[figure]style=plain,subcapbesideposition=top

# Experimental study on the principle of minimal work fluctuations

Wei Cheng  CAS Key Laboratory of Microscale Magnetic Resonance and School of
Physical Sciences, University of Science and Technology of China, Hefei
230026, China  CAS Center for Excellence in Quantum Information and Quantum
Physics, University of Science and Technology of China, Hefei 230026, China
Wenquan Liu  School of Science, Beijing University of Posts and
Telecommunications, Beijing, 100876, China  Yang Wu  CAS Key Laboratory of
Microscale Magnetic Resonance and School of Physical Sciences, University of
Science and Technology of China, Hefei 230026, China  CAS Center for
Excellence in Quantum Information and Quantum Physics, University of Science
and Technology of China, Hefei 230026, China  Zhibo Niu  CAS Key Laboratory of
Microscale Magnetic Resonance and School of Physical Sciences, University of
Science and Technology of China, Hefei 230026, China  CAS Center for
Excellence in Quantum Information and Quantum Physics, University of Science
and Technology of China, Hefei 230026, China  Chang-Kui Duan  CAS Key
Laboratory of Microscale Magnetic Resonance and School of Physical Sciences,
University of Science and Technology of China, Hefei 230026, China  CAS Center
for Excellence in Quantum Information and Quantum Physics, University of
Science and Technology of China, Hefei 230026, China
Jiangbin Gong<EMAIL_ADDRESS>Department of Physics, National University of
Singapore, Singapore 117551, Singapore  Center for Quantum Technologies,
National University of Singapore, Singapore 117543, Singapore  Joint School of
National University of Singapore and Tianjin University, International Campus
of Tianjin University, Binhai New City, Fuzhou 350207, China  MajuLab, CNRS-
UCA-SU-NUS-NTU International Joint Research Unit, Singapore  Xing Rong
<EMAIL_ADDRESS>CAS Key Laboratory of Microscale Magnetic Resonance and
School of Physical Sciences, University of Science and Technology of China,
Hefei 230026, China  CAS Center for Excellence in Quantum Information and
Quantum Physics, University of Science and Technology of China, Hefei 230026,
China  Hefei National Laboratory, University of Science and Technology of
China, Hefei 230088, China  Jiangfeng Du<EMAIL_ADDRESS>CAS Key Laboratory
of Microscale Magnetic Resonance and School of Physical Sciences, University
of Science and Technology of China, Hefei 230026, China  CAS Center for
Excellence in Quantum Information and Quantum Physics, University of Science
and Technology of China, Hefei 230026, China  Hefei National Laboratory,
University of Science and Technology of China, Hefei 230088, China  School of
Physics, Zhejiang University, Hangzhou 310027, China

###### Abstract

The central quantity in the celebrated quantum Jarzynski equality is
$e^{-\beta W}$, where $W$ is work and $\beta$ is the inverse temperature. The
impact of quantum randomness on the fluctuations of $e^{-\beta W}$ and hence
on the predictive power of the Jarzynski estimator is an important problem.
Working on a single nitrogen-vacancy center in diamond and riding on an
implementation of two-point measurement of non-equilibrium work with single-
shot readout, we have conducted a direct experimental investigation of the
relationship between the fluctuations of $e^{-\beta W}$ and adiabaticity of
non-equilibrium work protocols. It is observed that adiabatic processes
minimize the variance of $e^{-\beta W}$, thus verifying an early theoretical
concept, the so-called principle of minimal work fluctuations. Furthermore, it
is experimentally demonstrated that shortcuts-to-adiabaticity control can be
exploited to minimize the variance of $e^{-\beta W}$ in fast work protocols.
Our work should stimulate further experimental studies of quantum effects on
the bias and error in the estimates of free energy differences based on the
Jarzynski equality.

Figure 1:  Experimental pulse sequence to study the PMWF in the NV center
platform. The pulse sequence is used to obtain work statistics via the two-
point measurement scheme, which includes the thermal state preparation, the
first projective measurement, the work protocol and the second projective
measurement.

##  I I. Introduction

Thermal and quantum fluctuations are dominant features in non-equilibrium
thermodynamics [1, 2, 3, 4, 5] and can be quantified by fluctuation theorems
[6, 7, 8, 9, 10, 11, 12]. Fluctuation theorems are not only insightful for our
understanding of the second law of thermodynamics but also instrumental for us
to extract equilibrium thermodynamic quantities from non-equilibrium processes
[13, 14, 15, 16]. The celebrated Jarzynski equality [6] does apply to quantum
systems and assumes essentially the same form as in the classical case,
namely, $\langle e^{-\beta W}\rangle=e^{-\beta\Delta F}$, where
$\langle\cdot\rangle$ represents both thermal and quantum ensemble averaging.
This equality connects the mean of $e^{-\beta W}$ with the exponential
function of a free-energy difference $\Delta F$ at the inverse temperature
$\beta$. Interestingly, though suppression of work fluctuations is of great
interest in the context of nanoscale heat engines [17, 18, 19], there is no
experimental effort focusing on the fluctuations in the quantity $e^{-\beta
W}$ measured in quantum processes. Such fluctuations can be enhanced by
quantum randomness and are hence of crucial importance when applying the
Jarzynski relation to quantum settings.

For a finite number of measured work values, the Jarzynski free-energy
estimator of $\Delta F$ is given by $-\beta^{-1}\ln[\sum_{k=1}^{N}e^{-\beta
W_{k}}/N]$, where $W_{k}$ denotes work in the $k$th experimental run. This
estimator is a biased one, so when regarding practical applications the
fluctuations of $e^{-\beta W}$ are crucial. It determines whether the variance
of $e^{-\beta W}$ diverges, in which case the convergence of our estimate of
$\Delta F$ will be extremely slow. If the variance of $e^{-\beta W}$ does
converge, it still determines the number of experimental runs ($N$) needed to
reach a given estimate precision of $\Delta F$ [20]. Several theoretical
studies [21, 22, 23, 24] touched upon the suppression of the fluctuations in
$e^{-\beta W}$. In particular, the so-called principle of minimal work
fluctuations (PMWF) [22] states that for a quantum system initially at thermal
equilibrium but detached from its surrounding afterwards, the minimal variance
of $e^{-\beta W}$ can be achieved in an adiabatic process if the instantaneous
energy levels do not cross during the work protocol. This indicates if the
variance diverges for the adiabatic process, then it diverges for all work
protocols. Adiabatic processes call for a long evolution time, however, the
coherence time of realistic quantum systems is limited. Regarding this, the
PMWF also proved that one can equally use shortcuts-to-adiabatic (STA)
processes [25, 26] to achieve the minimal variance of $e^{-\beta W}$. PMWF can
thus guide us in the designing of work protocols in order to suppress the
variance of $e^{-\beta W}$ and hence reduce experimental efforts when applying
the Jarzynski equality to estimate $\Delta F$. The suppression of the variance
of $e^{-\beta W}$ also naturally leads to the suppression of the fluctuations
in the work itself [21].

Here we report an experimental investigation of the PMWF utilizing a single
nuclear spin of a nitrogen-vacancy ($\rm{NV}$) center in diamond. Our study
focuses on the variance of $e^{-\beta W}$ during a work protocol applied to a
thermal equilibrium state. The work statistic is directly obtained by the two-
point measurement of work [1, 2]. The realization of the two-point measurement
of work makes it possible to examine the fluctuations in $e^{-\beta W}$ and
hence verify the PMWF. Experimental results show that the variance of
$e^{-\beta W}$ decreases gradually as the work protocol approaches the
adiabatic limit. Furthermore, we demonstrate that the variance of $e^{-\beta
W}$ in the STA process, which is achieved by implementing a counter-diabatic
(CD) driving Hamiltonian, is the same as that in adiabatic work protocols.
Besides, the role of temperature on the variance of $e^{-\beta W}$ is also
examined. It is observed that a higher-temperature setting suppresses the
variance of $e^{-\beta W}$.

##  II II. Theory

We investigate the variance of $e^{-\beta W}$ when a system initially at
thermal equilibrium undergoes a non-equilibrium work protocol while being
isolated from the heat bath. During the protocol, an external control was
applied to tune the system Hamiltonian from $H(0)$ to $H(\tau)$. The work done
on the system can be obtained via the two-point measurement scheme, which
requires performing two projective measurements at the beginning and the end
of the protocol. This allows us to determine the trajectory, such as
$|E_{m}(0)\rangle\to|\tilde{E}_{n}(\tau)\rangle$, in each realization of the
work protocol, with $|E_{m}(0)\rangle$ and $|\tilde{E}_{n}(\tau)\rangle$
denoting the eigenstates of $H(0)$ and $H(\tau)$, respectively. The work
associated with the trajectory
$|E_{m}(0)\rangle\to|\tilde{E}_{n}(\tau)\rangle$ is
$W_{mn}=\tilde{E}_{n}-E_{m}$, where $E_{m}$ and $\tilde{E}_{n}$ are
eigenvalues corresponding to $|E_{m}(0)\rangle$ and
$|\tilde{E}_{n}(\tau)\rangle$, respectively. The work distribution is given by

$P(W)=\sum_{m,n}p^{0}_{m}p^{\tau}_{m\rightarrow
n}\delta[W-(\tilde{E}_{n}-E_{m})],$ (1)

where $p^{0}_{m}$ is the initial thermal population of $|E_{m}(0)\rangle$ and
$p^{\tau}_{m\rightarrow n}$ is the transition probability to
$|\tilde{E}_{n}(\tau)\rangle$ conditioned that the trajectory started from
$|E_{m}(0)\rangle$. The variance of $e^{-\beta W}$ is directly obtained from

$\sigma^{2}(e^{-\beta W})=\sum_{W}P(W)e^{-2\beta
W}-\left[\sum_{W}P(W)e^{-\beta W}\right]^{2}.$ (2)

The PMWF predicts that $\sigma^{2}(e^{-\beta W})$ achieves its minimal value
when $p^{\tau}_{m\rightarrow n}=\delta_{mn}$, which can be realized in an
adiabatic process or an STA process.

##  III III. Experiments

In our experiment, a single nuclear spin of an $\rm{NV}$ center system with
natural abundance of 13C (1.1$\%$) is used. The $\rm{NV}$ center is a type of
defect in diamond which consists of a substitutional nitrogen atom adjacent to
a carbon vacancy. The vacancy contains a spin-1 electron spin which couples to
the spin-1 nuclear spin of the adjacent 14N atom. Hamiltonian of the NV center
system can be written as
$H_{\rm{NV}}=2\pi(DS_{z}^{2}+\omega_{e}S_{z}+QI_{z}^{2}+\omega_{n}I_{z}+A_{zz}I_{z}S_{z})$,
where $S_{z}$ and $I_{z}$ are the spin operators of the electron spin and the
nuclear spin, respectively. Here, $D=2.87\,\rm{GHz}$ is the ground-state zero-
field splitting of the electron spin, $Q=-4.95\,\rm{MHz}$ is the quadrupolar
interaction of nuclear spin, and $A_{zz}=-2.16\,\rm{MHz}$ is the longitudinal
component of the hyperfine interaction between nuclear spin and electron spin.
The Zeeman frequencies of the electron and nuclear spin induced by the applied
static magnetic field are denoted by $\omega_{e}$ and $\omega_{n}$,
respectively. We select two energy levels of the nuclear spin,
$|\textnormal{-}1\rangle_{n}$ and $|0\rangle_{n}$, as the system, with the
electron spin playing an ancillary role in the study, as illustrated in Fig.
1.

To implement two-point measurement of work in the NV center system, we
realized non-demolition projective measurement of the nuclear spin using the
single-shot readout technique [27]. The photoluminescence rate of the NV
center for $|0\rangle_{e}$ is higher than that for
$|\textnormal{-}1\rangle_{e}$ with a contrast of about 30% [28]. To perform
projective measurement on the 14N nuclear spin in the computational basis, the
electron spin was first polarized into $|0\rangle_{e}$, then flipped to
$|\textnormal{-}1\rangle_{e}$ only if the nuclear spin state was
$|\textnormal{-}1\rangle_{n}$ with a selective microwave pulse. The selective
microwave pulse was designed via optimal control to be robust against quasi-
static noise. Subsequently, a 532-nm laser pulse with a time duration of 200
ns was applied to excite the NV center and the fluorescence photons emitted
were collected. Our experiment was executed at a magnetic field of about 7600
G with its direction along the NV center symmetry axis. This high external
magnetic field can suppress the relaxation of nuclear spin, so the application
of the 532-nm laser pulse almost does not alter the state of the nuclear spin.
By repeating the procedures above, we can accumulate the fluorescence signal
to distinguish between different nuclear spin states. Experimentally, the
procedure was repeated 1200 times and the fidelity of the single-shot readout
was $0.98(1)$. Projective measurement along an arbitrary energy basis can be
realized by applying appropriate rotations before the single-shot readout.

Figure 2:  Thermal state and joint probability for effective temperature
$\beta Z=0.6$ (a) and $\beta Z=0.8$ (b). Left panels show the population of
the initial thermal state. Right panels show the joint probability of
different trajectories. Dots with error bars are experimental results. Pillars
and lines show theoretical predictions.

The pulse sequence to study the PMWF consists of four steps, as depicted in
Fig. 1. In our experiment, the switching of the system Hamiltonian as a work
protocol is chosen as follows:

$H_{0}(t)=2\pi(ZS_{z}^{\prime}+X(t)S_{x}^{\prime})$ (3)

where $S_{z}^{\prime}=(|1\rangle\langle 1|-|0\rangle\langle 0|)/2$,
$S_{x}^{\prime}=(|1\rangle\langle 0|+|0\rangle\langle 1|)/2$,
$Z=5/\sqrt{3}~{}\rm{kHz}$, and $X(t)=5(1-\cos{(\pi t/\tau}))/2~{}\rm{kHz}$.
Here, energy levels $|0\rangle_{n}$ and $|\textnormal{-}1\rangle_{n}$ are
relabeled with $|1\rangle$ and $|0\rangle$, respectively. Firstly, we prepared
the thermal state of the initial Hamiltonian $H(0)$ in three substeps.
Initially, the nuclear spin was initialized into state $|0\rangle$ through the
single-shot readout and post-selection. Next, a radiofrequency (RF) pulse
$R_{1}$ was applied to prepare a coherent state
$\sqrt{P_{\rm{thm}}^{0}}|0\rangle+\sqrt{P_{\rm{thm}}^{1}}|1\rangle$, where
$P_{\rm{thm}}^{0}$ and $P_{\rm{thm}}^{1}$ are populations of the thermal
state. Then, the coherence of the nuclear spin was dissipated by applying two
selective microwave (MW) pulses separated by a free evolution time being 10
$\upmu$s. The selective MW pulse was applied to flip the electron spin from
$|0\rangle_{e}$ to $|\textnormal{-}1\rangle_{e}$ conditioned that the nuclear
spin state is $|0\rangle$. The fidelity of the prepared thermal state in this
way exceeds 0.9999 in our experiment. After the thermal state was prepared,
the first projective measurement was performed, projecting the nuclear spin
onto an eigenstate of $H_{0}(0)$, such as $|E_{m}(0)\rangle$, with a
probability of $p_{m}^{0}$. Subsequently, an RF pulse was applied to tune the
Hamiltonian of the nuclear spin from $H_{0}(0)$ to $H_{0}(\tau)$ to realize
the work protocol. Experimentally, the Rabi frequency and time duration of the
RF pulse were adjusted to realize the work protocol with different
adiabaticity. Finally, the second projective measurement was performed to
project the nuclear spin onto an eigenstate of $H_{0}(\tau)$, such as
$|\tilde{E}_{n}\rangle$, and gave the transition probability,
$p^{\tau}_{m\rightarrow n}$. Note that, the energy basis of the final
Hamiltonian $H(\tau)$ differs from the computational basis, so an RF pulse
$R_{2}$ was applied before the single-shot readout.

Figure 3:  Mean and variance of the exponential work versus the time durations
of the work protocol. Blue circle dots (with error bars) and lines depict the
experimental and theoretical mean values [denoted as $\langle e^{-\beta
W}\rangle$], respectively. Green square dots (with error bars) and lines show
the experimental and theoretical variances [denoted as $\sigma^{2}(e^{-\beta
W})$], respectively. Inverse temperatures were $\beta Z=0.6$ (a) and $\beta
Z=0.8$ (b).

The results of the thermal state preparation and the joint probabilities
$P_{m\tilde{n}}\equiv p^{0}_{m}p^{\tau}_{m\rightarrow n}$ are presented in
Fig. 2. Two different initial states were prepared according to two different
inverse temperatures: $\beta Z=$ 0.6 and 0.8. The measured populations are
shown in left panels of Fig. 2.(a) and (b), respectively. These populations
indicate that inverse temperatures of experimental prepared thermal states are
$\beta Z=0.62(2)$ and $0.78(2)$, which is consistent with our intended state
preparation. We varied the time duration $\tau$ of the work protocol from 50
$\upmu$s to 800 $\upmu$s to investigate how the fluctuation of $e^{-\beta W}$
changes as the protocol approaches the adiabatic limit. To characterize the
adiabaticity of work protocols, the adiabatic parameter was introduced as
$\Gamma=\min_{t\in[0,\tau]}|\langle E_{1}(t)|\partial H(t)/\partial
t|E_{2}(t)\rangle|/(E_{1}(t)-E_{2}(t))^{2}$ [29]. Here $|E_{1}(t)\rangle$,
$|E_{2}(t)\rangle$ are two instantaneous eigenstates of $H(t)$ with
$E_{1}(t)$, $E_{2}(t)$ being the corresponding energies. When $\Gamma$ is much
smaller than 1, the work protocol can be regarded as adiabatic. In our
experiment, $\Gamma$ decreases from 1.014 to 0.063 as $\tau$ increases. Joint
probabilities with different time durations are displayed in right panels of
Fig. 2.(a) and (b). These joint probabilities presented here have been
corrected to consider the influence of the infidelity of the single-shot
readout (see Appendix A). $P_{mn}$ with $m\neq n$ characterizes the
probability of non-adiabatic transitions that happened during the work
protocol. As seen in Fig. 2 (b) and (d), $P_{m\tilde{n}}$ with $m\neq n$
decays to zero as $\tau$ increases, which fulfills the anticipation of the
quantum adiabatic theorem.

Figure 4:  Variances of $e^{-\beta W}$ for normal and STA work protocols. Red
(dark gray) and blue (light gray) bars show $\sigma^{2}(e^{-\beta W})$ of
normal and STA work protocols with the same time duration, respectively. The
minimal variance of $e^{-\beta W}$ can be achieved in fast work protocols
assisted by STA. Inverse temperatures were $\beta Z=0.6$ (a) and $\beta Z=0.8$
(b).

To investigate the PMWF, the average values and fluctuations of $e^{-\beta W}$
are experimentally obtained and displayed in Fig. 3. The mean values of
$e^{-\beta W}$ for different time durations are plotted as blue circle dots in
Fig. 3 (a) and (b) for different values of $\beta$, with lines indicating the
theoretical values of $e^{-\beta\Delta F}$. All dots are consistent with lines
when error bars were considered, verifying the validity of the Jarzynski
equality. The variance of $e^{-\beta W}$ calculated via Eq. (2) is presented
by green square dots in Fig. 3 (a) and (b). The results clearly confirm that
$\sigma^{2}(e^{-\beta W})$ decreases as $\tau$ increases. In particular, it is
observed that the variance of $e^{-\beta W}$ acquires its minimum value when
the work protocol reaches adiabatic, as predicted by the PMWF.

Next we investigate the PMWF associated with fast work protocols. This is of
great interest since an adiabatic process can be impractical in some
situations, e.g., where inevitable decoherence becomes severe. In an adiabatic
process, the dynamics of the $n$th eigenstate follows the adiabatic path [25],
such that
$|\psi_{n}(t)\rangle=e^{-i\int^{t}_{0}dt^{\prime}E_{n}(t^{\prime})-\int^{t}_{0}dt^{\prime}\langle
E_{n}(t^{\prime})|\partial_{t^{\prime}}E_{n}(t^{\prime})\rangle}|E_{n}(t)\rangle$.
Here $|E_{n}(t)\rangle$ is the $n$th instantaneous eigenstate of $H_{0}(t)$.
To realize this evolution rapidly, a CD Hamiltonian is introduced such that
the solution of the Schrödinger equation
$i\partial_{t}|\varphi(t)\rangle=H_{{\rm{CD}}}(t)|\varphi(t)\rangle$ is
exactly $|\psi_{n}(t)\rangle$ given that
$|\varphi(0)\rangle=|E_{n}(0)\rangle$. In our model, the CD Hamiltonian takes
the following form (see Appendix B),

$H_{{\rm{CD}}}(t)=H_{0}(t)+Y(t)S_{y}^{\prime},$ (4)

where $Y(t)=Z\dot{X}(t)/(X^{2}(t)+Z^{2})$ and
$S_{y}^{\prime}=(-i|1\rangle\langle 0|+i|0\rangle\langle 1|)/2$. The variance
of $e^{-\beta W}$, denoted as $\sigma^{2}(e^{-\beta W})$, in such STA work
protocols versus that in bare work protocols, both with inverse temperature
$\beta Z=0.6$, is shown in Fig. 4(a). The $\sigma^{2}(e^{-\beta W})$ in the
STA work protocols are obviously less than those in bare and fast work
protocols ($\tau=50,100,200,300~{}\upmu$s) without STA. When
$\tau=800~{}\upmu$s, the bare work protocol can be regarded as adiabatic. The
obtained variance of $e^{-\beta W}$ in the STA work protocols matches with
that obtained in the bare and slow work protocol. These results confirm that
fast STA work protocols can exhibit the same minimal variance of $e^{-\beta
W}$ as in an adiabatic work protocol. This observation is further enhanced by
considering a lower temperature in Fig. 4(b) with $\beta Z=0.8$. It is also
seen that the lower temperature case yields appreciably larger variance of
$e^{-\beta W}$, indicating that for a fixed precision in estimating $\Delta
F$, more experimental runs are required as temperature decreases to suppress
the estimator bias.

##  IV IV. Conclusion

Quantum effects on the statistics of $e^{-\beta W}$, a central quantity in the
quantum Jarzynski equation, constitute an intriguing topic to understand the
predictive power of the Jarzynski estimator in the quantum domain. With the
successful implementation of two-time measurement of quantum work, we have
conducted a direct experimental investigation of the relationship between the
variance of $e^{-\beta W}$ and adiabaticity of non-equilibrium work protocols.
We experimentally observed that adiabatic processes minimized the variance of
$e^{-\beta W}$, and further studied the minimal variance of $e^{-\beta W}$ in
fast work protocols assisted by shortcuts-to-adiabaticity control. Our
experimental results verify the so-called principle of minimal work
fluctuations and shall stimulate future experimental studies on work
fluctuations. For example, work protocols involving systems with infinite-
dimensional Hilbert space tend to yield a diverging variance of $e^{-\beta W}$
[23, 24]. The convergence or divergence of the variance depends on many system
parameters and can exhibit interesting phase diagrams [24]. The predictive
power of the Jarzynski equality in such systems requires scrutiny.

##  V ACKNOWLEDGMENTS

This work was supported by the National Key R&D Program of China (Grants No.
2018YFA0306600 and No. 2016YFB0501603), the National Natural Science
Foundation of China (Grants No. 12174373), the Chinese Academy of Sciences
(Grants No. XDC07000000, No. GJJSTD2020000), Innovation Program for Quantum
Science and Technology (Grant No. 2021ZD0302200), Anhui Initiative in Quantum
Information Technologies (Grant No. AHY050000) and Hefei Comprehensive
National Science Center. X.R. thanks the Youth Innovation Promotion
Association of Chinese Academy of Sciences for their support. W.L. is funded
by Beijing University of Posts and Telecommunications Innovation Group.

W.C. and W.L. contributed equally to this work.

##  VI APPENDIX A: Correction to joint probabilities

In our experiments, we implemented the two-time measurement protocol to
measure work statistics. In particular, we performed projective measurements
at the start and the end of a switching process. The required projective
measurement was realized by single-shot readout and this procedure is not
perfect due to errors mainly caused by the longitudinal relaxation process of
the nuclear spin.[27]. To mitigate the influence of such errors on our
statistics, we considered a transition matrix (elaborated below) to correct
the bare state-to-state transition probabilities accordingly.

Let us assume that the state before the readout is in a measurement basis
$|j\rangle$. Consider now a non-zero probability $p(i|j)$ that the output of
the single-shot readout ends up with a different basis $|i\rangle$ (hence
errors in our readout). We can now define a transition matrix as
$\bm{T}_{ij}=p(i|j)$. Because our study involves a two-level system consisting
of $|\textnormal{-}1\rangle_{n}$ and $|0\rangle_{n}$, the transition matrix is
a two by two matrix:

$\bm{T}=\begin{bmatrix}p(\textnormal{-}1|\textnormal{-}1)&p(\textnormal{-}1|0)\\\
p(0|\textnormal{-}1)&p(0|0)\end{bmatrix},$ (5)

In our experiment, this transition matrix can be measured and was found to be

$\bm{T}^{\rm{exp}}=\begin{bmatrix}0.980&0.045\\\ 0.020&0.955\end{bmatrix},$
(6)

With this transition matrix that reflects our errors in state readout, we can
correct our state-to-state transition probabilities.

First of all, the initial populations of our true prepared thermal state are
$\bm{p}_{0}=(p_{0}(-1),p_{0}(0))^{T}$, where $p_{0}(-1)$ and $p_{0}(0)$
represent the populations in states $|\textnormal{-}1\rangle_{n}$ and
$|0\rangle_{n}$, respectively. The measured populations are slightly different
from $\bm{p}_{0}$ due to readout errors in the first projective measurement,
resulting in $\bm{p}_{0}^{\rm{exp}}=\bm{T}^{\rm{exp}}\bm{p}_{0}$. Thus, the
corrected initial populations are given by
$\bm{p}_{0}=(\bm{T}^{\rm{exp}})^{-1}\bm{p}_{0}^{\rm{exp}}$. Additionally, the
readout error of the second projective measurement can also affect the
measured state-to-state transition probabilities. Suppose the true state-to-
state transition probabilities due to the population transfer in our work
protocol are given by,

$\bm{P_{c}}=\begin{bmatrix}p_{c}(\textnormal{-}1|\textnormal{-}1)&p_{c}(\textnormal{-}1|0)\\\
p_{c}(0|\textnormal{-}1)&p_{c}(0|0)\end{bmatrix},$ (7)

Then, the measured transtion probabilities become
$\bm{P_{c}}^{\rm{exp}}=\bm{T}^{\rm{exp}}\bm{P}_{c}$. This indicates that the
corrected state-to-state transition probabilities are given by
$\bm{P}_{c}=(\bm{T}^{\rm{exp}})^{-1}\bm{P}_{c}^{\rm{exp}}$. With both the
initial populations and the state-to-state transition probabilities corrected,
the corrected joint probabilities for a sampled initial state making a
transition to a final state are
$(\bm{P})_{ij}=[(\bm{T}^{\rm{exp}})^{-1}\bm{P}_{c}]_{ij}[(\bm{T}^{\rm{exp}})^{-1}\bm{p}_{0}^{\rm{exp}}]_{j}$,
which can be expressed as the following matrix:

$\bm{P}=\begin{bmatrix}p_{c}(\textnormal{-}1|\textnormal{-}1)p_{0}(-1)&p_{c}(\textnormal{-}1|0)p_{0}(0)\\\
p_{c}(0|\textnormal{-}1)p_{0}(-1)&p_{c}(0|0)p_{0}(0)\end{bmatrix}.$ (8)

##  VII APPENDIX B: Counter-diabatic Hamiltonian to realize shortcuts-to-
adiabatic

In our experiment, the chosen Hamiltonian in a bare work protocol takes the
form as follows:

$H_{0}(t)=2\pi(ZS_{z}^{\prime}+X(t)S_{x}^{\prime}).$ (9)

The instantaneous eigenstates of $H_{0}(t)$ is given by
$H_{0}(t)|E_{n}(t)\rangle=E_{n}(t)|E_{n}(t)\rangle$. In the adiabatic
approximation, the dynamics of the $n$th eigenstate follows the adiabatic
path,
$|\psi_{n}(t)\rangle=e^{-i\int^{t}_{0}dt^{\prime}E_{n}(t^{\prime})-\int^{t}_{0}dt^{\prime}\langle
E_{n}(t^{\prime})|\partial_{t^{\prime}}E_{n}(t^{\prime})\rangle}|E_{n}(t)\rangle$[25].
Then the unitary operator is given by
$U(t)=\sum_{n}e^{-i\int^{t}_{0}dt^{\prime}E_{n}(t^{\prime})-\int^{t}_{0}dt^{\prime}\langle
E_{n}(t^{\prime})|\partial_{t^{\prime}}E_{n}(t^{\prime})\rangle}|E_{n}(t)\rangle\langle
E_{n}(0)|$. To realize this unitary evolution, the counter-diabatic
Hamiltonian is given by

$\displaystyle H_{\rm{CD}}(t)$ $\displaystyle=i\partial_{t}U(t)U^{\dagger}(t)$
(10) $\displaystyle=\sum_{n}E_{n}(t)|E_{n}(t)\rangle\langle E_{n}(t)|$
$\displaystyle+i\sum_{n}|\partial_{t}E_{n}(t)\rangle\langle E_{n}(t)|$
$\displaystyle-i\sum_{n}\langle
E_{n}(t)|\partial_{t}E_{n}(t)\rangle|E_{n}(t)\rangle\langle E_{n}(t)|.$

The instantaneous eigenstates of $H_{0}(t)$ are
$|E_{0}(t)\rangle=\cos(\theta(t)/2)|0\rangle+\sin(\theta(t)/2)|1\rangle$ and
$|E_{1}(t)\rangle=\sin(\theta(t)/2)|0\rangle-\cos(\theta(t)/2)|1\rangle$,
where $\theta(t)=\arctan(X(t)/Z)$. It is easy to calculate and find that
$i\sum_{n}|\partial_{t}E_{n}(t)\rangle\langle
E_{n}(t)|=\partial_{t}\theta(t)S_{y}^{\prime}=Z\dot{X}(t)/(X^{2}(t)+Z^{2})S_{y}^{\prime}$
and $\langle E_{n}(t)|\partial_{t}E_{n}(t)\rangle=0$ for $n=0,1$. Thus in our
experiment, the counter-diabatic Hamiltonian takes the following form:

$H_{\rm{CD}}(t)=2\pi(Z(t)S_{z}^{\prime}+X(t)S_{x}^{\prime}+Y(t)S_{y}^{\prime}),$
(11)

where $2\pi Y(t)=Z\dot{X}(t)/(X^{2}(t)+Z^{2})$. Taking the time duration
$\tau=50~{}\upmu$s as an example, Fig. 5 depicts the time dependence of all
specific components of $H_{\rm{CD}}(t)$.

Figure 5:  CD Hamiltonian to realize the STA process with switching time
$\tau=50~{}\upmu$s.

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[b,c]Juan Rojo

# NNPDF4.0 aN3LO PDFs with QED corrections

Andrea Barontini  Niccolo Laurenti

###### Abstract

We review recent progress in the global determination of the collinear parton
distributions (PDFs) of the proton within the NNPDF framework. This progress
includes NNPDF4.0 variants with QED effects and with missing higher order
uncertainties (MHOUs) via the theory covariance matrix formalism, as well as
PDFs based on QCD calculations at approximate N3LO (aN3LO) accuracy. We
present the combination of these theoretical developments resulting into
NNPDF4.0 aN3LO variants accounting for QED corrections and with a photon PDF.
We compare these aN3LO QED PDFs with analogous results obtained by the MSHT
group, and briefly quantify their implications at the level of representative
LHC cross-sections.

## Introduction.

The precise determination of the quark and gluon substructure of the proton,
as encoded by the parton distribution functions (PDFs) [1, 2], is a central
ingredient of the LHC physics program. This sensitivity to the proton
structure is highlighted by recent high-precision measurements of Standard
Model (SM) parameters at the LHC [3], where the achieved precision is limited
by PDF-related uncertainties. Relevant analyses include the measurements of
the strong coupling $\alpha_{s}(m_{Z})$ from the $Z$ $p_{T}$ spectrum from
ATLAS [4], of the effective leptonic mixing angle $\sin^{2}\theta_{\rm
eff}^{\ell}$ from CMS [5, 6], as well as the measurements of the $W$ boson
mass $m_{W}$ from ATLAS [7] and from the CDF experiment at the TeVatron [8].
Similar considerations apply to searches of new physics in the high-energy
tails of differential distribution, as demonstrated with the Drell-Yan
forward-backward asymmetry as a case study [9] as well as by simultaneous PDF
and SMEFT analyses of high-$p_{T}$ data from the LHC [10, 11].

In this contribution we summarise recent progress within the NNPDF global
analysis framework. This progress has focused on extending the NNPDF4.0
determination in three complementary directions: QED corrections, missing
higher order uncertainties (MHOUs) by means of the theory covariance matrix
approach, and PDFs determined at approximate N3LO (aN3LO) accuracy in the QCD
coupling expansion. Here, for the first time, we present the combination of
these theoretical developments into NNPDF4.0 aN3LO sets, with and without
MHOUs, accounting for QED corrections and with a photon PDF. We also quantify
the impact of this combination both at the level of PDFs and of benchmark LHC
cross-sections, and compare them with their counterparts obtained by the MSHT
group. These combined PDF sets are made available via the LHAPDF interface
[12] as well as via the NNPDF Collaboration website.111
https://nnpdf.mi.infn.it/for-users/unpolarized-pdf-sets/

## NNPDF4.0 with QED corrections.

The original NNPDF4.0 determination [13, 14] did not include neither QED
corrections to the DGLAP evolution nor a photon PDF $\gamma(x,Q^{2})$.
Building up on the approach deployed for NNPDF3.1QED [15, 16] and based on the
LuxQED formalism [17], we have recently presented the NNPDF4.0QED analysis
[18]. This variant of NNPDF4.0 is based on the new theory pipeline [19] of the
NNPDF framework and assembled around the EKO [20], YADISM [21], and PineAPPL
[22] programs. The NNPDF4.0QED variants are based on QCD$\otimes$QED DGLAP
evolution accounting for $\mathcal{O}\left(\alpha\right)$,
$\mathcal{O}\left(\alpha\alpha_{s}\right)$ and
$\mathcal{O}\left(\alpha^{2}\right)$ corrections implemented and benchmarked
in EKO. In [18] we investigate the impact of QED effects on NNPDF4.0, compare
our results to other recent PDF sets that include $\gamma(x,Q^{2})$, and
quantify the impact of photon-initiated processes and electroweak corrections
on a variety of LHC processes, finding that they can reach the several-percent
level at the high-energy tail of differential distributions.

## NNPDF4.0 with MHOUs.

While most PDF determinations consider only experimental uncertainties when
constructing the covariance matrix used for the $\chi^{2}$ loss function, the
precision of available and future measurements is such that theoretical
uncertainties entering the cross-section predictions cannot be neglected
anymore. One of the dominant sources of theory uncertainties are those
associated to the missing higher orders (MHOs) in the QCD perturbative
expansion. A first systematic study of MHOUs within a global NNPDF analysis
was carried out in [23, 24], based on the theory covariance matrix formalism,
where MHOU and their correlations are parameterised in terms of nuisance
parameters determined from scale variations. In this way, MHOUs are accounted
for on a comparable footing as experimental errors. This initial study was
restricted to NLO, and confirmed that MHOUs are potentially important given
the accuracy of modern PDF fits. Thanks to the new NNPDF theory pipeline
mentioned above, we have been able to produce variants of the NNPDF4.0 NNLO
global fit which include MHOUs for all input processes [25]. This analysis
finds that MHOU enable an improved overall consistency of the PDF
determination, with an ensuing moderate reduction of PDF uncertainties at
NNLO.

## NNPDF4.0 at aN3LO accuracy.

Calculations of hard-scattering cross-sections at
$\mathcal{O}(\alpha_{s}^{3})$ (relative to the Born term) in the strong
coupling (N3LO) have become available for a rapidly growing set of hadron
collider processes such as inclusive and differential Higgs and Drell-Yan
production, see [26] for an overview, in addition to the deep-inelastic
scattering coefficient functions which have been know for a long time. To
obtain predictions for LHC observables with this same accuracy, partonic
cross-sections need to be combined with PDFs determined also at N3LO. These
N3LO PDFs should be determined by comparing to experimental data theory
predictions computed also at N3LO in the QCD expansion.

With this motivation, in [27] we presented a first NNPDF fit based on
approximate N3LO (aN3LO) calculations. This result required constructing an
approximation to the N3LO splitting functions and anomalous dimensions
reproducing all available information from fixed-order computations and from
small and large-$x$ resummation; extending the FONLL general-mass scheme for
DIS structure functions [28] to N3LO; and accounting both for the remaining
MHOUs and for the incomplete higher order uncertainties (IHOUs) associated to
the derivation of aN3LO expressions for the splitting functions and the
massive DIS coefficient functions. The NNPDF4.0 aN3LO analysis of [27]
demonstrates that, within the NNPDF framework, aN3LO PDFs are consistent with
their NNLO counterparts, that aN3LO corrections improve the description of the
global dataset, and that the perturbative convergence of Higgs and Drell-Yan
inclusive cross-sections is enhanced when consistently combining aN3LO PDFs
with N3LO partonic matrix elements. We also find that, as expected, MHOUs on
the PDFs become smaller with the increase of perturbative order.

Fit ID | Perturbative accuracy | Theory cov. mat.
---|---|---
NNPDF40_nnlo_as_01180_qed_mhou |  NNLO${}_{\rm QCD}\otimes$NLOQED |  MHOU7pt
NNPDF40_an3lo_as_01180_qed |  aN3LO${}_{\rm QCD}\otimes$NLOQED |  IHOU+MHOU3pt
NNPDF40_an3lo_as_01180_qed_mhou |  aN3LO${}_{\rm QCD}\otimes$NLOQED |  IHOU+MHOU7pt
Table 1: The settings of the three combined PDF sets presented in this
contribution.

## Combining aN3LO, MHOUs, and QED corrections in NNPDF4.0.

Given that the three theoretical developments outlined above (aN3LO
corrections, MHOUs, and QED effects) are separately beneficial for the
precision and/or the accuracy of the fitted PDFs, it is only natural to unify
them into a combined global PDF determination, along the lines of the recent
study by the MSHT group [29, 30]. Therefore, benefiting from the flexibility
of the new NNPDF theory pipeline, we present here for the first time new
variants of the NNPDF4.0 global analysis combining these three theory
developments in various ways. Specifically, we have produced a variant of
NNPDF4.0QED NNLO with MHOUs, and then two variants of NNPDF4.0 aN3LO with QED
corrections, either with or without MHOUs, see Table 1. Recall that, as
discussed in [27], in the aN3LO fits we always include 3-point MHOUs in the
hard-scattering cross-sections of hadronic processes to compensate for the
missing N3LO $K$-factors.

Figure 1: The central values of the photon PDF $\gamma(x,Q^{2})$ in the
NNPDF4.0 and MSHT20 QED sets, both for the NNLO and the aN3LO fits. For the
NNPDF4.0QED fit we also include the variant with MHOUs in the covariance
matrix. Results are shown normalised to the central value of NNPDF4.0QED NNLO.

We first assess the impact of MHOUs and of aN3LO corrections in the resulting
photon PDF $\gamma(x,Q^{2})$. Fig. 1 compares the central values of the photon
PDF in the NNPDF4.0 and MSHT20 QED sets, both for the NNLO and the aN3LO fits.
For the NNPDF4.0QED fit, we also include the variant with MHOUs in the
covariance matrix: it is clear that MHOUs have a very small effect on the
photon PDF, unless one goes to very low scales. Concerning the effect of aN3LO
corrections, in NNPDF4.0 they lead to an increase of $\gamma(x,Q^{2})$ as
compared to the NNLO fit by up to 1% for $x\geq 0.01$, and the same trend is
found for their MSHT20 counterparts. For a smaller $x$ values, aN3LO effects
suppress the NNPDF4.0 photon while instead they enhance the MSHT20 photon.

We next compare in Fig. 2 the NNPDF4.0 NNLO and aN3LO fits, with and without
QED corrections. We display the gluon, quark singlet, up valence, and antidown
PDFs at $Q=100$ GeV. The qualitative impact of the QED corrections on the
quark and gluon PDFs is the same in the NNLO and aN3LO fits. The largest
impact is associated to the gluon, where QED effects lead to an overall
decrease of around 1% as a consequence of the photon PDF contributing to the
momentum sum rule. For the quark PDFs, the impact of QED effects in the aN3LO
fit is small, but nevertheless consistent with the results the corresponding
NNLO fits.

Figure 2: Comparison of the NNPDF4.0 NNLO and aN3LO fits, with and without QED
corrections, at $Q=100$ GeV and normalised to the central value of the aN3LO
set. Bands indicate the 68% CL uncertainties.

One can also briefly assess the implications of the combined NNPDF4.0 sets
listed in Table 1 for LHC phenomenology. To this end, in Fig. 3 we revisit the
perturbative convergence analysis of representative inclusive cross-sections
considered in [27] at the LHC with $\sqrt{s}=13.6$ TeV. We display Higgs
production in gluon fusion, in associated production with $Z$ bosons, and
$W^{+}$ production at NLO, NLO, and N3LO. For the N3LO calculations, we also
display predictions based on the QED variants of the NNPDF4.0 and MSHT20 aN3LO
sets. For this comparison, one observes that the qualitative impact of adding
QED corrections to the aN3LO sets is the same in the two determinations,
namely a moderate reduction in the central values of the cross-sections by an
amount well contained within the theory uncertainty band.

Figure 3: The perturbative convergence of Higgs production in gluon fusion and
in associated production with $Z$ bosons, and $W^{+}$ production at the LHC
$\sqrt{s}=13.6$ TeV. For the N3LO calculations, we also display predictions
based on the QED variants of the NNPDF4.0 and MSHT20 aN3LO sets.

## Additional NNPDF progress.

Complementing the developments summarised above, other studies by the NNPDF
Collaboration have either been presented in the past months or will be
completed soon. To begin with, following our study of the total intrinsic
charm (IC) content of the proton [31], we investigated the possible asymmetry
between intrinsic charm and anticharm quarks. The crucial observation is that
there is no perturbative mechanism in QCD that can generate such an asymmetry
between charm and anticharm quarks, and thus its observation represents the
ultimate smoking gun for IC in the proton. A global analysis of hard
scattering data indicates a preference for a non-zero, positive intrinsic
charm asymmetry [32], although its significance is smaller than for the total
component. To either confirm or falsify this IC asymmetry suggested by current
data, in [32] we propose two dedicated observables: $Z+$charm production at
LHCb at the HL-LHC and $F^{c}_{2}$ at the upcoming Electron Ion Collider (EIC)
[33]. In both cases, tagging the charge of the final-state charm quark enables
the calculation of asymmetries sensitive to the valence IC.

Second, Monte Carlo (MC) event generator programs modelling LHC processes
often require PDFs sets with additional constraints as compared to the
baseline fits, such as positive-definite PDFs down to $Q\sim 1$ GeV, smooth
and physical extrapolation down to very small- and large-$x$ as well as small
$Q^{2}$ values, and the inclusion of a photon PDF enabling event generation in
the presence of electroweak corrections. These so-called “MC PDFs” also enter
tunes of non-perturbative QCD physics, such as the underlying event and
multiple parton interactions, necessary for the description of LHC collisions.
With this motivation we are developing LO, NLO, and NNLO variants of NNPDF4.0
with tailored settings for LHC event generators. These NNPDF4.0MC sets make
possible a reliable description of high-$p_{T}$ cross-sections as well as,
subsequently to a dedicated tuning procedure, of processes sensitive to
low-$p_{T}$ QCD phenomena, while minimising the differences with respect to
the baseline PDF sets.

Third, new methodologies are being studied with the aim to stress-test the
NNPDF fitting methodology. These efforts include bypassing neural networks by
parametrizing PDFs in terms of Gaussian Processes followed by Bayesian
inference [34], the development of improved hyperparameter optimisation
procedures based on the features of the complete PDF probability distribution
(and not only of the first moments as in [13]), as well as the application of
closure tests to the determination of PDF-related SM parameters such as the
strong coupling constant.

## Outlook.

The progress along several directions in the NNPDF analysis framework outlined
in this contribution contributes to the development of NNPDF4.1, an updated
global fit to be released in 2025. In addition to accounting for QED, MHOUs,
and aN3LO effects, the NNPDF4.1 determination will include significantly more
data, in particular LHC high-precision measurements based on the complete Run
II luminosity. Thanks to extending the PineAPPL formalism to NNLO fixed-order
calculations, NNPDF4.1 will achieve a global PDF determination without any
reliance on the QCD $K$-factor approximation at NNLO. This upcoming release
will also benefit from methodological improvements such as robust
hyperparameter optimisation methods constructed from estimators based on the
information contained on the full PDF probability distributions. Furthermore,
the implementation of NNPDF4.1 in GPU architectures will result in a
significant reduction of the overall computational overheads involved,
enabling novel PDF studies to be carried out more efficiently.

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