# Quantum state resolved molecular dipolar collisions over four decades of
energy

Guoqiang Tang1, Matthieu Besemer1, Stach Kuijpers1, Gerrit C. Groenenboom1,
Ad van der Avoird1, Tijs Karman1∗∗, Sebastiaan Y.T. van de Meerakker1∗∗
1Radboud University, Institute for Molecules and Materials
Heijendaalseweg 135, 6525 AJ Nijmegen, the Netherlands
∗∗To whom correspondence should be addressed;
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
> Collisions between cold polar molecules represent a fascinating research
> frontier, but have proven hard to probe experimentally. We report
> measurements of inelastic cross sections for collisions between NO and ND3
> molecules at energies between 0.1 and 580 cm-1, with full quantum state
> resolution. At energies below the $\sim$ 100 cm-1 well depth of the
> interaction potential, we observed backward glories originating from
> peculiar U-turn trajectories. At energies below 0.2 cm-1, we observed a
> breakdown of the Langevin capture model, which we interpreted in terms of a
> suppressed mutual polarization during the collision, effectively switching
> off the molecular dipole moments. Scattering calculations based on an _ab
> initio_ NO-ND3 potential energy surface revealed the crucial role of near-
> degenerate rotational levels with opposite parity in low-energy dipolar
> collisions.

The study of molecular collisions at energies well below 1 kelvin has emerged
as a new and exciting research frontier (?). Interest in cold and ultracold
molecular collisions stems from the inherent quantum nature that govern
interactions at these low energies, offering a wealth of new opportunities
ranging from the production of novel molecular quantum gases and precision
measurements in fundamental physics to ultracold chemistry with complete
control over all degrees of freedom (?, ?, ?). Systems that exhibit electric
dipole-dipole interactions are of particular interest, as the long-range and
anisotropic dipole-dipole interaction offers an exquisite “handle” to control
and steer collisions. At sufficiently low temperatures, small external
electromagnetic fields can vastly change the interaction, yielding distinctive
opportunities to engineer interaction Hamiltonians and to steer collision
pathways (?, ?).

Two main approaches have been developed to experimentally study cold molecular
collisions, and to harvest their prospects for unlocking new scattering
phenomena. In the first, molecules are stored in traps, yielding long
interaction times up to seconds. Spectacular progress has been reported
recently for ultracold collisions between dipolar bialkali molecules produced
through association of ultracold atoms, yielding temperatures well below 0.1
mK (?, ?, ?, ?, ?). Techniques that use direct slowing or laser cooling of
pre-existing molecules before loading them into a 2D or 3D trap at
temperatures in the 0.1 – 500 mK range are also coming to the fore (?, ?, ?,
?, ?, ?, ?, ?). These samples may be cooled further using optical cooling
schemes, or by evaporative or sympathetic cooling schemes if a favorable ratio
of elastic to inelastic collision cross sections is available (?, ?).
Molecular collision dynamics inside the trap is usually inferred from trap
loss measurements, although significant progress has been made recently to
directly probe the collision products (?, ?).

The second approach relies on crossed molecular beam methods, in which beams
of atoms or molecules are made to interact within a well defined crossing
volume. Cooling of the molecules of interest into low rotational and
translational temperatures occurs within a supersonic expansion by entraining
them in a seed gas of noble atoms. Low interaction energies can be achieved by
studying intrabeam collisions (?, ?, ?, ?), by controlling the beam crossing
angle (?, ?) or by changing the reagent’s speeds using for instance molecular
decelerators (?). Energies as low as 10 mK have been reached using the merged
beam technique, in which one of the beams is manipulated to tangentially
overlap with the other beam’s path (?, ?, ?).

Whereas trapping offers record-low temperatures and presumably the ideal
platform to use (ultra)cold molecules in a variety of applications, crossed
beam experiments are ideally suited to probe the inherent collision properties
with unprecedented detail. One of the most appealing features of the crossed
beam technique is the possibility to probe the collisions isolated from any
environment, the possibility to tune the collision energy over a wide range,
and the availability of a suite of advanced detection techniques to directly
measure the collision products state-selectively. Recently, quantum features
in the energy dependent state-to-state integral cross sections (ICSs) such as
partial wave scattering resonances have been measured at energies down to 0.2
cm-1 (?). The combination with the velocity map imaging (VMI) technique
allowed for measurements of differential cross sections (DCSs) that directly
revealed the quantum waves underlying the scattering event (?, ?).

Despite this vast progress, there are still many outstanding questions in our
understanding of cold molecular collisions. Depending on the collision energy,
different theoretical approaches are generally followed to describe the
collision process. At sufficiently low energies, the collisions may be
described by universal laws and capture theory (?, ?), based on interaction
potentials that mainly contain long-range electrostatic interactions (?). At
higher energies, on the other hand, full quantum scattering calculations based
on accurate _ab initio_ potential energy surfaces (PESs) are required. It is
at present unclear how and at which energies these approaches connect to each
other. In addition, at low energies the collisions become extremely sensitive
to the exact topology of the PES; a small modification of interaction
potentials can change cross sections by orders of magnitude (?). This fact has
major implications for the success of evaporative and sympathetic cooling
schemes, as these schemes strongly rely on the exact collision properties of
the system across a relatively large range of energies. As a result,
theoretical models urgently need experimental data to become predictive. Last
but not least, in describing cold collisions between polar molecules, it is
important to realize that polar molecules in isolated quantum states formally
do not possess a permanent dipole moment. Effective dipole moments must be
induced, for instance by an external electric field or the proximity of
another polar molecule, but it is at present unclear how and under which
conditions this induced dipole moment starts affecting the scattering
behavior.

It has proven extremely difficult to experimentally probe low-energy molecular
collisions in the required detail, in particular for collisions between two
dipolar molecules. Ideally, one would like to experimentally track how two
molecules approach each other as a function of the collision energy, interact
through their evolving potential energy landscapes, and reemerge as final
products (?). Only then can a complete picture of collision events be
obtained, from start to finish and from hot to cold, and validate various
theoretical models along the way. Although all methodological ingredients to
achieve this result have in principle been developed, the combination of all
techniques in a single experiment has been deemed impedingly difficult. State-
to-state crossed beam experiments below $\approx$ 1 kelvin have thus far only
been realized for experiments in which a molecule of interest scatters with
beams of He atoms or H2 molecules, making use of the high densities available
in neat beams and the low mass of the collision partner to decrease the
reduced mass (?, ?). Merged beams have been successful in reaching lower
energies (?), but its application to state-selective bi-molecular systems has
remained elusive. A state-to-state collision experiment using two polar
molecules has been performed for the OH-NO system, but the low densities
resulting from the necessity of using two seeded molecular beams has hampered
the extension of these experiments to energies below $\approx$ 100 K (?).

Here we present a joint experimental and theoretical study of inelastic
collisions between NO ($X\,^{2}\Pi_{1/2},v=0,j=1/2f$ referred to hereafter as
NO($1/2f$)) radicals and ND3
($X\,^{1}A_{1}^{\prime},v_{2}=0,j_{k}^{p}=1_{1}^{-}$, referred to hereafter as
ND3($1_{1}^{-}$)) molecules. We used the combination of Stark deceleration of
NO, hexapole state selection of ND3 and VMI detection in both crossed and
merged beam arrangements to measure fully state-resolved integral and
differential cross sections at collision energies spanning nearly four orders
of magnitude between 0.1 cm-1 and 580 cm-1. This energy range encompasses the
temperature scales relevant to atmospheric processes and combustion ($T\geq
100$ K), astrochemistry ($T\approx 1$ K) and approaches the ultracold pure
quantum regime ($T\leq 1$ mK). The scattering results revealed three
distinctive energy regimes with different mechanisms governing the collisions.
At the highest energies, between 100 and 580 cm-1, we observed correlated NO -
ND3 excitations, which could be understood from the electrostatic multipole
contributions to the interaction potential. In the intermediate energy regime,
between 10 and 100 cm-1, the collision energy became similar in magnitude to
the well depth of the angularly averaged potential. In this region we observed
a peculiar scattering mechanism in which the collision partners experience
half orbiting or U-turn-like trajectories, resulting in a prominent backward
component in the scattering distribution. At the lowest energies, we found an
intriguing dependence of the ICS on collision energy, which was interpreted in
terms of a breakdown of the Langevin capture model when the dipole-dipole
interaction energy becomes similar to the energy splittings between near-
degenerate rotational levels with opposite parity, effectively switching off
the molecular dipole moments. Excellent agreement was obtained with the cross
sections derived from quantum coupled-channels scattering calculations based
on a new _ab initio_ NO-ND3 PES throughout the entire energy range.

Crossed and merged beam scattering

We used a crossed molecular beam apparatus that is schematically shown in Fig.
1 and explained in more detail in the Supporting Materials (SM). Packets of
NO($1/2f$) radicals with a tunable velocity were produced by passing a beam of
$\approx$ 5% NO seeded in either krypton, xenon or argon through a 2.6 meter
long Stark decelerator. Packets of state-selected ND3($1_{1}^{-}$) molecules
were produced by passing a beam of $\approx$ 5% ND3 seeded in rare gas atoms
through electrostatic hexapoles. To cover a large range of collision energies,
we used assemblies of straight hexapoles at 45 and 90 degree intersection
angle with the Stark decelerator axis, as well as a curved hexapole to merge
the ND3 and NO packets at near zero degree intersection angle. The relatively
small dipole moment of 0.16 D for NO compared to the 1.5 D moment for ND3
ensured that both beams could be merged without significantly affecting the NO
packet (SM). The straight hexapole beamlines featured a beamstop and diaphragm
located such that the rare gas atoms were effectively eliminated from the ND3
molecular beam pulse. The scattered NO radicals were state-selectively
detected using a two-color laser ionization scheme and velocity mapped on a
two dimensional detector (SM).

Velocity mapped ion images probing differential cross sections

We measured velocity mapped ion images for various final NO states at 21
different collision energies $E_{\mathrm{col}}$ (see SM for all images and
Fig. 2 for a selection), directly probing the DCSs of the scattering process.
At high energies, multiple concentric rings were observed that correspond to
the coincident excitation of ND3 into excited $j_{k}$ levels. By virtue of the
Stark decelerator, the resolution in most images was sufficient to probe
individual transitions up to $\Delta j=8$ and $\Delta k=6$, however,
excitation to near-degenerate parity components within a given ND3($j_{k}$)
level could not be resolved. As the energy was reduced, the images became
smaller and featured fewer rings. At sufficiently low energies, excitation of
ND3 became energetically inaccessible, and only a single ring was observed
that converged to a single point at the lowest probed collision energy of 0.1
cm-1. At all energies, most scattering was observed in the forward direction,
with a striking and rather intense narrow backscattered feature appearing for
collision energies below $\approx$ 100 cm-1.

We compared the experimental images to simulated images based on coupled-
channels (CC) scattering calculations of state-to-state cross sections using
our _ab initio_ NO-ND3 PES and the kinematics of the experiment (SM).
Unfortunately, the high density of rotational levels in both NO and ND3, in
combination with the dipole-dipole interaction extending to large radial
distances $R$, made full CC calculations computer intensive. To limit the
computational cost we truncated the channel basis, depending on the collision
energy (SM). The open-shell character and $\Lambda$-doublet splittings of NO
were taken into account as these factors were probed in the experiment, but
the inversion splittings in ND3 were neglected as these splittings were not
experimentally resolved. For the energies 158 and 580 cm-1, the NO radical was
treated as a closed-shell molecule, which further reduced the size of the
channel basis. The validity of this approximation at high energies was
verified by comparing the cross sections at 65 cm-1 obtained from the closed-
shell and open-shell calculations (SM).

The angular and radial distributions derived from the experimental and
simulated images showed excellent agreement (SM), validating theory to uncover
the mechanisms underlying these observations and the role of the dipole-dipole
interaction in them. Not surprisingly, our calculations revealed that the
scattering dynamics is dominated by rather different mechanisms across the
four orders of magnitude wide collision energy range probed in the
experiments. Trends in the energy dependence of the pair-correlated NO-ND3
excitations could be related to the electrostatic multipole contributions to
the interaction potential. The dipole-dipole and quadrupole-dipole interaction
dominate the NO-ND3 attractive forces at long range and govern ND3 excitations
with $\Delta k=0$, and these transitions were found strongest at energies
above $\sim 60$ cm-1. At lower energies, transitions with $|\Delta k|=3$
gained intensity, which require weaker and more short-ranged terms involving
the octupole moment of ND3. This counter-intuitive result originated from a
balance between the energy gap law that favored transitions with low
excitation energy, and the relative strength of the multipole interactions
(SM). Model calculations showed that if the final states with $\Delta k=0$ and
$|\Delta k|=3$ were degenerate, excitations with $\Delta k=0$ were then indeed
preferred throughout, confirming the role of the energy gap (SM).

Backward glories

The dominant forward scattering observed in almost all images was found to
originate from the dipole-dipole interaction, and disappeared when this term
was switched off in model calculations (SM). By contrast, the narrow
backscattered feature observed below collision energies of $\approx$ 100 cm-1
persisted, even in the absence of the dipole-dipole interaction. Using
classical trajectory calculations, we found that this feature originated from
peculiar scattering trajectories with impact parameters within a surprisingly
narrow range above the classical turning point (see Fig. 3(b)). Common wisdom
predicts that such soft or glancing collisions result in forward or side
scattering, yet, we observed half-orbiting trajectories that swung around the
scattering partner to exit at a near-exact backscattered angle of
$\chi=-180^{\circ}$ (we used the common convention of the deflection angle
$\chi$ which is related to the observable scattering angle $\theta=|\chi|$).
We found that this peculiar behavior was related to the well-known phenomenon
of rainbow scattering, resulting in a maximum deflection (rainbow) angle that
occured for trajectories with a specific (rainbow) impact parameter. We found
that for energies below the well depth of the angularly averaged interaction
potential (101.1 cm-1 for NO-ND3), the attraction could become sufficiently
strong to push the rainbow angle to $\chi=-180^{\circ}$, defining the onset of
orbiting (Fig. 3(a)). This fact resulted in a strong enhancement of the DCS
(that in a classical treatment diverges as $1/\sin\theta$) at
$\theta=180^{\circ}$, similar to the more familiar glory effect found in
forward scattering. Such glories do not represent a large fraction of the
scattering, but they accumulate intensity in a small region. The ‘soft-
collision backward glories’ as found here are known in optics (?), but their
observation in scattering has thus far been scarce and limited to high energy
collisions of alkali systems that have deep potential wells (?, ?). Yet, they
should be obiquitous in collisions between dipolar molecules characterized by
only modest well depths provided that the collision energy is sufficiently
low.

Local maximum in low-energy integral cross sections

At the lowest energies it was difficult to discern structure in the small
images. We therefore made complementary ICS measurements for collisions that
de-excite NO radicals from the $1/2f$ to the $1/2e$ level by scanning the
collision energy from 0.1 to 10 cm-1. The collision signals were converted
into relative cross sections taking the spatial and temporal overlap of both
beams into account, assuming that detectable collision products could only
emerge after the curved hexapole when the merging process had completed (SM).
We observed a rapidly increasing cross section as the energy was reduced (Fig.
4a). At low energies, one may expect the cross sections to follow the Langevin
capture model, which for a dipole-dipole interaction with long-range $1/R^{3}$
dependence predicts a cross section that scales with
$E_{\mathrm{col}}^{-2/3}$. We found that the cross section indeed followed
this energy dependence, but started to deviate for energies below $\sim 1$
cm-1.

The cross section that followed from CC calculations reproduced the Langevin
cross section at high energies, and furthermore revealed an intricate
dependence on collision energy and a large sensitivity to the exact parameters
used in the calculations (?). We first used the CC calculations described
above that take the $E_{\Lambda}=0.0119$ cm-1 $\Lambda$-doublet splitting
between the $1/2f$ and $1/2e$ states of NO into account but that neglected the
$E_{\mathrm{inv}}=0.053$ cm-1 inversion splitting between the $1_{1}^{-}$ and
$1_{1}^{+}$ states of ND3, and calculated the ICS for energies extending into
the ultracold regime (blue curve in Fig. 4a). Excellent agreement with the
experiment was obtained, except for energies below 0.2 cm-1 where we found
that the predicted ICS could change by orders of magnitude by manually
adjusting the value for $E_{\Lambda}$ between zero and 0.3 cm-1 (SM). Clearly,
the approximation to neglect $E_{\mathrm{inv}}$ in the calculations was no
longer valid at low energies, and we performed CC calculations that included
$E_{\mathrm{inv}}$ at the expense of a smaller basis set that was appropriate
at these energies (red curve in Fig. 4a). Both curves nearly coincided at high
energies and converged to the Langevin cross section, but at energies below
0.2 cm-1 they showed a local maximum (LM) that peaks at different energies
before they entered the Wigner regime where the cross section scaled with
$E_{\mathrm{col}}^{-1/2}$ following the Wigner threshold law.

To understand this intricate LM in the ICS, one should realize that the
initial states of NO and ND3 in the experiments —the $1/2f$ and $1_{1}^{-}$
state, respectively— do not have permanent dipole moments. Effective dipole
moments are induced by electric fields, but also by dipole-dipole
interactions, by mixing the $1/2f$ and $1/2e$ states of NO and the $1_{1}^{-}$
and $1_{1}^{+}$ states of ND3. Figure 4b shows the effective interaction
energy $V(R)$ as seen by the approaching NO and ND3 molecules calculated as a
function of $R$. We used a simple model with two two-level systems that
accounts only for two opposite parity states of each molecule, at energies
$\pm E_{\Lambda}/2$ and $\pm E_{\mathrm{inv}}/2$, and the dipole-dipole
interaction that couples collision channels where both molecules change their
parity, yielding off-diagonal elements that scale with
$\mu_{NO}\mu_{ND_{3}}/R^{3}$ (SM). At short distances, typically probed at
high energies, the dipole-dipole interaction is strong compared to the
inversion splittings, and mixes the opposite parity states of both molecules.
This mutual polarization leads to $1/R^{3}$ dipolar interactions also without
applying an external field to polarize the molecules (?, ?), which results in
the observed $E_{\mathrm{col}}^{-2/3}$ Langevin capture. As the collision
energy is reduced, this cross section increases reflecting a larger capture
impact parameter. At larger distances, however, the interaction potential
begins to deviate from $1/R^{3}$ and switches to a $1/R^{6}$ dependence when
the dipole-dipole interaction becomes weaker than the parity splittings, and
can no longer polarize the molecules. This transition occurs where the dipole-
dipole interaction energy is comparable to the sum of the energy splittings
between levels of opposite parity in both molecules, explaining the dependence
on the inclusion of $E_{\mathrm{inv}}$. Hence, we observe a breakdown of the
Langevin capture model, leading to the characteristic LM in the cross section.
The field-free dipole-dipole capture we observe towards higher energies may
also be interpreted in terms of nonadiabatic transitions as described in Ref.
(?, ?, ?).

The experimentally observed scattering behavior deviated significantly from
the theoretical prediction below 0.2 cm-1, and did not show the local maximum.
In this region, we found that the cross section not only depended critically
on $E_{\Lambda}$ and $E_{\mathrm{inv}}$, but also responded extremely
sensitively to external electric fields (?). Even fields of only 100 - 1000
V/cm caused the cross section to increase by an order of magnitude, even
though such fields hardly changed the energy splittings. By contrast, they
broke the parity selection rule of the dipole-dipole interaction, resulting in
a large cross section increase for collisions in which NO was transferred to
the 1/2 _e_ state but ND3 remained in the $1_{1}^{-}$ state (SM). At the
lowest collision energies accessible to our experiment, the NO and ND3
molecules travelled with near-matching velocities causing a partial overlap of
both beams already in the final section of the curved hexapole where large
inhomogeneous electric fields were present (SM). Although high-field seeking,
simulations showed that NO (1/2 _e_) collision products were not significantly
influenced by the hexapole field, and had a high probability to be detected by
the laser further downstream. In addition to changing the cross sections,
these in-hexapole collisions also increased the effective beam intersection
angle and collision energy for given mean velocities of the NO and ND3
packets. To account for both effects, we calculated the electric field
dependent ICSs for all possible $m$-resolved transitions in ND3 (SM), and
simulated for each channel the expected cross section to produce detectable NO
(1/2 _e_) collision products as a function of the collision energy probed in
the experiment (colored curves in Fig. 4c). We found excellent agreement
between the experiment and theoretical prediction when including the in-
hexapole collisions in our analysis (green curve in Fig. 4c), and could
interpret the slowed rate of increase in cross section observed at the lowest
energies as the onset of the local maximum. We found that at the lowest
energies up to 50% of the detectable collision events occured inside the
strong field of the hexapole where the elastic ND3 channel started dominating
the collision dynamics. The partial overlap inside the curved hexapole is
inherent to the merged beam approach with matching velocities, but may in
future experiments be circumvented by tagging one of the beams using an
optical excitation pulse when the merging is completed, effectively setting a
time zero for the beam overlap. This would also reduce the minimum collision
energy achievable, which would then solely be determined by the velocity
spreads of both packets.

The collision behavior including the LM found here should be ubiquitous in the
scattering of two polar molecules with near-degenerate energy levels of
opposite parity. It is exactly this kind of species that is most amenable to
Stark deceleration, hexapole state selection and electrostatic trapping alike
(?). The energy $E_{\mathrm{LM}}$ at which the LM peaks, as well the cross
section $\sigma_{\mathrm{LM}}$ found at this energy, can be estimated from
capture theory (SM) and are given for a selection of relevant systems in Table
1 (see SM for an extended table). The value for $E_{\mathrm{LM}}$ exclusively
depends on the sum of the parity splitting energies of both colliders,
shifting the LM to higher energies for systems with larger splittings.

Concluding remarks

The success attained here to measure completely state-resolved ICSs and DCSs
for polar bimolecular collisions at energies tunable by nearly four orders of
magnitude – down to 0.1 cm-1 – reveals detailed views on collisions between
cold polar molecules, opens new vistas to control the collision dynamics using
external fields, and offers new insights to assess the feasibility of future
experiments. The existence of backward glories in the DCSs, field-free dipolar
collisions by mutual polarization, the breakdown of the capture model, and the
extreme sensitivity to external fields at low energies, all provide
extraordinary opportunities to control the collision dynamics. The collision
energies we achieved are lower than the typical interaction energy of a polar
molecule with external fields, yielding distinctive opportunities to
manipulate the dipole-dipole interaction. Tailored external electric fields
could be applied in a dedicated section after the curved hexapole, in which
both the strength and direction of the field could be adjusted. As the
molecule’s dipole moment is either oriented along ($\uparrow$) or against
($\downarrow$) the electric field direction depending on the quantum state,
the interaction could be effectively switched from repulsive
($\uparrow\uparrow$) to attractive ($\uparrow\downarrow$), tuned in strength
by controlling the degree of polarization, or controlled by resonances
associated with field-linked bound states (?). Microwave fields could be
applied to dress the rotational levels, which could enhance or supress
collision channels (?, ?, ?, ?, ?). Collision induced recoil energies from
Stark-shifted energy levels could directly be recorded using VMI, yielding
pair-correlated $m$-resolved collision cross sections revealing the relative
orientation of the products. Application of various such external control
knobs to tune and follow the dynamics in collisions between cold polar
molecules, a Holy Grail in the field for two decades, are now within reach.

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Acknowledgement We thank Daan Snoeken for help in developig the NO-ND3 PES,
Youp Caris for assistance during the ICS measurements, and Zhi Gao for
assistance during the high energy DCS measurements. We thank Henrik Haak and
Gerard Meijer of the Fritz-Haber-Institut der Max-Planck-Gesellschaft for
providing the curved hexapole. We thank Frank Stienkemeier for fruitful
discussions on earlier work on backward glories. We thank Niek Janssen and
André van Roij for expert technical support. Funding: This work is part of the
research program of the Netherlands Organization for Scientific Research (NWO)
through an ENW-M and a VICI grant. S.Y.T.v.d.M. acknowledges support from the
European Research Council (ERC) under the European Union’s Horizon 2020
Research and Innovation Program (Grant Agreement No. 817947 FICOMOL). G.T.
acknowledges support from the China Scholarship Council under grant
201606340077. Author contributions: The experiments were performed and
analyzed by G.T.. The theoretical calculations were performed by M.B., T.K.,
G.C.G. and A.v.d.A.. Simulations on beam overlap were performed by S.K. The
project was conceived and supervised by S.Y.T.v.d.M.. The main manuscript was
written by S.Y.T.v.d.M.. The Supplement was written by G.T., M.B. and
S.Y.T.v.d.M.. All authors were involved in the interpretation of the data,
discussed the results, and commented on the manuscript. Competing interests:
None declared. Data and materials availability: All data needed to evaluate
the conclusions in the paper are present in the paper or the Supplementary
Materials, as well as online at DOI: 10.5281/zenodo.7561771 (?).

Supplementary Materials
Supplementary text
Figures S1 - S35
Tables S1- S4
References 55-81

Table 1: Prediction for the collision energy (upper entry, in cm-1) and maximal cross section (lower entry, in Å2) of the LM in the ICS for collisions between various combinations of the molecules NH3, ND3, NO ($X\,^{2}\Pi_{1/2}$), NH ($a\,^{1}\Delta$), OH ($X\,^{2}\Pi_{3/2}$) and CO ($a^{3}\Pi$). For each molecule, the rotational state with maximal Stark shift is assumed (see SM for details). |  ND3 |  NH3 |  NO |  NH |  OH |  CO
---|---|---|---|---|---|---
ND3 |  0.28 |  2.20 |  0.17 |  0.14 |  0.28 |  0.17
|  4457 |  1101 |  1061 |  8531 |  5319 |  5748
NH3 |  2.20 |  4.12 |  2.09 |  2.06 |  2.20 |  2.09
|  1101 |  714 |  195 |  1386 |  1333 |  1070
NO |  0.17 |  2.09 |  0.06 |  0.03 |  0.18 |  0.06
|  1061 |  195 |  355 |  3963 |  1254 |  1886
NH |  0.14 |  2.06 |  0.03 |  2.01$\times$10-5 |  0.14 |  0.03
|  8531 |  1386 |  3963 |  3.71$\times$106 |  1.00$\times$104 |  2.04$\times$104
OH |  0.28 |  2.20 |  0.18 |  0.14 |  0.29 |  0.18
|  5319 |  1333 |  1254 |  1.00$\times$104 |  6351 |  6795
CO |  0.17 |  2.09 |  0.06 |  0.03 |  0.18 |  0.07
|  5748 |  1070 |  1886 |  2.04$\times$104 |  6795 |  1.00$\times$104
Figure 1: Schematic representation of the experimental setup and the
rotational energy level scheme of the NO radical and para-ND3. A state-
selected and velocity tunable beam of NO radicals is produced using a
2.6-meter-long Stark decelerator, and crossed with a hexapole state-selected
beam of ND3 molecules using either a 0∘, 45∘ or 90∘ angle of incidence. Only
the last section of the decelerator is shown. Scattered NO radicals are
detected state-selectively using velocity map imaging. The rotational levels
of NO and ND3 are split into two components of opposite parity. The energy
splittings are largely exaggerated for reasons of clarity. The initial levels
selected by the Stark decelerator and hexapoles are indicated in red. Figure
2: Selection of the experimental (Expt) and simulated (Sim) velocity map ion
images for the scattering process NO ($1/2f$) + ND3 ($1_{1}^{-}$)
$\rightarrow$ NO ($j^{\prime}_{\mathrm{NO}}$) + ND3 ($j^{\prime}_{k}$) as a
function of collision energy. The images are presented such that the relative
velocity vector is oriented horizontally, with the forward direction on the
right side of the image. Small segments of the images around forward
scattering are masked due to imperfect state selection of the NO packet. The
final state $j^{\prime}_{\mathrm{NO}}$ of NO probed by the lasers is indicated
above each image. Note the three different velocity scales used to cover the
large range of collision energies. Figure 3: Illustration of the origin of
backward glories. (a) Deflection $\chi$ as a function of the impact parameter
$b$ resulting from classical trajectory calculations for elastic collisions
between NO and ND3 for selected collision energies. At energies below 101
cm-1, backscattering ($\chi$ = -180∘) can occur for impact parameters well
above the classical turning point. (b) Swarm of classical trajectries for a
narrow range of impact parameters above the classical turning point leading to
scattering in the backward direction at a collision energy of 65 cm-1. The
Soft Collision Backward Glory Scattering (SCBGS) trajectory leading to perfect
backscattering $\chi=-180^{\circ}$ is indicated in purple. Figure 4: Low-
energy scattering behavior. (a) Experimentally observed ICS for the scattering
process NO ($1/2f$) + ND3 ($1_{1}^{-}$) $\rightarrow$ NO ($1/2e$) + ND3
($j^{\prime}_{k}$) as a function of collision energy together with the
theoretically predicted ICS without (blue curve) and with (red curve) taking
the inversion doublet structure of ND3 into account. Data is accumulated using
a continuous cycle over collision energies. Vertical error bars represent
statistical uncertainties at 95% of the confidence interval (all panels).
Horizontal error bars represent uncertainties in energy calibration (all
panels). Scattering signals were corrected for flux-to-density effects taking
the spatial, temporal and velocity distributions of both the NO and ND3
packets into account, assuming that detectable collisions only occur when the
merging process has completed (see SM). Experimental cross sections are scaled
to the theoretical ones. (b) The NO-ND3 interaction strength as a function of
the radial distance $R$ between the molecules. The sum of the
$\Lambda$-doublet splitting of NO ($E_{\Lambda}$) and the inversion splitting
of ND3 ($E_{\mathrm{inv}}$) is indicated by the green dashed line. (c) As in
(a), but taking the partial beam overlap inside the curved hexapole and the
$m$-resolved electric-field dependent ICSs into account. The cross sections
resulting from numerical trajectory simulations for the elastic and inelastic
channels in ND3 that lead to detectable NO (1/2 _e_) products are indicated
(colored data points). The statistical errors associated with numerical
trajectory simulations are indicated by the colored areas. Experimental cross
sections are scaled to the sum of the individual channels (green). The
percentage of collisions that take place inside the curved hexapole is
indicated (magenta dashed line, right axis).


# Deep learning as a tool for quantum error reduction in quantum image
processing

Krzysztof Werner  Dept. of Computer Graphics, Vision and Digital Systems,
Silesian University of Technology, Gliwice, Poland<EMAIL_ADDRESS>Kamil
Wereszczyński<EMAIL_ADDRESS>Dept. of Computer Graphics, Vision
and Digital Systems, Silesian University of Technology, Gliwice, Poland;
Rafał Potempa  Krzysztof Cyran  Dept. of Computer Graphics, Vision and Digital
Systems, Silesian University of Technology, Gliwice, Poland;

###### Abstract

Despite the limited availability and quantum volume of quantum computers,
quantum image representation is a widely researched area. Currently developed
methods use quantum entanglement to encode information about pixel positions.
These methods range from using the angle parameter of the rotation gate (e.g.,
the Flexible Representation of Quantum Images, FRQI), sequences of qubits
(e.g., Novel Enhanced Quantum Representation, NEQR), or the angle parameter of
the phase shift gates (e.g., Local Phase Image Quantum Encoding, LPIQE) for
storing color information. All these methods are significantly affected by
decoherence and other forms of quantum noise, which is an inseparable part of
quantum computing in the noisy intermediate-scale quantum era. These phenomena
can highly influence the measurements and result in extracted images that are
visually dissimilar to the originals. Because this process is at its
foundation quantum, the computational reversal of this process is possible.
There are many methods for error correction, mitigation, and reduction, but
all of them use quantum computer time or additional qubits to achieve the
desired result. We report the successful use of a generative adversarial
network trained for image-to-image translation, in conjunction with Phase
Distortion Unraveling error reduction method, for reducing overall error in
images encoded using LPIQE.

##  1 Introduction

We propose the use of Generative Adversarial Networks (GANs) for error
reduction, as a new method of error reduction in image processing, where a
quantum computer is used to handle all calculations, and a classical computer
is used for error reduction. In the future, we plan to implement the error
reduction method on the quantum computer as well.

This method is intended for use in quantum object detection of aviation
instruments, in images taken on a Microsoft HoloLens 2 device inside the
flight simulator cockpit.

GAN is a class of deep learning algorithm proposed in 2014 by Goodfellow et
al. [7]. In this approach, two neural networks compete against each other in a
zero-sum game as follows.

  * •

Generator, produces candidates for output, using inputs original data.

  * •

Discriminator, evaluates candidates provided by the generator, recognizing
whether the generated output is artificial or original. So, it is trained to
distinguish generated output from the original samples.

Such a game forces the generator to produce an output similar to the desired
one. After the network is trained, it can be used to generate images or to
evaluate image correctness. Such networks have been used, for example, for
generating paintings in the desired style from images. The GAN networks can be
implemented using a wide variety of programming libraries) e.g., TensorFlow
[1] and Keras [3], pygan or torchgan [14]). In our experiment, after training,
we used the model for the generation of images with reduced error, based on
the images reconstructed from the quantum computers.

(LPIQE) or the Local Phase Image Quantum Encoding method was proposed in 2020
[19]. It was meant to be implemented in photonic quantum computers (which are
computers that use photons as the source of qubits). Computers like this are
currently in operation and are being offered to researchers. The most known
photonic processor from Xanadu, described by Madsen et al. in [12], uses the
squeezed states and measurement-based quantum computation model. QuiX Quantum
from the Netherlands (e.g., Somhorst et al. [15], Taballione et al. [16] or de
Goede et al. [5]) uses other sources such as single photons or quantum dots as
the supply of the quantum systems.

LPIQE uses controlled phase shift gates for color representation. Such
configurations are very natural for optic based systems, in contrary to other
methods, that need to be implemented using significantly more optical devices.

For image representation, this method uses ${\lceil log_{2}X\rceil+\lceil
log_{2}Y\rceil+1}$ qubits, where _X_ and _Y_ references the X and Y dimensions
of the encoded image, and the additional qubit is used for storing color
information (in 1 channel mode). For storing images represented in RGB
spectrum, an additional 2 qubits are needed for R, G, or B channel selection.
Encoding color information in local phase allows the method to be combined
with phase distortion unraveling (PDU) error reduction.

Quantum error correction and reduction is a widely researched area. Current
implementations of quantum computers are heavily affected by the impact of
decoherence. They require the temperatures of $0.015K$ (which is the
consequence of the lack of room temperature superconductors) and a perfect
vacuum. These conditions are nearly impossible to achieve on Earth, which
makes quantum systems volatile to the slightest changes of the surrounding
environment. Decoherence is a term that encompasses the interactions, between
the quantum computer and other particles, that are not expected by their
operators, and which destroy the quantum nature of the system [21].

Some types of quantum error correction that aim to protect the qubit or a
quantum system against decoherence and quantum noise are quantum redundancy
and measurement stabilization, which use the multidimensional Hilbert space on
which the quantum states are mapped [10], and quantum surface codes, which are
implemented on 2D qubit latices and comprehensively tackle the error of the
whole quantum system [2]. These methods of course have their shortcomings,
such as using additional qubits or severely restricting the architecture of
quantum circuits. Either way, they are always implemented on a quantum
computer alongside the algorithm.

Quantum error reduction focuses on minimizing the error. Methods that are part
of this family are implemented on classical machines, quantum computers, or as
a hybrid–classical-quantum systems. Some of the methods are Richardson
extrapolation error mitigation [11], which performs measurement with
artificially increased noise levels and then uses Richardson extrapolation to
approximate no noise in the system and the quasi-probability method [18, 4],
which tries to probabilistically inverse the noise process that occurred in
the system.

The PDU method of error reduction was proposed in 2022 [20]. It was intended
to be used in conjunction with quantum circuits using phase shift gates for
calculations. Error reduction is then conducted fully on a classical computer.
The PDU method is based on PDU functions. These are interpolated functions for
a set of points $(x_{d},\varepsilon_{d})$, where $x_{d}$ is the value for
which the function was measured, and $\varepsilon_{d}$ is the recorded error.
The method is then applied on the result acquired from the quantum computer.

Given the current state of quantum hardware, the goal of our research is to
overcome the quantum volume and noisy intermediate-scale quantum (NISQ) era
limitations and to allow researchers to reach meaningful conclusions, even
with imperfect systems. This work shows experimentally that generative
artificial intelligence could be used in error reduction in the NISQ era. In
the area of quantum computing, the usefulness of generative AI can be
conserved even beyond NISQ, because the fidelity of a transmitted state/of the
transmitted state(s) is expected to be on a much lower level than inside the
quantum computers (e.g., Tann [17]).

##  2 Materials and Methods

###  2.1 LPIQE method

LPIQE is the method of quantum image representation that uses the angle
parameter of the controlled phase gate. It was developed for implementation on
quantum computers. The basic building block of the method is the quantum
cosine sampling (QCoSamp) operator, as shown in Figure 1.

Figure 1: QCoSamp operator base of computation. Source: IBM’s quantum
composer.

The encoding circuit is composed of a base of computation and controlled phase
shift gates. The simplified version is shown in Figure 2.

Figure 2: QCoSamp operator encoding the angle of $-\frac{3}{4}\pi$. Source:
IBM’s quantum composer.

For the image encoding using this method, let’s take the image of size H × W
pixels. It could be represented as a matrix
$I_{m}=\big{[}\hat{p}_{r,c}\big{]}_{H\times W}$, where $\hat{p}_{r,c}$ is the
intensity of the pixel placed in the $r$-th row and $c$-th column. The image
then can be flattened to a single vector using vertical vectorization to
obtain

$\overset{\longrightarrow}{I_{m}}=[p_{j}]^{T},s.t.:j=rW+c.$ (1)

Then, it could be represented in the form of the state

$\displaystyle\ket{I_{m}}=\big{[}e^{ip_{0}}\dots
e^{ip_{J}},0^{M-J}\big{]}=\sum_{j=0}^{J}e^{ip_{j}}\ket{j},$ (2) $\displaystyle
J=WH-1,\;\;M=2^{\lceil log_{2}(WH)\rceil}$

where the first part of the above state is the vector of $WH$ exponential
functions of pixels intensities, and the second part is a complement to the
quantum state’s requirement of having the power of two coefficients.

The final state can then be obtained by an operator defined by the matrix

$\displaystyle\tilde{\mathcal{L}}(I_{m})=\mathbf{1}\overset{\longrightarrow}{I_{m}}=$
$\displaystyle\begin{bmatrix}e^{ip_{0}}&0&\ldots&0\\\
0&e^{ip_{1}}&0\ldots&0\\\ \vdots&0&\ddots&0\\\ 0&\ldots&0&e^{ip_{J}}\\\
\end{bmatrix}$ $\displaystyle\mathcal{L}(I_{m})=$
$\displaystyle\begin{bmatrix}\tilde{\mathcal{L}}(I_{m})_{J\times
J}&\\!\\!\\!\framebox{}\\!\\!\\!&\mathbf{0}\\\
\hline\cr\mathbf{0}&\\!\\!\\!\framebox{}\\!\\!\\!&\mathbf{1}\end{bmatrix}_{M\times
M}$ (3)

where $\mathbf{1}$ is the identity matrix/operator. At this point, the image
can be easily encoded using a unitary gate from the matrix function present in
most modern libraries.

###  2.2 Error reduction by the PDU method

PDU is the hybrid classical–quantum error reduction method. It comprises of
the set of PDU functions, that are calibrated for the quantum computer and
circuit, and then interpolated between calibrated values. The correlations
between qubits are important for achieving the desired level of correction.

The calibration consists of running the function on a quantum computer and
recording the differences between the expected values $\tilde{\gamma}(x)$ and
values received from the quantum computer $\gamma(x)$:

$\varepsilon(x)=\tilde{\gamma}(x)-\gamma(x),$ (4)

where $x$ is an arbitrary value. The function is interpolated for the set of
those arguments. These arguments are the probabilities following from local
phases of qubits, extracted using the phase-kickback technique. This can be
generalized such that $x$ can be any object, which is described as follows:

###### Definition 2.1

Let $\ket{\psi}_{x}$ be the local phase $e^{ix}$ of the eigen-state
$\ket{\psi}\in\mathbb{E}$ ($\mathbb{E}$ is a measurement basis) in a subspace
of an $n$-qubit state, and $\mathfrak{p}\ket{\psi}_{x_{k}}$ is the
experimentally designated probability amplitude for state $\ket{\psi}$ with
phase $x_{k}$. In that case the function

$\varepsilon:\mathbb{E}\times[-\pi,\pi]\cap\mathbb{R}\longrightarrow\mathbb{R}$

is a general PDU function if and only if its each projection on the eigen-
state is smooth due to phase and fulfills the following:

$\displaystyle\forall
k:\;\;\varepsilon(\ket{\psi},x_{k})=\mathfrak{p}\ket{\psi}_{x_{k}}-\langle\psi\rangle_{x_{k}}$
$\displaystyle\forall\ket{\psi}\;\neg\exists\tilde{\varepsilon}(\ket{\psi},x):$
$\displaystyle\;\;\int_{-\pi}^{\pi}dx\;\;\tilde{\varepsilon}(\ket{\psi},x)<\int_{-\pi}^{\pi}dx\;\;\varepsilon(\ket{\psi},x)$
(5)

###  2.3 GAN pix2pix network

GANs [7] are a class of AI algorithms. In this architecture, two networks
compete with each other in a zero-sum game of two players – the generator $G$
and the discriminator $D$. Given a random noise vector $z$ and ground truth
$y$, the generator learns to produce artificial outputs similar to the ground
truth:

$G:z\to y.$ (6)

The _PatchGAN_ (or _pix2pix_) model designed by Isola et al. [8], used for the
experiments in this work, belongs to a group of _conditional GANs_ proposed by
Mirza et al. [13]. This means that except for the random noise $z$, the GAN
takes a side input $x$ (e.g., the outline of an object used to generate its
outputs):

$G:\\{x,z\\}\to y.$ (7)

Generally, the GAN’s zero-sum game has an objective, also known as the value
function:

$G^{*}=\arg\underset{G}{\min}\ \underset{D}{\max}V(G,D),$ (8)
$V(D,G)=\mathbb{E}_{x\sim p_{\text{data}}}\log[D(x)]+\\\ \mathbb{E}_{x\sim
p_{\text{model}}}\log[1-D(G(z))],$ (9)

where $V$ is the value function, $x$ denotes the data, and $z$ is the input
noise for the generator. $D(x)$ represents the probability that $x$ came from
the ground truth data, rather than from the model. For a conditional GAN
(e.g., pix2pix model), the value function 9 takes the form of

$V(D,G)=\mathbb{E}_{x\sim p_{\text{data}}}\log[D(x,y)]+\\\ \mathbb{E}_{x\sim
p_{\text{model}}}\log[1-D(x,G(x,z))].$ (10)

From equations 9 and 10 comes the conclusion that the goal of the generator is
to minimize the objective, while the goal of the discriminator is to maximize
it. Namely, the two models compete with each other, where the discriminator
tries to catch the generator’s output, while the generator tries to outsmart
it. The only difference is that the conditional GAN has a side input $x$,
which is used by both the generator and discriminator. During training, the
two models compete but at the convergence the generator’s outputs are
indistinguishable from the ground truth by the discriminator (i.e.,
$D(x)\approx 0.5$). The discriminator can be discarded during the model’s
inference phase [6].

###  2.4 Experimental protocol

In the experiments, we explored the use of pix2pix GAN neural networks for
error correction in quantum image processing.
We devised the following experiments:

  1. 1.

Reducing the error in $16\times 16$ images encoded with a quantum computer
simulator, using images corrected with a PDU function as a reference.

  2. 2.

Reducing the error in $8\times 8$ images encoded on real quantum computers,
using images corrected with a PDU function as a reference.

  3. 3.

Reducing the error in $16\times 16$ images encoded with a quantum computer
simulator and corrected using a PDU function, with the original images as a
reference.

  4. 4.

Reducing the error in $8\times 8$ images encoded on real quantum computers and
corrected using a PDU function, with the original images as a reference.

Experiments 1 and 3 were done using the same initial dataset of 631 encoded
and reconstructed images, and experiments 2 and 4 were done using the initial
dataset of 61 encoded and reconstructed images.

For the experiments, the GAN network was implemented using TensorFlow and
Keras libraries for Python. Both generator and discriminator were designed to
accept $256\times 256$ pixel images, to increase the networks’ learning
capacities. The generator was implemented as a U-Net model with a total of
54,429,315 training parameters and 15 convolutional layers with LeakyReLU in
the initial 7 encoding blocks. The bottleneck and decoding blocks used ReLU as
the activation function. Between each of the consecutive encoding and decoding
blocks, there was also a batch normalization operation. The first three
decoding blocks additionally had a dropout with a value of 0.5. The network
contained skip connections from encoders to corresponding decoders. An
overview of the generator is shown in Figure 3.

The discriminator was defined as a model with 6 convolutional layers,
resulting in 6,968,257 training parameters. All the encoding blocks used
LeakyReLU as the activation layers, and between all of them the batch
normalization operation was performed. The output block used a sigmoid
function as the activation layer. An overview of the generator is shown in
Figure 4.

Figure 3: Overview of the generator network in the form of a U-Net used for
the experimental part. All encoding layers’ blocks consist of a convolutional
layer (not shown), batch normalization (BN), and LeakyReLU (LReLU) activation.
Decoding layers’ blocks consist of a transposed convolutional layer (not
shown), batch normalization (BN), dropout (DO), and ReLU activation. The
output block is a transpose convolutional layer with a tanh function
activation layer. The thickness dimension of each block corresponds to the
number of its output channels, whereas its width corresponds to the edge of
the resulting square image. Source: own compilation. Figure 4: Overview of the
discriminator network of pix2pix GAN used for the experimental part. All
encoding layers’ blocks consist of a convolutional layer (not shown), batch
normalization (BN), and LeakyReLU (LReLU) activation. The output block is a
convolutional layer with a sigmoid function activation layer. The thickness
dimension of each block corresponds to the number of its output channels,
whereas its width corresponds to the edge of the resulting square image.
Source: own compilation.

Each up- or down-scaling layer used a stride of $2\times 2$. The convolutional
and transpose convolutional layers used a kernel of size $4\times 4$. For all
LeakyReLU, the alpha parameter was set to 0.2.

The training was performed using an Adam optimizer [9] with learning rate of
$2\cdot 10^{-4}$ and exponential decay rate for the first moment estimates
$\beta_{1}=0.5$. The loss function was a weighted combination of binary cross-
entropy defined as

$H(y,\hat{y})=-y\log\hat{y}-(1-y)\log(1-\hat{y})$ (11)

and mean absolute error (MAE) defined as

$\text{MAE}(y,\hat{y})=\frac{1}{N}|y-\hat{y}|,$ (12)

where $y$ denotes the true output and $\hat{y}$ its estimate, and N is the
total number of performed estimations. The corresponding weights are equal to
1 and 100, respectively.

####  2.4.1 Steps for the experiments

Steps to reproduce the experiments are as follows:

  1. 1.

Obtain images:

    1. (a)

Implement the PDU function for the experiment, with the desired granularity of
measurements (for the experiment, the granularity used was 5).

    2. (b)

Calibrate the PDU function.

    3. (c)

Generate black-and-white images (sizes for the experiment were $16\times 16$
and $8\times$ pixels) with 256 shades of gray.

    4. (d)

Implement the quantum circuits for the generated images using LPIQE.

    5. (e)

Encode the images by running the circuits (for experiments 1 and 3, the
statevector_simulator was used, and for 2 and 4, IBMQ Nairobi was used).

    6. (f)

Reconstruct the images, and save the results alongside the original images.

    7. (g)

Use the calibrated PDU function on results to perform error reduction, and
save the resulting image with the previous ones.

  2. 2.

Create and train GAN network:

    1. (a)

Create GAN neural network for picture-to-picture translation.

    2. (b)

Upscale the images to $256\times 256$ pixels.

    3. (c)

Load training data appropriate for the experiment (for 1, use the
reconstructed images and original ones from the quantum simulator; for 2, use
the same but from the quantum computer; for 3, use the images after error
reduction and original ones from the quantum simulator; and for 4, use the
images after error reduction and original ones from a quantum computer).

    4. (d)

Divide the data into training and testing sets, and train the network on the
training set (for experiments 1 and 3, 250 epochs were performed, and for 2
and 4, 1000 epochs were performed in training).

  3. 3.

Evaluate the data:

    1. (a)

For the testing set, collect the results from the generator.

    2. (b)

Descale the resulting images into original resolution.

    3. (c)

Compare the resulting pictures with the original ones.

##  3 Results

###  3.1 Reducing the error in $16\times 16$ images encoded using a quantum
simulator, using original images as a reference.

For the experiment, we collected 631 distinct images reconstructed after
encoding on a quantum simulator.

An example of the correction performed is shown in Figure 5. For that
particular example, the Pearson’s correlation increased from -0.01935 to
0.99962 after using the generator from the GAN network.

|  |
---|---|---
Figure 5: Example results for the image encoded using the statevector
simulator. The figure represents the image after encoding on a simulator and
reconstruction (left), the image generated by the neural network (middle), and
the expected original image (right).

Detailed statistics are shown in Figures 6, 7 and Tables 1, 2.

---
Figure 6: The chart shows the difference in mean square error before and after
reduction.
---
Figure 7: The chart shows the difference in standard deviation of error before and after reduction. Table 1: Minimal, maximal, and mean correlation coefficients values, MSE, and STDEV of error for experiment 1 before error reduction. | min | mean | max
---|---|---|---
Pearson’s R | -0.15957 | 0.00708 | 0.16703
Mean square error | 0.17357 | 0.22277 | 0.28308
Std dev of error | 0.29144 | 0.32861 | 0.38558
Table 2: Minimal, maximal, and mean correlation coefficients values, MSE, and STDEV of error for experiment 1 after error reduction. | min | mean | max
---|---|---|---
Pearson’s R | 0.72075 | 0.96135 | 0.99993
Mean square error | 0.00003 | 0.00885 | 0.06843
Std dev of error | 0.00484 | 0.04444 | 0.24108

For this experiment, we observed a numerical improvement in all measured
statistics. The standard deviation of error dropped on average 7.4 times, and
the mean square error dropped on average 25.1 times. Pearson’s R also rose by
0.954.

###  3.2 Reducing the error in $8\times 8$ images encoded on real quantum
computers, using images corrected with a PDU function as a reference.

For the experiment, we collected 61 distinct images reconstructed after
encoding on a quantum computer and fed them to the neural network to obtain
images with reduced error.

An example is shown in Figure 8. For that particular example, the Pearson’s
correlation increased from 0.24326 to 0.89631.

|  |
---|---|---
Figure 8: Example results for the image encoded using an IBMQ Nairobi, 7 qubit
quantum computer. The figure represents the image after encoding and
reconstruction (left), the image generated by the neural network (middle), and
the expected original image (right).

The statistics for all collected samples are shown in Tables 3, 4, and Figures
9, 10.

---
Figure 9: The chart shows the difference in mean square error for the
experiment performed on a real quantum device.
---
Figure 10: The chart shows the difference in standard deviation of error for the experiment performed on a real quantum device. Table 3: Minimal, maximal, and mean correlation coefficients values, MSE, and STDEV of error for experiment 2 before error reduction. | min | mean | max
---|---|---|---
Pearson’s R | 0.01121 | 0.17262 | 0.31465
Mean square error | 0.12317 | 0.21375 | 0.35261
Std dev of error | 0.32898 | 0.37183 | 0.42898
Table 4: Minimal, maximal, and mean correlation coefficients values, MSE, and STDEV of error for experiment 2 after error reduction. | min | mean | max
---|---|---|---
Pearson’s R | -0.32539 | 0.56511 | 0.98399
Mean square error | 0.00410 | 0.10783 | 0.34393
Std dev of error | 0.05883 | 0.26588 | 0.55277

For this experiment, we still observed a numerical improvement in all measured
statistics, although the level of correction was much lower. The standard
deviation of error dropped on average 1.45 times, whereas the mean square
error dropped on average 1.26 times. Pearson’s R also rose by 0.392 to 0.565
with p-value of 0.005.

###  3.3 Reducing the error in $16\times 16$ images encoded with a quantum
computer simulator and corrected using a PDU function, with the original
images as a reference.

In this experiment, we used images encoded onto a quantum simulator, decoded
and then corrected with a PDU function, and fed to the GAN network. An example
is shown in Figure 11.

|  |
---|---|---
Figure 11: Example results for the image encoded using the statevector
simulator. The figure represents the image after encoding on a simulator,
reconstruction, and error reduction with a PDU method (left), the image
generated by the neural network (middle), and the expected original image
(right).

Detailed statistics are shown in Figures 12, 13 and Tables 5, 6.

---
Figure 12: The chart shows the difference in mean square error before and
after error reduction.
---
Figure 13: The chart shows the difference in standard deviation of error before and after error reduction. Table 5: Minimal, maximal, and mean correlation coefficients values, MSE, and STDEV of error for experiment 3 before error reduction. | min | mean | max
---|---|---|---
Pearson’s R | 0.87700 | 0.91805 | 0.94214
Mean square error | 0.01077 | 0.01779 | 0.02378
Std dev of error | 0.09665 | 0.12655 | 0.14971
Table 6: Minimal, maximal, and mean correlation coefficients values, MSE, and STDEV of error for experiment 3 after error reduction. | min | mean | max
---|---|---|---
Pearson’s R | 0.99639 | 0.99983 | 0.99993
Mean square error | 0.000014 | 0.00007 | 0.00106
Std dev of error | 0.00371 | 0.00769 | 0.02574

In this experiment, we observed a numerical improvement in Pearson’s R test,
which rose by 0.0818 to 0.999 with p-value of 3.25e-83. Meanwhile, the mean
square error dropped 270,14 times, whereas the standard deviation of error
dropped 16,46 times.

For the collected mean square error data, we also calculated the standard
deviation for mean square error ($\sigma$) and calculated statistics about the
percentage of data that fit the equation $mse\pm n*\sigma$. We found that for
$n=1$, 75.71% of data fell into this category; for $n=2$, it was 97.62%; and
for $n=3$, it was 99.52%. We also calculated that for 95% of the data to meet
the criteria of the equation, $n$ should be equal to 1.845; for 90%, $n$
should be equal to 1.716; for 75%, $n$ should be equal to 0.756; and for 50%,
$n$ should be equal to 0.679.

###  3.4 Reducing the error in $8\times 8$ images encoded on real quantum
computers and corrected using a PDU function, with the original images as a
reference.

In this experiment, to reduce the error after encoding and decoding the image
on a quantum computer, first the PDU method was used, and then the trained GAN
network was used to reduce the error further. An example of such a correction
is shown in Figure 14.

|  |
---|---|---
Figure 14: Example results for the image encoded using an IBMQ Nairobi, 7
qubit quantum computer. The figure represents the image after encoding,
decoding, and correction with a PDU method (left), the image generated by the
neural network (middle), and the expected original image (right).
---
Figure 15: The chart shows the difference in mean square error before and
after reduction for the experiment performed on a real quantum computer.
---
Figure 16: The chart shows the difference in standard deviation of error
before and after reduction for the experiment performed on a real quantum
computer.

The statistics for all collected samples are shown in Tables 7, 8, and Figures
15, 16.

Table 7: Minimal, maximal, and mean correlation coefficients values, MSE, and STDEV of error for experiment 4 before error reduction. | min | mean | max
---|---|---|---
Pearson’s R | 0.79338 | 0.88266 | 0.96455
Mean square error | 0.00906 | 0.03122 | 0.04825
Std dev of error | 0.09512 | 0.15817 | 0.19945
Table 8: Minimal, maximal, and mean correlation coefficients values, MSE, and STDEV of error for experiment 4 after error reduction. | min | mean | max
---|---|---|---
Pearson’s R | 0.99805 | 0.99970 | 1.00000
Mean square error | 0.0000006 | 0.00016 | 0.00065
Std dev of error | 0.00069 | 0.00993 | 0.02175

In terms of measured statistics, we observed further improvement noticed
mostly in the drop of mean square error, which was 226 times smaller after
using the neural network for error reduction. Also, the standard deviation of
error dropped by 15.9 times, and the average Pearson’s R rose by 0.117 to the
value of 0.9997 with p-value of 6.2e-78.

For the collected mean square error data, we also calculated the standard
deviation for mean square error ($\sigma$) and calculated statistics about the
percentage of data that fit the equation $mse\pm n*\sigma$. We found that for
$n=1$, 85.71% of data fell into this category; for $n=2$, it was 95.23%; and
for $n=3$, it was 100%. We have calculated that for 95% of the data to meet
the criteria of the equation, $n$ should be equal to 1.168; for 90%, $n$
should be equal to 1.001; for 75%, $n$ should be equal to 0.975; and for 50%,
$n$ should be equal to 0.877.

We suspect that this noticeable error reduction was possible because the PDU
function already corrected the bulk of errors induced by quantum noise,
leaving the result less random (compared to the original) and making it easier
for the neural network to observe patterns.

Also, the observed statistics for the standard deviation of the combined mean
square error seemed to be higher than those obtained in experiment 3. We
suspect that the reason for that could be the difference in quantum computer
architecture for which the simulator was constructed. The simulator was made
for simulating ion trap computers, and phase elements of qubits seemed to be
simulated in a less precise way.

##  4 Discussion & Conclusions

In this paper, we showed that GANs can be used to further reduce the error
from quantum noise in quantum calculations that use quantum sampling. Existing
error reduction methods (such as PDU presented in this paper) can lower the
quantum error significantly enough for neural networks to be effectively used
to further reduce the error to a point where visual identification of objects
on the picture is possible. With the further development of quantum computers,
encoded images will be significantly larger, which would require the neural
network to be more complex. This would require special equipment to be able to
train them effectively. Although we proved that GAN networks can reduce the
quantum noise error of images, they are much less effective, especially on
normal computers. Perhaps with the use of supercomputers, this method would be
able to reduce the quantum error in a similar fashion, but as researchers, we
did not have access to such a machine. The combined PDU and GAN error
reduction was statistically significant, which was confirmed by visual
similarity between the expected result and one produced by the neural network.
The Pearson coefficients were on the level of $0.99$ with p-values near zero,
which proves the very high linear correlation between the original and
corrected images. We are especially happy with the results obtained from the
real quantum computers in experiment 4. The mean square error of 0.00014 and
standard deviation of error of 0.00993 showed that this method could be used
with existing, imperfect quantum computers from the NISQ era and produce
results that – at least for us – are visually indistinguishable from the
original images. The results obtained from experiments 1 and 3 also showed
that these networks can be used beyond the NISQ era, for quantum error
reduction. We would like to develop this method further so it can be used as
part of a quantum image processing pipeline, where images will be encoded onto
a quantum computer, and then measured and decoded, and image transformations
performed in the encoded state. We focused on quantum image processing because
we are interested in developing methods in this area. However, these methods
could be used for other types of data. In conclusion, this paper presented how
neural networks can enhance the existing PDU error reduction method for
quantum encoded images. The method will be utilized in the image processing in
photonic quantum solutions and for quantum communication in faster-then-light
communication. It could also be used to pursue the goal of quantum object
detection. This could be utilized in detecting aviation instruments on
pictures taken inside a flight simulator cockpit, which could be used for
pilot training.

##  5 Version

This is quantumarticle version v.

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## Author Contributions

Conceptualization, Krz.W. and Kam.W.; methodology, R.P. and Krz.W; validation,
Kam.W.; formal analysis, R.P., Krz.W.; investigation, R.P. and Krz.W.;
writing—original draft preparation, R.P. and Krz.W.; writing—review and
editing, Krz.W.; visualization, R.P. and Krz.W.; supervision, Kam.W and K.C.;
project administration, K.C.; funding acquisition, K.C. and Kam.W. All authors
have read and agreed to the published version of the manuscript.

## Funding

The authors would like to acknowledge that this paper has been written based
on the results achieved within the WrightBroS project. This project has
received funding from the European Union’s Horizon 2020 research and
innovation programme under the Marie Skłodowska-Curie grant agreement No
822483. Supplementarily, this research work has been co-financed from Polish
financial resources for science in 2019-2023 conferred for implementation of
the co-financed international project. Disclaimer. The paper reflects only the
author’s view and the Research Executive Agency (REA) is not responsible for
any use that may be made of the information it contains.

## Conflict of Interests

The authors declare no conflict of interest.