# Exponential precision by reaching a quantum critical point

Louis Garbe  Vienna Center for Quantum Science and Technology, Atominstitut,
TU Wien, 1040 Vienna, Austria  Obinna Abah  School of Mathematics, Statistics
and Physics, Newcastle University, Newcastle upon Tyne NE1 7RU, United Kingdom
Centre for Theoretical Atomic, Molecular and Optical Physics, Queen’s
University Belfast, Belfast BT7 1NN, United Kingdom  Simone Felicetti
Istituto di Fotonica e Nanotecnologie, Consiglio Nazionale delle Ricerche
(IFN-CNR),
Via Cineto Romano 42, 00156 Rome, Italy  Ricardo Puebla  Instituto de Física
Fundamental, IFF-CSIC, Calle Serrano 113b, 28006 Madrid, Spain  Centre for
Theoretical Atomic, Molecular and Optical Physics, Queen’s University Belfast,
Belfast BT7 1NN, United Kingdom<EMAIL_ADDRESS>
###### Abstract

Quantum metrology shows that by exploiting nonclassical resources it is
possible to overcome the fundamental limit of precision found for classical
parameter-estimation protocols. The scaling of the quantum Fisher information–
which provides an upper bound to the achievable precision– with respect to the
protocol duration is then of primarily importance to assess its performances.
In classical protocols the quantum Fisher information scales linearly with
time, while typical quantum-enhanced strategies achieve a quadratic
(Heisenberg) or even higher-order polynomial scalings. Here we report a
protocol that is capable of surpassing the polynomial scaling, and yields an
exponential advantage. Such exponential advantage is achieved by approaching,
but without crossing, the critical point of a quantum phase transition of a
fully-connected model in the thermodynamic limit. The exponential advantage
stems from the breakdown of the adiabatic condition close to a critical point.
As we demonstrate, this exponential scaling is well captured by the new bound
derived in [1], which in turn allows us to obtain approximate analytical
expressions for the quantum Fisher information that agree with exact numerical
simulations. In addition, we discuss the limitations to the exponential
scaling when considering a finite-size system as well as its robustness
against decoherence effects. Hence, our findings unveil a novel quantum
metrological protocol whose precision scaling goes beyond the paradigmatic
Heisenberg limit with respect to the protocol duration.

The existence of quantum fluctuations limits the precision that can be
achieved in parameter-estimation protocols when using a limited number of
physical resources. While the development of quantum mechanics has unveiled
this fundamental limitation, it also provides us with the solution: By
exploiting quantum resources, such as squeezing or entanglement, it is
possible to overcome the fundamental limit of precision found for classical
protocols [2, 3, 4, 5]. Accordingly, quantum sensing is one of the most
promising applications of current quantum technologies [6]. In the context of
quantum metrology, in order to assess the performances of a parameter-
estimation strategy it is of utmost relevance to understand the scaling of the
precision with respect to the amount of used resources, such as number of
probes or total measurement time. At theoretical level, the most common figure
of merit to quantify the estimation precision is the quantum Fisher
information (QFI), which sets the ultimate achievable precision according to
the Cramér-Rao bound [7, 4]. A paradigmatic example consists in the celebrated
Heisenberg limit, valid under a series of very general assumptions [3], for
which the QFI scales quadratically with the number of particles of the probe
system and with the protocol duration time. This corresponds to a quadratic
enhancement over the classical case, where the QFI scales linearly with those
resources. It has also been shown that even super-Heisenberg scalings can be
achieved when allowing for $k$-body interaction terms [8, 9], in which case
scalings $N^{2k}$, or potentially even exponential scaling in $N$, could be
achieved. Another possibility is to enable time-dependent Hamiltonian
evolutions [10], which allows one to surpass the quadratic scaling in time.

In this context, systems undergoing critical phase transitions are ideal
candidates for sensing applications, thanks to their high sensitivity to
external perturbations. In particular, quantum phase transitions (QPT) [11]
represent a compelling resource [12, 13, 14] for quantum metrology due to
highly nonclassical properties developed in proximity of the critical point.
It has also been shown [15, 16] that, in spite of the critical slowing down,
the framework of critical quantum metrology makes it possible to achieve the
Heisenberg scaling, where the QFI grows quadratically in time and number of
probes. Recent works [16, 17, 18, 19, 20, 21, 22, 23, 24] have shown that the
framework of critical quantum metrology can be applied to a broad class of
quantum optical models. Current solid-state and atomic technology allow for
the implementation of these models in a controllable way, where their
parameters can be tuned in real time. Intense research efforts [1, 25, 26, 27,
28, 29, 30, 31, 32] are now dedicated to identifying the optimal control
strategies that maximize the estimation precision and mitigate potential
errors.

In this article, we present the design of a quantum sensing protocol that goes
beyond the typical power law scaling of the QFI, achieving an exponential
scaling with respect to protocol duration time. The protocol is based on a
quantum-control strategy that exploits the critical nature of quantum phase
transition, by bringing cyclically the probe system in proximity of the
critical point. This modulation is intrinsically nonadiabatic, as at the
critical point the gap closes. We provide an explanation to the exponential
time-scaling of the QFI using the bound recently introduced in Ref. [1],
showing that it is made possible by the exponential growth of the number of
excitations generated by the control strategy. This general understanding also
covers the results of a recent work where exponential scaling of the QFI is
achieved [25]. Furthermore, in order to characterize the performances of the
proposed protocols for practical applications, we analyze the effect of
dissipative processes and of finite-size corrections. We find the protocol to
be resilient to thermal and dissipative effects, and the exponential scaling
is preserved when the decay rate is comparable to or smaller than the single-
cycle time. Finite-size corrections fix the saturation limit, and so they
constrain the maximum number of cycles allowed. The proposed protocol can be
advantageous when the estimation time is the most relevant resource, and when
using quantum platforms where ground-state cooling is challenging.

##  1 Critical fully-connected model

Among the different families of quantum critical systems, we focus on fully-
connected models such as the quantum Rabi, the Lipkin-Meshkov-Glick [33], and
Dicke model [34]. In the thermodynamic limit, these systems feature a QPT that
divides the phase diagram in a normal and a symmetry-broken phase [35, 36, 37,
38, 39, 40, 41, 42, 43]. In the Dicke and Lipkin-Meshkov-Glick models, the
thermodynamic limit refers to the standard notion of infinitely many
components. In the quantum Rabi model, and related finite-component critical
systems [44, 45, 46, 47, 48, 49], it rather refers to a certain ratio of the
system parameters. For instance in the quantum Rabi model, a QPT emerges when
the qubit frequency becomes much larger than the field frequency. The
frequency ratio acts as an effective system size, and finite-size critical
exponents can also be defined in this case. These fully-connected systems
admit a simple description in terms of an effective bosonic mode whose
potential depends on a rescaled and dimensionless coupling strength $g$. These
models constitute a suitable test-bed for the exploration of different aspects
of quantum critical phenomena [50, 51, 52, 53, 54, 55, 56, 44, 45, 57, 58, 59,
60, 61, 48, 47, 62, 63, 64]. Within the normal phase $0\leq g\leq 1$ and in
the thermodynamic limit, the effective Hamiltonian describing these systems
can be written as [37, 62, 1]

$\displaystyle H=\omega
a^{\dagger}a-\frac{g^{2}\omega}{4}(a+a^{\dagger})^{2},$ (1)

where $[a,a^{\dagger}]=1$ and $\omega$ denotes the frequency of the bosonic
mode. Here, $g$ is a rescaled coupling parameter which can typically be
written as $g=\lambda/\lambda_{c}$, where $\lambda$ is the dimensionful
coupling strength of the model and $\lambda_{c}$ its critical value. From Eq.
(1) it follows that the energy gap vanishes at the QPT as
$\Delta(g)\propto|g-g_{c}|^{1/2}$ so that the critical exponents in this case
are of a mean-field type $z\nu=1/2$ [11]. It is worth stressing that the
critical traits in these systems have been experimentally observed [65, 66,
67, 68, 69, 70]. Importantly, the critical coupling value $\lambda_{c}$
typically depends on $\omega$. In the Lipkin-Meshkov-Glick and quantum Rabi
model, for example, we have a dependence as $\lambda_{c}\propto\sqrt{\omega}$.
Therefore, the rescaled coupling depends itself on $\omega$ as $g\propto
1/\sqrt{\omega}$. In the following, we assume this dependence, which makes the
sensing protocol critically dependent on the value of $\omega$ [1].

##  2 Quantum Fisher Information

The ultimate precision for the estimation of a parameter $x$ is given by the
QFI [4], denoted as $I_{x}$, such that the variance for the estimated
parameter $x$ is bounded as $(\delta x)^{2}\geq I_{x}^{-1}$ for a single
measurement, which is known as the quantum Cramér-Rao bound. This result is
obtained optimizing over all possible positive operator-valued measurements
and classical data processing. Thus, the scaling of $I_{x}$ with respect to
the experimental resources, such as the duration of the metrological protocol,
is of key importance. Let us denote by $\rho_{x}$ the system state in which
the unknown value of $x$ has been encoded. The QFI is related to the Bures
distance between two infinitesimally closed states, $\rho_{x}$ and
$\rho_{x+\epsilon}$, which can be written as $d_{{\rm B},x}^{2}=2(1-{\rm
Tr}[\sqrt{\sqrt{\rho_{x}}\rho_{x+\epsilon}\sqrt{\rho_{x}}}])$, so that [71]

$\displaystyle I_{x}=4\left(\left.\frac{\partial d_{{\rm
B},x}}{\partial\epsilon}\right|_{\epsilon=0}\right)^{2}.$ (2)

The signal-to-noise ratio can be then written as $Q_{x}=x^{2}I_{x}$.

Since the Hamiltonian (1) is quadratic in $a$ and $a^{\dagger}$, any initial
Gaussian state evolving under (1) is also a Gaussian state [72]. Recall that a
Gaussian state $\rho$ is that whose Wigner function is Gaussian, and thus
$\rho$ can be fully determined in terms of its first and second moments in the
two-dimensional phase space ${\bf X}^{\top}=(x,p)$ [72], where here we employ
the convention $x=a+a^{\dagger}$ and $p=i(a^{\dagger}-a)$. The first moments
of the state is simply $\langle{\bf X}^{\top}\rangle=({\rm Tr}[\rho\ x],{\rm
Tr}[\rho\ p])$. The second moments are given by the covariance matrix ${\bf
R}$, which is real and symmetric, and its matrix elements read as

$\displaystyle R_{i,j}=\frac{1}{2}\langle X_{i}X_{j}+X_{j}X_{i}\rangle-\langle
X_{i}\rangle\langle X_{j}\rangle.$ (3)

As shown in Ref. [71], the QFI adopts the following form for Gaussian states,

$\displaystyle I_{x}=\frac{1}{2}\frac{{\rm Tr}[({\bf R}^{-1}\partial_{x}{\bf
R})^{2}]}{1+P^{2}}+2\frac{(\partial_{x}P)^{2}}{1-P^{4}}+L_{x},$ (4)

with $L_{x}={\bf\Delta X}^{\prime\top}_{x}{\bf R}^{-1}{\bf\Delta
X}^{\prime}_{x}$ and ${\bf\Delta X}^{\prime}_{x}=\partial\langle{\bf
X}_{x+\epsilon}-{\bf X}_{x}\rangle/\partial\epsilon|_{\epsilon=0}$, and
$P={\rm det}[{\bf R}]^{-1/2}$ denotes the purity of the Gaussian state $\rho$.
Throughout the article we will consider Gaussian states with $\langle
x\rangle=\langle p\rangle=0$, and therefore $L_{x}=0$ in Eq. (4). In this
manner, the QFI and the corresponding signal-to-noise ratio $Q_{x}$ can be
computed from the covariance matrix ${\bf R}$.

In the following we will focus on the estimation of the bosonic frequency
$\omega$, although similar results can be found for the estimation of $g$ in
Eq. (1).

Figure 1: Schematic illustration of the protocol. At $t=0$ the state is
assumed to be in a vacuum or thermal state, which is brought to the critical
point $g_{\tau}=g_{c}=1$ in a time $\tau$ by tuning $g(t)$. The cycle is
completed at time $2\tau$ such that $g(2\tau)=g(0)=0$. After one cycle $m=1$
the state becomes squeezed, with a squeezing parameter and angle $|s|$ and
$\theta$, respectively, so that the state is squeezed along the direction
${\bf x}_{s}^{\top}=(\cos(\theta/2),\sin(\theta/2))$ in the phase space ${\bf
X}^{\top}=(x,p)$, reducing the variance by a factor $e^{-2|s|}$. Performing
another cycle, $m=2$, the state can be further squeezed such that
$|s_{m}|=m|s|$ where $m=1,2,\ldots$, under suitable parameters. This squeezing
amplification leads to an exponential precision for the estimation of the
system parameters (see main text for further details).

##  3 Non-adiabatic cycles: Exponential scaling

The Hamiltonian in Eq. (1) displays a QPT at the critical point $g_{c}=1$,
which is accompanied by a vanishing energy gap, among other features. As a
consequence, by tuning $g(t)$ towards the QPT in a finite time the system will
unavoidably depart from adiabaticity [73, 74]. Such non-adiabaticity
translates in the formation of quantum excitations in the system, which can be
harnessed and beneficial in different contexts [75]. In the following, we show
that the non-adiabaticity caused by the QPT can be exploited to lead in an
exponential scaling of the QFI with respect the protocol duration.

In particular, we choose a protocol $g(t)$ that completes a cycle in a time
$2\tau\gtrsim 1/\omega$ as (cf. Fig. 1)

$\displaystyle g(t)=\begin{cases}g_{\tau}\frac{t}{\tau}&\ {\rm for}\ \ 0\leq
t\leq\tau\\\ g_{\tau}\left(2-\frac{t}{\tau}\right)&\ {\rm for}\ \ \tau\leq
t\leq 2\tau,\end{cases}$ (5)

so that $g(0)=0$ and $g(\tau)=g_{\tau}$. The state at any time $t$ follows
from $\dot{\rho}=-i[H(t),\rho(t)]$, with the initial and final state upon the
completion of the cycle given by $\rho(0)$ and $\rho(2\tau)$, respectively.
Note that the condition $2\tau\gtrsim 1/\omega$ rules out fast cyclic
transformation ($\omega\tau\rightarrow 0$) for which the initial state remains
trivially unchanged. If we perform a cycle away from the critical point, i.e.
for $g_{\tau}<g_{c}=1$, and for sufficiently slow cycles, $\tau\gg
1/\Delta(g_{\tau})$, the protocol is able to meet the adiabatic condition.
Hence, by virtue of the adiabatic theorem, the state upon the cyclic
transformation is simply $\rho(2\tau)=\rho(0)$. By contrast, if we bring the
system all the way to the critical point, i.e. if $g_{\tau}=g_{c}=1$, the
adiabatic condition will break down at some point, since $\Delta(g_{c})=0$.
Thus, $\rho(2\tau)\neq\rho(0)$ regardless of how slow the cycle is performed,
as studied in Ref. [74]. Indeed, the transformation (5) reaching the critical
point produces squeezing, that is,
$\rho(2\tau)=\mathcal{S}(s)\rho(0)\mathcal{S}^{\dagger}(s)$ with
$\mathcal{S}(s)={\rm exp}[(s(a^{\dagger})^{2}-s^{*}a^{2})/2]$ and
$s=|s|e^{i\theta}$, with $|s|$ and $\theta$ the squeezing parameter and its
angle, respectively. In this manner, the state is squeezed along ${\bf
x}_{s}^{\top}=(\cos(\theta/2),\sin(\theta/2))$ in the phase space. To a good
degree of approximation (cf. App. A), the acquired squeezing after completing
the protocol $g(t)$ with duration $2\tau$ (and $\tau\gtrsim 1/\omega$) is
given by [74, 75]

$\displaystyle|s|=\frac{\log(3)}{2},$ (6)

while the angle $\theta$ depends on $\tau$ (see Fig. 1). Note that this is
caused solely by the presence of the QPT.

Although setting $g_{\tau}=1$ allows to automatically break adiabaticity, the
resulting squeezing is robust against small deviations from $g_{\tau}=1$, i.e.
for cycles with $|g_{\tau}-g_{c}|\ll 1$ Eq. (6) still holds for
$1/\omega\lesssim\tau\lesssim 1/\Delta(g_{\tau})$ (cf. App. A).

Figure 2: Noiseless dynamics in the thermodynamic limit for an initial vacuum
state. Panel (a) shows the number of bosons after $m$ cycles $N_{m}$ for
$\omega\tau=8$ (solid red circles) $\omega\tau=9$ (open green triangles) and
$\omega\tau=8.2$ (open orange circles). The phase-matching condition is
achieved for $\omega\tau=8$ which leads to an exponential growth of $N_{m}$,
while for $\omega\tau=9$ it follows that $N_{2m+1}\approx 1/3$ (dotted black
line) (see main text). The results for $\omega\tau=8$ follow the trend
predicted in Eq. (8) (solid black line), which is closely followed by small
deviations from the phase-matching condition, as illustrated for
$\omega\tau=8.2$. Panel (b) shows the computed signal-to-noise ratio
$Q_{\omega}=\omega^{2}I_{\omega}$ together with the bound computed numerically
using Eq. (7) (solid black line) as well as its approximated expression
(dotted black line), given in Eq. (10), i.e. $I_{\omega}^{\rm B}\approx
4\tau^{2}3^{T/\tau}$. The exponential scaling is robust against deviations
from the phase-matching condition. For $\omega\tau=9$ there is no exponential
scaling but rather $Q_{\omega}\propto(2m\tau)^{2}=T^{2}$, which is also well
captured by the bound (not explicitly shown).

This cyclic transformation can be carried out $m$ times by a concatenation of
the protocol $g(t)$ in (5), that is, $g(t+2m\tau)=g(t)$ with $t\in[0,2\tau]$
and $m=1,2,\ldots$, in a total time $T=2m\tau$. By doing so, the produced
squeezing can be amplified to yield $|s_{m}|=m|s|=m\log(3)/2$, where $|s_{m}|$
denotes the squeezing produced on the initial state $\rho(0)$ after the $m$th
cycles, i.e. $\rho(2m\tau)=\mathcal{S}(s_{m})\rho(0)\mathcal{S}(s_{m})$ with
$s_{m}=|s_{m}|e^{i\theta_{m}}$ (cf. Fig. 1). Such amplification requires a
phase-matching condition for subsequent cycles, i.e.
$\theta_{m+1}=\theta_{m}$. However, if $\theta_{m+1}=\theta_{m}+\pi$ (modulo
$2\pi$) the $m+1$th cycle compensates the squeezing generated in the previous
cycle, and thus we expect $|s_{2m+1}|\approx|s|=\log(3)/2$ and
$|s_{2m}|\approx 0$, so that $N_{2m+1}\approx N_{m=1}=1/3$ while
$N_{2m}\approx 0$.

It is worth mentioning that other schemes can also achieve a linear
amplification of squeezing, such as the one reported in Ref. [32] based on a
suitable periodic modulation of the oscillator frequency [76] which differs
from Eq. (5).

###  3.1 Bound to the quantum Fisher information

As aforementioned, Eq.(4) allows us to compute exactly the QFI. However,
before presenting the numerical results in the next section, we show here that
the main features of the QFI behavior can be predicted analytically. For that
we rely on a recently derived bound to the QFI, which is valid for active
interferometric protocols and Gaussian states. This bound is denoted as
$I^{\rm B}_{\omega}$ such that $I_{\omega}\leq I_{\omega}^{\rm B}$, and in our
case is given by (see Ref. [1] for the details of the derivation):

$\displaystyle I_{\omega}^{\rm
B}=8(\chi^{2}+\phi^{2})\left[\int_{0}^{T}dt(2N(t)+1)\right]^{2},$ (7)

where $N(t)$ denotes the number of bosons at time $t$, while $\chi$ and $\phi$
are the eigenvalues of the matrix $\partial_{\omega}H(t)$ in the phase space
${\bf X}$, which here take a time-independent value $\chi=\phi=1/2$ [1]. If
the number of probes $N(t)$ is time-independent, Eq.(7) returns the Heisenberg
scaling $N^{2}T^{2}$. By contrast, if $N(t)$ is time-dependent, which is the
case here, we can achieve more exotic scalings in $T$. For simplicity, let us
consider an initial vacuum state, $\rho(0)=\left|0\right\rangle\left\langle
0\right|$, although we remark that the results are robust against finite-
temperature initial states (see App. B). From Eq. (6) and since $N={\rm
Tr}[\mathcal{S}(s)\rho(0)\mathcal{S}^{\dagger}(s)a^{\dagger}a]=\sinh^{2}(|s|)$,
we expect the number of bosons $N_{m}={\rm Tr}[\rho(2m\tau)a^{\dagger}a]$
after $m$ cycles, assuming a phase-matching condition, to obey

$\displaystyle N_{m}=\sinh^{2}\left(\frac{m}{2}\log(3)\right),$ (8)

which grows exponentially with $m$, $N_{m}\sim 3^{m}/4$ for $m\gg 1$. Hence,
we can already anticipate that $I_{\omega}^{\rm B}$ will show a similar
exponential scaling. Indeed, by approximating $N(t)$ during
$t\in[2(m-1)\tau,2m\tau]$ by $N_{m}$ so that $\int_{(2m-1)\tau}^{2m\tau}dt\
(2N(t)+1)\approx 2\tau(2N_{m}+1)$, we find

$\displaystyle I_{\omega}^{B}$ $\displaystyle\leq 16\tau^{2}(2N_{m}+1)^{2}$
$\displaystyle\approx 64\tau^{2}\sinh^{4}(m\log(3)/2)$ (9)

where in the last step we have assumed $N_{m}\gg 1$. In this manner, for $m\gg
1$ we find

$\displaystyle I_{\omega}^{B}\approx 4\tau^{2}3^{2m}=4\tau^{2}3^{T/\tau},$
(10)

where $T=2m\tau$ is the total time of the protocol after $m$ cycles. That is,
the bound to the QFI scales exponentially with $T$. This is the central result
of the article. In the reminder of the article we will employ numerical
simulations to corroborate the validity of the previous expression (cf. Sec.
3.2), while in Sec. 4 we investigate the noise robustness of such exponential
scaling, as well as the potential impact of finite-size effects.

Figure 3: Robustness of the exponential scaling in $Q_{\omega}$ versus
decoherence effects. The results have been obtained for
$\rho(0)=\left|0\right\rangle\left\langle 0\right|$, $g_{\tau}=g_{c}=1$ and
$\omega\tau=8$ as in Fig. 2, although equivalent results can be found for
different $\omega\tau\gtrsim 1$. Panel (a) shows $Q_{\omega}$ for increasing
number of cycles $m$, (b) illustrates the variance along the squeezing axes,
while (c) depicts the prefactor $\alpha$ such that $Q_{\omega}\propto
3^{\alpha m}$, obtained as a best fit in the interval $m\in[5,10]$, as a
function of $2\tau\kappa$. In (b) the solid red line corresponds to $3^{\pm
m}$, as expected theoretically for $\kappa=0$. Open (full) points correspond
to $(\Delta{\bf x}_{s})^{2}$ ($(\Delta{\bf p}_{s})^{2}$) with the same format
as in (a). In (b), the solid red line corresponds to $\alpha$ obtained from a
fit to the bound in Eq. (7). The exponential advantage holds even for non-zero
dissipation interaction strength $\kappa$, although the prefactor $\alpha$ in
the scaling decreases. For $2\tau\kappa=1$ (open yellow triangle in (a) and
(b)) the scaling shifts to the standard quadratic scaling $Q_{\omega}\propto
T^{2}$, while for $2\tau\kappa>1$ the signal-to-noise ratio $Q_{\omega}$
saturates and $\alpha=0$.

###  3.2 Noiseless dynamics in the thermodynamic limit

The dynamics in the thermodynamic limit can be computed exploiting the
quadratic nature of the Hamiltonian (1). Assuming that the initial state is
such that $\langle x\rangle=\langle p\rangle=0$ and that it undergoes a
noiseless evolution under the protocol $g(t)$, the state at time $t$ is
completely characterized by the following time-dependent Lyapunov equation of
motion for the covariance matrix (see App. C)

$\displaystyle\dot{{\bf R}}(t)={\bf W}(t){\bf R}(t)+{\bf R}(t){\bf
W}^{\dagger}(t),$ (11)

with

$\displaystyle{\bf W}(t)=\begin{bmatrix}0&\omega(1-g^{2}(t))\\\
-\omega&0\end{bmatrix}$ (12)

while $\langle x\rangle=\langle p\rangle=0\ \forall t$. The number of bosons
at time $t$ is then given by $N(t)=\frac{1}{4}({\rm Tr}[{\bf R}(t)]-2)$, while
the QFI can be computed exactly using Eq. (4), and compared with the bound
prediction (10). In order to test the prediction of the exponential scaling
for $I_{\omega}$ we consider $\rho(0)=\left|0\right\rangle\left\langle
0\right|$ so that ${\bf R}(0)=\mathbb{I}$. Numerically solving Eq. (11) under
the protocol $g(t)$ given in Eq. (5), we can compute $N_{m}=N(2m\tau)$ and the
corresponding signal-to-noise ratio $Q_{\omega}=\omega^{2}I_{\omega}$. The
phase-matching condition is achieved here when $\omega\tau=2n$ with
$n=1,2,\ldots$, so that $\theta_{m}=\pm\pi/2\ \forall m$, while for
$\omega\tau=2n+1$ the $m+1$th cycle counteracts the generated squeezing in the
previous one (cf. App. C). Thus, for the phase-matching condition the state is
squeezed in the direction ${\bf x}_{s}^{\top}=(1,\pm 1)/\sqrt{2}$ in the phase
space. The number of bosons after $m$ cycles is plotted in Fig. 2(a) for
$\omega\tau=8$ (phase-matching condition), $\omega\tau=8.2$ and
$\omega\tau=9$. The results clearly show the exponential growth of $N_{m}$,
which is well captured by Eq. (8), for $\omega\tau=8$. Finally, the signal-to-
noise ratio $Q_{\omega}$ is plotted in Fig. 2(b), together with the bound and
its approximated value given in Eq. (7) and (10), respectively. The numerical
results show that the QFI shows an exponential scaling with the protocol
duration $T$, $Q_{\omega}\propto 3^{T/\tau}$, as predicted by Eq. (10). The
exponential scaling still holds, although the prefactor is reduced, in
situations when $\omega\tau\sim 2n$ but $\omega\tau\neq 2n$, as exemplified by
$\omega\tau=8.2$. To the contrary, for $\omega\tau=2n+1$ the number of bosons
is bounded by $1/3$ which leads to the standard $Q_{\omega}\propto T^{2}$
scaling. We stress again that, although the previous results have been
computed considering $\rho(0)=\left|0\right\rangle\left\langle 0\right|$, i.e.
a zero-temperature initial state, initial thermal states with an arbitrary
temperature also lead to an exponential advantage (cf. App. B).

##  4 Noise robustness and finite-size effects

Let us now analyze the robustness of the exponential scaling in the QFI with
respect to decoherence. For that, we model the interaction of the system with
an environment at an inverse temperature $\beta$ such that $N_{\rm
th}=(e^{\beta\omega}-1)^{-1}$ through the standard Lindblad master equation
[77]

$\displaystyle\dot{\rho}(t)=-i[H(t),\rho(t)]+\mathcal{D}[\rho(t)]$ (13)

where the dissipator reads as

$\displaystyle\mathcal{D}[\rho]=$ $\displaystyle\kappa\frac{(N_{\rm
th}+1)}{2}\left(2a\rho a^{\dagger}-\\{a^{\dagger}a,\rho\\}\right)$ (14)
$\displaystyle+\kappa\frac{N_{\rm th}}{2}\left(2a^{\dagger}\rho
a-\\{aa^{\dagger},\rho\\}\right),$ (15)

and the parameter $\kappa$ accounts for the system-environment interaction
strength. As the master equation is still quadratic in $a$ and $a^{\dagger}$,
the time-dependent Lyapunov equation for the covariance matrix modifies to

$\displaystyle\dot{{\bf R}}(t)={\tilde{\bf W}}(t){\bf R}(t)+{\bf
R}(t){\tilde{\bf W}}^{\dagger}(t)+{\bf F},$ (16)

where now ${\bf F}=\kappa(2N_{\rm th}+1)\mathbb{I}$ and ${\tilde{\bf W}}={\bf
W}-\kappa/2\mathbb{I}$. As before, we compute the $Q_{\omega}$ from Eq. (4)
solving Eq. (16). The results are gathered in Fig. 3(a) for $N_{\rm th}=2$,
which shows that the exponential scaling is robust against decoherence
provided $2\tau\kappa\ll 1$. There we show $Q_{\omega}$ for different values
of $\kappa$ starting from $\rho(0)=\left|0\right\rangle\left\langle 0\right|$.
For $2\tau\kappa\ll 1$ the dissipation still permits an exponential scaling
$Q_{\omega}\propto 3^{\alpha m}=3^{\alpha T/(2\tau)}$ but with smaller
prefactor, $0<\alpha\leq 2$, which is reduced as $\kappa$ increases. To the
contrary $2\tau\kappa\gg 1$ the state relaxes to the thermal equilibrium
before the cycle is completed and thus $Q_{\omega}$ does not increase with the
protocol duration. For $2\tau\kappa\approx 1$ both mechanisms are balanced and
$Q_{\omega}\propto T^{2}$ is still possible. Since $Q_{\omega}\propto
4\tau^{2}3^{2m}$, for a fixed number of cycles $m$, the QFI is maximal when
$2\tau\kappa=1$ provided $\omega\tau$ ensures the phase-matching condition,
while for fixed total evolution time $T$ it is more beneficial to perform the
largest allowed number of cycles $m=T/(2\tau)$, and hence to take the shortest
possible $\tau$, with the constraints $2\tau\gtrsim 1/\omega$ and
$2\tau\kappa\ll 1$. After $m$ cycles the system finds itself in a squeezed
thermal state, so that the variance along the squeezing axis reads as
$(\Delta{\bf x}_{s})^{2}=(2n_{\kappa}+1)e^{-2|s_{m}|}$, while $(\Delta{\bf
p}_{s})^{2}=(2n_{\kappa}+1)e^{2|s_{m}|}$ is enlarged, where $n_{\kappa}$ is
the number of thermal bosons acquired by the state due to the interaction with
the environment. Moreover, the total number of bosons after $m$ cycles is
$N(2\tau m)=(2n_{\kappa}+1)\sinh^{2}|s_{m}|+n_{\kappa}$. Although $|s_{m}|$
increases with $m$ for $2\tau\kappa\lesssim 1$, so does the number of thermal
bosons. This leads to a saturation in the variance $(\Delta{\bf x}_{s})^{2}$,
while $(\Delta{\bf p}_{s})^{2}$ keeps increasing with $m$, which still allows
$Q_{\omega}$ to grow. This is shown in Fig. 3(b). For $2\tau\kappa>1$, both
quadratures saturate, and therefore also $Q_{\omega}$. In addition, we also
show the behavior of the prefactor $\alpha$ as a function $2\tau\kappa$, which
is illustrated in Fig. 3(c). There we show the fitted $\alpha$ such that
$Q_{\omega}\propto 3^{\alpha m}$. For a noiseless evolution, $\alpha\approx 2$
(see Fig. 2(b) and App. A for a comment regarding the expected deviation with
respect $\alpha=2$), while as $\kappa$ increases $\alpha\rightarrow 0$. Yet,
there is a wide range of values for $\kappa$ in which the exponential scaling
holds. Similar results can be found for other $\omega\tau$ fulfilling (or
close to) a phase-matching condition. For comparison, we also compute $\alpha$
using the bound in Eq. (7). As shown in Fig. 3(b), although the bound still
captures the exponential scaling for $2\tau\kappa\lesssim 1$, it becomes loose
and overestimates the prefactor $\alpha$. It is worth mentioning that the
transition from $\alpha=2$ to $1$ can be understood as a departure from the
Heisenberg to the standard quantum limit. Indeed, we can re-express
$Q_{\omega}$ in terms of both the maximum number of bosons and the protocol
duration time. Then, $\alpha=2$ corresponds to $Q_{\omega}\sim N_{m}^{2}T^{2}$
(Heisenberg limit), while $\alpha=1$ to $Q_{\omega}\sim N_{m}T$, which is the
scaling of the standard quantum limit. This expression is useful if we are in
a situation in which both the protocol duration and the number of photons are
limited. By contrast, if the only relevant resource is the time, we can
express again $N_{m}\sim 3^{m}=3^{T/\tau}$. Then both $\alpha=1$ and
$\alpha=2$ lead to an exponential scaling of the QFI with $T$, yet $\alpha>1$
means a scaling beyond the standard quantum limit.

Finally, we turn our attention to finite-size effects. As commented in Sec. 1,
the Hamiltonian $H$ in Eq. (1) is valid in the normal phase and in the
thermodynamic limit. Yet, any realistic exploration of a critical system is
unavoidably accompanied by finite-size corrections. Let us denote by $\eta$
the system size, which in Lipkin-Meshkov-Glick and quantum Rabi model refers
to the number of spins and a ratio between spin and bosonic frequencies,
respectively. Indeed, for $\eta<\infty$ the leading-order correction to Eq.
(1) leads to a Hamiltonian of the form
$H_{\eta}=H+\frac{f(g)}{\eta}(a+a^{\dagger})^{4}$ where $f(g)$ is a function
of $g$ [62, 1]. In this manner, the $1/\eta$ correction introduces a confining
potential and lifts the vanishing energy gap at $g_{c}=1$. In our case, such
correction can become significant since $N$ grows exponentially, and so does
${\rm Tr}[\rho(t)(a+a^{\dagger})^{4}]$. Hence, we expect that the exponential
scaling reported in Sec. 3 holds as long as $f(g){\rm
Tr}[\mathcal{S}(s_{m})\rho(0)\mathcal{S}^{\dagger}(s_{m})(a+a^{\dagger})^{4}]\ll\eta$,
as otherwise the correction can no longer considered as a perturbation and
will significantly modify the Gaussian nature of the state and, consequently,
the generated squeezing. The previous conditions allow us to define a maximum
number of cycles $m^{*}$ before the $1/\eta$ correction becomes relevant, i.e.
$m^{*}\approx\log_{3}\eta$. Hence, for $\eta\approx 10^{6}$ we expect that the
exponential scaling holds up to $m^{*}\approx 10$.

##  5 Conclusions

In this article we have reported a quantum metrological scheme that yields a
quantum Fisher information that scales exponentially with the protocol
duration time $T$. Such scheme is rooted in the non-adiabaticity of a cycle in
the control parameter reaching a quantum critical point in fully-connected
models. In one cycle, the state acquires squeezing, which can be amplified
under a suitable choice of parameters by subsequent cycles. As we show, after
$m$ cycles of a duration $2\tau$ each, the quantum Fisher information scales
as $I_{\omega}\propto 3^{T/\tau}$. This scaling is well captured by the recent
bound put forward in Ref. [1], which in turn allows us to find approximated
expressions for the quantum Fisher information. We discuss the potential
deviations to this ideal scenario, such as finite-size effects or the effect
of decoherence mechanisms. Supported by numerical simulations, we find that
the exponential precision is robust against decoherence effects. In addition,
as we argue, finite-size effects pose a limit to the maximum number of cycles
that can be performed before the exponential scaling breaks down.

Our results, together with those reported recently by Gietka et al. [25],
highlight that systems featuring a quantum phase transition are a valuable
resource in quantum metrology as they can yield an exponential advantage for
parameter estimation.

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##  Appendix A Robustness of the squeezing against finite-time cycles and
variations in $g_{\tau}$

As commented in the main text, the state becomes squeezed after one cycle,
whose squeezing parameter is given by $|s|=\log(3)/2$ (cf. Eq. (6)). Here we
provide a brief derivation of this expression. For that, we follow [74]. The
solution to the dynamics under Eq. (1) can be written in terms of the Ermakov-
Milne equation, $\ddot{\xi}(t)+\omega^{2}(t)\xi(t)=1/(4\xi^{3}(t))$, where
$\omega^{2}(t)=\omega^{2}(1-g^{2}(t))$ is the frequency of the harmonic
oscillator at time $t$, and $\xi(t)$ an effective width of the state, whose
equilibrium value is $\xi(t)=(2\omega(t))^{-1/2}$. In the
$\omega\tau\rightarrow\infty$ limit it is possible to find solutions to the
above equation as a combination of Airy functions [74]. Indeed, for
$\omega\tau\rightarrow\infty$, one can find the overlap between the ground and
the evolved state upon a cycle $g(t)$, which is given by
$f(2\tau)=\sin(\pi/3)$ (see Ref. [74] for more details). Since for $\langle
x\rangle=\langle p\rangle=0$ the Hamiltonian in Eq. (1) only produces
squeezing in the state, and because $|\langle
0|\mathcal{S}(s)|0\rangle|^{2}=\cosh^{-1}(|s|)$, it follows that
$\cosh^{-1}(|s|)=\sin(\pi/3)$ which finally leads to $|s|=\log(3)/2$.

This result holds to a good degree of approximation for $\omega\tau\gtrsim 1$
and $g_{c}-g_{\tau}\ll 1$. Let us denote $|s(\tau)|$ the squeezing parameter
after one cycle $g(t)$ (cf. Eq. (5)) with duration $2\tau$ (see App. C for
details on how to obtain $|s|$ from ${\bf R}$). From numerical simulations we
find that $|s(\tau)|\approx|s|$ provided $g_{c}-g_{\tau}\ll 1$ and
$\omega\tau_{q}\gtrsim 1$. Indeed, Fig. 4(a) shows the robustness of
$|s(\tau)|$ when $g_{\tau}\neq g_{c}=1$ but $|g_{c}-g_{\tau}|\ll 1$. On the
one hand, if $g_{\tau}\ll g_{c}=1$ the dynamics is not influenced by the
critical point, so $|s|\approx 0$. In addition, for $g_{\tau}\lesssim g_{c}$
the larger $\tau$, the more sensitive $|s(\tau)|$ becomes to deviations in
$g_{\tau}$. This is due to the finite energy gap at $g<g_{c}$, that is, for
$\tau\gg 1/\Delta(g_{\tau})$ the evolution becomes adiabatic and Eq. (6) no
longer holds. On the other hand, from numerical simulation we find that
finite-time corrections when $g_{\tau}=1$ obey
$|s|-|s(\tau)|\approx(27\omega\tau)^{-2/3}$ for $\omega\tau\gtrsim 1$ (cf.
Fig. 4(b)). Hence, by increasing $\omega\tau$ the resulting squeezing gets
closer to the expected $|s|=\log(3)/2$ at the price of loosing robustness
against potential deviations from $g_{\tau}=1$. For $\omega\tau\rightarrow 0$
one trivially obtains $|s(\tau)|\approx 0$. For the numerical results
presented in the main text we consider $\omega\tau\sim O(10)$. In particular,
for $\omega\tau=8$, the difference between $|s(\tau)|$ and $|s|$ amounts to
$\approx 0.03$, and thus, the expected exponential factor $\alpha$, such that
$Q_{\omega}\propto 3^{\alpha m}$, is $\alpha\approx 1.94$ (cf. Fig. 3(b))
rather than $\alpha=2$ (cf. Eq. (10)), which would be reached in the limit
$\omega\tau\rightarrow\infty$.

##  Appendix B Finite-temperature initial state

Let us consider an initial thermal state at inverse temperature
$\beta=(k_{B}T)^{-1}$, $\rho_{\beta}=e^{-\beta H_{0}}/{\rm Tr}[e^{-\beta
H_{0}}]$ where $H_{0}=\omega a^{\dagger}a$ and $N_{\beta}={\rm
Tr}[\rho_{\beta}a^{\dagger}a]=(e^{\beta\omega}-1)^{-1}$. As discussed in the
main text, the state upon $m$ cycles can acquire a $m$fold squeezing
$|s_{m}|=m|s|$ where $|s|=\log(3)/2$, so that
$\rho(2m\tau)=\mathcal{S}(s_{m})\rho_{\beta}\mathcal{S}^{\dagger}(s_{m})$,
whose occupation number $N_{m}={\rm Tr}[\rho(2m\tau)a^{\dagger}a]$ is given by

$\displaystyle N_{m}=\frac{1}{2}\left((2N_{\beta}+1)\cosh(2|s_{m}|)-1\right)$
(17)

For $N_{\beta}=0$ one recovers the expression in Eq. (8) when substituting
$|s_{m}|=m\log(3)/2$. From the bound (7) and again approximating
$2N(t)+1\approx 2N_{m}+1$ for $t\in[2(m-1)\tau,2m\tau]$ and assuming
$|s_{m}|\gtrsim 1$, we arrive to

$\displaystyle I_{\omega}^{B}\approx
16\tau^{2}(2N_{\beta}+1)^{2}\cosh^{2}(2|s_{m}|)\sim\tau^{2}3^{T/\tau},$ (18)

which is equivalent to Eq. (10) up to prefactors. The bound captures the
exponential scaling. Yet, it becomes loose as it overestimates $I_{\omega}$,
since the $N_{\beta}$ bosons originally contained in $\rho_{\beta}$ do not
participate actively in the parameter estimation.

Figure 4: Robustness of the squeezing parameter $|s(\tau)|$ achieved after one
cycle under $g(t)$ in a total time $2\tau$, such that $g(0)=0$ and
$g(\tau)=g_{\tau}$, and for an initial vacuum state. Panel (a) shows the
squeezing $|s(\tau)|$ as a function of the deviation to the critical point
$g_{c}-g_{\tau}$. The solid horizontal line corresponds to the expected
$|s|=\log(3)/2$, while the points have been obtained numerically for different
$\omega\tau$ and $g_{\tau}$ values. Panel (c) shows the finite-time
corrections to the squeezing $|s(\tau)|$ when $g_{\tau}=1$. The solid line
corresponds to a best fit, $|s|-|s(\tau)|\approx(27\omega\tau)^{-2/3}$.

##  Appendix C Squeezing and quantum Fisher information of a Gaussian state

A Gaussian state $\rho$ is characterized by a Gaussian Wigner function in the
phase space ${\bf X}^{\top}=(x,p)$, such that $W({\bf X})=P/(2\pi)\ e^{-({\bf
X}-\langle{\bf X}\rangle)^{\top}{\bf R}^{-1}({\bf X}-\langle{\bf X}\rangle)}$,
where $\langle{\bf X}^{\top}\rangle=(\langle x\rangle,\langle p\rangle)$ and
${\bf R}$ is the covariance matrix whose matrix elements are
$R_{i,j}=\frac{1}{2}\langle X_{i}X_{j}+X_{j}X_{i}\rangle-\langle
X_{i}\rangle\langle X_{j}\rangle$. In the previous $P={\rm det}[{\bf
R}]^{-1/2}$ denotes the purity of $\rho$ [71]. Under the Hamiltonian
$H(t)=\omega a^{\dagger}a-g^{2}(t)\omega(a+a^{\dagger})^{2}/4$ and the master
equation given in Eq. (13), it is straightforward to find the time-dependent
Lyapunov equation for ${\bf R}$, which is given in Eq. (16). Recall that we
employ $x=a+a^{\dagger}$ and $p=i(a^{\dagger}-a)$, so that $N={\rm Tr}[\rho
a^{\dagger}a]=({\rm Tr}[{\bf R}]-2)/4$, while we consider initial states with
$\langle{\bf X}^{\top}\rangle=(0,0)$ so that $\langle x\rangle=\langle
p\rangle=0\ \forall t$. For a decoherence-free evolution, the covariance
matrix ${\bf R}$ fulfills ${\rm det}[{\bf R}]=1$. Moreover, as the evolution
produces only squeezing, it can be diagonalized at any time, $\tilde{{\bf
R}}={\bf V}{\bf R}{\bf V}^{\top}$ yielding ${\rm diag}[{\tilde{{\bf
R}}}]=(e^{2|s|},e^{-2|s|})$, while the angle $\theta$ follows from the
eigenvectors of ${\bf R}$ in the phase space ${\bf X}$. That is, the
eigenvector with eigenvalue $e^{-2|s|}$ is of the form ${\bf v}^{\top}={\bf
x}_{s}^{\top}=(\sin(\theta/2),\cos(\theta/2))$.

In all the simulations presented in the main text, the derivatives
$\partial_{\omega}{\bf R}$ and $\partial_{\omega}P$ have been computed
numerically setting $\epsilon/\omega\sim 10^{-8}$ which ensured the
convergence of the results.

The phase $\theta$ upon one cycle can be estimated as follows. First, the
phase gained during the evolution introduces a factor $e^{i\beta}$ where
$\beta$ reads

$\displaystyle\beta=-\int_{0}^{2\tau}dt\
\sqrt{1-g^{2}(t)}=-2\omega\tau\int_{0}^{1}d\tilde{t}\sqrt{1-g^{2}(\tilde{t})}$
(19)

where $g(\tilde{t})$ is the protocol with a rescaled time $\tilde{t}=t/\tau$.
That is, the state at time $2\tau$ can be written as
$\left|\psi(2\tau)\right\rangle=\sum_{n=0}c_{n}e^{i(n+1)\beta}\left|n\right\rangle$.
In this manner, we can write $\beta=\omega\tau\nu$ with
$\nu=-2\int_{0}^{1}d\tilde{t}\ \sqrt{1-g^{2}(\tilde{t})}$. Then, the phase
$\theta$ changes linearly with $\omega\tau$, i.e.

$\displaystyle\theta=\nu\omega\tau+\theta_{0},$ (20)

where $\theta_{0}=\pi/2$ since for $\omega\tau\rightarrow 0$ the evolution
squeezes the state in the $x$ direction. For a linear ramp, we find
$\nu=-\pi/2$, so that $\theta=-\omega\tau\pi/2+\pi/2$. For $\omega\tau=2n$
with $n=1,2,\ldots$ one obtains $\theta=\pm\pi/2$. For subsequent cycles, the
choice $\omega\tau=2n$ leads to the desired phase-matching condition
$\theta_{m+1}=\theta_{m}$.


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