figure table

# Achromatic design of a photonic tricoupler and phase shifter for broadband
nulling interferometry

Teresa Klinner-Teo  Sydney Institute for Astronomy, School of Physics,
University of Sydney, NSW 2006, Australia  Sydney Astrophotonic
Instrumentation Laboratories, University of Sydney, NSW 2006, Australia  AAO-
USyd, School of Physics, University of Sydney, NSW 2006, Australia  Marc-
Antoine Martinod  Sydney Institute for Astronomy, School of Physics,
University of Sydney, NSW 2006, Australia  Sydney Astrophotonic
Instrumentation Laboratories, University of Sydney, NSW 2006, Australia  AAO-
USyd, School of Physics, University of Sydney, NSW 2006, Australia  Peter
Tuthill  Sydney Institute for Astronomy, School of Physics, University of
Sydney, NSW 2006, Australia  Sydney Astrophotonic Instrumentation
Laboratories, University of Sydney, NSW 2006, Australia  AAO-USyd, School of
Physics, University of Sydney, NSW 2006, Australia  Simon Gross  MQ Photonics
Research Centre, School of Engineering, Macquarie University, NSW 2006,
Australia  Barnaby Norris  Sydney Institute for Astronomy, School of Physics,
University of Sydney, NSW 2006, Australia  Sydney Astrophotonic
Instrumentation Laboratories, University of Sydney, NSW 2006, Australia  AAO-
USyd, School of Physics, University of Sydney, NSW 2006, Australia  Sergio
Leon-Saval  Sydney Institute for Astronomy, School of Physics, University of
Sydney, NSW 2006, Australia  Sydney Astrophotonic Instrumentation
Laboratories, University of Sydney, NSW 2006, Australia  AAO-USyd, School of
Physics, University of Sydney, NSW 2006, Australia

###### Abstract

Nulling interferometry is one of the most promising technologies for imaging
exoplanets within stellar habitable zones. The use of photonics for carrying
out nulling interferometry enables the contrast and separation required for
exoplanet detection. So far, two key issues limiting current-generation
photonic nullers have been identified: phase variations and chromaticity
within the beam combiner. The use of tricouplers addresses both limitations,
delivering a broadband, achromatic null together with phase measurements for
fringe tracking. Here, we present a derivation of the transfer matrix of the
tricoupler, including its chromatic behaviour, and our 3D design of a fully
symmetric tricoupler, built upon a previous design proposed for the GLINT
instrument. It enables a broadband null with symmetric, baseline-phase-
dependent splitting into a pair of bright channels when inputs are in anti-
phase. Within some design trade space, either the science signal or the fringe
tracking ability can be prioritised. We also present a tapered-waveguide
$180^{\circ}$-phase shifter with a phase variation of $0.6^{\circ}$ in the
$1.4-1.7~{}\mu$m band, producing a near-achromatic differential phase between
beams for optimal operation of the tricoupler nulling stage. Both devices can
be integrated to deliver a deep, broadband null together with a real-time
fringe phase metrology signal.

###### keywords:

broadband nulling interferometry, integrated-optics, photonics, tricoupler,
fringe tracking, high contrast imaging

*Email<EMAIL_ADDRESS>
##  1 Introduction

Methods for exoplanet direct detection and characterisation are often limited
by the inherent high contrast ratio and small angular separation intrinsic to
exoplanetary systems, even when found orbiting the most favorable nearby
stars. Unlike indirect detection methods, nulling interferometry enables
direct imaging of dim stellar companions, extincting the bright starlight by
arranging a condition of destructive interference from two (or more) apertures
[1]. Interferometry can also exploit sparse or separated apertures and is not
bound by the formal diffraction limit of conventional monolithic telescopes,
and so can observe companions closer to the host star compared with techniques
such as coronagraphy[2]. The contrast ratios required for exoplanet detection
span a range from $10^{-4}$ for self-luminous hot exoplanets observed in the
mid-infrared [3] to $10^{-10}$ for Earth-like exoplanets imaged in reflected
light from their host star [4]. Performance levels over this range lie within
theoretical limits for nulling interferometers (“nullers” hereafter). While
most have hitherto been built using bulk optics (for example, the Keck
interferometer[5] and the Large Binocular Telescope Interferometer[6]),
integrated optics and photonic methods are increasingly becoming the desired
technology for nullers[7]. The use of photonics provides single-mode
waveguides, which perform modal-filtering (where phase aberrations from the
wavefront translate coupling efficiencies into single-moded waveguides, in
essence converting phase into intensity fluctuations), and their monolithic
designs ensure stability against environmental conditions and compactness for
scalability[2].

The Guided-Light Interferometric Nulling Technology (GLINT) instrument[2, 8]
is a nuller currently integrated into the Subaru Coronagraphic Extreme
Adaptive Optics (SCExAO) system at the Subaru Telescope, and is operated by
our team. It is able to combine light in the astronomical $H$ band from
multiple sub-apertures coupled into separate single-mode waveguides, nulling
multiple baselines simultaneously. In its present configuration, nulling is
carried out within a photonic chip, which is able to combine light channeled
from four sub-apertures on the telescope, producing six non-redundant
baselines. Light is combined using directional couplers, with output flux
directed into two channels: (1) the null channel in which on-axis starlight
destructively interferes and is therefore heavily suppressed (potentially,
together with off-axis planet-light not similarly extincted), and (2) the
anti-null or “bright” channel which contains constructively interfered
starlight. The aim of the instrument is to tune the interference to maximise
destructive interference of the starlight, extincting it from the null channel
and routing it instead into the bright channel. The most commonly used
observable quantity, the null depth, is derived from the ratio of these two
outputs. The GLINT device also features photometric monitoring outputs that
give an instantaneous measurement of the flux contributions of each input beam
to the interference signals. This configuration of the instrument has been
characterised on-sky and demonstrated[8] to reach a null depth of $10^{-3}$
with a precision of $10^{-4}$. However, to be truly scientifically
competitive, a nuller targeting planets in the infrared[9] will require a
significantly improved null depth of around $10^{-6}$. The current limitations
preventing GLINT from reaching deeper nulls have been identified: the
chromatic behavior of the directional coupler that induces a wavelength-
dependent null depth and the residual phase fluctuations of light entering the
instrument. Replacing the directional coupler with a symmetric tricoupler,
which splits input beams into three outputs, rather than two, has been shown
in simulation to be capable of overcoming these limitations[10]. Assuming that
the waveguides in the interaction region of the tricoupler are arrayed in an
equilateral triangle and that the injected beams are in perfect anti-phase and
with equal flux, the central channel will produce a completely nulled output,
regardless of the wavelength of incoming light. Underlying this powerful
configuration are arguments based on simple symmetry: anti-symmetric input
beams are unable to couple to an even-symmetry mode field required to overlap
with the central null channel[11]. A further advantage is provided by the
output fluxes from the two bright channels, which can be used to measure the
incoming differential phase thereby providing a signal for active cophasing
correction in real time. Simulations of a tricoupler able to deliver this
achromatic null have been shown to reach a null depth 45 times better than
with a single directional coupler [10]. The use of a tricoupler has been
previously mooted as a combiner for nulling and stellar interferometry [12,
11]. To exploit the equal coupling ratios and phase delays between outputs,
early devices made use of three waveguides in a planar, rather than
triangular, configuration. In order to produce the desired performance,
waveguides were tapered to adiabatically alter the coupling ratios[13]. Planar
tricouplers following this methodology have been designed for nulling
interferometry[14]. However, devices with an equilateral triangular
configuration in the interaction region are able to exploit further
symmetries: namely, an equal splitting ratio between bright channels[10].
Early tricouplers made use of fused fibre twisted together to produce this
triangular configuration in the central region, which introduced asymmetries,
altering the desired coupling ratio [15]. Today, the ultrafast laser
inscription (ULI)[16, 17, 18, 19] process enables efficient and precise
inscription of tricouplers and other photonic devices requiring three-
dimensional design freedom. The use of tricouplers for space interferometry is
currently being investigated[20, 21].

Here, we present a full chromatic treatment of a new design for a tricoupler.
This design incorporates fully symmetric triangular inputs and outputs, unlike
the design presented in Ref. 10, which eliminates slight asymmetries in the
splitting ratios and allows for equal splitting between non-null channels
across the band. This greatly simplifies the treatment of interactions using
the tricoupler’s transfer matrix and offers more flexibility with choice of
waveguide injected into, while still being fully fabricable using the ULI
technique. Additionally, we investigate the potential for this tricoupler to
maximise the signal from a faint stellar companion, showing that adjustments
to the physical properties of the device can either maximise science
throughput or provide better fringe tracking. Furthermore, any instrument
incorporating a tricoupler for the nulling beam combination requires a perfect
$180^{\circ}$ differential phase between injected beams across the working
spectral band. Therefore, an achromatic phase shift has to be implemented
preceding every tricoupler in the chip to enforce this anti-phase condition
between beams. To date, simulations of nulling with tricouplers have assumed
inputs arranged to be in anti-phase using a (chromatic) delay in air: a simple
piston term applied to one beam[10] which only meets the ideal anti-phase
condition at one wavelength within the observing band. This limits the maximum
number of simultaneously nulled baselines for any given number of apertures
and wastes the signal-to-noise advantages that a truly achromatic null would
deliver over all spectral channels in the band. We introduce an achromatic
$180^{\circ}$ photonic phase shifter using tapered waveguides, which can be
used in concert with the tricoupler to achieve an achromatic null across the
working band.

In Section 2, we derive the full transfer matrix for the current proposed
tricoupler, taking chromatic behaviour into account. We introduce our new
proposed completely symmetric equilateral tricoupler in Section 3, and
simulate the throughput of a planet light signal, as well as examine the
effects of changing the design to maximise this signal, in Section 4. In
Section 5, we advance our design for an achromatic phase shifter which can be
added at the input side of each tricoupler. The combination of both devices
can be replicated for each combiner in the case of a multi-baseline nuller, as
shown in Section 6. These photonic components can be realistically fabricated
using ULI inside a single integrated-optics chip.

##  2 Modelling beam combination with transfer matrices

The tricoupler proposed in Ref. 10 exhibits asymmetric coupling ratios. While
we expect similar coupling effects regardless of the waveguide in which the
light is injected, some discrepancies arise depending on which waveguide is
fed. These come from the waveguides’ shape, which transitions from a linear
array at both the inputs and outputs to an equilateral triangular
configuration in the central region. These transition areas result in two
asymmetrically arranged zones at both ends of the interaction region, where
the waveguides converge on approach to the interaction region, and again where
they diverge (Fig. 1).

Figure 1: Top (left) and 3D (right) view of the asymmetric tricoupler. While
the interaction region follows an equilateral triangular pattern, some
coupling effects happen in the remapping from and to a linear array before and
after the interaction region, breaking the full symmetry of the interactions
between the waveguides inside the tricoupler. The light enters the device from
the bottom and leaves at the top of the tricoupler.

The transfer matrix for a tricoupler with symmetry between left and right
waveguides, but not three-fold equilateral symmetry as described above, would
have at least two independent coupling coefficients. Hence mode field
orthogonality is generally needed, in addition to energy conservation and
symmetry arguments, to determine the phase shift. However, by assuming mode
orthogonality principle[22], the transfer matrix can be written as

$M=\begin{pmatrix}\tilde{T}_{1}&\tilde{C}_{1}&\tilde{C}_{2}\\\
\tilde{C}_{1}&\tilde{T}_{2}&\tilde{C}_{1}\\\
\tilde{C}_{2}&\tilde{C}_{1}&\tilde{T}_{1}\\\ \end{pmatrix},$ (1)

where $\tilde{C}_{1}$, $\tilde{C}_{2}$, $\tilde{T}_{1}$ and $\tilde{T}_{2}$
are the complex coupling coefficients from the input waveguide to the central
and outer output waveguides, and transmission coefficients in the outer and
central waveguides, respectively (Fig. 2).

Figure 2: Diagram of a tricoupler. Thick black lines represent waveguides.
Section A-A shows a cross-section of the interaction region in the $xy$-plane,
where the three waveguides are arranged in an equilateral triangular
formation. The interaction region has length $L$. The top waveguide is
illuminated with intensity $I_{\text{1,in}}$, and couples into the other
waveguides with coupling coefficients $C_{1}$ and $C_{2}$, and transmission
coefficients $T_{1}$ and $T_{2}$.

$M$ describes the relationship between the input and output beams of the
tricoupler, such that if $I$ is a vector of the complex amplitudes of the
input waveguides, then the vector of output waveguide amplitudes $O$ is given
by $O=MI$. The transmission and coupling coefficients are treated as complex
phasors which act on the incoming beams $\tilde{E}_{\text{in}}$:

$\displaystyle\tilde{T}_{i}=T_{i}e^{i\psi_{T_{i}}}$ (2)
$\displaystyle\tilde{C}_{i}=C_{i}e^{i\psi_{C_{i}}}$ (3)

$T_{i},\ C_{i}$ are defined to be the square modulus of these phasors such
that

$|\tilde{T_{i}}|^{2}=\frac{|\tilde{E}_{\text{{out,i}}}|^{2}}{|\tilde{E}_{\text{{in,i}}}|^{2}}=T_{i}^{2}.$
(4)

A tricoupler with asymmetric coupling or tapered waveguides[14, 22] requires
the treatment of coupling to the centre and unfed outer waveguides, $C_{1}$
and $C_{2}$, separately. As a result, expressions for the differential phases
between channels become complicated to compute in terms of the coupling and
transmission coefficients, which in turn makes the transfer matrix complicated
and potentially impacts the efficiency of phase calculations for fringe
tracking. Instead, we focus here on a design of a fully symmetric tricoupler.
Equal coupling of light simplifies the transfer matrix to

$\displaystyle M=$
$\displaystyle\begin{pmatrix}\tilde{T}&\tilde{C}&\tilde{C}\\\
\tilde{C}&\tilde{T}&\tilde{C}\\\ \tilde{C}&\tilde{C}&\tilde{T}\\\
\end{pmatrix}$ $\displaystyle=e^{i\psi_{c}}$
$\displaystyle\begin{pmatrix}Te^{i\Delta\psi}&C&C\\\ C&Te^{i\Delta\psi}&C\\\
C&C&Te^{i\Delta\psi}\\\ \end{pmatrix},$ (5)

where, as above, the transmission and coupling coefficients $T$ and $C$ are
wavelength-dependent and experimentally measurable by injecting light into one
waveguide at the time, $\psi_{t}$ and $\psi_{c}$ are the arguments of the
transmission and coupling coefficients respectively, and
$\Delta\psi=\psi_{t}-\psi_{c}$.

The conservation of energy between the inputs and the outputs implies[22]

$T^{2}+2C^{2}=1.$ (6)

Hence the transmission coefficient is expressed with respect to the coupling
coefficient as:

$T=\sqrt{1-2C^{2}}.$ (7)

Furthermore, the conservation of energy inside the device[22] itself implies
that

$\displaystyle|\tilde{T}+2\tilde{C}|^{2}=1.$ (8)

Developing this and injecting the result of Equation 7 gives

$\Delta\psi=\arccos{\frac{-C}{2T}}.$ (9)

Thus, the change of phase between waveguides can be characterised entirely by
the measurement of the transmission and coupling coefficients.

Interferometric signals collected in the three outputs can be obtained by
considering two wavefronts $\tilde{a_{1}}$ and $\tilde{a_{2}}$ so that

$\displaystyle O=T_{\text{tricoupler}}$
$\displaystyle\begin{pmatrix}a_{1}e^{i\phi_{1}}\\\ 0\\\ a_{2}e^{i\phi_{2}}\\\
\end{pmatrix},$ (10)

where $O$ is the vector of the three output wavefronts, $a_{i}$ and $\phi_{i}$
are the amplitude and the phase of the incoming wavefront $i=1,2$. The
intensities of the interferometric signals from the three outputs $I_{A}$,
$I_{B}$ and $I_{C}$ are deduced from the development of the squared modulus of
Eq. (10) so that

$\displaystyle\begin{pmatrix}I_{A}\\\ I_{B}\\\ I_{B}\\\ \end{pmatrix}=$
$\displaystyle\begin{pmatrix}T^{2}I_{1}+C^{2}I_{2}+2TC\sqrt{I_{1}I_{2}}\cos\left(\phi_{1}-\phi_{2}+\Delta\psi\right)\\\
C^{2}\left(I_{1}+I_{2}+2\sqrt{I_{1}I_{2}}\cos\left(\phi_{1}-\phi_{2}\right)\right)\\\
C^{2}I_{1}+T^{2}I_{2}+2TC\sqrt{I_{1}I_{2}}\cos\left(\phi_{1}-\phi_{2}-\Delta\psi\right)\\\
\end{pmatrix},$ (11)

where $I_{i}=a_{i}^{2}$ ($i=1,2$). We can see that the tricoupler indeed
provides a nulled signal in the central output (second line of Eq. (11))
regardless the wavelength, providing $\phi_{1}-\phi_{2}=\pi$ and
$I_{1}=I_{2}$.

Designing the tricoupler so that $C=T=\frac{1}{\sqrt{3}}$ leads to a phase-
shift $\Delta\psi=\frac{2\pi}{3}$ between all waveguides. This configuration
is ideal for simultaneous broadband null depth measurement and fringe tracking
capability[10]. While careful design of 3-way planar waveguides has been shown
capable of approaching this ideal[14], doing so requires considerable control
over many parameters governing the detailed waveguide properties. Here,
instead, we adopt the strategy of enforcing equal coupling by way of
geometrical symmetry.

##  3 Designs for a symmetric equilateral tricoupler

In this Section and Section 4, we present two tricouplers, Tricoupler A and
Tricoupler B, which are built upon previous designs and investigate the
effects of tuning various parameters to optimise for equal splitting, fringe
tracking or science signal performance.

As mentioned above, the tricoupler contains several exploitable symmetries
when arranged in an equilateral triangular configuration. The ideal tricoupler
couples light with equal splitting ratios between all three waveguides, and
outputs have a phase delay of $120^{\circ}$ between them. Figure 3 shows the
output intensity of each channel as a function of wavelength and input
differential phase for Tricoupler A, introduced in this section.

Figure 3: Map of the intensities in the three outputs of the tricoupler tuned
for equal splitting at 1.55$~{}\mu$m, against wavelength and differential
phase of incoming beams, when light of equal amplitude and phase is injected
into the two input waveguides. Flux at 180∘ differential phase is shown by the
white dashed line. Total intensity is normalised to 1.

When inputs have a differential phase of $180^{\circ}$, the central channel
remains dark across the band. Furthermore, the left and right outputs are
symmetric, meaning that the input differential phase – even when not in
perfect antiphase – can be recovered.

Our proposed design for a completely equilateral tricoupler, Tricoupler A, was
created in the RSoft numerical simulation environment and is shown in Figure
4. Unlike the device in Ref. 10, the branches of this tricoupler, as well as
the central interaction region, are all arrayed in an equilateral triangular
configuration. The input beams are fed into two of the tricoupler’s arms (the
bottom left and right in Figure 4), interfere with each other in the
interaction region in the centre and are output at the top. The interaction
region of Tricoupler A has a length of $566.3~{}\mu$m, which allows for an
equal 3-way ($\frac{1}{3}/\frac{1}{3}/\frac{1}{3}$) splitting ratio at the
central wavelength of $1.55~{}\mu$m. Waveguides are labelled using the same
convention as in Ref. 10. As in the previous design, the index difference
between core and cladding is $0.005$, and the waveguide cross-section is
circular, with a diameter of $7~{}\mu$m.

Figure 4: Schematic of the proposed equilateral tricoupler. The three inputs
are at the bottom and direction of propagation is upward. Left: top view of
the design. Right: 3D view of the design. Axes are not to scale. The
separation between waveguides at input and output is 75$~{}\mu$m.

In common with the earlier-generation tricoupler (symmetry only in the left
and right waveguides[10]), weak interaction between waveguides still occurs
outside the interaction region, however the coupling ratios now remain equal
(Fig. 5(a)) due to the equilateral configuration, unlike the previous design
which only manifests reflection symmetry (Fig. 5(b)). Light injected at the
central wavelength in the new design therefore splits equally between the two
remaining waveguides. This equal splitting allows interactions in the
tricoupler to be described by the transfer matrix shown in Equation 5, which
greatly simplifies calculations for both simulations of the chip and phase
recovery for fringe tracking. The design additionally exhibits greater
flexibility in the choice of waveguide injection, as the equilateral symmetry
allows for any of the three waveguides to be treated as the science channel,
with the remaining two used as the bright channels for fringe tracking.

(a) Equilateral symmetry tricoupler

(b) Left-right symmetry tricoupler[10]

Figure 5: Intensity splitting ratios between the three outputs for injection
of light into the left (left) and right (right) waveguide. The sum of the
intensities is normalised to $1$ for each wavelength. Top: splitting ratios
for the tricoupler presented here. Bottom: splitting ratios for the tricoupler
presented in Ref. 10.

The change in tricoupler design leading to this additional symmetry comes at
little cost to the convenience of its manufacture and integration into the
nulling chip. Although this design sacrifices the convenience of the planar
configuration of input and output beams, its integration into the photonic
circuitry can easily be accommodated. At input, only two waveguides are
illuminated before the tricoupler, so no significant symmetry-enforced changes
are needed at the pupil remapper side with respect to the existing chip. At
the output side, a modest extension of the triangular waveguides was found to
be sufficient to separate them beyond any residual parasitic interaction.
After this, reconfiguring them into a linear array (now without the constraint
of optical pathlength matching) can be trivially accomplished. The ultrafast
laser inscription technique, allowing the creation of 3-dimensional circuitry
of almost arbitrary configuration, can easily meet all these practical design
requirements.

##  4 Considerations for science signal throughput

In addition to the ability to carry out achromatic nulling of starlight and
fringe tracking, the capability of the tricoupler to transmit off-axis light
from potential stellar companions (for convenience hereafter denoted “planet
light”) must also be considered. The nuller can be conceived as broadcasting a
fringe pattern on-sky representing the transmission, with the central, dark
fringe on the axis where it extincts flux from the target host star (Figure
6). In an ideal case, an exoplanet will be located in the bright fringe of the
nuller, and therefore its light arrives at the nulling instrument with a
differential phase of 180∘.

Figure 6: Diagram of the fringe pattern created by a monochromatic nuller with
candidate locations for the target star and planets overplotted. The yellow
star denotes the targeted host star; the blue dot is a planet located in the
central dark fringe of the nuller, the orange one is a planet located an
eighth of a fringe away from the star, the green dot is a planet located a
quarter of fringe away and the red dot is a planet in the centre of a bright
fringe (half a fringe away from the star), which is the ideal situation for
detecting its light. The bottom panel shows the ideal throughput of each
planet as given by the nuller’s fringe pattern.

As this phase shift is due to a geometrical delay, there is a wavelength
dependence in the differential phase across the band for the science object.
On top of this astrometric effect, the tricoupler exhibits a chromatic
throughput when the incoming phase is no longer equal to 180∘ (Fig. 3).
Therefore, the spectrum of the off-axis source is expected to be changed by
the off-axis chromatic response of the tricoupler. An ideal tricoupler would
exclude all incoming on-axis starlight while transmitting all incoming (off-
axis) planet light in the central null output channel. It is highly desirable
to have an off-axis throughput as achromatic as possible to ensure sufficient
signal to noise ratio at all wavelengths, as desired for spectroscopic
measurements on the object. This property is of little consequence to the
fringe tracking and phase stabilisation of the tricoupler, as the contribution
of the planet light is negligible when compared with the starlight, due to the
high-contrast nature of these observations.

The signal of an exoplanet located in the first bright fringe of the nuller’s
throughput was simulated using RSoft. The angular offset $\theta$ of the
planet from its host star was first calculated using the condition that the
planet occurs in a bright fringe of the nuller, and has a differential phase
of 180∘:

$\displaystyle\theta=\frac{\lambda_{0}}{2B},$ (12)

where $\lambda_{0}$ is the wavelength in the centre of the band,
$1.55~{}\mu$m, and $B$ is the baseline, which is set to 5.55m[2]. From this,
the planet’s wavelength-dependent differential phase at input was found:

$\displaystyle\Delta\psi=\frac{2\pi}{\lambda}B\theta,$ (13)

where $\lambda$ is the wavelength of the incoming planet light. For this
simulation, wavelengths in the $1.4-1.7~{}\mu$m range were spanned. This phase
offset planet signal was then fed into three different candidate tricouplers
in order to explore the trade space driving the design process. Two key
parameters were tuned: the pitch (separation between waveguides in the
interaction region) and interaction length (length of the interaction region).
As the starlight cancellation property of the tricoupler is affected only by
the symmetry of the interaction region, by retaining the equilateral
triangular configuration of all the tested devices, the broadband nulling
capability was preserved.

The throughputs of simulated planet light for three tricouplers with different
pitches were computed, and the result is shown in Fig. 7. The interaction
length for each is optimised such that equal splitting occurs at
1.55$~{}\mu$m. For all three tricouplers, planet light in the centre output
has the same throughput: 0.66 at 1.55$~{}\mu$m. Moreover, the intensities in
the centre output for Tricoupler A vary by 55%, indicating a strongly
chromatic signal across the band. While changing the pitch marginally affects
the throughput at either end of the waveband, tuning the interaction length to
balance left and right outputs at 1.55$~{}\mu$m has the effect of fixing the
throughput to the same intensity in the centre of the band. This limits both
the maximum throughput at the central wavelength, and the maximum achievable
achromaticity of the signal: the curves in Figure 7 imply that substantial
gains for the planet light signal might require trades against other desired
properties of the tricoupler.

Figure 7: Throughput of an off-axis source located half a fringe away from the
phase centre for the bright (left, right) and nulled (centre) outputs and
considering three different tricouplers based on the design given in Section
3. While the original, Tricoupler A, has a pitch of $10~{}\mu$m, here we add
tricouplers with pitches of 9$~{}\mu$m and 12$~{}\mu$m. The interaction
lengths of all three tricouplers are designed such that equal splitting occurs
at $1.55~{}\mu$m. Total injected intensity is normalised to 1.

A higher planet light throughput was observed when a tricoupler was tuned such
that the equal splitting condition occurred at a different wavelength, through
changing the interaction length for a given pitch. Figure 8 shows a comparison
between the planet light output of Tricoupler A and a tricoupler whose equal
splitting occurs at 1.7$~{}\mu$m instead of 1.55$~{}\mu$m, with a pitch of
$12~{}\mu$m and a interaction length of 1122$~{}\mu$m, named Tricoupler B.
Here, peak planet light transmission is in the centre of the observing band,
with a value at the centre of the band of 0.88. The total planet light
throughput integrated across the band is 28% higher than for the previous
1.55$~{}\mu$m-tuned tricoupler, and the signal intensity varies by 23%: more
than a factor of two improvement. This latter metric – the yielding of a
flatter spectral response to planet light across the band – makes it better
suited for recovering spectral information for any off-axis stellar companion.

Figure 8: Throughput of an off-axis source located half a fringe away from the
phase centre in the left, centre and right outputs for two simulated
tricouplers: (Blue dashed line) Tricoupler A from Section 3 designed for
1.55$~{}\mu$m, and (Orange solid line) Tricoupler B, which is tailored for
equal splitting at $1.7~{}\mu$m. Total injected intensity is normalised to 1.

The design of the tricoupler for increased planet light throughput also holds
for different positions of the planet relative to its host star, allowing
multiple planet candidates at different separations from their host star to be
observed. Figure 9 shows the throughputs of both tricouplers for a planet
located at different angular separations from the star: at the first bright
fringe, as in Figure 8, corresponding to a 180∘ geometrical phase delay in the
tricoupler; halfway between (ie. located a quarter of a fringe away),
corresponding to a 90∘ delay; an eighth of a fringe away (45∘), and in the
plane of the star (0∘). The closer the planet is to its host star, the less
light is detected in the tricoupler. The planet located on the same axis as
its star, corresponding to the blue dot in Figure 6, is completely nulled
along with the starlight. For planets located on the other side of the star
(to the left of the star in Figure 6), the sign of the incoming phase delay is
flipped, so the curves in Figure 9 are flipped horizontally. Hence, this
design of the tricoupler allows for planet light throughput to be maximised
compared with Tricoupler A, regardless of the location of the planet relative
to its host star.

Figure 9: Throughputs in Tricouplers A (left) and B (right) of light from
planetary companions located at different positions relative to their host
star: on the same axis as the star ($0^{\circ}$ phase delay, blue curve),
$\frac{1}{8}$ of a fringe away ($45^{\circ}$ phase delay, orange curve),
$\frac{1}{4}$ of a fringe away ($90^{\circ}$ phase delay, green curve) and
half a fringe away ($180^{\circ}$ phase delay, red curve). Figure 10: Output
intensities of all three channels of Tricoupler A at three selected
wavelengths: 1.45$~{}\mu$m, 1.55$~{}\mu$m and 1.65$~{}\mu$m. Figure 11: Top:
Map of the intensities in the three outputs of Tricoupler B, against
wavelength and differential phase of incoming beams. Flux at 180∘ differential
phase is shown by the white dashed line. Bottom: Output intensities of all
three channels at three selected wavelengths: 1.45$~{}\mu$m, 1.55$~{}\mu$m and
1.65$~{}\mu$m.

However, the useful extra throughput of Tricoupler B and flattening of the
spectral response for an off-axis source was found to come at the price of
losses in the sensitivity of the phase measurement, ultimately impacting the
fringe tracking capability of the tricoupler. The output intensities for three
selected wavelengths at each end and in the centre of the waveband,
1.45$~{}\mu$m, 1.55$~{}\mu$m and 1.65$~{}\mu$m, are shown in Figure 10. In an
ideal case, the left and right outputs (blue and green) will cross when their
gradients are the steepest, meaning that the differential output flux linearly
and significantly changes with respect to the differential phase. The left and
right outputs of Tricoupler A cross when almost at their steepest, although
less so at shorter wavelengths (Fig. 10).

Figure 11 shows the equivalent output intensities for Tricoupler B, where
equal splitting is designed to occur at 1.7$~{}\mu$m and planet light
throughput is maximised. The differential flux between left and right is
overall shallower than for Tricoupler A when the differential phase is around
180∘ and becomes nonexistent at the centre of the spectral bandwidth. Thus,
phase metrology needed to track the fringes and therefore maintain a stable
null has been compromised compared to Tricoupler A. This sacrifice has yielded
gains as discussed above for the recovery of the planet light signal. For any
given nulling interferometry experiment, tuning between these various
tradeoffs must be informed by practical considerations of the desired science
outcomes, the nature of the targets under study, and the expected performance
metrics of the wider instrumental system.

##  5 Designs for an achromatic phase shifter

Achieving a null fringe in any spectral channel of a beam combiner requires
the control of the relative phase of the input beams by way of some adjustable
piston term. GLINT currently uses a segmented deformable mirror, with each
segment matched to a single input waveguide (in effect, an in-air delay line)
to find and maintain this destructive interference null condition. Such delay
lines are intrinsically chromatic unless they are operated at an optical path
difference of zero. On the other hand, the fundamentals of our design, as
discussed above, requires injected beams to be in anti-phase: a path
difference of $180^{\circ}$. No simple delay system can polychromatically
achieve this non-zero phase relationship.

We propose to replace the air-delay lines with photonic achromatic phase
shifters (APS hereafter) inscribed inside the chip upstream of each
tricoupler. In-guide delays can be effected by adjusting the effective
refractive index, for example by changing the physical properties of the
material or by changing the diameter of the waveguide. These alterations
produce a chromatic effect, as varying delay functions are imposed across the
band. A phase shifter, altering the differential phase between a pair of
waveguides, can be engineered by imposing different changes to one guide
compared to the other. With careful optimisation, the functional form of these
chromatic delays can be balanced to yield a net overall achromatic delay. This
method was successfully used by the European Southern Observatory’s GRAVITY’s
ABCD beam combiner[23], with the design and calculations required extensively
described in Ref. 24.

Following the method used in Ref. 24, each waveguide is fabricated with three
tailored sections where diameter and length are tuned for refractive index
changes. The target outcome for the pair of waveguides is an achromatic phase
shift across the entire $H$ band, with our best solution illustrated in (Fig.
13). The tapered (conical) transitions between sections were added to minimise
the loss of light that would otherwise occur at step-function changes in the
guide structure. The criterion for the minimum taper angle[24], $\theta$, for
light to pass adiabatically between segments is

$\theta(z)<\frac{r(z)}{\lambda_{\text{{max}}}}\Delta n,$ (14)

where $r$ is the radius of the waveguide, $\lambda_{\text{min}}$ the smallest
wavelength in the band and $\Delta n$ the difference between background and
waveguide refractive index. Figure 12 shows a schematic of a waveguide, where
a tapered section transitions between two different radii. From this, the
following relationship can also be found:

$\tan(\theta(z))=\frac{r_{2}-r_{1}}{L},$ (15)

with $r_{2}>r_{1}$ the radii of the two waveguide sections.

Figure 12: Schematic of a waveguide with two different radii, $r_{1}$ and
$r_{2}$, with a tapered section transitioning between these. The angle of the
taper is given by $\theta$.

Using this relationship, a threshold minimum length for each taper section is
given as:

$L>\frac{r_{2}-r_{1}}{\tan\left(\frac{r_{1}}{\lambda_{\text{{max}}}\Delta
n}\right)}.$ (16)

As $L$ depends on the waveguide radii, the minimum length of each taper
depends on the widths of the two waveguide sections on either side of it.
While these taper sections do mitigate losses, they also have the unfortunate
effect of introducing further phase chromaticity[24]. To combat this, each
taper segment was implemented on both guides so that both wavefronts undergo
identical taper-induced chromatic phase effects. The way this is implemented
is shown in Figure 13: for each guide taper sections belonging to the other
guide were added onto the end.

Figure 13: Schematic of the proposed APS. Light is injected at the bottom with
the direction of propagation upward. Final sections uppermost in each
waveguide contain tapers corresponding to those needed by the other guide,
replicated to balance chromatic effects.

To optimise device throughput, light loss over the phase shifter was plotted
while varying the total length as computed by RSoft for the central
wavelength, as well as the upper- and lower-bound wavelengths of the band, and
is shown in Figure 14. Beyond a length of roughly $2.5$ times the minimum
taper length, the loss in the phase shifter decreases with increasing taper
length (corresponding to increasing overall APS length) for all wavelengths,
however it plateaus at a value with diminishing returns past four times the
minimum length. Hence, the optimum taper length adopted was a scaling of four
times the minimum from Equation 16. The total length of the APS including
these taper lengths is $18$ mm. This design is a first step toward a
functional APS made with ULI, hence some parameters may need to change based
on experimental constraints while keeping an achromatic phase shift of 180∘
and for shortening the device.

Figure 14: Total loss of light due to propagation in the APS as a function of
lengths of taper sections in multiples of the minimum length, as defined by
Equation 16. This loss is plotted for the central wavelength ($1.55~{}\mu$m)
as well as the upper and lower-bound wavelengths, $1.4~{}\mu$m and
$1.7~{}\mu$m.

Simulations of the performance of the full, chromatically-symmetrised device
with the lengths of tapered sections at four times the minimum length were
performed in RSoft. The key performance indicator, the phase difference at the
output over the waveband of $1.4-1.7~{}\mu$m, was found to vary by less than
0.6∘, as illustrated in Fig. 15. This allows for an effectively achromatic
phase shift under the expected working conditions.

Figure 15: Differential phase between wavefronts on the output side of the APS
device with taper lengths of four times the minimum length, as a function of
wavelength.

##  6 Integrating components into a fully-functioning device

Since it is designed to be fabricated using the same ULI process as required
by the 3D tricoupler design proposed in Section 3, the APS design above can be
incorporated into the same chip as the tricoupler. A layout of both components
sequentially integrated into a single device is shown in Figure 16. Two beams
of starlight are fed into the left and right inputs at the bottom of the
diagram, firstly undergoing the $180^{\circ}$ achromatic phase shift in the
APS section, and are subsequently combined in the tricoupler section. Finally,
the left, central and right outputs exit the chip at the top, where they would
progress through to a sensor.

Figure 16: Schematic of the proposed combined APS (lower regions of the
diagram) and Tricoupler A (upper region). Figure 17: Power in each output for
the combined APS and tricoupler device, as a function of wavelength. Total
power is shown on a logarithmic scale and normalised to 1.

The full optimised architecture of an integrated device consisting of the APS
and Tricoupler A (as shown in Figure 16) was then numerically tested in the
RSoft package. Results are plotted in Fig. 17, where the simulation has been
framed to illustrate the ideal working condition for the overall device:
injection by way of twin beams of equal intensity and phase. Performance as
shown in Fig. 17 can be seen to be close to ideal: very little light is found
in the central (null) channel, with the maximum output intensity in this
channel at $10^{-4}$ with respect to total input flux. Output signals are
nearly equally split over the left and right guides. Encouragingly, these
design outcomes appear to be nearly achromatic with the same splitting
persisting across the operational waveband.

The total length of the combined device is $24.6$ mm. In RSoft simulations,
the device has a total throughput of $94.5\%$, which takes into account loss
due to waveguide bending, but not propagation through the material. Using an
upper bound for waveguide propagation losses of $0.14\pm 0.04$dB/cm[2], the
total throughput including bend and propagation losses is calculated to be
$87.3\%$.

Once parameters to fabricate a working model have been tuned, a compact laser-
written device implementing the combined phase shifter/tricoupler in Figure 16
can be replicated multiple times within a single chip. This enables nulling
interferometry to be performed on several baselines at once. As the
$180^{\circ}$ phase shift is introduced directly before the tricoupler –
rather than outside the photonic chip as in the current design of GLINT – all
baselines can be nulled simultaneously.

##  7 Conclusion

With further development, nullers like the GLINT instrument have the potential
to reach the scales of contrast and angular resolution needed for the direct
imaging of exoplanets. The use of a photonic tricoupler on the nulling chip
overcomes several significant challenges in reaching this contrast, including
the chromaticity inherent in traditional directional couplers and real-time
metrology to combat phase fluctuations. A new design of the tricoupler has
been proposed, which incorporates full equilateral symmetry for simplicity in
interactions and flexibility in waveguide injection. The benefits of
optimising for either the off-axis light throughput or the fringe tracking
capability have also been explored, with the trade space between these
highlighted and clarified for optimisation as desired specific to the details
of any science application.

To implement the achromatic $180^{\circ}$ phase shift required for nulling, a
photonic phase shifter has been designed, making use of tapered waveguides of
differing diameters to impose tailored variations in the effective index. This
method allows the phase shifter to be integrated into a single chip, directly
upstream of the tricoupler. The phase shifter achieves a $180^{\circ}$ phase
shift with a variation of $0.6^{\circ}$ across the waveband when simulated.
Work remains to be done to shorten the length of the phase shifter in order to
ease the scalability and improve the throughput. The phase shifter and
tricoupler presented here, integrated into a single component, can readily be
replicated several times on a single photonic chip for use in simultaneous
multi-baseline nulling.

### Disclosures

The authors declare no conflicts of interest.

### Code, Data, and Materials Availability

Data underlying the results presented in this paper are not publicly available
at this time but may be obtained from the authors upon reasonable request.

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# A Review of the Applications of Quantum Machine Learning in Optical
Communication Systems

Ark Modi41(A.M.), Alonso Viladomat Jasso42(A.V.J.), Roberto Ferrara1(R.F.),
Christian Deppe1(C.D.),
Janis Nötzel2(J.N), Fred Fung3(F.F.), Maximilian Schädler3(M.S.)  1 Institute
for Communications Engineering (LNT),
2 Emmy Noether Group for Theoretical Quantum Systems Design
Technical University of Munich, D-80333 Munich, Germany
3 Optical and Quantum Laboratory, Munich Research Center
Huawei Technologies Düsseldorf GmbH, Riesstr. 25-C3,80992 Munich, Germany
4 These authors contributed equally to the work
Email: {ark.modi, viladomat.jasso, roberto.ferrara, christian.deppe,
<EMAIL_ADDRESS>
{fred.fung<EMAIL_ADDRESS>
###### Abstract

In the context of optical signal processing, quantum and quantum-inspired
machine learning algorithms have massive potential for deployment. One of the
applications is in error correction protocols for the received noisy signals.
In some scenarios, non-linear and unknown errors can lead to noise that
bypasses linear error correction protocols that optical receivers generally
implement. In those cases, machine learning techniques are used to recover the
transmitted signal from the received signal through various estimation
procedures. Since quantum machine learning algorithms promise advantage over
classical algorithms, we expect that optical signal processing can benefit
from these advantages. In this review, we survey several proposed quantum and
quantum-inspired machine learning algorithms and their applicability with
current technology to optical signal processing.

###### Index Terms:

Quantum Machine Learning, Quantum Algorithms, Quantum Computing, 6G
Communication, Quantum-Classical Hybrid Algorithms, Optical Communication

##  I Summary

Artificial intelligence has made significant progress thanks to modern large-
scale machine learning (henceforth referred to as ML), leading to the
deployment of weakly intelligent cognitive systems in various aspects of daily
and professional life. ML involves adjusting software agent parameters through
training processes, allowing them to develop problem-solving skills. This
progress relies on analyzing large amounts of task-specific training data to
learn desired input-output behaviours. The success of modern ML is largely
attributed to working with domain-agnostic models and training algorithms,
with deep learning being especially successful. Deep learning utilizes
artificial neural networks with billions of adjustable parameters, making them
flexible and effective in various computational intelligence tasks. However,
training deep neural networks requires vast amounts of representative data and
considerable computational resources. To train state-of-the-art systems
effectively, like OpenAI’s GPT-3, dedicated compute clusters and high-
performance computing hardware are necessary due to the immense computational
demands. As a result, the practical feasibility and success of current ML
applications are highly dependent on access to such advanced computing
resources.

Researchers are increasingly exploring quantum computing as a potential
solution to the computational demands of modern ML systems. A “quantum
advantage" can manifest in various ways, primarily affecting time complexity
or execution time, and accuracy. Quantum computing has made significant
strides and promises faster computations in scientific and industrial
applications. A number of works such as [1, 2, 3, 4, 5, 6, 7] claim to achieve
a time advantage while works such as [8, 9, 10, 11, 12] show accuracy and
convergence gains. Quantum computers operate on qubits, which exist in
superposition and can carry more information than classical bits. Computation
with qubits is probabilistic, and measurements cause decoherence, collapsing
the qubit to a specific state. Quantum bits can be entangled, meaning the
state of one qubit affects the state of others. We see (so far) that “quantum”
advantage arises from these two key properties of quantum systems -
entanglement and sampling. Sampling advantage is noticeable in linear
algebraic quantum machine learning (henceforth contracted to QML) procedures;
however, in the Noisy Intermediate-Scale Quantum (NISQ) era, classical
replication of this advantage is possible with only linear slowdowns [13]. The
paradigm changes when Quantum Random Access Memory (QRAM) becomes available
since it allows for the amortization of state preparation costs over multiple
iterations, making QML more efficient. On the other hand, entanglement, a
quintessential quantum phenomenon, endows quantum systems with two primary
advantages: (a) complex correlations: quantum entanglement enables the storage
of intricate correlations within the data, facilitating the efficient
representation of complex relationships in quantum machine learning models.
(b) quantum parallelism: entanglement allows for quantum parallelism, enabling
the simultaneous processing of multiple data points or states, substantially
accelerating some computations. Entanglement allows quantum computers to work
with exponentially larger search spaces, making them particularly useful for
combinatorial optimization in artificial intelligence and certain ML
techniques.

Adiabatic quantum computers, like those produced by D-Wave, are designed for
solving combinatorial optimization problems known as QUBOs, which have
applications in ML tasks like data clustering and support vector machine
training. Adiabatic quantum computing formulates problems as energy
minimization tasks and uses Hamiltonian operators to find their ground states,
which represent solutions to the problems. This approach utilizes the
adiabatic theorem, allowing the system to transition from a known initial
Hamiltonian to the problem Hamiltonian, finding the ground state in the
process. This method also benefits from quantum tunnelling, enabling it to
overcome local minima and potentially solve problems exponentially faster than
classical optimization. Adiabatic quantum computing shares similarities with
the classical paradigm of Hopfield neural networks, making it a viable quantum
analogue of this classical approach.

On the other hand, quantum gate computing manipulates qubits using quantum
mechanical operators, resembling classical digital computing in terms of
gates. Quantum circuits, composed of quantum gates, perform computations on
qubits to achieve specific input/output behaviours. Designing effective
quantum circuits is a challenging task, and classical ML is increasingly used
as a tool for quantum circuit design. Quantum gate computing is attractive for
ML due to its mathematical foundation in complex linear algebra and its
ability to handle exponentially larger state spaces compared to classical
computing. This makes it appealing for tasks involving large high-dimensional
data vectors, as it is expected to offer quantum speedup and potentially solve
intractable problems.

A QML “mini-revolution” has occurred recently, with numerous scientific
reports proposing quantum circuits for various ML tasks, such as linear
algebra routines, regression, and classification. Present-day approaches for
QML involve variational quantum computing algorithms or hybrid quantum-
classical methods. These methods use parameterized quantum circuits with
tunable quantum gates and rely on classical optimization techniques to adjust
the gate parameters for the desired computation.

Variational and hybrid quantum-classical algorithms are appealing because they
reduce the quantum computing resources needed for successful QML. Researchers
also consider parameterized quantum circuits as a quantum analogue of
classical deep neural networks, but there are important differences. Quantum
gates implement unitary operators, not non-linear functions like in classical
neural networks, and reading out the internal states of a quantum circuit
destroys their quantum coherence. As a result, variational or hybrid quantum-
classical algorithms are currently the primary approach for optimizing
parameterized quantum circuits in QML.

The growing literature on QML shows promising potential for mainstream
applications. However, it is essential to temper overly optimistic
expectations, especially given the limitations of current NISQ computing.

Quantum algorithm design abstracts away the limitations of physical qubits
present in current NISQ devices. These devices have less than a hundred
qubits, limited coherence times, and low fault tolerance due to noise and
fluctuations. Creating and maintaining quantum states reliably over longer
periods is challenging but expected to improve with technological
advancements. Promising candidates to look out for include superconducting
qubits and topological qubits. Quantum error correction mechanisms, similar to
those used in classical computing, are crucial for fault tolerance.

Present-day quantum computers face difficulties in handling large quantum
circuits due to error-prone quantum gate operations. Adiabatic quantum
computers offer more reliable manipulation of larger qubit systems, but they
are limited to specific energy minimization problems and lack the universality
of quantum gate computers. Despite theoretical equivalence, emulating quantum
circuits on adiabatic quantum computers requires unrealized qubit connectivity
structures.

There are several practical limitations and challenges that need to be
considered when applying quantum computing to ML tasks. They can be summarised
as follows:

1\. _Encoding and decoding data_ : Quantum algorithms require careful
consideration of how classical data is encoded into quantum states and decoded
back to classical representations. The effort for preparing quantum states and
reading them back into classical memory greatly impacts the quantum advantage,
especially if it becomes exponential.

2\. _Bit-level computing_ : Present-day quantum computing is primarily focused
on bit-level computing, lacking abstract data structures and control
structures found in classical programming. This means that certain ML
algorithms relying on these constructs are not realizable on current quantum
computers.

3\. _Quantum compilers and APIs_ : Efforts are being made to develop quantum
compilers and high-level application programming interfaces (APIs) for quantum
computing. However, these tools are still in their early stages, and users
need to think at the linear algebraic level of quantum computing.

4\. _Simulated quantum processors_ : Some APIs allow for efficient digital
simulations of quantum information processing, but these simulations are based
on universal quantum processors. Algorithms that work on simulated quantum
computers may not necessarily work on existing physical quantum computers.

5\. _Probabilistic nature_ : Quantum computations involving measurements are
inherently probabilistic, requiring repeated runs to obtain results. Any
outcome needs to be interpreted in terms of expectations rather than
deterministic outcomes.

Taking these limitations into account is crucial before making claims about
the superiority of quantum algorithms in ML. Practical implementation
challenges and the current state of quantum computing technology need careful
consideration.

QML is a relatively new field, and its best practices and standards are still
evolving. Similar to the early days of classical ML, QML is currently facing
challenges related to verifiability and reproducibility. In classical ML’s
early stages, practical results were often reported without disclosing
implementation details, data collection or processing protocols, and
experimental procedures, leading to issues with the validity of claimed
capabilities.

Presently, the field of QML often exhibits similar omissions in reporting
crucial details of practical results. There is a lack of transparency in the
scientific literature, making it difficult to evaluate rigorously the methods
and reproducibility of reported outcomes. While this might be somewhat
acceptable for a nascent field, it is essential to consider that the
performance of current QML methods may not scale or generalize well to larger
or different application settings.

To establish trust and credibility in QML, it will be vital for researchers to
adopt practices similar to those in classical ML, providing code, data, and
experimental protocols in their publications. By doing so, the field can
progress towards ensuring the reliability and reproducibility of reported
results and accelerate its growth.

Despite the caveats and limitations of QML, recent technological progress
justifies serious engagement with the topic. While quantum computers and
algorithms are not yet mature enough to impact ML practically, ongoing
development and substantial investments suggest rapid improvement is likely.
As the underlying technology advances, viable solutions may emerge, leading to
unexpected developments and disruptions.

However, potential risks related to QML have not received as much attention as
their benefits. Ethics, reliability, trustworthiness, and safety have been
recognized as important topics in classical ML, but similar scrutiny is yet to
be applied to QML. As QML may significantly impact artificial cognitive
systems, it is crucial to assess potential security issues. Further studies
investigating the reliability, vulnerability, and potential new forms of
attacks or defense mechanisms for critical digital infrastructures in the
context of QML are required. The assessment of QML from a cybersecurity
perspective and a determination of measures to address security challenges is
required.

In summary, QML is a promising, cutting-edge, and complex field that requires
further development and exploration to unlock its full potential. As with any
transformative technology, it will take time, research, and advancements in
hardware and algorithms to fully understand its capabilities and limitations.

##  II Application of QML in Optical Communication

QML is an emerging field that combines quantum computing principles with ML
algorithms to solve complex problems. When applied to optical communication
systems, QML can offer several advantages and applications. Here are some of
the key areas where QML can be beneficial in optical communication systems:

  1. 1.

_Channel Estimation and Equalization_ : In optical communication, signal
distortions can occur due to various factors like dispersion and noise. QML
techniques can be used to estimate and equalize the channel conditions,
enabling more reliable data transmission and improved communication
performance [14].

  2. 2.

_Fault Detection and Error Correction_ : QML algorithms can be applied to
detect and correct errors that arise during data transmission in optical
communication systems. This can enhance the overall reliability and robustness
of the communication network [15, 10].

  3. 3.

_Optimal Resource Allocation_ : QML can optimize the allocation of resources
in optical communication networks, such as determining the best routes for
quantum signals or optimizing the placement of quantum repeaters to extend the
communication distance [16, 17].

  4. 4.

_Adaptive Photonics_ : QML can be applied to adaptive photonics, where the
properties of photons are optimized in real-time to maximize the communication
performance. This can lead to adaptive and self-optimizing optical
communication systems [18].

As mentioned before, it is important to note that QML is still a developing
area, and its practical implementation in optical communication systems is an
active research field. Currently, most of the research output is theoretical
and sometimes not in the NISQ context, making the determination of the state-
of-the-art difficult. However a comparison with well-known methods for
decoding M-QAM optical fibre signals is presented in [10]. As quantum
technologies advance, the integration of QML with optical communication holds
great promise for revolutionizing the way we transmit and process information.

Machine Learning applications in optical communications Proactive fault
detection Optical Performance Monitoring Nonlinearity compensation Software-
defined Networking Quality of transmission estimation Physical layer design
$\bullet$ SVM with DES $\bullet$ SVM $\bullet$ Naive Bayes-based method
$\bullet$ ANN $\bullet$ ANN with shape-based clustering $\bullet$ ANN
$\bullet$ Kernel-based methods $\bullet$ PCA $\bullet$ DNN $\bullet$ LSTM
$\bullet$ CNN $\bullet$ SVM $\bullet$ SVM $\bullet$ ELM $\bullet$ K-Means
$\bullet$ EM algorithm $\bullet$ ANN $\bullet$ BP algorithm $\bullet$ DNN
$\bullet$ DNN with PCA $\bullet$ ANN $\bullet$ Q-learning $\bullet$ Q-network
$\bullet$ SVM $\bullet$ Logic Regression $\bullet$ kNN Clustering $\bullet$
ANN $\bullet$ Random forest $\bullet$ DNN-based end-to-end system optimisation
$\bullet$ ANN-based receiver for NFDM systems $\bullet$ ANN-based receiver for
IM/DD systems $\bullet$ ANN-based Raman amplifier design

Figure 1:  Source: [19]. Summary of Machine Learning applications in Optical
Communication Systems.

Fig. 1 summarises the application of ML in optical communication systems. In
general, in most applications where ML can be used, there exists a competing
QML algorithm. As with any real-world application, the general algorithm has
to be engineered, at an algorithmic, software and hardware level to achieve
the best possible results. A case in point is k-means clustering - if one
replaces the k-means algorithm with a hybrid quantum-classical implementation
of the general quantum k-means algorithm proposed in [6], the ‘advantage’ is
quite questionable, as shown in [15]. However, as demonstrated in [10], with
some engineering modifications and innovation, one can outperform the
classical algorithm. Another important thing to consider is the emerging class
of quantum-inspired algorithms [20, 13, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 4, 31, 32, 33, 1], which use methods and ideas inspired from quantum
computing to optimise classical algorithms. For most of the applications
mentioned in Fig. 1, there exist competing quantum or quantum-inspired methods
that show promise in providing advantage over classical methods. Some of these
algorithms are listed as follows:

  * •

_Neural Networks (NN) / Deep Learning_ \- Quantum Neural Networks (QNN) [34,
35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52]

  * •

_Kernel-based methods_ \- Quantum kernels [12, 53, 54, 55, 8, 56, 57, 58, 59,
9]

  * •

_Principal Component Analysis (PCA)_ \- Quantum PCA[60], Quantum Inspired PCA
[13, 22, 21, 61, 62]

  * •

_Support Vector Machine (SVM)_ \- Quantum SVM (QSVM) [8, 9, 63, 64, 65, 66,
67, 68]

  * •

_Expectation-Maximisation Algorithm_ \- Quantum Expectation-Maximization
Algorithm [69, 70, 71, 72]

  * •

_Classical kNN clustering_ \- Quantum and Quantum-inspired kNN clustering [10,
73, 74, 13]

  * •

_Logic regression_ \- Quantum Logic regression [75, 76, 77, 78]

  * •

_Random Forest_ \- Quantum and Quantum-inspired random forest algorithms [79,
80, 62]

##  III Conclusions

Quantum and Quantum-Inspired Machine Learning is an emerging field with
significant potential and promise. However, like any nascent area of research,
it faces several challenges and limitations. The following points describe the
current state of QML.

  * •

_Early Stage of Development_ : QML is still in its early stages, and
researchers are actively exploring its potential applications and limitations.
Many of the algorithms and techniques are still being developed and refined.

  * •

_Hardware Limitations_ : Building and maintaining quantum computers with a
sufficient number of qubits and low decoherence rates is a very challenging
task. As of now, quantum computers have not reached the level of efficiency
and scalability to outperform classical computers for most ML tasks,
especially when considering an industrial deployment.

  * •

_Questionable Quantum Advantage_ : The field of quantum computing often sees
claims of "quantum advantage," which refers to the act of quantum computers
solving problems faster than classical computers, or even solving problems
that are practically infeasible for classical computers. While there have been
some demonstrations of quantum advantage in specific tasks such as prime
factorisation, it is still a questionable claim in many cases, especially in
the context of NISQ devices. The source of this "quantum advantage" is also
often unclear, and in some cases, the quantum ideas can be replicated
classically to achieve great advantage - as can be seen in the case of
sampling-based quantum-inspired algorithms.

  * •

_Interplay between Quantum and Classical_ : Many ML tasks will always be
performed efficiently using classical methods. QML is expected to have the
most impact only in specific areas where quantum properties provide a
computational advantage, such as solving certain optimization problems or
simulating quantum systems. Quantum systems can be used to compute and store
complex correlations between data, and to parallelise certain computations -
this promises some advantage but the applications have to be engineered
carefully.

  * •

_Algorithm Complexity_ : Implementing and optimizing QML algorithms can be
challenging and computationally expensive.

  * •

_Data Requirements_ : QML algorithms may require a large amount of high-
quality quantum data, which is currently difficult to obtain. Obtaining and
preparing such data for QML tasks can be a significant hurdle. The classical
data loading problem, a result of the unavailability of stable quantum memory,
is a significant issue that often introduces an exponential slowdown in the
hybrid quantum-classical implementations of QML procedures.

While QML faces these challenges, current research and technological
development in this field are ongoing. As quantum computing technology, the
potential for QML to impact various fields, including optimization,
cryptography, and material science, remains a subject of active investigation.

## Acknowledgement

This work was funded by the TUM-Huawei Joint Lab on Algorithms for Short
Transmission Reach Optics (ASTRO). J.N. was funded from the DFG Emmy-Noether
program under grant number NO 1129/2-1 and Munich Center for Quantum Science
and Technology (MCQST). C.D., and J.N. were funded by the Federal Ministry of
Education and Research of Germany in the joint project 6G-life, project
identification number: 16KISK002. C.D., J.N., and A.V.J. were funded the
Munich Quantum Valley (MQV) which is supported by the Bavarian state
government with funds from the Hightech Agenda Bayern Plus. C.D., J.N., and
R.F. were funded by the Bavarian State Ministry for Economic Affairs, Regional
Development and Energy in the project 6G and Quantum Technology (6GQT). A.M.,
and C.D. were funded by the Federal Ministry of Education and Research of
Germany in the project QR.X with the project number 16KISQ028. We acknowledge
useful discussions with Kareem H. El-Safty, and the use of ChatGPT 3.0 as a
language tool.

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