††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# Experimental super-Heisenberg quantum metrology with indefinite gate order

Peng Yin  CAS Key Laboratory of Quantum Information, University of Science and
Technology of China, Hefei 230026, People’s Republic of China  CAS Center For
Excellence in Quantum Information and Quantum Physics, University of Science
and Technology of China, Hefei, Anhui 230026, China  Xiaobin Zhao  QICI
Quantum Information and Computation Initiative, Department of Computer
Science, The University of Hong Kong, Pokfulam Road, Hong Kong 999077, China
Yuxiang Yang  QICI Quantum Information and Computation Initiative, Department
of Computer Science, The University of Hong Kong, Pokfulam Road, Hong Kong
999077, China  Institute for Theoretical Physics, ETH Z$\ddot{u}$rich, 8093
Z$\ddot{u}$rich, Switzerland  Yu Guo  CAS Key Laboratory of Quantum
Information, University of Science and Technology of China, Hefei 230026,
People’s Republic of China  CAS Center For Excellence in Quantum Information
and Quantum Physics, University of Science and Technology of China, Hefei,
Anhui 230026, China  Wen-Hao Zhang  CAS Key Laboratory of Quantum Information,
University of Science and Technology of China, Hefei 230026, People’s Republic
of China  CAS Center For Excellence in Quantum Information and Quantum
Physics, University of Science and Technology of China, Hefei, Anhui 230026,
China  Gong-Chu Li  CAS Key Laboratory of Quantum Information, University of
Science and Technology of China, Hefei 230026, People’s Republic of China  CAS
Center For Excellence in Quantum Information and Quantum Physics, University
of Science and Technology of China, Hefei, Anhui 230026, China  Yong-Jian Han
CAS Key Laboratory of Quantum Information, University of Science and
Technology of China, Hefei 230026, People’s Republic of China  CAS Center For
Excellence in Quantum Information and Quantum Physics, University of Science
and Technology of China, Hefei, Anhui 230026, China  Bi-Heng Liu  CAS Key
Laboratory of Quantum Information, University of Science and Technology of
China, Hefei 230026, People’s Republic of China  CAS Center For Excellence in
Quantum Information and Quantum Physics, University of Science and Technology
of China, Hefei, Anhui 230026, China  Jin-Shi Xu  CAS Key Laboratory of
Quantum Information, University of Science and Technology of China, Hefei
230026, People’s Republic of China  CAS Center For Excellence in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei, Anhui 230026, China  Giulio Chiribella<EMAIL_ADDRESS>QICI
Quantum Information and Computation Initiative, Department of Computer
Science, The University of Hong Kong, Pokfulam Road, Hong Kong 999077, China
Department of Computer Science, University of Oxford, Parks Road, Oxford OX1
3QD, United Kingdom  Perimeter Institute for Theoretical Physics, Caroline
Street, Waterloo, Ontario N2L 2Y5, Canada  Geng Chen<EMAIL_ADDRESS>CAS
Key Laboratory of Quantum Information, University of Science and Technology of
China, Hefei 230026, People’s Republic of China  CAS Center For Excellence in
Quantum Information and Quantum Physics, University of Science and Technology
of China, Hefei, Anhui 230026, China  Chuan-Feng Li<EMAIL_ADDRESS>CAS Key
Laboratory of Quantum Information, University of Science and Technology of
China, Hefei 230026, People’s Republic of China  CAS Center For Excellence in
Quantum Information and Quantum Physics, University of Science and Technology
of China, Hefei, Anhui 230026, China  Guang-Can Guo  CAS Key Laboratory of
Quantum Information, University of Science and Technology of China, Hefei
230026, People’s Republic of China  CAS Center For Excellence in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei, Anhui 230026, China

###### Abstract

The precision of quantum metrology is widely believed to be restricted by the
Heisenberg limit, corresponding to a root mean square error that is inversely
proportional to the number of independent processes probed in an experiment,
$N$. In the past, some proposals have challenged this belief, for example
using non-linear interactions among the probes. However, these proposals
turned out to still obey the Heisenberg limit with respect to other relevant
resources, such as the total energy of the probes. Here, we present a photonic
implementation of a quantum metrology protocol surpassing the Heisenberg limit
by probing two groups of independent processes in a superposition of two
alternative causal orders. Each process creates a phase space displacement,
and our setup is able to estimate a geometric phase associated to two sets of
$N$ displacements with an error that falls quadratically with $N$. Our results
only require a single-photon probe with an initial energy that is independent
of $N$. Using a superposition of causal orders outperforms every setup where
the displacements are probed in a definite order. Our experiment features the
demonstration of indefinite causal order in a continuous-variable system, and
opens up the experimental investigation of quantum metrology setups boosted by
indefinite causal order.

††preprint: APS/123-QED

##  I Introduction

Quantum metrology giovannetti2004quantum ; giovannetti2006quantum ;
giovannetti2011advances  is one of the most promising near-term quantum
technologies. Its core promise is that quantum resources, such as entanglement
and coherence, can improve the precision of measurements beyond the limits of
classical setups. The prime example is the reduction of the root mean square
error (RMSE) from the standard quantum limit (SQL) $N^{-1/2}$ to the
Heisenberg limit $N^{-1}$ for the estimation of a parameter from $N$
independent processes, or from $N$ independent applications of the same
process. The Heisenberg scaling has been demonstrated in a variety of setups,
e.g., by preparing $N$ entangled probes bollinger1996optimal ;
walther2004broglie ; afek2010high , or by letting a single probe evolve
sequentially through the $N$ processes under consideration chen2018heisenberg
; chen2018achieving . Generally, the Heisenberg limit $N^{-1}$ is broadly
regarded as the ultimate quantum limit for independent processes Anisimov .
Scalings beyond the Heisenberg limit are known in the presence of inefficient
detectors beltran2005breaking , nonlinear interactions among $N$ particles
boixo2007generalized ; roy2008exponentially ; choi2008bose ; zwierz2010general
; napolitano2011interaction , or non-Markovian correlations between $N$
processes yang2019memory . However, such enhancements typically do not apply
to the basic scenario of $N$ independent processes. Moreover, some of these
enhancements have been subject to dispute zwierz2010general ;
zwierz2012ultimate ; berry2012optimal ; hall2012does . For example, setups
based on nonlinear interactions have been found to obey the Heisenberg limit
in terms of relevant resources such as the energy of the probes hall2012does ;
berry2012optimal , and universal resource count zwierz2010general ;
zwierz2012ultimate .

Recently, research in the foundations of quantum mechanics identified a new
quantum resource that could be exploited in quantum metrology, potentially
breaking the $N^{-1}$ barrier of the Heisenberg limit zhao2020quantum . The
resource is the ability to combine quantum processes in a superposition of
alternative orders, using a primitive known as the quantum SWITCH
chiribella2009beyond ; chiribella2013quantum . The quantum SWITCH is a
hypothetical machine that operates on physical processes, taking two processes
as input and letting them act in an order controlled by the state of a quantum
particle. Originally motivated by foundational questions about causality in
quantum mechanics hardy2007towards ; oreshkov2012quantum , the research on the
applications of the quantum SWITCH has revealed a number of information-
processing advantages in various tasks, including quantum channel
discrimination chiribella2012perfect ; araujo2014computational , distributed
computation guerin2016exponential , quantum communication ebler2018enhanced
and quantum thermodynamics felce2020quantum . The application to quantum
metrology was recently explored in a series of theoretical works
frey2019indefinite ; mukhopadhyay2018superposition ; zhao2020quantum ;
chapeau2021noisy . In particular, Ref. zhao2020quantum  identified a scenario
where the ability to coherently control the order of $2N$ independent
processes on an infinite dimensional system enables the estimation of a
geometric phase with a super-Heisenberg scaling $N^{-2}$. This enhancement was
shown to be due to the indefinite order in which the unknown processes are
probed: any setup that probes the same $2N$ processes in a definite order (or
in a classical mixture of definite orders) while using a finite amount of
energy will necessarily be subject to the Heisenberg scaling $N^{-1}$.

While the theory indicates that the ability to coherently control the order is
a resource for quantum metrology, the benefits of this resource had not been
observed experimentally until now. The main challenge is that the existing
experimental setups procopio2015experimental ; rubino2017experimental ;
goswami2018indefinite ; wei2019experimental ; goswami2020increasing ;
taddei2021computational ; guo2020experimental  inspired by the quantum SWITCH
control operations on finite dimensional systems, while the setup achieving
super-Heisenberg scaling requires coherent control over the order of
operations in an infinite dimensional, continuous-variable system. Such
control needs to be particularly accurate in order to demonstrate demonstrate
high metrological precision promised by the theory in Ref. zhao2020quantum ,
and achieving high accuracy while controlling the order of continuous-variable
operations is far from straightforward.

Figure 1: Geometric phase associated to two sets of phase space displacements.
The displacements take place in the plane associated to two canonically
conjugated variables $x$ and $p$. The first set consists of displacements
$\\{\alpha_{1},\dots,\alpha_{N}\\}$ (in blue), while the second consists of
displacements $\\{\beta_{1},\dots,\beta_{N}\\}$ (in grey). When the two sets
of displacements are performed in two alternative orders, they give rise to
two paths that enclose an area $\mathcal{A}$, corresponding to the geometric
phase associated to a continuous-variable quantum system subject to two
sequences of phase space displacements in two alternative orders.

Here, we initiate the application of indefinite gate order in experimental
quantum metrology, by building a photonic quantum SWITCH that accurately
controls the order of operations on a continuous-variable system. The setup is
used to estimate the geometric phase aharonov1959significance ;
berry1984quantal  associated to two groups of $N$ phase space displacements,
as illustrated in Fig. 1. By combining the two groups of displacements in a
coherent superposition of orders, we experimentally observe an $N^{-2}$
scaling of the root mean square error, which beats the scaling achievable by
every setup where the given displacements are arranged in a definite gate
order. Our setup could be used to perform precision tests of the canonical
commutation relations, and to detect tiny deviations from the standard quantum
values, such as those envisaged in certain theories of quantum gravity
garay1995quantum ; szabo2003quantum ; pikovski2012probing ; kempf1995hilbert .
Overall, this proof-of-principle demonstration opens up the experimental
exploration of new schemes for quantum metrology boosted by coherent control
over the order.

Figure 2:  Experimental setup for coherently controlling gate orders in a
continuous-variable photonic system. The experimental setup consists of three
parts: (1) a polarization controller of heralded signal photons, (2) a Mach-
Zehnder (M-Z) interferometer that provides control on the order, and (3) a
polarization analyser (PA) to perform projective measurements on the control
qubit. The signal and idler photon pairs are generated through a spontaneous
parametric down conversion process by pumping Beta Barium Borate crystal (not
shown in the figure) with 390$nm$ light, and the signal photons are
initialized to $\frac{1}{\sqrt{2}}\left(|H\rangle+|V\rangle\right)$
($|H\rangle$ and $|V\rangle$ denote horizontal and vertical polarization) by
PBS1 and HWP0. This coherent superposition in polarization is transformed into
the coherent superposition between the two arms (denoted as yellow and blue
arms) of the M-Z interferometer, which is composed of beam displacers (BD1 and
BD2) and the optical components in between. The $x$ and $p$ displacements are
caused by the ${\rm MgF}_{2}$ plates and the wedge pairs, respectively. The
two arms constitute the control qubit and the transmission of a single photon
in a superposition of the two arms induces a a coherent superposition of two
alternative orders of displacements on the transverse modes. At the end of the
M-Z interferometer, the two arms are recombined by BD2 and transformed to a
polarization qubit, which is projected to
$|\pm\rangle:=\frac{1}{\sqrt{2}}\left(|H\rangle\pm|V\rangle\right)$ by the PA
(consisting of SHWP3 and PBS2). The two exits of PBS are connected to two
single photon detectors (SPDs), and the corresponding clicks coincident with
those generated by the idler photons are recorded by ID800.

##  II Results

Our experiment is conducted on a single-photon quantum system, with a discrete
variable degree of freedom corresponding to the photon’s polarization, and a
continuous variable degree of freedom associated to the photon’s transverse
spatial modes. We consider a scenario where the continuous variable degree of
freedom undergoes two sets of phase space displacements, as shown in Fig. 1,
with set $S_{1}$ consisting of displacements $\\{D_{\alpha_{j}}\\}_{j=1}^{N}$
and set $S_{2}$ consisting of displacements $\\{D_{\beta_{k}}\\}_{k=1}^{N}$,
where $\\{\alpha_{j}\\}$ and $\\{\beta_{k}\\}$ are arbitrary complex numbers
and $D(\alpha)=\exp[\alpha\,a^{\dagger}-\alpha^{*}\,a]$, $\alpha^{*}$ denoting
the complex conjugate of $\alpha$, and $a$ ($a^{{\dagger}}$) denoting the
annihilation (creation) operator. This scenario extends the settings of Ref.
zhao2020quantum  from displacements in two orthogonal directions to arbitrary
displacements.

The displacements in $S_{1}$ and $S_{2}$ determine two alternative paths in
phase space: one path generated by applying first the displacements in $S_{1}$
and then those in $S_{2}$, and the other path generated by applying first the
displacements in $S_{2}$ and then those in $S_{1}$. Together, the two paths
identify a loop, which we assume to be simple, i.e. to have no self-
intersections. The loop encircles an area $\mathcal{A}=2\,N^{2}\,\left|{\sf
Im}(\overline{\alpha}^{*}\overline{\beta})\right|$, where $\sf Im$ denotes the
imaginary part, and $\overline{\alpha}$ and $\overline{\beta}$ are the average
displacements $\overline{\alpha}=\sum_{j}\alpha_{j}/N$ and
$\overline{\beta}=\sum_{k}\,\beta_{k}/N$, respectively (see Supplementary Note
1 for the derivation). The associated phase factor $e^{i\mathcal{A}}$ is an
example of a geometric phase aharonov1959significance ; berry1984quantal .

We now provide an experimental setup for measuring the geometric phase
approaching super-Heisenberg limit precision. Specifically, our setup provides
an accurate estimate of the regularized area $A:=\mathcal{A}/N^{2}$. For
simplicity, we will focus on the case where set $S_{1}$ consists of position
displacements $D_{x_{j}}:=e^{-ix_{j}P}$, while set $S_{2}$ consists of
momentum displacements $D_{p_{k}}:=e^{ip_{k}X}$, with
$X=(a+a^{\dagger})/\sqrt{2}$ and $P=-i(a-a^{\dagger})/\sqrt{2}$. This case
amounts to setting $\alpha_{j}=\sqrt{2}\,x_{j}$, $\beta_{k}=i\sqrt{2}\,p_{k}$,
and $\hbar=1$. With these settings, the task is to estimate the product
$A=\bar{x}\bar{p}$, where $\overline{x}$ and $\overline{p}$ are the average
$x$-displacement and the average $p$-displacement.

It is worth stressing that the theory demonstrated in this paper is valid for
arbitrary displacements, not only for displacements generated by two
canonically conjugate observables. This point is important for experiments,
because it guarantees robustness with respect to fluctuations of the
generators of the displacements, which, in general are not exactly canonically
conjugate.

To demonstrate the precise estimation of the area $A=\bar{x}\bar{p}$, one
crucial prerequisite is that both the $x$ and $p$ displacements should have
small magnitude, so that the total phase does not exceed $2\pi$. Our setup is
shown in Fig. 2 (see also Methods for more discussion). A major challenge in
the experiment is that generating small $x$ displacements requires thin
birefringent plates, which are hard to manufacture and handle. To reach an
appropriate balance, we utilized customized 2 mm thick ${\rm MgF}_{2}$ plates,
which generate displacements of $x_{j}\approx 18.6$ $\mu m$ for horizontally
polarized ($|H\rangle$) photons. The effects of indefinite causal order are
reproduced by controlling the application of the $x$ displacements, switching
them on-and-off by rotating the polarization of photons. This approach also
enables an initial calibration of the optical setup that eliminates stochastic
phases before execution of our measurements (see the Methods for more
details).

For the $p$ displacements, the experimental implementation is trickier. The
$p$ displacement $e^{ip_{j}X}$ is equivalent to a linear phase modulation
$e^{ip_{j}x}$ for each eigenvector of the operator $X$. This phase modulation
has to be combined with the phase factor $e^{ikz}$ associated to the
propagation of the photon along the $z$ axis (in the paraxial approximation),
yielding a total phase factor $e^{ip_{j}x+ikz}$. This observation shows that a
$p$ displacement is equivalent to a deflection of the incident photon, which
can be achieved by introducing a wedge pair. The usage of wedge pairs causes
gradually varying deflections without any pixelation, enabling the accurate
implementation of small $p$ displacements, calculated as $p_{j}\approx
2\pi\theta_{j}^{\rm eff}/\lambda$ aguilar2020robust , where $\theta^{\rm
eff}_{j}\approx 2.8\times 10^{-4}$ ${\rm rad}$ is the deflection angle
associated to the wedge pair. With these settings, the geometric phases in our
experiment are always smaller than 2$\pi$, which is necessary to observe the
true scaling of precision. The $x$ ($p$) displacements take slightly different
values with fluctuations no more than $\pm 5\%$ around a reference value, as
per factory specifications.

As shown in Fig. 2, The order of the displacements is controlled by the
polarization of a single photon, which is initialized in the state
$(|H\rangle+|V\rangle)/\sqrt{2}$, where $|H\rangle$ and $|V\rangle$ denote the
states of horizontal and vertical polarization. The preparation of control
qubit is achieved by a polarization controller for heralded single photons,
implemented by a polarizing beam splitter (PBS1) and a half-wave plate (HWP0).
The superposition of polarizations then induces a superposition of paths
through a M-Z interferometer, with the two paths traversing the two groups of
displacements in opposite orders. Finally, at the output of the
interferometer, the photon polarization is measured in the basis
$\\{|+\rangle,|-\rangle\\}$, with
$|\pm\rangle=(|H\rangle\pm|V\rangle)/\sqrt{2}$. Then a polarization analyser
(PA) consisting of a phase compensation plate (PCP), a small half-wave plate
(SHWP3) and a polarizing beam splitter (PBS2) is used to implement projective
measurement on the polarization degree of freedom.

Our setup generates a coherent superposition of different orders of
continuous-variable gates. Precisely, the setup reproduces the overall action
of the quantum SWITCH, making the photon experience the $x$ and $p$
displacements in a coherent superposition of two alternative orders. The state
of the photon right before the measurement is the superposition
$\frac{1}{\sqrt{2}}\,(|H\rangle\otimes D_{p_{N}}\dots D_{p_{1}}D_{x_{N}}\dots
D_{x_{1}}|\psi\rangle+e^{i\phi_{0}}|V\rangle\otimes D_{x_{N}}\dots
D_{x_{1}}D_{p_{N}}\dots D_{p_{1}}|\psi\rangle)$, where $|\psi\rangle$ is the
state of the transverse modes of a Gaussian beam and is set to the ground
state of a harmonic oscillator, and $\phi_{0}$ is a constant offset for
varying $N$, caused by the unevenness of the four full-sized HWPs (HWP1, HWP2,
HWP3, and HWP4) (see Methods for more details).

Figure 3:  $P_{-}$ for varying $N$. The black dots are the experimentally
measured data of $P_{-}$ for varying $N$. The red lines is the fitting of
these dots with the fitting function $\frac{1}{2}(1-\cos(N^{2}A+\phi_{0}))$.
In each trial of experiment, $P_{-}$ is calculated from the results of
projective measurement on $\nu=1000$ single photons, and the data points
denote the mean values ($\pm$ RMSE) of 30 trials of experiment. As we can see,
all the data points fall at the fitting curve within the error-bar.

When the photon’s polarization is measured in the basis
$\\{|+\rangle,|-\rangle\\}$, the probabilities of the possible outcomes are

$\displaystyle P_{\pm}=\frac{1}{2}\left(1\pm\cos(N^{2}A+\phi_{0})\right)\,.$
(1)

Crucially, the phase exhibits a quadratic dependence on the number of
displacements, leading to a super-Heisenberg sensitivity to small changes of
$A$. In the experiment, we estimated the probability $P_{-}$ for
$N=0,1,2,\cdots,8$, as shown in Fig. 3. The data is in agreement with Eq. 3
with parameters $A=0.042$ ${\rm rad}$ and $\phi_{0}=0.307$ ${\rm rad}$.

The statistics is obtained by repeating the experiment for $\nu$ times. From
the obtained data, the phase $A$ is then estimated with a maximum likelihood
estimate $\hat{A}:=\arg\max_{A}\log p(m_{1},\dots,m_{\nu}|A)$, where
$m_{j}\in\\{+,-\\}$ is the $j$-th measurement outcome, and
$p(m_{1},\dots,m_{\nu}|A)$ is the probability of obtaining the measurement
outcomes $\\{m_{1},\dots,m_{\nu}\\}$ conditioned on the parameter being $A$.
Theoretically, the RMSE of this parameter is zhao2020quantum

$\displaystyle\delta A=\frac{1}{\sqrt{\nu}N^{2}}.$ (2)

Figure 4: Super-Heisenberg scaling of the RMSE of $A$. The black data points
are the experimentally measured RMSE of $A$ for varying $N$. The black line is
the fitting of the data with fitting function $\frac{1}{cN^{2}}$ where
$c\approx 30.65$ is the parameter for fitting. The red line represents the
theoretical super-Heisenberg limit $\delta A=\frac{1}{\sqrt{\nu}N^{2}}$ with
$\nu=1000$. The dashed blue line is the theoretical lower bound on the RMSE
for an experimentally feasible strategy with fixed order, analyzed in
Supplementary Information.

To indicate that indefinite causal order can boost the precision of quantum
metrology, we use $\nu=1000$ counts to calculate $P_{-}$ in each trial of
measurement, and the precision is defined as the RMSE of the results from 30
trials of measurement. The RMSE for different $N$ are plotted in Fig. 4, and
it is shown that $\delta A$ decreases with super-Heisenberg scaling $\delta
A\propto\frac{1}{N^{2}}$, as predicted by Ref. zhao2020quantum . Furthermore,
the quadratically fitted line (color black) approaches the theoretical super-
Heisenberg limit $\delta A=1/(\sqrt{\nu}N^{2})$ (color red). The super-
Heisenberg limit approached in the experiment is in stark contrast with the
ultimate limit achievable by probing the displacements in a fixed order, using
probes with the same amount of energy as used in our experiment: as shown in
Ref. zhao2020quantum  every setup with fixed order and minimum probe energy
will necessarily lead to an RMSE with Heisenberg limited scaling $\propto
1/N$. In Fig. 4, we also compare the precision achieved in our experiment with
the precision of an alternative, experimentally feasible strategy that
achieves Heisenberg scaling by probing the displacements in a fixed order. The
comparison shows that the quantum SWITCH not only offers an enhanced scaling,
but also an absolute improvement in precision in the applied range of $N$.

Figure 5: The scaling of the total phase against $N$. (a) Total phase
$\Phi_{\rm tot}=N^{2}A+\phi_{0}$ derived from Fig. 3. The black squares are
the experimental data and the red line is the quadratic fitting. (b) Total
phase $\Phi_{\rm tot}$ for varying number of $x$-displacements when only one
$p$-displacement is applied. The black squares are the experimental data and
the red line is the linear fitting. Data are presented as the mean values
($\pm$ RMSE) of 30 trials of experiment, and in each trial we use $\nu$=1000
counts to estimate the total phase.

To further investigate the origin of the super-Heisenberg scaling in our
experiment, we implemented another experimental test by keeping only one fixed
$p$ displacement while merely varying the number of $x$ displacements. To be
specific, we insert only one wedge pair into the setup and keep
$\theta_{single}^{\rm eff}$ fixed, and then we change the number of ${\rm
MgF}_{2}$ plates $N_{x}$ from 0 to 8. The corresponding total phase is linear
in $N_{x}$, and therefore satisfies the Heisenberg limit, as shown in Fig.
5(b). In contrast, the total phase obtained with the original indefinite gate
order protocol is quadratic in $N$, which yields a RMSE approaching super-
Heisenberg limit, as shown in Fig. 5(a). From this point of view, solely
increasing the $x$ ($p$) displacements cannot constitute the effective
resources of an indefinite gate order protocol, and the resulted RMSE is
Heisenberg-limited.

##  III Discussion

Our experimental setup demonstrates a precision approaching super-Heisenberg
scaling, which is impossible to achieve with any scheme that probes the same
set of gates in a fixed order using the same amount of energy as our scheme.
It is interesting to compare this result with previous investigations on
nonlinear quantum metrology boixo2007generalized ; roy2008exponentially ;
choi2008bose ; zwierz2010general ; napolitano2011interaction . A typical
scenario is that of a probe consisting of $N$ interacting particles, with a
total energy growing nonlinearly with $N$. In this scenario, it has been
observed that the root mean square error for various parameters can scale more
favourably than the “Heisenberg scaling” $N^{-1}$. However, the scaling of the
precision in terms of the energy of the probes still satisfies the Heisenberg
limit. In contrast, our experiment demonstrates super-Heisenberg scaling
$N^{-2}$ with a single photon probe, whose initial energy is independent of
$N$. A theoretical work advocates that by defining the expectation value of
the generator as the universal resource count, any unitary evolution in
definite causal order is necessarily Heisenberg-limited zwierz2012ultimate .
In our scheme, the quadratic scaling advantage is accessed through a
nontrivial quantum loop shown in Fig.1, which combines two distinct unitary
operations in a superposition of alternative orders chiribella2009beyond ;
chiribella2013quantum . In this sense, it is an interesting open question
whether the universal resource count, which is initially formalized for fixed-
order unitary evolution, can be generalized to schemes with indefinite causal
order. Moreover, we remark that our results do not rely on the assumption of a
prior probability distribution over the unknown displacements: the only
assumption on the displacements is that they are contained in a finite
interval, which is independent of $N$ zhao2020quantum .

For practical implementation, the duration of scaling is inevitably restricted
by some realistic factors. In our experiment, one major factor undermining the
scaling is the photon loss rafal2012nc , and $N$ pairs of displacement could
reduce the number of measurements from $\nu$ to $\eta^{N}\nu$ and then the
RMSE scales with $1/\sqrt{\eta^{N}\nu}N^{2}$ ( 1-$\eta\approx$0.004 is the
loss due to the optical elements to implement each pair of $x$ and $p$
displacements). With such a level of photon loss, the observed super-
Heisenberg scaling in Fig. 4 could be maintained even to a modest value of
$N$, e.g., $N=100$.

Our setup achieves the scaling $N^{-2}$ using a superposition of two
alternative orders. A natural question is whether this scaling could be
improved with more elaborated setups involving more than two orders. For the
estimation problem considered in this work, the answer is negative: any setup
using a superposition of alternative orders and a finite amount of energy in
the probes is necessarily bound to the $N^{-2}$ scaling, no matter how many
orders are superposed zhao2020quantum . On the other hand, it is in principle
possible that the superposition of multiple causal orders may offer scaling
advantages in other quantum metrology problems, and general strategies with
indefinite causal order may offer scalings beyond $N^{-2}$. Determining
whether these scenarios exist remains as an interesting direction of future
research.

Another interesting open question is whether indefinite causal order could be
used to achieve super-Heisenberg scaling for discrete-variable systems. For
example, one could consider a setting where multiple non-commuting rotations
are performed on a quantum spin, and the goal is to estimate parameters
associated to the amount of non-commutativity of the given rotations. While it
is natural to expect an advantage of indefinite causal order in this problem,
the analysis is considerably more complicated, due to the non-Abelian nature
of the rotation group.

Finally, an important direction of future research is to bridge the gap
between proof-of-principle demonstrations and more practical applications. An
intriguing possibility is that schemes like the one presented in this work, or
like its finite dimensional versions may have applications in quantum
magnetometry or in the estimation of geometric quantities associated to
electromagnetic and gravitational fields.

Acknowledgments . This work was supported by the Innovation Program for
Quantum Science and Technology (Nos. 2021ZD0301200), National Natural Science
Foundation of China (Grant Nos. 12122410, 92065107, 11874344,
61835004,11821404), Anhui Initiative in Quantum Information Technologies
(AHY060300), the Fundamental Research Funds for the Central Universities
(Grant No. WK2030000038, WK2470000034), the Hong Kong Research Grant Council
through grant 17300918 and though the Senior Research Fellowship Scheme
SRFS2021-7S02, the Croucher Foundation, and the John Templeton Foundation
through grant 61466, The Quantum Information Structure of Spacetime (qiss.fr).
Research at the Perimeter Institute is supported by the Government of Canada
through the Department of Innovation, Science and Economic Development Canada
and by the Province of Ontario through the Ministry of Research, Innovation
and Science. The opinions expressed in this publication are those of the
authors and do not necessarily reflect the views of the John Templeton
Foundation.

Author Contributions Peng Yin, Xiaobin Zhao, Yuxiang Yang and Giulio
Chiribella proposed the framework of the theory and made the calculations.
Peng Yin, Wen-Hao Zhang, Gong-Chu Li and Geng Chen planned and designed the
experiment. Peng Yin and Geng Chen carried out the experiment assisted by Bi-
Heng Liu, Jinshi Xu, Yongjian Han and Yu Guo. Peng Yin, Geng Chen, Xiaobin
Zhao Yuxiang Yang and Giulio Chiribella analyzed the experimental results and
wrote the manuscript. Guangcan Guo, Chuanfeng Li and Giulio Chiribella
supervised the project. All authors discussed the experimental procedures and
results.

Competing interests: The authors declare no competing financial interests.

##  IV Methods

Implementation of $x$ and $p$ displacements. One of the key challenges in our
experiment is to accurately realize displacement operations, and coherently
control their order. As shown in Fig. 2, the $x$ displacement is generated by
the birefringence of a 2 $mm$-thick ${\rm MgF}_{2}$ plate whose optical axis
is at a $45^{\circ}$ angle with $x$-axis in the $x$-$z$ plane (the $z$-axis is
defined as the propagating direction of photons and $x$-axis is defined as the
direction of horizontal polarization). This configuration results into a
reference displacement of approximately $18.6$ $\mu m$ along the $x$-axis when
the photon is horizontally polarized ($|H\rangle$). The $p$ displacement
corresponds to a linear phase modulation of the photon along the
$x$-direction, and is thus equivalent to a small deflection angle in the
$x$-$z$ plane (under the paraxial approximation). It is realized by a pair of
wedges with the same wedge angle of $1^{\circ}$. Here the usage of the wedge
pair enables a precise control of the $p$ displacement by mechanically
rotating the stages on which the wedges are mounted. By rotating one of the
two wedges, we introduce a deflection angle $\theta^{\rm eff}\approx 2.8\times
10^{-4}$ ${\rm rad}(0.016^{\circ})$ of the photon, which in turn induces a
linear phase modulation $e^{ipX}$ with $p\approx 2\pi\theta^{\rm
eff}/\lambda$. This special design of the interferometer guarantees that the
two paths maintain the same signs (directions) for both $x$ and $p$
displacements, which is necessary to produce the geometric phase as required.

The realization of the $p$ displacement involves a few additional experimental
issues that are crucial to a successful demonstration of super-Heisenberg
metrology. To ensure that the paraxial approximation is well justified, we
calibrated the beams (the yellow arm between HWP3 and HWP4, and the arms
between the wedge pairs in Fig. 2) so that they strictly propagates along $z$
axis before the insertion of $MgF_{2}$ crystal and wedge pairs. The
calibration needs to be extremely precise, and we achieved this goal by
adopting a combination of beam displacers and CCD camera to measure the
distances between the beam under calibration and the reference beam (the blue
arm after BD1 in Fig. 2, which is chosen as the $z$ axis) $d_{1}$ and $d_{2}$
at two places with a distance $\sim 4$ m. When the beam under calibration is
propagating along $z$ axis, we should have $d_{1}=d_{2}$. The CCD camera can
measure the distances $d_{1}$ and $d_{2}$ with a precision of $\sim 5$ $\mu$m,
so that the alignment between the beams and $z$ axis can be up to the
precision $\sim\frac{5\mu m}{4m}=1.25\times 10^{-6}$ rad, which is two orders
of magnitude lower than the deflection angle corresponding to the $p$
displacement, thus reaching sufficient accuracy for the requirements of our
experiment.

Initialization of the optical setting. To demonstrate the advantage of
coherent control on the order, an initialization of the M-Z interferometer is
performed before the execution of quantum SWITCH. Here, the coherence of the
control qubit is transformed back and forth between the path degree of freedom
(yellow and blue arms in Fig. 2) and polarization degree of freedom
(horizontal and vertical polarization). The main purpose of the initialization
is to eliminate the initial phase between the two arms (denoted as yellow and
blue ) of the M-Z interferometer in Fig. 2, as such phase changes
stochastically with the number of optical elements inside the M-Z
interferometer. In the initialization process, we firstly prepare the single
photons into $|+\rangle:=\frac{1}{\sqrt{2}}\left(|H\rangle+|V\rangle\right)$
by orienting HWP0 to $22.5^{\circ}$, and then we use BD1 to coherently split
$|H\rangle$ and $|V\rangle$ components into two arms of the M-Z interferometer
(denoted by yellow and blue lines), which serves as the control qubit. With
the two small HWPs (SHWP1 and SHWP2) orienting at $45^{\circ}$ , and the four
full-sized HWPs (HWP1, HWP2, HWP3, and HWP4) oriented at $0^{\circ}$, the
photons on the two arms are both $|V\rangle$ when passing the $MgF_{2}$ plates
and no $x$ displacement occurs thereby. With these settings, the two arms are
recombined by BD2 without creating a geometric phase. By tilting the PCP, we
can achieve a vanishing projection probability to
$|-\rangle=\frac{1}{\sqrt{2}}\left(|H\rangle-|V\rangle\right)$ at the PA, the
initial phase difference between the two arms is thus eliminated by the PCP.
For a given $N$, to create a geometric phase, we only need to orient the four
full-sized HWPs (HWP1, HWP2, HWP3, and HWP4) to $45^{\circ}$; and thus, the
polarization of the photon is the same as that in the initialization step for
all the elements in the setup except $MgF_{2}$ plates. To be specific, HWP1
and HWP3 tune the polarization of photons from $|V\rangle$ to $|H\rangle$,
which implements the $x$ displacement due to the birefringence of $MgF_{2}$
plates, HWP2 and HWP4 keep the polarization of photons unchanged for all other
optical elements except the $MgF_{2}$ plates. From Fig. 2, we can see that the
photon first experiences the $p$ displacements and then the $x$ displacements
for the yellow arm, while it experiences the opposite order for the blue arm.
Thanks to this mechanism, the polarization coherently controls the order of
the $x$ and $p$ displacements, resulting in a superposition of alternative
orders whenever the photon is prepared in a superposition of the $|H\rangle$
and $|V\rangle$ states. To probe the scaling of the precision of the geometric
phase with the number of displacements, the above procedure (initialization
and generation of the geometric phase) is repeated for different values of
$N$.

##  V References

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Data Availability: The data that support the findings of this study are
presented in the article and the Supplementary Information, and are available
from the corresponding authors upon reasonable request. Source data are
provided with this paper.

##  VI Supplementary Information

###  VI.1 Estimation of the phase space area associated to arbitrary
displacements

Here we show that the regularized phase space area associated to two set of
displacements can be estimated with super-Heisenberg scaling using the quantum
SWITCH. This result extends the theory of Ref. zhao2020quantum  from
displacements in two orthogonal directions to arbitrary displacements.

Let $(z_{1},\dots,z_{S})$ be a sequence of $S$ complex numbers, each
representing a displacement in a two-dimensional plane, with Cartesian axes
$x$ and $y$ associated to the real and imaginary parts, respectively. The
sequence identifies a path $(z_{1},\dots,z_{S})$, and the signed area
delimited by this path and by the $y$-axis is

$\displaystyle{\cal
A}_{z_{1},\dots,z_{S}}=\frac{1}{2}\sum_{j=1}^{S}\sum_{l\geq j}{\sf
Im}[z_{j}]\,{\sf Re}[z_{l}]\,.$ (3)

In particular, consider the case of a sequence $(z_{1},\dots,z_{2N})$, with
$z_{j}:=\alpha_{j}$, $j\in\\{1,\dots,N\\}$ and $z_{N+j}:=\beta_{j}$,
$j\in\\{1,\dots,N\\}$. In this case, the area becomes

$\displaystyle{\cal A}_{z_{1},\dots,z_{2N}}=$
$\displaystyle\sum_{j=1}^{N}\sum_{l\leq j}\left({\sf Im}[\alpha_{j}]\,{\sf
Re}[\alpha_{l}]+{\sf Im}[\beta_{j}]\,{\sf Re}[\beta_{l}]\right)$
$\displaystyle+N^{2}{\sf Im}[\bar{\alpha}]{\sf Re[\bar{\beta}]}\,,$ (4)

with $\bar{\alpha}=\sum_{j}\alpha_{j}/N$ and
$\bar{\beta}=\sum_{j}\,\beta_{j}/N$. Similarly, for the sequence
$(z^{\prime}_{1},\dots,z^{\prime}_{2N})$, with $z^{\prime}_{j}:=\beta_{j}$,
$j\in\\{1,\dots,N\\}$ and $z^{\prime}_{N+j}:=\alpha_{j}$,
$j\in\\{1,\dots,N\\}$, the area is

$\displaystyle{\cal A}_{z^{\prime}_{1},\dots,z^{\prime}_{2N}}=$
$\displaystyle\sum_{j=1}^{N}\sum_{l\leq j}\left({\sf Im}[\alpha_{j}]\,{\sf
Re}[\alpha_{l}]+{\sf Im}[\beta_{j}]\,{\sf Re}[\beta_{l}]\right)$
$\displaystyle+N^{2}{\sf Im}[\bar{\beta}]{\sf Re[\bar{\alpha}]}\,.$ (5)

The area delimited by the two paths $(z_{1},\dots,z_{2N})$ and
$(z^{\prime}_{1},\dots,z^{\prime}_{2N})$ is then

$\displaystyle{\cal A}$
$\displaystyle=\left|A_{z_{1},\dots,z_{2N}}-A_{z^{\prime}_{1},\dots,z^{\prime}_{2N}}\right|$
$\displaystyle=N^{2}\,\left|{\sf Im}[\bar{\alpha}]{\sf Re[\bar{\beta}]}-{\sf
Im}[\bar{\beta}]{\sf Re[\bar{\alpha}]}\right|$
$\displaystyle=N^{2}\,\left|{\sf Im}[\bar{\alpha}\bar{\beta}^{*}]\right|\,,$
(6)

as stated in the main text.

We now show that the area $A$ can be estimated with super-Heisenberg precision
by combining two sets of displacement operations
$S_{1}=\\{D_{\alpha_{k}},k=1,\cdots,N\\}$ and
$S_{2}=\\{D_{\beta_{k}},k=1,\cdots,N\\}$ into the quantum SWITCH.

Using the notation
$D_{\boldsymbol{\alpha}}:=D_{\alpha_{N}}D_{\alpha_{N-1}}\cdots D_{\alpha_{1}}$
and $D_{\boldsymbol{\beta}}:=D_{\beta_{N}}D_{\beta_{N-1}}\cdots
D_{\beta_{1}}$, the combination of the operations $D_{\boldsymbol{\alpha}}$
and $D_{\boldsymbol{\beta}}$ in the quantum SWITCH, with the control initially
in the state $(|H\rangle+|V\rangle)/\sqrt{2}$ and the target in a generic
state $|\psi\rangle$, gives rise to the output state

$\displaystyle|\Psi\rangle$ $\displaystyle=\frac{|H\rangle\otimes
D_{\boldsymbol{\beta}}D_{\boldsymbol{\alpha}}|\psi\rangle+|V\rangle\otimes
D_{\boldsymbol{\alpha}}D_{\boldsymbol{\beta}}|\psi\rangle}{\sqrt{2}}$
$\displaystyle=\frac{(I\otimes
D_{\boldsymbol{\beta}}D_{\boldsymbol{\alpha}})\,(|H\rangle\otimes|\psi\rangle+|V\rangle\otimes
U_{\boldsymbol{\alpha},\boldsymbol{\beta}}|\psi\rangle)}{\sqrt{2}}\,,$ (7)

with
$U_{\boldsymbol{\alpha},\boldsymbol{\beta}}:=D_{\boldsymbol{\alpha}}^{\dagger}D_{\boldsymbol{\beta}}^{\dagger}D_{\boldsymbol{\alpha}}D_{\boldsymbol{\beta}}$.
By repeatedly applying the relation
$D_{\gamma_{1}}D_{\gamma_{2}}=D_{\gamma_{1}+\gamma_{2}}\exp\left[i\text{Im}(\gamma_{1}\gamma_{2}^{*})\right]$,
valid for arbitrary $\gamma_{1}$ and $\gamma_{2}$, we obtain

$\displaystyle U_{\boldsymbol{\alpha},\boldsymbol{\beta}}=I\,\exp\left\\{i{\sf
Im}\left[\sum_{l=2}^{4N}\gamma_{l}\left(\sum_{k=1}^{l-1}\gamma_{k}\right)^{*}\right]\right\\}\,,$
(8)

where $\vec{\gamma}=(\gamma_{1},\cdots,\gamma_{4N})$ is the vector
$(\beta_{1},\cdots,\beta_{N},\alpha_{1},\cdots,\alpha_{N},-\beta_{N},\cdots,-\beta_{1},-\alpha_{N},\cdots,-\alpha_{1})$.

By explicit evaluation, we then obtain
$U_{\boldsymbol{\alpha},\boldsymbol{\beta}}=I\,e^{2iN^{2}\,{\sf
Im}[\overline{\alpha}\overline{\beta}^{*}]}$. Hence, the final state of the
control and target systems is

$\displaystyle|\Psi\rangle=\frac{|H\rangle+e^{2iN^{2}\,{\sf
Im}[\overline{\alpha}\overline{\beta}^{*}]}|V\rangle}{\sqrt{2}}\otimes
D_{\boldsymbol{\beta}}D_{\boldsymbol{\alpha}}\,|\psi\rangle\,,$ (9)

The phase shift $e^{2iN^{2}\,{\sf Im}[\overline{\alpha}\overline{\beta}^{*}]}$
can be estimated by measuring the control system on the orthonormal basis
$|\pm\rangle=(|{\rm H}\rangle\pm{\rm V})/\sqrt{2}$. Since the phase shift
grows as $N^{2}$, the parameter ${\sf
Im}[\overline{\alpha}\overline{\beta}^{*}]$ (and therefore the regularized
area ${\cal A}/N^{2}=|{\sf Im}[\overline{\alpha}\overline{\beta}^{*}]|$) can
be estimated with super-Heisenberg scaling precision following the procedure
described in the main text.

###  VI.2 An explicit setup that achieves Heisenberg scaling with fixed gate
order

The most general setup for probing the $2N$ unknown displacements in a fixed
order is a sequential setup using auxiliary systems chiribella2008memory ;
chiribella2012optimal ; yang2019memory , as shown in Fig. 6. In Ref.
zhao2020quantum  it was shown that every setup with fixed order and bounded
probe energy will necessarily lead to an RMSE with Heisenberg scaling $\propto
1/N$. However, the specific constant in Ref. zhao2020quantum  may not be
achievable by any concrete setup. On the practical side, it is interesting to
compare the precision achieved by our experiment with actual setups that can
be built in a quantum optics laboratory, using resources that are comparable
to those used in our setup.

Here we analyze a concrete setup with the following characteristics: (1) it
achieves the Heisenberg scaling $1/N$, (2) it only uses quadrature
measurements, and (3) like our setup, it does not require intermediate
operations (the gates in $V_{1},\dots,V_{2N}$ in Fig. 6). In this setup, the
$x$-displacements and the $p$-displacements are measured separately, achieving
Heisenberg scaling in the estimation of the two average displacements
$\overline{x}$ and $\overline{p}$. For fairness of the comparison with the
setup presented in the main text, we will assume that the initial state of the
probe is the lowest energy state (with energy $1/2$), as in our experiment.

In the following we use the dimensionless variable
$x_{j}^{\prime},p_{j}^{\prime},\bar{x^{\prime}}$, and $\bar{p^{\prime}}$ to
denote the dimensionless $j$-th $x$ and $p$ displacements and the average $x$
and $p$ displacements in zhao2020quantum , and their relation with their
dimensional counterparts are given by
$x_{j}^{\prime}=\frac{x_{j}}{\sqrt{2}\sigma_{x}},p_{j}^{\prime}=\frac{p_{j}}{\sqrt{2}\sigma_{p}},\bar{x^{\prime}}=\frac{\bar{x}}{\sqrt{2}\sigma_{x}},\bar{p^{\prime}}=\frac{\bar{p}}{\sqrt{2}\sigma_{p}}$,
where $\sigma_{x}=989.9\mu m$ and $\sigma_{p}=\frac{1}{2\sigma_{x}}$ are the
standard deviations of intensity in the $x$ and $p$ coordinates. Under the two
sets of definitions, the value of $A$ does not change. We consider the
following standard scheme where $A$ is estimated by separately measuring
$\bar{x^{\prime}}:=\sum_{j=1}^{N}x_{j}^{\prime}/N$ and
$\bar{p^{\prime}}:=\sum_{j=1}^{N}p_{j}^{\prime}/N$:

  1. 1.

Apply $x_{1}^{\prime}$, $x_{2}^{\prime}$, $\dots$, $x_{N}^{\prime}$
sequentially on the lowest-energy state $|0\rangle$ of the transverse modes.

  2. 2.

Measure the output state with homodyne measurement
$\\{dx^{\prime}\,|x^{\prime}\rangle\langle x^{\prime}|\\}$. Use MLE to produce
an estimate $\hat{\bar{x^{\prime}}}$ of $\bar{x^{\prime}}$.

  3. 3.

Apply $p_{1}$, $p_{2}$, $\dots$, $p_{N}$ sequentially on the vacuum state
$|0\rangle$.

  4. 4.

Measure the output state with homodyne measurement
$\\{dp^{\prime}\,|p^{\prime}\rangle\langle p^{\prime}|\\}$. Use MLE to produce
an estimate $\hat{\bar{p^{\prime}}}$ of $\bar{p^{\prime}}$.

  5. 5.

Output $\hat{A}=\hat{\bar{x^{\prime}}}\cdot\hat{\bar{p^{\prime}}}$.

Figure 6:  General scheme with fixed gate order. The $2N$ unknown
displacements ${\bf
d}=(\alpha_{1},\dots,\alpha_{N},\beta_{1},\dots,\beta_{N})$ are arranged in a
fixed order, corresponding to a permutation $\pi\in S_{2N}$, so that for
$j\in\\{1,\dots,2N\\}$ the $j$-th displacement is $z_{j}:=d_{\pi(j)}$. To
probe the displacements, the input system is initialized in a general state
$|\Psi\rangle$, possibly entangled with an auxiliary system. The unknown
displacement operations are interspersed with fixed unitary gates
$(V_{1},\dots,V_{2N})$, and eventually a measurement with operators $(M_{i})$
is performed on the output.

The performance of the above strategy can be readily evaluated. First, the
probability distribution of $x^{\prime}$ conditioned on a specific value of
the mean $x$-displacement is
$P(x^{\prime}|\bar{x^{\prime}})=(1/\sqrt{\pi})e^{-|N(\bar{x^{\prime}}-x^{\prime})|^{2}}$
(similarly for $\bar{p^{\prime}}$). The Fisher information of
$\bar{x^{\prime}}$ and $\bar{p^{\prime}}$ are thus both $2N^{2}$, and by the
Cramér-Rao inequality we can bound the RMSE of this strategy as

$\displaystyle\delta
A\geq\frac{\sqrt{\bar{x^{\prime}}^{2}+\bar{p^{\prime}}^{2}}}{\sqrt{2\nu}N}.$
(10)

Notice that another similar strategy would have been to apply all
displacements and then use heterodyne measurement
$\\{(1/\pi)|\beta\rangle\langle\beta|\\}$ that simultaneously measures
$\bar{x^{\prime}}$ and $\bar{p^{\prime}}$. However, such a strategy would have
had larger overall error, representing the price to be paid for the joint
measurement of two conjugate quadratures.

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