t)}}},$
(5)

where $\ket{\psi(t)}=U(t)\ket{\psi_{0}}$.

A few comments in order: First, Eq. (4) is our first main result, which holds
universally for all initial states, both time-independent and driven quantum
systems. Second, $\Gamma(t)$ depends on the control Hamiltonian $H_{1}(t)$ and
the initial state $\ket{\psi_{0}}$. Optimizing $\Gamma(t)$ over all possible
unitary dynamics and initial states lead to

$\Gamma(t)\leq 2\|\partial_{\lambda}H_{\lambda}^{\text{(S)}}(t)\|.$ (6)

By combining this bound with
$I(t)\leq\left(\int_{0}^{t}\Gamma(\tau)d\tau\right)^{2}$, which can be
obtained by integrating both sides of Eq. (4), one immediately reobtains the
bound given in previous works [15, 38, 37]. Compared to these studies, our
bound (4) provides a feasible approach to study the scaling behavior of the
QFI when the initial state $\ket{\psi_{0}}$ is restricted to a specific set of
initial states.

SNL for short-range local interactions.—Here we will demonstrate that our
bound, characterizing the growth of QFI, is closely linked to the Lieb-
Robinson bound, which characterizes operator complexity in quantum many-body
with short-range local interactions. We consider time-independent many-body
Hamiltonian of the following form:

$H_{\lambda}=\lambda\sum_{i=1}^{N}h_{X_{i}}+H_{1},$ (7)

where $h_{X_{i}}$ is supported on the set $X_{i}$ with cardinality $|X_{i}|=R$
and diameter $\text{diam}(X_{i})=\max_{k,\,l\in X_{j}}|k-l|$ and $H_{1}$
representing the interactions between the spins. We require $H_{\lambda}$ only
contains local and short-range interactions, imposing that
$\text{diam}(X_{j})$ is independent of $N$ [35, 33, 40] and $h_{X_{j}}$ is a
local operator. Equation (7) is the model used in magnetometry, where
$\lambda$ represents the magnetic field [41]. We note that the bound (4) can
be written in terms of dynamic correlation matrices of local operators, i.e.,

$\Gamma(t)=2\sqrt{\sum_{jk}\text{Cov}[h_{X_{j}}^{(\text{H})}(t)h_{X_{k}}^{(\text{H})}(t)]{}_{\ket{\psi_{0}}}},$
(8)

where
$\text{Cov}[AB]\big{|}_{\ket{\psi_{0}}}\equiv\frac{1}{2}\langle\\{A,\,B\\}\rangle-\langle
A\rangle\langle B\rangle$ and

$h_{X_{i}}^{(\text{H})}(t)\equiv
e^{\text{i}H_{\lambda}t}h_{X_{i}}e^{-\text{i}H_{\lambda}t}.$ (9)

As indicated by Eq. (6), we observe that $\Gamma(t)\leq 2N,$ implying
$I(t)\leq 4N^{2}t^{2}$ [37, 36]. Such an HL can be saturated by utilizing GHZ-
like initial states [15, 16, 13], while also ensuring that $H_{1}$ commutes
with $\sum_{i=1}^{N}h_{X_{i}}$. As we have mentioned before, it is challenging
to prepare GHZ state experimentally and estimate the corresponding time cost
theoretically.

In general, the many-body interaction $H_{1}$ can generate the entanglement
between the probes except that $H_{1}$ commutes with
$\sum_{i=1}^{N}h_{X_{i}}$, where the states will remain separable throughout
the process of signal sensing. In this special case, the precision remains at
SNL at all times, i.e., $I(t)=4Nt^{2}$. Generically, $H_{1}$ does not commute
with $\sum_{i=1}^{N}h_{X_{i}}$, Thus one naturally ask the question: For
separable initial states, what is the tight bound that limits the precision?
Is it possible to surpass the SNL using many-body interactions?

To address this question, we analyze the scaling of $\Gamma(t)$ with respect
to $N$ with the Lieb-Robinson bound for local Hamiltonians [34, 32, 33, 35],
which imposes a restriction on the connected correlation functions.
Specifically, if the sensing Hamiltonian (7) only contains local or short-
range interactions, the static correlation
$\text{Cov}[h_{X_{j}}h_{X_{k}}]\big{|}_{\ket{\psi_{0}}}$ between two disjoint
local operators $h_{X_{j}}$ and $h_{X_{k}}$ decays exponentially, provided the
initial state $\ket{\psi_{0}}$ is separable or the non-degenerate ground state
of some local and gapped Hamiltonians, not necessarily the same as Eq. (7). In
this case, the dynamic correlation function also decays exponentially,

$\big{|}\text{Cov}[h_{X_{j}}^{(\text{H})}(t)h_{X_{k}}^{(\text{H})}(t)]{}_{\ket{\psi_{0}}}\big{|}\leq\mathcal{C}\exp(-[d(X_{j},\,X_{k})-v_{\text{LR}}t]/\xi),$
(10)

where $\mathcal{C}$ and $\xi$ are constants that solely depend on the topology
of the sites, $d(X_{j},\,X_{k})$ is the distance between $X_{j}$ and $X_{k}$,
and $v_{\text{LR}}$ is the celebrated Lieb-Robinson velocity. Using Eq. (10),
we demonstrate that the scaling of $\Gamma(t)$ is lower bounded by $\sqrt{N}$
[39]:

$\Gamma(t)\leq 2\gamma(t)\sqrt{N},$ (11)

where $\gamma(t)$ is only a function of time and independent of $N$. It
remains finite as long as $t$ is finite and behaves as
$e^{v_{\text{LR}}t/\xi}$ as $t\to\infty$.

Eq. (11) is the main result of this work. Clearly, for finite but fixed times
it implies that the QFI is limited by SNL. On the other hand, at sufficiently
long times, for time-independent systems, one can show that $I(t)/t^{2}$ is
independent of time [39, 42] and is only a function of $N$. We further show
that in [39] the time scale to reach this regime corresponds to the case where
$t$ is much larger than the inverse of the minimum energy gap for the system.
In this regime, when $N$ is large, $I(t)\sim t^{2}N^{\alpha}$. Since Eq. (11)
is valid for all times and all $N$, combined with Eq. (4) we conclude
$\alpha\leq 1$. Therefore, in local short-range models where operator growth
is constrained by the Lieb-Robinson bound, SNL cannot be surpassed.

The spread of the generator of the metrological bound. — We can characterize
the structure $\left[\partial_{\lambda}H_{\lambda}(t)\right]^{(\text{H})}$,
the generator of the bound (4). We note that although
$h_{X_{i}}^{\text{(H})}(t)$ spreads over the lattice, the metrological
generator $\left[\partial_{\lambda}H_{\lambda}(t)\right]^{(\text{H})}$, being
a sum of these non-local operators, may still remain local as a whole, thus
keeping the precision limited to the SNL. This observation provides an
alternative perspective to understand the SNL for separable initial states in
local or short-range models. A trivial example is when $H_{1}$ commutes with
$\sum_{i}h_{X_{i}}$ while $H_{\lambda}$ does not commute with each individual
$h_{X_{i}}$, in which case $\Gamma(t)$ remains the SNL.

Now we present a non-trivial example: we assume
$\left[\partial_{\lambda}H_{\lambda}(t)\right]^{\text{(}\text{H})}$ can be
expanded in terms of two-body basis operators

$\left[\partial_{\lambda}H_{\lambda}(t)\right]^{(\text{H})}=\sum_{i=1}^{N}\sum_{j\geq
i}^{N}\sum_{\alpha}\eta_{ij}^{\alpha}\mathcal{O}_{ij}^{\alpha}$ (12)

where we have suppressed the time dependence for simplicity and for spin
systems $\mathcal{O}_{ij}^{\alpha}$ is a basis spin operator, such as
$\sigma_{i}^{x}\sigma_{j}^{y}$ while for fermionic systems
$\mathcal{O}_{ij}^{\alpha}$ is a Hermitian basis fermionic operator, such as
$c_{i}^{\dagger}c_{j}+\text{h.c.}$or
$\text{i}(c_{i}^{\dagger}c_{j}-\text{h.c.})$. It should be emphasized that the
number of different types of operators indexed by $\alpha$ are finite and does
not scale with $N$. If the initial state is separable and
$\left[\partial_{\lambda}H_{\lambda}(t)\right]^{(\text{H})}$ is described by
fast-decaying long-range two-body interactions, i.e.,

$\lim_{k\to\infty}\lim_{N\to\infty}\sum_{i\leq k}\sum_{j\geq
k}|\eta_{ij}^{\alpha}|=\lim_{k\to\infty}\int_{1}^{k}dx\int_{k}^{\infty}dy|\eta_{xy}^{\alpha}|<\infty,$
(13)

then the SNL can not be surpassed. The proof can be found in the Supplement
Materials [39]. We can express
$\left[\partial_{\lambda}H_{\lambda}(t)\right]^{(\text{H})}=\sum_{i=1}^{N}\tilde{\mathscr{O}}_{i}$,
where $\tilde{\mathscr{O}}_{i}\equiv\frac{1}{2}\sum_{\alpha}\left(\sum_{j\geq
i}^{N}\eta_{ij}^{\alpha}\mathcal{O}_{ij}^{\alpha}+\sum_{j\leq
i}^{N}\eta_{ji}^{\alpha}\mathcal{O}_{ji}^{\alpha}\right)$ has support across
the whole chain [39]. Essentially, the condition (13) ensures that
$\tilde{\mathscr{O}}_{i}$ does not scale with $N$ so that it behaves
effectively as a local operator, although it may have a support across the
entire chain. It is crucial to note that $\tilde{\mathscr{O}}_{i}$ can be
different from $h_{X_{i}}^{(\text{H})}(t)$, which is generically non-local. We
will further elaborate this observation with the example of magnetometry using
the TFI model.

SNL in the TFI model.—We consider the integrable TFI chain

$H_{\lambda}^{{\rm
TFI}}=-\left(J\sum_{i=1}^{N}\sigma_{i}^{x}\cdot\sigma_{i+1}^{x}+\lambda\sum_{i=1}^{N}\sigma_{i}^{z}\right),$
(14)

with the periodic boundary condition $\sigma_{1}^{z}=\sigma_{N+1}^{z}$, where
$J,\,\lambda>0$. In the thermodynamic limit $N\to\infty$, when $J\gg\lambda$
the ground state is ferromagnetic and degenerate, represented by
$|++\cdots+\rangle$ or $|--\cdots-\rangle$, while for $J\ll\lambda$ the ground
state is paramagnetic $|\uparrow\uparrow\cdots\uparrow\rangle$.

For any initial separable state, Eq. (4) predicts that the QFI cannot surpass
the SNL. On the other hand, this model can be exactly solved by mapping it to
a free fermion model [43, 44] and therefore one can compute
$\left[\partial_{\lambda}H_{\lambda}(t)\right]^{(\text{H})}$ explicitly. We
show in the Supplemental Materials [39] that indeed in this case
$\left[\partial_{\lambda}H_{\lambda}(t)\right]^{(\text{H})}$ has the structure
of Eq. (12) with four types of ferimonic operators:
$\mathcal{O}_{ij}^{(1)}=(c_{i}^{\dagger}c_{j}+{\rm h.c.})$,
$\mathcal{O}_{ij}^{(2)}=(c_{i}c_{j}^{\dagger}+{\rm h.c.)}$,
$\mathcal{O}_{ij}^{(3)}=(c_{i}^{\dagger}c_{j}^{\dagger}+{\rm h.c.}),$ and
$\mathcal{O}_{ij}^{(4)}=\text{i}(c_{i}^{\dagger}c_{j}^{\dagger}-{\rm h.c.})$.
The expression for the $\eta$-functions, characterizing the weights of
different operators spreading from the $i$-th site to the $j$-th site, can be
found in [39]. In the thermodynamic limit, $\eta_{ij}^{\alpha}$ behaves like
$p^{j-i}$ for $j\geq i$, where $p=J/\lambda$ for $J<\lambda$, $p=\lambda/J$
for $\lambda<J$, and $p=0$ for $J=\lambda$ [39]. The power-law decay of
$\eta$-functions indicates that the evolved operator remains extremely local,
as shown in Fig. 2(b), which ensures the condition (13), i.e.,

${\text{{\hbox{\int_{1}^{k}dx\int_{k}^{\infty}dy|\eta_{xy}^{\alpha}|\sim\text{}}<\infty}}},$
(15)

as $k\to\infty.$ Therefore, the locality of evolved operator suggests that QFI
beyond the SNL cannot be achieved by initial separable probe states in this
integrable TFL model. Fig. 2(a) characterizes the diffusion of the
correlators, suggesting that the numerical choices of $t=0.5$ and $t=5\times
10^{4}$ can be considered as the time scales for the part and full spread of
local operators, respectively. Fig. 2(c-d) numerically verify that only SNL
can be achieved for the different initial separable spin coherent states
parameterized by
$\ket{\psi_{0}}=\bigotimes_{i=1}^{N}[\cos(\theta/2)\ket{\uparrow}_{i}+\sin(\theta/2)e^{{\rm
i\phi}}\ket{\downarrow}_{i}]$.

Furthermore, if we consider the initial state as the ground state of the TFI
model with known values of parameters $\lambda_{*}$ and $J$, which can be
prepared by cooling. The asymptotic behavior of the QFI with respect to the
unknown parameter $\lambda$ under the the Hamiltonian (14) is

$\lim_{t,N\to\infty}\frac{I(t)}{Nt^{2}}=f(\lambda,J,\lambda_{*})\sim O(1),$
(16)

where the expression of $N$-independent function $f$ is given in the
Supplemental Materials [39], confirming the claim that only SNL can be
achieved even with the ground state of local and gapped Hamiltonians. Taking
$\lambda_{*}\to+\infty$, where the ground state becomes the spin coherent
state with $\theta=\phi=0$, we find

$\lim_{t,N\to\infty}\frac{I(t)}{Nt^{2}}=\frac{J^{2}(4\lambda^{2}-3J^{2})}{\lambda^{4}}$
(17)

for $J<\lambda$ and $\lim_{t,N\to\infty}I(t)/(Nt^{2})=1$ for $J\geq\lambda$,
which is also verified in Fig. 2(d).

Figure 2: (a) Numerical calculation of the operator diffusion in the TFI chain
with $N=10$. (b) Coefficient $|\eta_{ij}^{(1)}|$ characterizing the decay of
the two-body interactions. (c-f) Scaling of the QFI with respect to the number
of spins at different times for differential initial separable spin coherent
states
$\ket{\psi_{0}}=\bigotimes_{i=1}^{N}[\cos(\theta/2)\ket{\uparrow}_{i}+\sin(\theta/2)e^{{\rm
i\phi}}\ket{\downarrow}_{i}]$. Here numerical data is obtained by directly
diagonalizing the Hamiltonian of the TFI model, while theoretical data is
derived using results by mapping the TFI model to the free fermion model. The
analytical result refers to Eq. (17). Other parameters used for the
calculations are $J=2,$ $\lambda=5$, and $\phi=0$.

SNL in the chaotic Ising model.—Different from the integrable models, the
operator complexity in chaotic models grows very rapidly [27, 28, 30, 31, 29].
Nevertheless, Eq. (4) predict that even if the model is chaotic, where local
operators are expected to growth fast than integrable models as long as the
model only contains local interactions, the SNL cannot be surpassed by using
separable states. To see such an example, we consider the Ising model with
both transverse and longitudinal fields and the Hamiltonian is given by

$H_{\lambda}^{{\rm
CI}}=-\sum_{i=1}^{N}(J\sigma_{i}^{x}\sigma_{i+1}^{x}+h\sigma_{i}^{x}+\lambda\sigma_{i}^{z}),$
(18)

where open boundary conditions are adopted. Energy-level spacing statistics
indicate that this model is quantum chaotic for $J=h=\text{$\lambda$}$ [29,
45]. Fig. 3(c-f) verify the prediction by Eq. (11) that separable states
cannot surpass the SNL even in such chaotic short-range systems. To surpass
the SNL, we are thus motivated to explore long-range models. The effect of
quantum chaos in quantum metrology has been studied in Ref. [46] in the
context of kick top, which is a single-spin model. Here, we show that quantum
chaos plays no role in local chaotic many-spin models.

Beyond SNL with LRI model.—As we have shown above, it is only possible to
break the SNL in long-range and nonlocal systems, which violates the Lieb-
Robinson inspired bound (11). Thus, we consider the long-range Ising model
with power-law decay,

$H_{\lambda}^{{\rm
LRI}}=-\left(J\sum_{i<j}\frac{\sigma_{i}^{x}\sigma_{j}^{x}}{|i-j|^{\alpha}}+\lambda\sum_{i}\sigma_{i}^{z}\right),$
(19)

which reduces to the TFI model as $\alpha\to\infty.$ For $\alpha=0$, this
model corresponds to the Lipkin-Meshkov-Glick model [47]. In this long-range
model, the breakdown of exponential decay in connected correlation function
Eq. (10) will result in the failure of bound presented in (11). Consequently,
we expect that for small $\alpha$ where the long-range interactions decay
sufficiently slowly, it is possible to surpass the SNL with separable initial
states. As depicted in Fig. 3(g-h), we have identified specific instances of
this scenario.

Figure 3: Numerical calculation of the operator diffusion in (a) the CI model
(b) the LRI model with $N=10$. The scaling of the QFI with respect to the
number of spins at different times for differential initial separable spin
coherent states
$\ket{\psi_{0}}=\bigotimes_{i=1}^{N}[\cos(\theta/2)\ket{\uparrow}_{i}+\sin(\theta/2)e^{{\rm
i\phi}}\ket{\downarrow}_{i}]$ in (c-f) the CI model and (g-h) the LRI model.
Other parameters used for the calculations are $J=\lambda=1,$ $\phi=0$ in the
CI model, and $J=1$, $\lambda=0.5,$ $\alpha=3$ in the LRI model.

Conclusion and outlook.—In conclusion, we have derived a universal bound on
the growth of the QFI under arbitrary dynamics and initial states. We apply
our bound to the case of separable initial states or the non-degenerate ground
state of a gapped and local sensing Hamiltonian. We prove that with these
particular set of initial states, the QFI cannot surpass the SNL, as we have
explicitly demonstrated with TFI and CI models. This indicates that either
initial entanglement or long-range interactions are essential resource to
bring the advantage of quantumness, as we have demonstrated with the LRI.

These findings suggest an intriguing connection between operator growth and
the growth of the QFI. As such, our results shed light on many aspects on the
interplay between many-body physics, quantum control theory, and information
scrambling and open the door to investigate many intricate questions, such as
driven many-body sensing, the optimal control metrology over a restricted set
of initial states, the connection between the growth of the QFI and the
measures characterizing quantum chaos and operator complexity. We leave these
studies for the future.

Acknowledgement. —We thank Adolfo del Campo for useful comments on the
manuscript. It is a pleasure to acknowledge discussions with Federcio Balducci
and Xingze Qiu. XWG was supported by the NSFC key grant No. 12134015, the NSFC
grant No. 12121004, and partially supported by the Innovation Program for
Quantum Science and Technology 2021ZD0302000. JY was funded by the Wallenberg
Initiative on Networks and Quantum Information (WINQ) and would like to thank
Hui Zhai for the hospitality to host his long-term visit at the Institute of
Advanced Study in Tsinghua University, during which this work was completed.

Notes added.—When completing this work, we noted that a bound similar with Eq.
(4) also appears in Ref. [48] with the focus on non-Hermitian sensing.

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Supplemental Materials

###### Contents

  1. I Saturation condition of the bound on the growth of the QFI
  2. II Analysis of the growth bound of the QFI from Lieb-Robinson bound
  3. III Proof of $\lim_{t\to+\infty}G(t)/t$ being time-independent for time-independent Hamiltonians
  4. IV Proof of the SNL for fast-decaying two-body metrological operator
    1. IV.1 Two-body spin operators
    2. IV.2 Two-body fermionic operators
  5. V Metrological generator for the TFI periodic chain
  6. VI QFI for the short-range TFI model

##  I Saturation condition of the bound on the growth of the QFI

We rewrite Eq. (3) in the main text as

$I(t)=4\big{\langle}\left[G(t)-\langle G(t)\rangle\right]^{2}\big{\rangle},$
(S1)

where $\langle\bullet\rangle$ denotes the average on the initial state
$\ket{\psi_{0}}$. Taking derivatives on both sides, we find

$\displaystyle\dot{I}(t)$
$\displaystyle=4\big{\langle}\\{\dot{G}(t)-\langle\dot{G}(t)\rangle,\,G(t)-\langle
G(t)\rangle\\}\big{\rangle}$ $\displaystyle=\text{{\rm
8Re}$\langle$}[\dot{G}(t)-\langle\dot{G}(t)\rangle]\left[G(t)-\langle
G(t)\rangle\right]\rangle$ $\displaystyle\leq
8|\langle[\dot{G}(t)-\langle\dot{G}(t)\rangle]\left[G(t)-\langle
G(t)\rangle\right]\rangle|.$ (S2)

Using the Cauchy-Schwarz inequality, we find

$\big{|}\big{\langle}[\dot{G}(t)-\langle\dot{G}(t)\rangle]\left[G(t)-\langle
G(t)\rangle\right]\big{\rangle}\big{|}\leq\sqrt{\text{Var}[\dot{G}(t)]\text{Var}[G(t)]}.$
(S3)

Thus, we find

$\dot{I}(t)\leq 4\sqrt{\text{Var}[\dot{G}(t)]I(t)},$ (S4)

which is equivalent to the universal bound (4) in the main text. We define

$\ket{\Phi_{0}(t)}\equiv\left(G(t)-\langle G(t)\rangle\right)\ket{\psi_{0}},$
(S5)

with initial values

$\ket{\Phi_{0}(0)}=0,\quad\ket{\dot{\Phi}_{0}(0)}=\left[\partial_{\lambda}H_{\lambda}(0)-\langle\partial_{\lambda}H_{\lambda}(0)\rangle\right]\ket{\psi_{0}}.$
(S6)

The inequality is saturated at some instant time $t$ if

$\text{Im}\braket{\dot{\Phi_{0}}(t)\big{missing}}{\Phi_{0}(t)}=0,\,\text{Re}\braket{\dot{\Phi_{0}}(t)\big{missing}}{\Phi_{0}(t)}\geq
0,\,,\,c(t)\ket{\dot{\Phi}_{0}(t)}=\ket{\Phi_{0}(t)},$ (S7)

which is equivalent to

$c(t)\ket{\dot{\Phi}_{0}(t)}=\ket{\Phi_{0}(t)},\,c(t)\geq 0.$ (S8)

Now we require the inequality is satisfied from the initial time $t=0$ up to
time $t$, we should integrate Eq. (S8). Note that initial time $t=0$ may be
not appropriate to be used as the reference point for the integration since
$\ket{\Phi_{0}(0)}=0$. Instead, we can choose a reference point
$t_{0}\in(0,\,t]$ such that $\ket{\Phi(t_{0})}\neq 0$ and then obtain

$\ket{\Phi_{0}(t)}=e^{\int_{t_{0}}^{t}\frac{d\tau}{c(\tau)}}\ket{\Phi_{0}(t_{0})}.$
(S9)

By further considering the requirement $c(\tau)\geq 0$, we can conclude that
$\ket{\Phi_{0}(t)}$ should not change its direction as time evolves, while
exponentially increasing in its length. To satisfy the initial condition (S6),
we require that

$\lim_{\epsilon\to
0}e^{-\int_{\epsilon}^{t_{0}}\frac{d\tau}{c(\tau)}}=0,\,\lim_{\epsilon\to
0}\frac{e^{-\int_{\epsilon}^{t_{0}}\frac{d\tau}{c(\tau)}}}{c(\epsilon)}=\alpha(t_{0}),\,\alpha(t_{0})\in\mathbb{R},\,$
(S10)

where

$\ket{\dot{\Phi}_{0}(0)}=\alpha(t_{0})\ket{\Phi(t_{0})}.$ (S11)

The first condition of Eq. (S10) implies that $c(\epsilon)$ should vanish as
$\epsilon\to 0$. Thus we assume $c(\epsilon)\sim\epsilon^{\delta}$ as
$\epsilon\to 0$ with $\delta\geq 1$. A straightforward analysis show that the
only possible case that is consistent with the second equation of Eq. (S10) is
$\delta=1$. Therefore, Eq. (S9) may be rewritten as

$\ket{\Phi_{0}(t)}=\frac{e^{\int_{t_{0}}^{t}\frac{d\tau}{c(\tau)}}}{\alpha(t_{0})}\ket{\dot{\Phi}_{0}(0)}$
(S12)

To summarize, Eq. (S5) and Eq. (S9) must be consistent. Taking derivatives on
sides, we obtain the condition for the saturation of the maximum growth rate
of the QFI:

$\left(U^{\dagger}(t)\partial_{\lambda}H_{\lambda}(t)U(t)-\langle
U^{\dagger}(t)\partial_{\lambda}H_{\lambda}(t)U(t)\rangle\right)\ket{\psi_{0}}=\frac{e^{\int_{t_{0}}^{t}\frac{d\tau}{c(\tau)}}}{\alpha(t_{0})c(t)}\ket{\dot{\Phi}_{0}(0)},\,\forall
t$ (S13)

which is obviously satisfied at $t=0$.

##  II Analysis of the growth bound of the QFI from Lieb-Robinson bound

It is straightforward to show that

$\displaystyle\frac{1}{4}\Gamma^{2}(t)$
$\displaystyle=\sum_{j=1}^{N}\text{Var}[h_{X_{j}}^{(\text{H})}]{}_{\ket{\psi_{0}}}+\sum_{j,k=1,\,j\neq
k}^{N}\text{Cov}[h_{X_{j}}^{(\text{H})}(t)h_{X_{k}}^{(\text{H})}(t)]{}_{\ket{\psi_{0}}}$
$\displaystyle=\sum_{j=1}^{N}\text{Var}[h_{X_{j}}^{(\text{H})}(t)]{}_{\ket{\psi_{0}}}+\sum_{j=1}^{N}\sum_{k\in\mathcal{B}(j,\,R)}\text{Cov}[h_{X_{j}}^{(\text{H})}(t)h_{X_{k}}^{(\text{H})}(t)]{}_{\ket{\psi_{0}}}+\sum_{j=1}^{N}\sum_{k\notin\mathcal{B}(j,\,R)}\text{Cov}[h_{X_{j}}^{(\text{H})}(t)h_{X_{k}}^{(\text{H})}(t)]{}_{\ket{\psi_{0}}}$
(S14)

where $R$ is the cardinality of the support $X_{j}$ and $\mathcal{B}(j,R)$
represents the closed neighborhood of site $j$ with a radius $R$.
Specifically, this implies that the intersection between the sets $X_{j}$ and
$X_{k}$ is empty if and only if the site $k$ does not belong to the closed
neighborhood $\mathcal{B}(j,R)$, i.e.,

$X_{j}\cap X_{k}=\varnothing\,\iff\,k\notin\mathcal{B}(j,\,R).$ (S15)

The first term on the right-hand side of Eq. (S14) describes the self-
correlation of local operators. The second term accounts for the correlation
between pairs of local operators that overlap from the initial time. The last
term represents the correlation among the pairs of local operators whose
supports are initially disjoint but may overlap later due to the Lieb-Robinson
type of diffusion.

Note that $\|h_{X_{j}}\|\sim R$, which does not scale with $N$. As a result,
we find that

$\text{Var}[h_{X_{j}}^{(\text{H})}(t)]{}_{\ket{\psi_{0}}}\text{{\hbox{\lesssim
R^{2},}}}$ (S16)

and

$\big{|}\sum_{k\in\mathcal{B}(j,\,R)}\text{Cov}[h_{X_{j}}^{(\text{H})}(t)h_{X_{k}}^{(\text{H})}(t)]{}_{\ket{\psi_{0}}}\big{|}\lesssim
R^{3}.$ (S17)

On the other hand, if $k\notin\mathcal{B}(j,\,R)$, then the set $X_{j}$ and
$X_{k}$ are disjoint, allowing us to estimate
$\text{Cov}[h_{X_{j}}^{(\text{H})}(t)h_{X_{k}}^{(\text{H})}(t)]{}_{\ket{\psi_{0}}}$
by using the Lieb-Robinson bound (10). Consequently, we find [34, 32, 33, 35],

$\big{|}\sum_{k\notin\mathcal{B}(j,\,R)}\text{Cov}[h_{X_{j}}^{(\text{H})}(t)h_{X_{k}}^{(\text{H})}(t)]{}_{\ket{\psi_{0}}}\big{|}\lesssim
R^{2}\sum_{k\notin\mathcal{B}(j,\,R)}\exp\left(-[d(X_{j},\,X_{k})-v_{\text{LR}}t]/\xi\right),$
(S18)

which holds for initial separable states or the non- degenerate ground state
of local and gapped Hamiltonians. Here, $d(X_{j},X_{k})$ is the distance
connecting the set $X_{j}$ with the set $X_{k}$, $\xi$ is a constant that
solely depends on the topology of the sites, and $v_{\text{LR}}$ is the
celebrated Lieb-Robinson velocity. By substituting Eqs. (S16-S18) into Eqs.
(S14), we find

$\displaystyle\frac{1}{4}\Gamma^{2}(t)$ $\displaystyle\lesssim
NR^{2}+NR^{3}+R^{2}\sum_{j=1}^{N}\sum_{k\notin\mathcal{B}(j,\,R)}\exp\left(-[d(X_{j},\,X_{k})-v_{\text{LR}}t]/\xi\right)$
$\displaystyle\lesssim N+Ne^{v_{\text{LR}}t/\xi}\sum_{n=1}^{N}e^{-n/\xi}$
$\displaystyle\lesssim N+Ne^{v_{\text{LR}}t/\xi}\int_{1}^{N}e^{-x/\xi}dx$
$\displaystyle\lesssim N+Ne^{v_{{\rm LR}}t/\xi}\xi(e^{-1/\xi}-e^{-N/\xi})$
$\displaystyle\leq\gamma^{2}(t)N,$ (S19)

where $\gamma(t)$ is a function of only time, independent of $N$, and we have
used the results $R\sim O(1)$ and $\lim_{N\to\infty}e^{-N/\xi}=0$.

##  III Proof of $\lim_{t\to+\infty}G(t)/t$ being time-independent for time-
independent Hamiltonians

For time-independent Hamiltonians $H_{\lambda}(t)=H_{\lambda}$, the generator
for quantum sensing $\lambda$, Eq. (2) is given by

$\displaystyle G(t)$ $\displaystyle=\int_{0}^{t}e^{{\rm
i}H_{\lambda}\tau}\partial_{\lambda}H_{\lambda}e^{-{\rm
i}H_{\lambda}\tau}d\tau$
$\displaystyle\overset{\tau=t(1-s)}{=}t\int_{0}^{1}e^{{\rm
i}H_{\lambda}t(1-s)}\partial_{\lambda}H_{\lambda}e^{{\rm-i}H_{\lambda}t(1-s)}ds.$
(S20)

By using the Baker–Campbell–Hausdorff formula [49], the above equation can be
rewritten as

$\displaystyle\frac{G(t)}{t}$
$\displaystyle=\sum_{n=0}^{\infty}\int_{0}^{1}(1-s)^{n}ds\frac{({\rm
i}t\mathcal{L})^{n}}{n!}\partial_{\lambda}H_{\lambda}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{({\rm
i}t\mathcal{L})^{n}}{(n+1)!}\partial_{\lambda}H_{\lambda}$
$\displaystyle\equiv g({\rm i}t\mathcal{L})\partial_{\lambda}H_{\lambda},$
(S21)

where the Liouvillian is defined as $\mathcal{L}(O)\equiv[H_{\lambda},O]$
whose eigenvalues are $L_{mn}=E_{m}-E_{n}$ and

$g(x)\equiv\sum_{n=0}^{\infty}\frac{x^{n}}{(n+1)!}=\frac{e^{x}-1}{x}.$ (S22)

By the Choi-Jamio$\l$kowski isomorphism, we can rewrite operator
$|\partial_{\lambda}H_{\lambda})$ as a vector in the space
$\mathcal{H\otimes\mathcal{H}}$ where $\mathcal{H}$ is the original Hilbert
space. Due to the Hermiticity of $\mathcal{L}$, we can assume that it can be
diagonalized as $\mathcal{L}=\sum_{mn}L_{mn}|L_{mn})(L_{mn}|$ so that

$\frac{G(t)}{t}\equiv\sum_{mn}g({\rm
i}tL_{mn})|L_{mn})(L_{mn}|\partial_{\lambda}H_{\lambda}).$ (S23)

We observe that in the limit $t\to\infty$

$\lim_{t\to\infty}g({\rm i}tL_{mn})=0,\,\text{for},\,L_{mn}\neq 0.$ (S24)

Therefore, we conclude

$\lim_{t\to+\infty}\frac{G(t)}{t}=\sum_{mn}\delta_{L_{mn},0}|L_{mn})(L_{mn}|\partial_{\lambda}H_{\lambda}).$
(S25)

which is time-independent. Apparently, the condition to reach this limit is

$t\gg 1/\min_{m,\,n}\\{L_{mn}\\}$ (S26)

which is the minimum energy gap of the system. Since
$I(t)=\text{4Var}[G(t)]{}_{\ket{\psi_{0}}}$ then we can assume $I(t)\sim
t^{2}N^{\alpha}$ for sufficient long-time and large $N$.

##  IV Proof of the SNL for fast-decaying two-body metrological operator

Then it is straightforward to verify that

$[\partial_{\lambda}H_{\lambda}(t)]^{(\text{H})}=\sum_{i}\sum_{j\geq
i}\sum_{\alpha}\eta_{ij}^{\alpha}\mathcal{O}_{ij}^{\alpha}=\sum_{i}\sum_{j\leq
i}\sum_{\alpha}\eta_{ji}^{\alpha}\mathcal{O}_{ji}^{\alpha}$ (S27)

So one can find

$[\partial_{\lambda}H_{\lambda}(t)]^{(\text{H})}=\frac{1}{2}\sum_{i}\sum_{\alpha}\left(\sum_{j\geq
i}\eta_{ij}^{\alpha}\mathcal{O}_{ij}^{\alpha}+\sum_{j\leq
i}\sum_{\alpha}\eta_{ji}^{\alpha}\mathcal{O}_{ji}^{\alpha}\right)\equiv\sum_{i}\tilde{\mathscr{O}}_{i}$
(S28)

where $\tilde{\mathscr{O}}_{i}$ is defined in the main text. To facilitate the
following discussions, we also introduce

$\mathscr{O}_{i}\equiv\sum_{j\geq
i}^{N}\sum_{\alpha}\eta_{ij}^{\alpha}\mathcal{O}_{ij}^{\alpha}.$ (S29)

Then it is straightforward to show that

$\frac{1}{4}\Gamma^{2}(t)=\sum_{k}\text{Cov}[\mathscr{O}_{k}\mathscr{O}_{k}]{}_{\ket{\psi_{\text{sep}}}}+2\sum_{k}\sum_{i<k}\text{Cov}[\mathscr{O}_{i}\mathscr{O}_{k}]{}_{\ket{\psi_{\text{sep}}}},$
(S30)

where $\ket{\psi_{{\rm sep}}}$ is a separable state. Thus, we find

$\frac{1}{4}\Gamma^{2}(t)=\sum_{k}S_{N}(k),$ (S31)

where $S_{N}(k)\equiv S_{N}^{(1)}(k)+S_{N}^{(2)}(k)$ and

$\displaystyle S_{N}^{(1)}(k)$
$\displaystyle\equiv\text{$\sum_{\alpha\beta}$$\sum_{\ell\geq k}\sum_{j\geq
k}\eta_{k\ell}^{\alpha}$$\eta_{kj}^{\beta}$Cov}[\mathcal{O}_{k\ell}^{\alpha}\mathcal{O}_{kj}^{\beta}]{}_{\ket{\psi_{\text{sep}}}},$
$\displaystyle S_{N}^{(2)}(k)$
$\displaystyle=2\sum_{\alpha\beta}\sum_{i<k}\sum_{j\geq i}\sum_{\ell\geq
k}\eta_{ij}^{\alpha}\eta_{k\ell}^{\beta}\text{Cov}[\mathcal{O}_{ij}^{\alpha}\mathcal{O}_{k\ell}^{\beta}]{}_{\ket{\psi_{\text{sep}}}}.$
(S32)

Our aim is to show that

$\lim_{N\to\infty}\frac{\Gamma^{2}(t)}{N}<\infty,$ (S33)

which suggests that SNL cannot be surpassed by separable states. To show Eq.
(S33), it suffices to demonstrate that

$\lim_{k\to\infty}\lim_{N\to\infty}\left[S_{N}^{(1)}(k)+S_{N}^{(2)}(k)\right]<\infty\text{,}$
(S34)

Due to $\|\mathcal{O}_{jk}^{\alpha}\|\sim O(1),$ we have

${\rm Var}[\mathcal{O}_{k\ell}^{\alpha}])_{\ket{\psi_{{\rm
0}}}}\sim\|\mathcal{O}_{jk}^{\alpha}\|^{2}\sim O(1),$ (S35)

as $N\to\infty$ where $\ket{\psi_{0}}$ can be any state and not restricted to
separable states. Also, by using the Cauchy-Schwartz inequality, we can show
that as $N\to\infty$

$\big{|}\text{Cov}[\mathcal{O}_{ij}^{\alpha}\mathcal{O}_{k\ell}^{\beta}]{}_{\ket{\psi_{\text{0}}}}\big{|}\leq\sqrt{\text{Var}[\mathcal{O}_{ij}^{\alpha}]_{\ket{\psi_{\text{0}}}}}\sqrt{\text{Var}[\mathcal{O}_{k\ell}^{\beta}]_{\ket{\psi_{\text{0}}}}}\sim
O(1).$ (S36)

Furthermore, we have the following condition to characterize the locality of
the fast decay operator:

$\lim_{k\to\infty}\lim_{N\to\infty}C_{N}^{\alpha}(k)<\infty,$ (S37)

where

$C_{N}^{\alpha}(k)\equiv\sum_{i\leq k}\sum_{j\geq k}|\eta_{ij}^{\alpha}|$
(S38)

Apparently

$\sum_{j\geq k}|\eta_{kj}^{\alpha}|\leq C_{N}^{\alpha}(k)$ (S39)

In the limit, $N\to\infty$ and $k\to\infty$, Euler-Maclaurin formula allows us
to rewrite the series on r.h.s. of Eq. (S38) in terms of a double integral

$\lim_{k\to\infty}\lim_{N\to\infty}C_{N}^{\alpha}(k)=\text{{\hbox{\lim_{k\to\infty}}}}\int_{1}^{k}dx\int_{k}^{\infty}dy|\eta_{xy}^{\alpha}|.$

which is easier to evaluate than the double series.

Now we are in a position to show that

$\displaystyle\lim_{k\to\infty}\lim_{N\to\infty}|S_{N}^{(1)}(k)|$
$\displaystyle\lesssim\sum_{\alpha\beta}\lim_{k\to\infty}\lim_{N\to\infty}\left(\sum_{\ell\geq
k}\sum_{j\geq k}|\eta_{k\ell}^{\alpha}\eta_{kj}^{\beta}|\right)$
$\displaystyle\lesssim\sum_{\alpha\beta}\lim_{k\to\infty}\lim_{N\to\infty}C_{N}^{\alpha}(k)C_{N}^{\beta}(k)$
$\displaystyle<\infty,$ (S40)

where we have utilized Eq. (S36) in the first inequality, Eq. (S39) in the
second inequality, and the fact that the number of terms of the sum over
$\alpha$ is finite in the last inequality. Next, we will discuss two cases
where the $\mathcal{O}$ is two-body spin or fermionic operators, respectively,
to show that

$\lim_{k\to\infty}\lim_{N\to\infty}|S_{N}^{(2)}(k)|<\infty.$ (S41)

###  IV.1 Two-body spin operators

When $\mathcal{O}_{ji}^{\alpha}$ and $\mathcal{O}_{kl}^{\beta}$ are two-body
spin operators, we note that if $\\{j,\,i\\}$ does not overlap with
$\\{k,\,l\\}$ then the correlation is zero since $\ket{\psi_{{\rm sep}}}$ is a
separable state. With this observation, we find

$\displaystyle|S_{N}^{(2)}(k)|$
$\displaystyle=2\sum_{\alpha\beta}\sum_{i<k}\sum_{j\geq i}\sum_{\ell\geq
k}|\eta_{ij}^{\alpha}\eta_{k\ell}^{\beta}\text{Cov}[\mathcal{O}_{ij}^{\alpha}\mathcal{O}_{k\ell}^{\beta}]{}_{\ket{\psi_{\text{sep}}}}\delta_{k,j}|$
$\displaystyle=2\sum_{\alpha\beta}\sum_{i<k}\sum_{\ell\geq
k}|\eta_{ik}^{\alpha}\eta_{k\ell}^{\beta}\text{Cov}[\mathcal{O}_{ik}^{\alpha}\mathcal{O}_{k\ell}^{\beta}]{}_{\ket{\psi_{\text{sep}}}}|$
$\displaystyle\lesssim\sum_{\alpha\beta}\sum_{i<k}\sum_{\ell\geq
k}|\eta_{ik}^{\alpha}\eta_{k\ell}^{\beta}|,$ (S42)

where the appearance of $\delta_{k,j}$ in the first equality comes from the
initial state is restricted to separable states. Using (S39) ,

$\displaystyle|S_{N}^{(2)}(k)|$
$\displaystyle\lesssim\sum_{\alpha\beta}C_{N}^{\alpha}(k)C_{N}^{\beta}(k),$

Taking the limit $N\to\infty$ and $k\to\infty$, we conclude Eq. (S41).

If we do not require separable initial states then

$\displaystyle|S_{N}^{(2)}(k)|$
$\displaystyle=2\sum_{\alpha\beta}\sum_{i<k}\sum_{j\geq i}\sum_{\ell\geq
k}|\eta_{ij}^{\alpha}\eta_{k\ell}^{\beta}\text{Cov}[\mathcal{O}_{ij}^{\alpha}\mathcal{O}_{k\ell}^{\beta}]{}_{\ket{\psi_{\text{0}}}}|$
$\displaystyle\lesssim\sum_{\alpha\beta}\sum_{i<k}\sum_{j\geq i}\sum_{\ell\geq
k}|\eta_{ij}^{\alpha}\eta_{k\ell}^{\beta}|$
$\displaystyle\lesssim\sum_{\alpha\beta}C^{\beta}_{N}(k)\sum_{i<k}\sum_{j\geq
i}|\eta_{ij}^{\alpha}|,$ (S43)

which suggests that $\lim_{k\to\infty}\lim_{N\to\infty}S_{N}^{(2)}(k)<\infty$
is not guaranteed since $\sum_{i<k}\sum_{j\geq i}|\eta_{ij}^{\alpha}|$ may not
be bounded given Eqs. (S37) and (S38). Actually, the HL can be achieved for
GHZ-like entangled initial states and thus Eq. (S34) is not expected to hold.

###  IV.2 Two-body fermionic operators

When $\mathcal{O}_{ij}^{\alpha}$ and $\mathcal{O}_{kl}^{\beta}$ are two-body
fermionic operators, see the below:

$\displaystyle c_{i}^{\dagger}c_{j}$
$\displaystyle=\left\\{\begin{array}[]{cc}\sigma_{i}^{-}\sigma_{i+1}^{z}\sigma_{i+2}^{z}\cdots\sigma_{j-1}^{z}\sigma_{j}^{+},&i<j,\\\
(1-\sigma_{i}^{z})/2,&i=j,\\\
\sigma_{j}^{+}\sigma_{j+1}^{z}\sigma_{j+2}^{z}\cdots\sigma_{i-1}^{z}\sigma_{i}^{-},&i>j.\end{array}\right.$
(S47) $\displaystyle c_{i}c_{j}^{\dagger}$
$\displaystyle=\left\\{\begin{array}[]{cc}-\sigma_{i}^{+}\sigma_{i+1}^{z}\sigma_{i+2}^{z}\cdots\sigma_{j-1}^{z}\sigma_{j}^{-},&i<j,\\\
(1+\sigma_{i}^{z})/2,&i=j,\\\
-\sigma_{j}^{-}\sigma_{j+1}^{z}\sigma_{j+2}^{z}\cdots\sigma_{i-1}^{z}\sigma_{i}^{+},&i>j.\end{array}\right.$
(S51) $\displaystyle c_{i}^{\dagger}c_{j}^{\dagger}$
$\displaystyle=\left\\{\begin{array}[]{cc}\sigma_{i}^{-}\sigma_{i+1}^{z}\sigma_{i+2}^{z}\cdots\sigma_{j-1}^{z}\sigma_{j}^{-},&i<j,\\\
0,&i=j,\\\
-\sigma_{j}^{-}\sigma_{j+1}^{z}\sigma_{j+2}^{z}\cdots\sigma_{i-1}^{z}\sigma_{i}^{-},&i>j,\end{array}\right.$
(S55)

which suggests that if $[i,j]\cap[k,\ell]=\text{$\varnothing$}$ then ${\rm
Cov}[\mathcal{O}_{ji}^{\alpha}\mathcal{O}_{k\ell}^{\beta}]_{\ket{\psi_{{\rm
sep}}}}=0.$ Thus, we can find

$\displaystyle S_{N}^{(k)}(k)$
$\displaystyle=2\sum_{\alpha\beta}\sum_{i<k}\sum_{j\geq i}\sum_{\ell\geq
k}\eta_{ij}^{\alpha}\eta_{k\ell}^{\beta}\text{Cov}[\mathcal{O}_{ij}^{\alpha}\mathcal{O}_{k\ell}^{\beta}]{}_{\ket{\psi_{\text{sep}}}}$
(S56)

Using the condition that the initial state is separable, one can replace
$j\geq i$ in above summation with $j\geq k$, leading to

$\displaystyle S_{N}^{(2)}(k)$
$\displaystyle=2\sum_{\alpha\beta}\sum_{i<k}\sum_{j\geq k}\sum_{\ell\geq
k}\eta_{ij}^{\alpha}\eta_{k\ell}^{\beta}\text{Cov}[\mathcal{O}_{ij}^{\alpha}\mathcal{O}_{k\ell}^{\beta}]{}_{\ket{\psi_{\text{sep}}}}$
(S57)

Thus, we find

$|S_{N}^{(2)}(k)|\lesssim\sum_{\alpha\beta}\sum_{i<k}\sum_{j\geq
k}\sum_{\ell\geq
k}|\eta_{ij}^{\alpha}\eta_{k\ell}^{\beta}|\leq\sum_{\alpha\beta}C_{N}^{\alpha}(k)\sum_{\ell\geq
k}|\eta_{k\ell}^{\beta}|\lesssim\sum_{\alpha\beta}C_{N}^{\alpha}(k)C_{N}^{\beta}(k).$

Taking $N\to\infty$ and $k\to\infty$ and using Eq. (S37), we prove Eq. (S41).

##  V Metrological generator for the TFI periodic chain

The integrable Ising model Eq. (14) can be diagonalized as a free fermion
model [43, 44]. Defining $\sigma_{i}^{\pm}=(\sigma_{i}^{x}\pm{\rm
i}\sigma_{i}^{y})/2$, we can construct fermion creation and annihilation
operators by the Jordan-Wigner transformation:

$c_{i}^{\dagger}=\prod_{j=1}^{i-1}\sigma_{j}^{z}\sigma_{i}^{-}.$ (S58)

The inverse transformation is then given by

$\sigma_{i}^{z}=1-2c_{i}^{\dagger}c_{i},\quad\sigma_{i}^{x}=\prod_{j=1}^{i-1}(1-2c_{j}^{\dagger}c_{j})(c_{i}+c_{i}^{\dagger}).$
(S59)

In terms of fermionic operators, the Hamiltonian can be expressed as

$\displaystyle H_{\lambda}^{{\rm TFI}}$
$\displaystyle=-J\sum_{i=1}^{N}\sigma_{i}^{x}\cdot\sigma_{i+1}^{x}-\lambda\sum_{i=1}^{N}\sigma_{i}^{z}$
$\displaystyle=-J\sum_{i=1}^{N-1}(c_{i}^{\dagger}-c_{i})(c_{i+1}^{\dagger}+c_{i+1})-\lambda\sum_{i=1}^{N}(1-2c_{i}^{\dagger}c_{i})+J\eta(c_{N}^{\dagger}-c_{N})(c_{1}^{\dagger}+c_{1}),$
(S60)

where $\eta=\exp({\rm i}\pi\sum_{i=1}^{N}c_{i}^{\dagger}c_{i})$ is the parity
operator commuting with the Hamiltonian. Thus, the above Hamiltonian can be
rewritten according to the parity symmetry, i.e.,

$H_{\lambda}^{{\rm TFI}}=\text{(}H_{\lambda}^{{\rm TFI}})_{{\rm
odd}}\oplus\text{(}H_{\lambda}^{{\rm TFI}})_{{\rm even}},$

where “odd” and “even” correspond to the Hamiltonian acting on the subspaces
of the Fock space with an odd or even number of fermions, which is also
related with periodic ($\eta=-1$) or antiperiodic ($\eta=1$) boundary
conditions.

It can be shown that $\text{(}H_{\lambda}^{{\rm TFI}})_{{\rm
odd}}=(H_{\lambda}^{{\rm TFI}})_{{\rm even}}$ at the thermodynamic limit
$N\to\infty$. Thus, for convenience, we work on the subspace with an even
number of fermions. Introducing the Fourier transformation

$f_{k}=\frac{1}{\sqrt{N}}\sum_{j=1}^{N}c_{j}{\rm e}^{{\rm i}kj},\quad
k=\frac{2\pi(n+1/2)}{N},\quad
n=-\frac{N}{2},-\frac{N}{2}+1,\ldots,\frac{N}{2}-1,$ (S61)

and the Bogoliubov fermions

$f_{k}=\cos\text{{\hbox{\frac{\theta_{k}}{2}}}}\alpha_{k}+{\rm
i}\sin\frac{\theta_{k}}{2}\alpha_{-k}^{\dagger},$ (S62)

the Hamiltonian reduces to the diagonalized form

$H_{\lambda}^{{\rm TFI}}=\sum_{k}\epsilon_{k}\alpha_{k}^{\dagger}\alpha_{k},$
(S63)

where $\epsilon_{k}=2\sqrt{J^{2}+\lambda^{2}-2J\lambda\cos k}$ and
$\theta_{k}$ is determined by

$\sin\theta_{k}=\frac{-2J\sin
k}{\epsilon_{k}},\quad\cos\theta_{k}=\frac{2(\lambda-J\cos
k)}{\text{{\hbox{\epsilon_{k}}}}}.$ (S64)

In terms of $\alpha_{k}^{\dagger}$ and $\alpha_{k}$,
$\partial_{\lambda}H_{\lambda}^{{\rm TFI}}=\sum_{i=1}^{N}\sigma_{i}^{z}$ can
be expressed as

$\partial_{\lambda}H_{\lambda}^{{\rm
TFI}}=N-2\sum_{k}\left(\cos^{2}\frac{\theta_{k}}{2}\alpha_{k}^{\dagger}\alpha_{k}+\sin^{2}\frac{\theta_{k}}{2}\alpha_{k}\alpha_{k}^{\dagger}+\frac{{\rm
i}}{2}\sin\theta_{k}\alpha_{k}^{\dagger}\alpha_{-k}^{\dagger}-\frac{{\rm
i}}{2}\sin\theta_{k}\alpha_{-k}\alpha_{k}\right).$ (S65)

Thus, we obtain

$\displaystyle[\partial_{\lambda}H_{\lambda}^{{\rm TFI}}(\tau)]^{{\rm(H)}}$
$\displaystyle=U^{\dagger}(\tau)\partial_{\lambda}H_{\lambda}^{{\rm
TFI}}U(\tau)$
$\displaystyle=N-2\sum_{k}\left(\cos^{2}\frac{\theta_{k}}{2}\alpha_{k}^{\dagger}\alpha_{k}+\sin^{2}\frac{\theta_{k}}{2}\alpha_{k}\alpha_{k}^{\dagger}+\frac{{\rm
i}}{2}{\rm e}^{2{\rm
i}\epsilon_{k}\tau}\sin\theta_{k}\alpha_{k}^{\dagger}\alpha_{-k}^{\dagger}-\frac{{\rm
i}}{2}{\rm e}^{-2{\rm
i}\epsilon_{k}\tau}\sin\theta_{k}\alpha_{-k}\alpha_{k}\right).$ (S66)

By using Eq. (S62), the Eq. (S66) can be expressed as

$[\partial_{\lambda}H_{\lambda}^{{\rm
TFI}}(\tau)]^{{\rm(H)}}=N-\sum_{k}\left[A(k)f_{k}^{\dagger}f_{k}+B(k)f_{k}f_{k}^{\dagger}+D(k)f_{k}^{\dagger}f_{-k}^{\dagger}+D^{*}(k)f_{-k}f_{k}\right],$
(S67)

where

$\displaystyle A(k)$
$\displaystyle=1+\cos^{2}\theta_{k}+\sin^{2}\theta_{k}\cos(2\epsilon_{k}\tau),$
$\displaystyle B(k)$
$\displaystyle=1-\cos^{2}\theta_{k}-\sin^{2}\theta_{k}\cos(2\epsilon_{k}\tau),$
$\displaystyle D(k)$ $\displaystyle={\rm
i}\sin\theta_{k}\cos\theta_{k}[\cos(2\epsilon_{k}\tau)-1]-\sin\theta_{k}\sin(2\epsilon_{k}\tau).$
(S68)

By further using the Fourier transformation (S61), we obtain

$[\partial_{\lambda}H_{\lambda}^{{\rm
TFI}}]^{{\rm(H)}}=\sum_{i=1}^{N}\mathscr{O}_{i}(\tau),$ (S69)

where

$\mathscr{O}_{i}(\tau)=1-\sum_{j=1}^{N}\left[\tilde{A}(j-i)c_{i}^{\dagger}c_{j}+\tilde{B}(i-j)c_{i}c_{j}^{\dagger}+\tilde{D}(j-i)c_{i}^{\dagger}c_{j}^{\dagger}+\tilde{D^{*}}(j-i)c_{i}c_{j}\right],$
(S70)

and $\tilde{A}(\ell)$ is the Fourier transformation of the function $A(k)$,
i.e.,

$\tilde{A}(\ell)\equiv\frac{1}{N}\sum_{k}A(k){\rm e}^{{\rm i}k\ell}.$ (S71)

We can further rewrite Eq. (S70) in terms of Eq. (12) in the main text, i.e.,

$\mathscr{O}_{i}=\sum_{j\geq
i}\sum_{\alpha}\eta_{ij}^{\alpha}\mathcal{O}_{ij}^{(\alpha)}$ (S72)

where we have suppressed the time-dependence,
$\mathcal{O}_{ij}^{(1)}=(c_{i}^{\dagger}c_{j}+{\rm h.c.})$,
$\mathcal{O}_{ij}^{(2)}=(c_{i}c_{j}^{\dagger}+{\rm h.c.)}$,
$\mathcal{O}_{ij}^{(3)}=(c_{i}^{\dagger}c_{j}^{\dagger}+{\rm h.c.}),$ and
$O_{ij}^{(4)}={\rm i}(c_{i}^{\dagger}c_{j}^{\dagger}-{\rm h.c.})$. The
corresponding $\eta$-functions are given by

$\displaystyle\eta_{ij}^{(1)}$
$\displaystyle=-(1-\delta_{ij})\tilde{A}(j\\!-\\!i)-\delta_{ij}\frac{\tilde{A}(0)}{2},$
$\displaystyle\eta_{ij}^{(2)}$
$\displaystyle=-(1-\delta_{ij})\tilde{B}(j\\!-\\!i)-\delta_{ij}\frac{\tilde{B}(0)}{2},$
$\displaystyle\eta_{ij}^{(3)}$ $\displaystyle=-2{\rm
Re}[\tilde{D}(j\text{\\!}-\\!i)],$ $\displaystyle\eta_{ij}^{(4)}$
$\displaystyle=-2{\rm Im}[\tilde{D}(j\\!-\\!i)].$ (S73)

The expressions for $\tilde{A}(\ell)$, $\tilde{B}(\ell)$, $\tilde{C}(\ell)$
and $\tilde{D}(\ell)$ are dramatically simplified in the limit $N\to\infty$
and $t\to\infty$. In the limit, $N\to\infty$, we can rewrite $\tilde{A}(\ell)$
as an integral

$\displaystyle\tilde{A}(\ell)$
$\displaystyle=\frac{1}{2\pi}\int_{-\pi}^{\pi}A(k)e^{ik\ell}dk$
$\displaystyle=\delta_{\ell,0}+\frac{1}{2\pi}\int_{-\pi}^{\pi}\cos^{2}\theta_{k}e^{ik\ell}dk+\frac{1}{2\pi}\int_{-\pi}^{\pi}\sin^{2}\theta_{k}\cos(2\epsilon_{k}\tau)e^{ik\ell}dk,$
(S74)

By the Riemann-Lebesgue lemma [50], the last integral in the right hand of Eq.
(S74) tends to zero as $\tau\to\infty$. The second integral in the right hand
of Eq. (S74) can be evaluated by introducing $z=e^{ik}$:

$I(\ell)\equiv\frac{1}{2\pi}\int_{-\pi}^{\pi}\cos^{2}\theta_{k}e^{ik\ell}dk=\frac{1}{2\pi{\rm
i}}\int_{\mathscr{C}}\frac{\left(J\frac{z^{2}+1}{2}-\lambda
z\right)^{2}}{(J^{2}+\lambda^{2})z-J\lambda(z^{2}+1)}z^{\ell-2}dz.$ (S75)

where the contour $\mathscr{C}$ is along the counterclockwise direction of the
unit circle on the complex plane.

We denote $I(\ell)=\frac{1}{2\pi{\rm i}}\int_{\mathscr{C}}f(z)dz$, where

$f(z)\equiv\frac{\left(J\frac{z^{2}+1}{2}-\lambda
z\right)^{2}}{(J^{2}+\lambda^{2})z-J\lambda(z^{2}+1)}z^{\ell-2}$ (S76)

The residues of $f$ are listed as follows:

$\displaystyle{\rm Res}(f,0)$
$\displaystyle=\frac{2\lambda^{2}-J^{2}}{4\lambda^{2}}\quad{\rm
for}\quad\ell=0;$ $\displaystyle{\rm Res}(f,0)$
$\displaystyle=-\frac{J}{4\lambda}\quad{\rm for\quad\ell=1;}$
$\displaystyle{\rm Res}\left(f,\frac{J}{\lambda}\right)$
$\displaystyle=\left(\frac{J}{\lambda}\right)^{\ell}\frac{\lambda^{2}-J^{2}}{4\lambda^{2}},\quad\text{for}\quad\text{all}\quad\ell$
$\displaystyle{\rm Res}\left(f,\frac{\lambda}{J}\right)$
$\displaystyle=\left(\frac{\lambda}{J}\right)^{\ell-2}\frac{J^{2}-\lambda^{2}}{4J^{2}},\quad\text{for}\quad\text{all}\quad\ell.$
(S77)

Then, by using the residue theorem, we obtain $I(\ell)={\rm Res}(f,0)$ for
$J=\lambda$, $I(\ell)={\rm Res}(f,0)+{\rm
Res}\left(f,\frac{\lambda}{J}\right)$ for $J>\lambda>0$, and $I(\ell)={\rm
Res}(f,0){\rm+Res}\left(f,\frac{J}{\lambda}\right)$ for $0<J<\lambda$.
Finally, we have

$\tilde{A}(\ell)=\left\\{\begin{array}[]{cc}3/2,&\ell=0,\\\ -1/4&\ell=1,\\\
0,&\ell=2,3,\ldots,N-1,\end{array}\right.\quad{\rm for}\quad J=\lambda>0;$
(S78)

$\tilde{A}(\ell)=\left\\{\begin{array}[]{cc}3/2,&\ell=0,\\\
-\lambda/4J&\ell=1,\\\
\frac{J^{2}-\lambda^{2}}{4J^{2}}\left(\frac{\text{$\lambda$}}{J}\right)^{\ell-2},&\ell=2,3,\ldots,N-1,\end{array}\right.\quad{\rm
for}\quad J>\lambda>0;$ (S79)

$\tilde{A}(\ell)=\left\\{\begin{array}[]{cc}2-\frac{J^{2}}{2\lambda^{2}},&\ell=0,\\\
-\frac{J^{3}}{4\lambda^{3}}&\ell=1,\\\
\frac{\lambda^{2}-J^{2}}{4\lambda^{2}}\left(\frac{\text{$J$}}{\lambda}\right)^{\ell},&\ell=2,3,\ldots,N-1,\end{array}\right.\quad{\rm
for}\quad 0<J<\lambda.$ (S80)

Similarly, we can obtain the other terms under the limit $\tau\to\infty$ and
$N\to\infty$

$\displaystyle\tilde{B}(\ell)$
$\displaystyle=\left\\{\begin{array}[]{cc}1/2,&\ell=0,\\\ 1/4,&\ell=1,\\\
0,&\ell=2,3,\ldots,N-1,\end{array}\right.\quad{\rm for}\quad
J=\lambda>0,\quad\tilde{B}(\ell)=\left\\{\begin{array}[]{cc}1/2,&\ell=0,\\\
\lambda/4J,&\ell=1,\\\
-\frac{J^{2}-\lambda^{2}}{4J^{2}}\left(\frac{\text{$\lambda$}}{J}\right)^{\ell-2},&\ell=2,3,\ldots,N-1,\end{array}\right.\quad{\rm
for}\quad J>\lambda>0,$ (S87) $\displaystyle\tilde{B}(\ell)$
$\displaystyle=\left\\{\begin{array}[]{cc}\frac{J^{2}}{2\lambda^{2}},&\ell=0,\\\
\frac{J^{3}}{4\lambda^{3}},&\ell=1,\\\
-\frac{\lambda^{2}-J^{2}}{4\lambda^{2}}\left(\frac{\text{$J$}}{\lambda}\right)^{\ell},&\ell=2,3,\ldots,N-1,\end{array}\right.\quad{\rm
for}\quad 0<J<\lambda,$ (S91) $\displaystyle\tilde{D}(\ell)$
$\displaystyle=\left\\{\begin{array}[]{cc}0,&\ell=0,\\\ -1/4&\ell=1,\\\
0,&\ell=2,3,\ldots,N-1,\end{array}\right.\quad{\rm for}\quad
J=\lambda>0,\quad\tilde{D}(\ell)=\left\\{\begin{array}[]{cc}0,&\ell=0,\\\
-\lambda/4J&\ell=1,\\\
\frac{J^{2}-\lambda^{2}}{4J^{2}}\left(\frac{\text{$\lambda$}}{J}\right)^{\ell-2},&\ell=2,3,\ldots,N-1,\end{array}\right.\quad{\rm
for}\quad J>\lambda>0,$ (S98) $\displaystyle\tilde{D}(\ell)$
$\displaystyle=\left\\{\begin{array}[]{cc}0,&\ell=0,\\\
\frac{J^{3}-2J\lambda^{2}}{4\lambda^{3}}&\ell=1,\\\
\frac{J^{2}-\lambda^{2}}{4\lambda^{2}}\left(\frac{\text{$J$}}{\lambda}\right)^{\ell},&\ell=2,3,\ldots,N-1,\end{array}\right.\quad{\rm
for}\quad 0<J<\lambda,$ (S102)

and $\tilde{D^{*}}(\ell)=-\tilde{D}(\ell)$.

##  VI QFI for the short-range TFI model

According to Eq. (S66), the generator $G(t)$ given in Eq. (2) of the
integrable Ising model is given by

$\displaystyle\frac{G(t)}{t}$
$\displaystyle=\frac{1}{t}\int_{0}^{t}\left[\partial_{\lambda}H_{\lambda}^{{\rm
Ising}}(\tau)\right]^{(\text{H})}d\tau$
$\displaystyle=N-2\sum_{k}\left(\cos^{2}\frac{\theta_{k}}{2}\alpha_{k}^{\dagger}\alpha_{k}+\sin^{2}\frac{\theta_{k}}{2}\alpha_{k}\alpha_{k}^{\dagger}+\frac{{\rm
i}}{2}g(2{\rm
i}\epsilon_{k}t)\sin\theta_{k}\alpha_{k}^{\dagger}\alpha_{-k}^{\dagger}-\frac{{\rm
i}}{2}g(-2{\rm i}\epsilon_{k}t)\sin\theta_{k}\alpha_{-k}\alpha_{k}\right),$
(S103)

where function $g(x)$ is given in Eq. (S22).

For the numerical calculation of QFI, we need to express $G(t)$ in the spin
representation. Thus, by using Eq. (S62) and (S61), we can rewrite Eq. (S103)
as

$\frac{G(t)}{t}$
$\displaystyle=N-\sum_{k}\left[\mathcal{A}(k)f_{k}^{\dagger}f_{k}+\mathcal{B}(k)f_{k}f_{k}^{\dagger}+\mathcal{D}(k)f_{k}^{\dagger}f_{-k}^{\dagger}+\mathcal{D}^{*}(k)f_{-k}f_{k}\right]$
$\displaystyle=N-\sum_{i,j=1}^{N}\left[\tilde{\mathcal{A}}(j-i)c_{i}^{\dagger}c_{j}+\tilde{\mathcal{B}}(i-j)c_{i}c_{j}^{\dagger}+\tilde{\mathcal{D}}(j-i)c_{i}^{\dagger}c_{j}^{\dagger}+\tilde{\mathcal{D}^{*}}(j-i)c_{i}c_{j}\right]$
(S104)

where

$\displaystyle\mathcal{A}(k)$
$\displaystyle=1+\cos^{2}\theta_{k}+\frac{1}{2}\sin^{2}\theta_{k}[g(2{\rm
i}\epsilon_{k}t)+g(-2{\rm i}\epsilon_{k}t)],$ $\displaystyle\mathcal{B}(k)$
$\displaystyle=1-\cos^{2}\theta_{k}-\frac{1}{2}\sin^{2}\theta_{k}[g(2{\rm
i}\epsilon_{k}t)+g(-2{\rm i}\epsilon_{k}t)],$ $\displaystyle\mathcal{D}(k)$
$\displaystyle={\rm i}\sin\theta_{k}\cos\theta_{k}\left[\frac{1}{2}g(2{\rm
i}\epsilon_{k}t)+\frac{1}{2}g(-2{\rm i}\epsilon_{k}t)-1\right]+\frac{{\rm
i}}{2}\sin\theta_{k}[g(2{\rm i}\epsilon_{k}t)-g(-2{\rm i}\epsilon_{k}t)],$
(S105)

and $\tilde{\mathcal{A}}(\ell)$ is the Fourier transformation of the function
$\mathcal{A}(k)$ like Eq. (S71). Equations (S104), (S105), and (S55) provide a
numerical approach to express $G(t)$ in terms of the spin operators. Finally,
QFI can be easily calculated by using
$I(t)=\text{4Var}[G(t)]{}_{\ket{\psi_{0}}}$, Eq. (3).

Now we consider a special case where the initial state $\ket{\psi_{0}}$ is
chosen to be the paramagnetic product state
$\ket{\uparrow\uparrow\cdots\uparrow}$, which can be viewed as the ground
state of the Ising chain $H_{\lambda_{*}}^{{\rm Ising}}$ (14) by taking
$\lambda_{*}\to+\infty$ and we denote it as $\ket{g(\lambda_{*})}$. We denote
the Bogoliubov fermionic operators and the Bogoliubov angle of
$H_{\lambda_{*}}^{{\rm Ising}}$ by $\alpha_{*k}$ and $\theta_{*k}$,
respectively. We now consider the QFI for the evolved state
$U(t)\ket{g(\lambda_{*})}$ with respect to the parameter $\lambda$ where
$U(t)=\exp(-{\rm i}H_{\lambda}^{{\rm Ising}})$. Here we focus on the long-time
limit and thus Eq. (S103) reduces to

$\displaystyle\lim_{t\to+\infty}\frac{G(t)}{t}$
$\displaystyle=N-2\sum_{k}\left(\cos^{2}\frac{\theta_{k}}{2}\alpha_{k}^{\dagger}\alpha_{k}+\sin^{2}\frac{\theta_{k}}{2}\alpha_{k}\alpha_{k}^{\dagger}\right)$
$\displaystyle=-2\sum_{k}\cos\theta_{k}n_{k},$ (S106)

where $n_{k}\equiv\alpha_{k}^{\dagger}\alpha_{k}$ and we use the result
$\lim_{t\to+\infty}g(\pm 2{\rm i}\epsilon_{k}t)=0$ for $J\neq\lambda$. A
constant is ignored in the last equation since it does not affect the value of
QFI. It can be checked that the ground state $\ket{g(\text{$\lambda_{*}$})}$
can be expressed as

$\ket{g(\lambda_{*})}=\frac{1}{\mathcal{N}}\exp\left[{\rm
i}\sum_{k>0}R(k)\alpha_{-k}^{\dagger}\alpha_{k}^{\dagger}\right]\ket{g(\lambda)},$
(S107)

where $\mathcal{N}$ is a normalization factor, $\ket{g(\lambda)}$ is the
ground state of Hamiltonian $H_{\lambda}^{{\rm Ising}}$, and

$R(k)=\tan\left(\frac{\theta_{k}-\theta_{*k}}{2}\right).$ (S108)

Then we have

$\displaystyle\lim_{t\to+\infty}\frac{\langle G^{2}(t)\rangle}{t^{2}}$
$\displaystyle=4\sum_{k}\cos^{2}\theta_{k}\langle
n_{k}^{2}+n_{k}n_{-k}\rangle+4\sum_{k}\sum_{k^{\prime}(\neq\pm
k)}\cos\theta_{k}\cos\theta_{k^{\prime}}\langle n_{k}n_{k^{\prime}}\rangle$
$\displaystyle=4\sum_{k}\cos^{2}\theta_{k}\langle
n_{k}^{2}+n_{k}n_{-k}\rangle+4\sum_{k}\sum_{k^{\prime}(\neq\pm
k)}\cos\theta_{k}\cos\theta_{k^{\prime}}\langle n_{k}\rangle\langle
n_{k^{\prime}}\rangle$
$\displaystyle=8\sum_{k}\cos^{2}\theta_{k}\frac{R^{2}(k)}{1+R^{2}(k)}+4\sum_{k}\sum_{k^{\prime}(\neq\pm
k)}\cos\theta_{k}\cos\theta_{k^{\prime}}\frac{R^{2}(k)R^{2}(k^{\prime})}{[1+R^{2}(k)][1+R^{2}(k^{\prime})]},$
$\displaystyle\lim_{t\to+\infty}\frac{\langle G(t)\rangle^{2}}{t^{2}}$
$\displaystyle=4\sum_{k,k^{\prime}}\cos\theta_{k}\cos\theta_{k^{\prime}}\langle
n_{k}\rangle\langle n_{k^{\prime}}\rangle$
$\displaystyle=4\sum_{k,k^{\prime}}\cos\theta_{k}\cos\theta_{k^{\prime}}\frac{R^{2}(k)R^{2}(k^{\prime})}{[1+R^{2}(k)][1+R^{2}(k^{\prime})]},$
(S109)

where $\langle O\rangle\equiv\langle g(\lambda_{*})|O|g(\lambda_{*})\rangle$
we use the results $\langle n_{k}\rangle=\langle
n_{k}n_{-k}\rangle=R^{2}(k)/[1+R^{2}(k)]$. Substituting Eqs. (S64), (S108),
and (S109) into the definition of QFI, we obtain

$\displaystyle\lim_{t\to+\infty}\frac{I(t)}{t^{2}}$
$\displaystyle=\lim_{t\to+\infty}4\frac{\langle G^{2}(t)\rangle-\langle
G(t)\rangle^{2}}{t^{2}}$
$\displaystyle=32\sum_{k}\cos^{2}\theta_{k}\frac{R^{2}(k)}{1+R^{2}(k)}\left[1-\frac{R^{2}(k)}{1+R^{2}(k)}\right]$
$\displaystyle=8\sum_{k}\cos^{2}\theta_{k}(\sin\theta_{k}\cos\theta_{*k}-\cos\theta_{k}\sin\theta_{*k})^{2}$
$\displaystyle=8J^{2}(\lambda-\lambda_{*})^{2}\sum_{k}\frac{(\lambda-J\cos
k)^{2}\sin^{2}k}{(J^{2}+\lambda^{2}-2\lambda J\cos
k)^{2}(J^{2}+\lambda_{*}{}^{2}-2\lambda_{*}J\cos k)}.$ (S110)

In the thermodynamic limit $N\to\infty$, the above equation becomes

$\displaystyle\lim_{t,N\to+\infty}\frac{I(t)}{Nt^{2}}$
$\displaystyle=8J^{2}(\lambda-\lambda_{*})^{2}\times\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{(\lambda-J\cos
k)^{2}\sin^{2}k}{(J^{2}+\lambda^{2}-2\lambda J\cos
k)^{2}(J^{2}+\lambda_{*}^{2}-2\lambda_{*}J\cos k)}dk$
$\displaystyle=8J^{2}(\lambda-\lambda_{*})^{2}\times\frac{1}{2\pi}\int_{\mathscr{C}}\frac{\left(\lambda-J\frac{z^{2}+1}{2z}\right)^{2}\left[1-\left(\frac{z^{2}+1}{2z}\right)^{2}\right]}{\left(J^{2}+\lambda^{2}-2\lambda
J\frac{z^{2}+1}{2z}\right)^{2}\left(J^{2}+\lambda_{*}^{2}-2\lambda_{*}J\frac{z^{2}+1}{2z}\right)}\frac{dz}{iz},$
(S111)

where the contour $\mathscr{C}$ is the the unit circle on the complex plane
and $z=e^{ik}$. Finally, by using the residue theorem, we obtain

$\lim_{t,N\to\infty}\frac{I(t)}{Nt^{2}}=\left\\{\begin{array}[]{cc}\frac{(\lambda-\lambda{}_{*})}{(J^{2}-\lambda\lambda_{*})^{2}}(J^{2}-2\lambda\lambda_{*}+\lambda{}_{*}^{2}),&0<\lambda,\lambda_{*}<J,\\\
\frac{J^{2}(\lambda-\lambda_{*})^{2}}{\lambda_{*}{}^{2}\lambda^{3}(J^{2}-\lambda\lambda_{*})^{2}}(\lambda
J^{4}+2\lambda_{*}J^{4}-4\lambda^{2}\lambda_{*}J^{2}-3\lambda\lambda_{*}J^{2}+4\lambda^{3}\lambda_{*}{}^{2}),&J<\lambda,\lambda_{*},\\\
\frac{1}{\lambda^{3}}[2\lambda_{*}J^{2}-3\lambda
J^{2}+\lambda(\lambda_{*}-2\lambda)^{2}],&0<\lambda_{*}<J<\lambda,\\\
\frac{1}{\lambda_{*}{}^{2}}(J^{2}-2\lambda\lambda_{*}+\lambda_{*}{}^{2}),&0<\lambda<J<\lambda_{*}.\end{array}\right.$
(S112)

Since the ferromagnetic product state $\ket{\uparrow\uparrow\cdots\uparrow}$
is the ground state $\ket{g(\lambda_{*})}$ by taking $\lambda_{*}\to+\infty$,
then we obtain

$\lim_{t,N\to\infty}\frac{I(t)}{Nt^{2}}=\left\\{\begin{array}[]{cc}\frac{J^{2}(4\lambda^{2}-3J^{2})}{\lambda^{4}},&J<\lambda,\\\
1,&0<\lambda<J.\end{array}\right.$ (S113)


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URL
https://proceedings.neurips.cc/paper/2016/file/148510031349642de5ca0c544f31b2ef-
Paper.pdf

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