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# Targeting low micro-roughness for 3D printed aluminium mirrors using a hot
isostatic press

Carolyn Atkins  Younes Chahid  UK Astronomy Technology Centre, Royal
Observatory, Edinburgh, EH9 3HJ, UK  Gregory Lister  UK Astronomy Technology
Centre, Royal Observatory, Edinburgh, EH9 3HJ, UK  Dept of Mechanical,
Aerospace & Civil Engineering, University of Manchester, M13 9PL, UK  Rhys
Tuck  UK Astronomy Technology Centre, Royal Observatory, Edinburgh, EH9 3HJ,
UK  Dept of Mechanical, Materials & Manu. Engineering, Uni. of Nottingham, NG7
2RD, UK  Richard Kotlewski  UK Astronomy Technology Centre, Royal Observatory,
Edinburgh, EH9 3HJ, UK  Robert M. Snell  Dept of Materials Science and
Engineering, University of Sheffield, Sheffield, S1 3JD, UK  Elaine R. Livera
Dept of Materials Science and Engineering, University of Sheffield, Sheffield,
S1 3JD, UK  Mariam Faour  Dept of Materials Science and Engineering,
University of Sheffield, Sheffield, S1 3JD, UK  Iain Todd  Dept of Materials
Science and Engineering, University of Sheffield, Sheffield, S1 3JD, UK
Robert Deffley  Dept of Materials Science and Engineering, University of
Sheffield, Sheffield, S1 3JD, UK  Royce Translational Centre, Sheffield
Business Park, Europa Ave, Sheffield S9 1ZA, UK  James Shipley  Quintus
Technologies AB, Västerås, Sweden  Tom Walsh  Quintus Technologies AB,
Västerås, Sweden  Johannes Gårdstam  Quintus Technologies AB, Västerås, Sweden
Cyril Bourgenot  Durham University, NETPark Research Institute, Sedgefield,
TS21 3FB, UK  Paul White  Durham University, NETPark Research Institute,
Sedgefield, TS21 3FB, UK  Spencer Davies  Durham University, NETPark Research
Institute, Sedgefield, TS21 3FB, UK  Samuel Tammas-Williams  School of
Engineering, University of Edinburgh, Edinburgh, EH9 3FB, UK

###### Abstract

Additive manufacturing (AM; 3D printing) in aluminium using laser powder bed
fusion provides a new design space for lightweight mirror production. Printing
layer-by-layer enables the use of intricate lattices for mass reduction, as
well as organic shapes generated by topology optimisation, resulting in
mirrors optimised for function as opposed to subtractive machining. However,
porosity, a common AM defect, is present in printed aluminium and it is a
result of the printing environment being either too hot or too cold, or gas
entrapped bubbles within the aluminium powder. When present in an AM mirror
substrates, porosity manifests as pits on the reflective surface, which
increases micro-roughness and therefore scattered light. There are different
strategies to reduce the impact of porosity: elimination during printing,
coating the aluminium print in nickel phosphorous, or to apply a heat and
pressure treatment to close the pores, commonly known as a hot isostatic press
(HIP).

This paper explores the application of HIP on printed aluminium substrates
intended for mirror production using single point diamond turning (SPDT). The
objective of the HIP is to reduce porosity whilst targeting a small grain
growth within the aluminium, which is important in allowing the SPDT to
generate surfaces with low micro-roughness. For this study, three disks, 50 mm
diameter by 5 mm, were printed in AlSi10Mg at $None$, $None$, and $None$ with
respect to the build plate. X-ray computed tomography (XCT) was conducted
before and after the HIP cycle to confirm the effectiveness of HIP to close
porosity. The disks were SPDT and the micro-roughness evaluated. Mechanical
testing and electron backscatter diffraction (EBSD) was used to quantify the
mechanical strength and the grain size after HIP.

###### keywords:

Additive manufacturing, 3D printing, Mirror manufacture, Lightweight mirrors,
Hot isostatic press, Porosity, Aluminium

##  1 INTRODUCTION

Aluminium alloys have a long heritage in the manufacture of low mass
(lightweight) mirrors for astronomy [1] and Earth observation [2]. There are a
number of benefits for using aluminium in mirror production, for example,
aluminium has a low density
($\sim$$2.7\text{\,}\mathrm{g}\text{\,}{\mathrm{cm}}^{-3}$) in comparison to
other metals commonly used in structures for telescope and instrument
fabrication; aluminium is soft and easy to machine; and when an aluminium
mirror is used in conjunction with aluminium mounts it allows the system to
approach being athermal, where changes in temperature do not cause stress
between mismatching materials. The fabrication of low mass aluminium mirrors
is no longer restricted to subtractive (mill, drill & lathe) or formative
(casting) manufacturing methods, additive manufacture (AM; 3D printing) is a
proven manufacturing method that can offer improved functionally through the
layer-by-layer production, which increases design freedom. The two primary
design benefits of AM in mirror fabrication are: the use of lattices and
optimised geometries to create structures more tailored for the physical
processes and environments that the mirror will encounter; and the ability to
combine multiple, previously individual, components into one, for example,
printing a mirror and the mount as one component, thereby eliminating
interfaces and reducing the system part count. Further practical benefits
include, reduced waste as only the required part is printed, and reduced time
for manufacture of an intricate low mass structure.

To date, AM aluminium mirrors have been printed exclusively using laser powder
bed fusion (L-PBF); in this AM method, a laser is used to melt powdered
aluminium and then fuse it to the previous layer. When the printing is
complete the solid printed part is excavated from the volume of loose un-fused
aluminium powder, which is then recycled in the next print. After printing,
process chains for mirror production diverge depending whether the aluminium
substrate is destined as the reflective surface [3, 4, 5, 6, 7, 8, 9, 10], or
if the substrate is coated in nickel phosphorus (NiP) [11, 12, 13, 4, 14, 15]
if the goal is short wavelength applications and therefore, very low micro-
roughness is required; however, the scope of this paper is where the AM
aluminium substrate is used to create the reflective surface. Typical good
quality micro-roughness values for AM aluminium are between
$4\text{\,}\mathrm{nm}5\text{\,}\mathrm{nm}$ root mean square (RMS, Sq) for
single point diamond turning [15, 4, 5] (SPDT) and $<$$4\text{\,}\mathrm{nm}$
where additional polishing steps have been used [7, 8]. However, poor quality
surfaces are also frequent where the micro-roughness is
$\geq$$10\text{\,}\mathrm{nm}$ RMS [3, 4, 5, 6, 14], the common causes for
poor quality are porosity within the AM substrate, and inclusions which
generate scratches. Porosity is a common defect in AM aluminium and it is a
result of the printing environment being either too hot or too cold, or gas
entrapped bubbles within the aluminium powder [16]. Porosity can be minimised,
or locally eliminated, by optimising the printer parameters to influence the
thermal environment and this is an active area of research [16, 17]. However,
the thermal environment is not only reliant upon the printer parameters, it is
also impacted by the mirror geometry, which adds a further layer of complexity
within the optimisation. Approaching AM aluminium mirrors from the point of
view of lowering cost and increasing accessibility, is there a commercially
available heat treatment that can deliver a defect free reflective AM
aluminium surface?

Hot isostatic press (HIP) is a heat treatment which is commonly applied to AM
metal components to reduce porosity. HIP involves holding the component at
high temperature and high pressure for a prolonged period of time, typically
hours. HIP has been demonstrated to deliver pore free reflective surfaces with
micro-roughness values between $6\text{\,}\mathrm{nm}8\text{\,}\mathrm{nm}$
[9, 10, 18] (average, Sa), but HIP also leads to a degradation from the as-
printed optimum, in addition, incorrect HIP settings can further degrade the
micro-roughness [19]. As HIP successfully removes the AM defects, the increase
in the roughness is likely due to a reduction in the mechanical hardness of
the material caused by a change in the microstructure.

The microstructure in L-PBF is formed during the transition from a liquid melt
pool, when the laser melts the aluminium powder, into a solid structure and,
in metals, crystallographic structures are formed in this transition. These
crystal structures are comprised of an ordered, repeating, three dimensional
array of atoms, and further, many metals are poly-crystalline, with regions of
near-perfect crystal structure separated by boundary regions. Therefore the
microstructure of L-PBF aluminium is an assortment of individual crystalline
structures, commonly termed grains. The grain growth in L-PBF for the common
AM aluminium alloy AlSi10Mg is columnar due to its percentage of silicon (10%)
making it hypoeutectic111Note, high silicon content AM aluminium alloys, such
as AlSi40 which is commonly paired with NiP in mirror fabrication [11, 12,
13], are hypereutectic and do not exhibit the same grain growth., where the
column extends in the build direction (z), this results in a smaller cross
section at $None$ (horizontal; x-y plane), than at $None$ (vertical, z) [20].
The impact of the columnar grain growth is anisotropic properties with higher
mechanical strength in the x-y plane, as mechanical strength is linked to the
grain diameter through the Hall-Petch relationship
($\sigma_{y}~{}\propto~{}\frac{1}{\sqrt{d}}$) [20], where the yield strength
($\sigma_{y}$) is inversely proportional to the square root of the grain
diameter ($d$). The effect on mirror fabrication is the expectation that
reflective surfaces printed normal to the build direction at $None$
orientation would exhibit smaller grains, higher mechanical strength, and
therefore, lower micro-roughness, than alternative orientations - confirmation
of this expectation forms a foundation of this study.

The aim of this paper is to investigate the dependence of print/grain
orientation when using HIP to remove AM defects. The paper builds upon earlier
studies in 2022 [16] and 2023 [19], where the micro-roughness of small
($50\text{\,}\mathrm{mm}$ diameter) mirror samples were evaluated against
print orientation and the application of HIP. This new study utilises
commercially available HIP settings and incorporates an analysis of the impact
of HIP on reducing porosity, the mechanical strength, and the grain size. The
goal of this paper is to identify optimum routes for mirror fabrication using
commercially accessible processes through acknowledging the role that the
microstructure has in optical quality. The experimental methodology is
described in Section 2 with the corresponding results presented in Section 3,
the paper closes with a conclusion and future work in Section 4.

##  2 Method

The 2024 study uses the same sample dimensions and print parameters as those
from the previous studies. The HIP of the samples is undertaken by an industry
partner to ensure that the first iteration of HIP parameters are commercially
available. To assess the impact of HIP in removing porosity within the
samples, X-ray computed tomography (XCT) is used to non-destructively evaluate
the internal structure before and after HIP. SPDT and optical metrology of the
resultant reflective surface after the post-HIP XCT, provides a quantitative
description of how HIP has influenced reflectively. In the final measurements,
the mirror samples are destructively tested to evaluate mechanical strength
through indentation and samples are cut into smaller segments to allow
electron backscatter diffraction (EBSD) to quantify the grain size and crystal
orientation. The process chain is enumerated below.

  1. 1.

Sample design & additive manufacturing

  2. 2.

Rough machining & pre-HIP XCT

  3. 3.

HIP & post-HIP XCT

  4. 4.

SPDT & micro-roughness evaluation

  5. 5.

Mechanical testing & EBSD

###  2.1 Sample design & additive manufacturing

To provide continuity with the previous trial reported in 2022 [16] and 2023
[19], the sample dimensions and the printer settings were identical. The
sample design is a $50\text{\,}\mathrm{mm}$ diameter disk, with a height of
$5\text{\,}\mathrm{mm}$ and a $1\text{\,}\mathrm{mm}$ chamfer applied to the
edge of the optical surface. The sample has three ‘tabs’, which are included
for ease of handling and mounting. The samples were orientated at $None$ to
the build plate surface (horizontal), $None$ and $None$ (vertical); support
material was added to the $None$ and $None$ orientations to remove overhangs,
as shown in Figure 1. The samples were printed using aluminium alloy AlSi10Mg,
on an AconityLAB L-PBF printer (Aconity3D GmbH, Germany) using an aluminium
base plate and in an argon atmosphere. The printer operated using with a
$390\text{\,}\mathrm{W}$ laser power,
$1.1\text{\,}\mathrm{mm}\text{\,}{\mathrm{s}}^{-1}$ laser speed and a
$150\text{\,}\mathrm{\SIUnitSymbolMicro m}$ hatch spacing (the distance
between neighbouring laser paths). Two sets of the three orientations were
printed.

###  2.2 Rough machining & pre-HIP XCT

Post-print, the samples were wire electrical discharge machined from the build
plate and the support material removed through machining. The base and the
eventual optical surface were machined to provide a smooth interface to the
SPDT machine and to remove the rough post-print surface. Fiducials, drilled
markers, were added to each sample to provide a unique identifier relating to
its location upon the build plate and its orientation. Not only do the
fiducials provide a unique identifier, they also act as alignment features
when comparing the pre- and post-HIP XCT datasets. The rough machined samples
are shown in Figure 1.

Figure 1: Upper \- the STL print files highlighting where support was
required. Lower \- the post-HIP rough machined samples prior to SPDT.

XCT measurements of each sample were taken after machining, these measurements
provide a non-destructive method to assess the location, size and quantity of
pores within the printed samples. During XCT, the samples are situated upon a
rotating platform and positioned at a distance from the X-ray source that
allows full illumination on the detector. As the platform rotates in steps,
the resultant images are recorded on the detector and once the sample has
rotated $None$, the detector images are used to reconstruct a volumetric
dataset of the sample. In this study, the samples were measured using a Nikon
MCT 225 (Nikon, Japan) XCT machine with a voltage of
$180\text{\,}\mathrm{kV}$, filament current of
$150\text{\,}\mathrm{\SIUnitSymbolMicro A}$, a $0.5\text{\,}\mathrm{mm}$
copper filter, and a $40\text{\,}\mathrm{\SIUnitSymbolMicro m}$ voxel size.
Only one volumetric dataset per sample was recorded during XCT, which limits
statistical significance.

###  2.3 Hot isostatic press & post-HIP XCT

The 2024 samples underwent HIP with a constant maximum temperature of
$520\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and a maximum pressure of
$150\text{\,}\mathrm{MPa}$ over a period of two hours, this was followed by a
rapid quench in the HIP furnace using URQ® (Uniform Rapid Quenching) followed
by a constant $175\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and
$110\text{\,}\mathrm{MPa}$ over four hours. Figure 2 highlights the
temperature and pressure profiles that the samples were subject to and the
baseline for this study. The HIP atmosphere was argon, and clean HIP
processing was used by the industrial partner, Quintus Technologies, using
patented Quintus Purus® technology. This was to ensure a clean argon
atmosphere to reduce surface contamination. Note, in contrast to the Quintus
HIP parameters, the 2022 & 2023 study (RTC-HIP) used incorrect parameters,
where a higher than standard temperature was recorded during the HIP
($\sim$$570\text{\,}\mathrm{\SIUnitSymbolCelsius}$),

Following the HIP, XCT measurements were repeated to assess the impact of the
HIP parameters. Identical scan parameters were used and the sample fiducials
provided confidence comparing the pre- and post-HIP measurements.

Figure 2: Temperature and pressure as a function to time for the applied HIP
cycle.

###  2.4 Single point diamond turning & micro-roughness

The samples were SPDT at the Centre for Advanced Instrumentation at Durham
University using a Moore Nanotech 250 UPL precision lathe. Approximately
$300\text{\,}\mathrm{\SIUnitSymbolMicro m}$ was removed from the optical
surface to generate the reflective surface, ensuring the removal of open pores
close to the surface, between $0\text{\,}\mathrm{\SIUnitSymbolMicro
m}300\text{\,}\mathrm{\SIUnitSymbolMicro m}$. HIP is only effective at closing
pores if the pore is fully encapsulated by surrounding material, pores
intersecting the surface would not be closed. Figure 3 presents the three
samples after SPDT, qualitatively all the samples demonstrated good
reflectively without the presence of pits (pores) or scratches.

An evaluation of the micro-roughness was performed using a Zygo Zegage Plus
whitelight interferometer (Zygo, USA) sampling 12 locations on each surface.
The measurements represented a square area of length
$417\text{\,}\mathrm{\SIUnitSymbolMicro m}$, to remove low-spatial frequency
distortions associated with form and waviness, an eighth order polynomial
subtracted from the data.

Figure 3: The reflective surface of the three orientations after SPDT.

###  2.5 Mechanical testing & electron backscatter diffraction

Mechanical testing and EBSD was conducted at the University of Sheffield.
Mechanical testing measured the ultimate tensile strength (UTS) of the
reflective surface of the samples. The UTS was measured via profilometry-based
indentation plastometry (PIP) [21] using a PLX-Benchtop (Plastometrex, UK). In
this process an indentation is made using a given load and to a defined depth,
the shape of the indentation is measured using profilometry [22]. The shape of
the indentation is iteratively matched using the finite element method to
determine the plasticity parameters, which in turn provides the stress-strain
curve to determine the yield strength and the UTS. The hardness of the
material can be inferred from the UTS through the linear relationship between
the two characteristics, where Vickers hardness ($H_{V}$) has an approximate
relationship of three times the UTS ($\sigma_{UTS}$) ($H_{V}\approx
3~{}\sigma_{UTS}$) [23]. Due to the physical indentation, PIP is an optically
destructive testing procedure for mirrors.

To evaluate the size and orientation of the individual grains EBSD is used. In
this measurement, incident electrons are scattered off the atoms close to the
surface of the sample, if a crystal structure is present in the sample, the
interactions will produce Kikuchi bands on the EBSD detector, from which
information on the crystal structure and orientation is inferred. As grain
boundaries typically separate regions of different crystallographic
orientation, EBSD can be used to show the position of grain boundaries, and
therefore provide information on the underlying grain size and morphology.

In this study, EBSD was performed in a JEOL 7900F scanning electron microscope
(SEM; JEOL Ltd., Japan) with a focus depth of 21, a probe current (spot size)
of 14, an accelerating voltage of $20\text{\,}\mathrm{kV}$, a specimen tilt of
$None$, and step size of $0.8\text{\,}\mathrm{\SIUnitSymbolMicro m}$. Grain
image and orientation analysis was performed on Matlab (Mathworks, USA) using
the MTex algorithm [24] with a threshold orientation of $>$$None$ to define
the high angle boundaries, and $<$$None$ to define the low angle grain
boundaries. Grains smaller than three pixels were removed before grain
reconstruction, and the resulting grains smoothed. Inverse pole figure (IPF)
maps were created to represent the output from the MTex algorithm.

To extend the EBSD measurement, energy dispersive X-ray spectroscopy (EDS) was
used to provide spatially resolved information on the elemental composition of
a sample to investigate if any regions or features are rich or deficient in
particular elements. In this case, EDS measurements are performed alongside an
EBSD scan using a step size of $0.18\text{\,}\mathrm{\SIUnitSymbolMicro m}$;
all other SEM settings remained constant. In this study, EDS is used clarify
artefacts seen in the EBSD IPF maps.

##  3 Metrology results

This section describes not only the results from the samples processed in 2024
(Quintus-HIP), but also those presented in 2023 (non-HIP & RTC-HIP). In the
case of micro-roughness (Section 3.2) previous data is drawn from 2023 [19],
whereas for mechanical testing (Section 3.3) and microstructure analysis
(Section 3.4) samples from 2022 [16] were tested and new data obtained.

###  3.1 X-ray computed tomography

The analysis of the pre- and post-HIP XCT data was undertaken using Avizo
2022.1 (Thermo Fisher Scientifc, Germany). Accurate quantification of size and
location of pores within an AM material is challenging due how a threshold
value is determined between what is a void and what is the AM material; small
variations in the threshold value can have a significant impact on both the
number and size of the pores [25]. The objective of this XCT comparison is to
validate the success of the HIP to close the pores within the aluminium
samples, rather than an accurate quantification of size and location.

Adaptive thresholding was used to identify the pores, this method determines
local threshold values across the dataset and therefore is less affected by
XCT reconstruction artefacts that locally vary the greyscale of the AM
aluminium - trials with an Otsu threshold and manual thresholds could not
account for a varying global greyscale. The output from the adaptive
thresholding was filtered to remove ‘pores’ that were less than five voxels,
as there is low confidence in identifying pores from measurement noise at this
scale. The analysis is shown in Figure 4, where the upper row highlights the
pre-HIP data and the lower row the post-HIP data, porosity is shown as dark
pixels within the structure. The individual images represent the superposition
of pores within semi-transparent 3D representation. Within the figure, $None$
and $None$ have been orientated so that the top of the image correlates with
the top of the sample with respect to the print orientation.

Figure 4: Porosity identification within pre- (upper row) and post-HIP (lower
row) XCT data. Pores are identified as dark pixels within the semi transparent
3D representation of the sample.

In comparing the pre- and post-HIP data, it is clear that the HIP has been
successful in removing the majority of pores from the sample; however, as
observed, there are features present in the pre-HIP data. A line of pixels is
present in both the $None$ and $None$ pre-HIP data, these lines are seen in
the original reconstructed slices and are artefacts from the reconstruction of
the X-ray images into a volumetric dataset and therefore, are not porosity.
However, the increase in porosity observed in the top portions of the pre-HIP
$None$ and $None$, is seen in the original slices and is considered a real
effect. The $None$ sample provides an indication of the cause given the
increase in porosity occurs between successive print layers. Between the two
layers an unknown error occurred either within the printer itself, or as a
result of a local change in environment. One conjecture is a change in gas
flow, which impacted fume extraction and as a result, led to obscuration of
the laser output and thereby limiting the power delivered to the AlSi10Mg
powder. The $None$ pre-HIP sample does not demonstrate a clear differential
between porosity levels within the orientation shown in Figure 4, this is in
part due to the $None$ print orientation that creates a larger print cross
section ($\frac{5~{}mm}{\sin{45}}$), which when viewed ‘face on’ leads to a
cross section projected with a width of $5\text{\,}\mathrm{m}\mathrm{m}$ (the
disk height), this results in a linear increase in porosity density for
$5\text{\,}\mathrm{mm}$ followed by a uniform density thereafter. In addition,
the support material used for $None$ (Figure 1 b)), does change the thermal
environment in comparison to $None$, which would impact the generation of
pores. Regardless, the objective of this study is the effect of HIP and, as
shown, HIP has been successful at removing porosity.

###  3.2 Micro-roughness

The micro-roughness measurement locations and output are presented in Figure
5. In Figure 5 a) the centre is avoided due to the SPDT residual artefact
commonly produced at the centre of the spindle. Figure 5 c) highlights a
representative micro-interferogram of the optical surface. Table 1 collates
the individual interferograms for comparison and the average, standard
deviation (Std), minimum and maximum for each sample are highlighted in the
final four rows. Three roughness parameters were extracted from the
measurements: Sa - average roughness, Sq - root mean square roughness (RMS),
and Sz - the peak to valley.

The results highlight that HIP has degraded the micro-roughness in comparison
to defect-free as-printed AM AlSi10Mg [19]; however, HIP has successfully
removed the effect of porosity and scratches frequently encountered after
SPDT. In comparing the three build orientations, the $None$ orientation
provides a lower micro-roughness, Sq = $9.3\text{\,}\mathrm{nm}$ RMS, than the
$None$ (Sq = $11.5\text{\,}\mathrm{nm}$) and $None$ (Sq =
$11.3\text{\,}\mathrm{nm}$) orientations.

Figure 5: micro-roughness evaluation - a) the measurement locations, b) a sample under measurement, and c) an representative micro-interferogram. Image credit Durham University; photo credit P. White. Table 1: micro-roughness parameters extracted from the micro-interferograms for the three Quintus-HIP samples printed in orientations $None$, $None$, and $None$. |  $None$ (Horizontal) | $None$ |  $None$ (Vertical)
---|---|---|---
Meas.# | Sa [nm] | Sq [nm] | Sz [nm] | Sa [nm] | Sq [nm] | Sz [nm] | Sa [nm] | Sq [nm] | Sz [nm]
1 | 6.6 | 8.8 | 166.8 | 8.5 | 11.2 | 177.0 | 8.4 | 11.1 | 124.6
2 | 7.1 | 9.3 | 135.0 | 8.4 | 11.1 | 126.7 | 8.5 | 11.1 | 128.2
3 | 7.3 | 9.8 | 427.1 | 8.5 | 11.1 | 116.8 | 8.5 | 11.2 | 140.8
4 | 7.1 | 9.4 | 160.3 | 8.7 | 11.4 | 143.9 | 8.8 | 11.5 | 180.0
5 | 6.9 | 9.2 | 221.1 | 8.7 | 11.5 | 126.0 | 8.6 | 11.3 | 202.2
6 | 6.7 | 9.0 | 348.3 | 9.1 | 12.2 | 264.2 | 8.7 | 11.5 | 205.3
7 | 7.0 | 9.2 | 135.5 | 8.8 | 11.6 | 132.5 | 8.8 | 11.6 | 422.2
8 | 6.9 | 9.0 | 163.3 | 8.7 | 11.5 | 155.7 | 8.7 | 11.3 | 158.9
9 | 6.7 | 9.0 | 248.4 | 9.0 | 11.9 | 237.9 | 8.9 | 11.6 | 144.8
10 | 7.2 | 9.5 | 259.7 | 8.6 | 11.3 | 157.8 | 8.6 | 11.3 | 129.4
11 | 6.9 | 9.4 | 332.1 | 8.4 | 11.1 | 125.2 | 8.8 | 11.4 | 135.2
12 | 7.0 | 9.5 | 362.4 | 8.7 | 11.5 | 119.8 | 8.6 | 11.2 | 138.0
Ave. | 7.0 | 9.3 | 246.7 | 8.7 | 11.5 | 157.0 | 8.7 | 11.3 | 175.8
Std | 0.2 | 0.3 | 95.8 | 0.2 | 0.3 | 45.7 | 0.1 | 0.2 | 79.0
min. | 6.6 | 8.8 | 135.0 | 8.4 | 11.1 | 116.8 | 8.4 | 11.1 | 124.6
max. | 7.3 | 9.8 | 427.1 | 9.1 | 12.2 | 264.2 | 8.9 | 11.6 | 422.2

###  3.3 Mechanical Testing

PIP was used to evaluate the UTS of three post-processed mirror variations:
the non-HIP and RTC- (Royce Translational Centre) HIP mirrors presented in
2023 [19]; and the Quintus-HIP mirrors created in this study. For each
variant, three print orientations were measured, resulting in 9 samples for
comparison. The objective for comparing the three different post-processing
methods was to link the micro-roughness with empirical hardness data. Table 2
presents the average UTS values for the different post-processing and build
orientations; the statistical values, standard deviation (std), minimum and
maximum, are provided to highlight variation across the sample. The number of
measurements per sample is provided in the final row. Figure 6 left provides a
graphical comparison between the different build orientations and post-
processing methods as a function of UTS, and right a photo presents the sample
after test.

Table 2: The ultimate yield strength (UTS) as a function of post-processing (HIP) and build orientation | non-HIP | RTC-HIP | Quintus-HIP
---|---|---|---
UTS [MPa] | 0∘ | 45∘ | 90∘ | 0∘ | 45∘ | 90∘ | 0∘ | 45∘ | 90∘
Ave. | 483 | 496 | 445 | 194 | 166 | 151 | 350 | 341 | 343
Std | 69 | 8 | 15 | 3 | 7 | 4 | 5 | 6 | 3
Min. | 342 | 490 | 430 | 191 | 156 | 147 | 341 | 336 | 339
Max. | 519 | 509 | 464 | 196 | 173 | 155 | 354 | 352 | 348
# meas. | 6 | 5 | 4 | 3 | 5 | 4 | 7 | 7 | 7
Figure 6: left The average UTS measurements, with standard deviations, for the
three post-processing variations and three build orientations, and right a
photo highlighting a sample after indentation (photo credit: R. M. Snell).

Table 2 and Figure 6 demonstrate that the non-HIP mirrors have a higher UTS,
and therefore hardness, than the HIP mirrors. This data is in agreement with
the micro-roughness data that indicate non-HIP mirrors, ignoring the effect of
porosity, exhibit higher reflectivity than HIP mirrors. HIP has the effect of
reducing the UTS/hardness which leads to an increase in micro-roughness and
therefore a decrease in reflectivity. Selecting HIP parameters has a clear
effect on the UTS and the resultant micro-roughness. As mentioned in Section
2.3, the RTC-HIP experienced a higher than standard HIP temperature
($\sim$$570\text{\,}\mathrm{\SIUnitSymbolCelsius}$), which resulted in a
‘milky’ appearance upon the optical surface and a high micro-roughness value.
It can now be shown that the UTS values for these samples have been reduced by
over half the from the equivalent non-HIP values. In contrast, the Quintus-
HIP, which reached a maximum temperature of
$\sim$$520\text{\,}\mathrm{\SIUnitSymbolCelsius}$, has an approximate factor
of two increase in the UTS than the RTC-HIP and this is replicated as an
increase in micro-roughness within the RTC samples.

Statistically, the standard deviation of the measurements indicate, with the
exception of non-HIP $None$, that the UTS is relatively constant across the
samples. The origin of the anomalous data point in non-HIP $None$ is unknown,
it could be a measurement error, but it has the effect of reducing the average
UTS in that sample. Excluding the anomalous point results in an average UTS of
$511\text{\,}\mathrm{MPa}$ with a standard deviation of
$\sim$$7\text{\,}\mathrm{MPa}$.

Comparing between build orientations, there is evidence that the UTS is higher
at $None$ and $None$ in the non-HIP, and this trend is replicated in the RTC-
HIP. The Quintus-HIP has reduced the variation between orientations; $None$
Quintus-HIP is $\sim$$10\text{\,}\mathrm{MPa}$ greater than $None$ and $None$.
However, contrasting with the micro-roughness (Table 1), the $None$ Quintus-
HIP demonstrates a clear improvement in micro-roughness (Sq
$\sim$$9\text{\,}\mathrm{nm}$) versus the $None$ and $None$ (Sq
$\sim$$11\text{\,}\mathrm{nm}$). Due to the marginal difference in UTS
Quintus-HIP data, alternative data sources are required to provide further
insight into the difference seen in micro-roughness between the orientations.

###  3.4 Grain size and orientation analysis

EBSD scans evaluated the three $None$ mirrors; non-HIP, RTC-HIP and Quintus-
HIP; measurements are on-going for $None$ and $None$ orientations. Samples,
approximately $10\text{\,}\mathrm{mm}$ length, $5\text{\,}\mathrm{mm}$ width
and $5\text{\,}\mathrm{mm}$ depth, were extracted from the middle of the
mirror surfaces and polished using a standard metallurgy preparation
procedure. A secondary polishing step was required to remove deformation from
the samples and this was achieved using argon ion-polishing using a PECS II
(Gatan, Inc.). EBSD scans were taken using the JEOL 7900F SEM and the
resultant IPF maps were orientated with respect to the AM build direction (z).

Figure 7: EBSD IPF maps with respect to the build direction for the $None$
mirrors: a) non-HIP, b) RTC-HIP, and c) Quintus-HIP. In the bottom right, d),
the inverse pole figure key is shown, showing the relationship between grain
colour on the plots and the grain orientation. The orientation [001]
represents the AM build direction z.

AM AlSi10Mg shows columnar grain growth in the build direction, the cross
section of which is evidenced by a line of fine grain structure, as
demonstrated in Figure 7 a) of the non-HIP sample. The red coloured grains
indicate that the preferential growth direction of the crystal is oriented
parallel to the build direction, out of the mirror surface. These fine grains
form in the middle of the laser path, where there is the maximum temperature
gradient. Next to the fine grains are elongated grains that have formed at the
edge of the melt pool; these grow within the plane of the mirror surface.

The RTC-HIP shows significant increase in grain size in comparison to the non-
HIP, as shown in Figure 7 b). The majority of grains in this map are an order
of magnitude greater in cross sectional area than the non-HIP. As observed,
the grain characteristics from the AM process, the columnar growth and melt
pool edge grains, have been removed during the RTC-HIP. In contrast, the
Quintus-HIP (Figure 7 c)) contains grains closer in appearance to the non-HIP
sample, with evidence of columnar fine grains (red) and melt pool edges. There
are fewer fine grains than in the non-HIP sample, but the grain coarsening is
minimal in comparison to the RTC-HIP.

EDS was used to identify the elemental composition of the small granular
artefacts observed within the large grains in the RTC-HIP IPF map (Figure 7
b). The dominant elemental compositions are shown in Figure 8 and it should be
noted that the brightness of each image cannot be used to provide comparative
information on the amount of each element present. From the EDS data and
analysis using Thermo-Calc Software, there is confidence that the artefacts
observed in the RTC-HIP grains are Mg2Si or Si precipitates mis-indexed to the
aluminium phase. The precipitates are visible as dark regions in the SEM image
in Figure 8 and the majority are silicon rich and aluminium deficient; this is
expected given the composition of the alloy (Al balance & Si $\sim$10%) and
the elevated temperatures involved in the HIP process. Literature suggests
that the EDS observed precipitates have a high likelihood of being Si
precipitates [26, 27, 28], especially given the low magnitude of Mg observed
(Figure 8 bottom right).

Figure 8: EDS images of the precipitates from the RTC-HIP mirror surface. The
top right and bottom row images represent element scans, as labelled. High
concentrations of elements are shown as bright colours, with lower
concentrations as black. The top left represents a standard SEM image, showing
where the element distributions originated.

###  3.5 Discussion

By using a multi-metrology approach, the impact of build orientation and HIP
has been investigated towards developing a strategy for aluminium AM mirrors.
XCT demonstrated the effectiveness of HIP in removing internal porosity and
loose material, which led to SPDT mirrors with no pits or scratches. However,
although a consistent surface quality was achieved, evidenced by the low
standard deviation in the micro-roughness measurements, the roughness was
higher than that exhibited in the areas with no defects within the non-HIP.
Mechanical testing and microstructure analysis (EBSD & EDS) highlighted the
loss in mechanical strength (UTS) through the increase in grain size after HIP
and demonstrated the importance of correct HIP parameters to minimise grain
growth and to avoid precipitates.

To date, the impact of build orientation is only partially determined;
however, it is hoped that future EBSD measurements will provide microstructure
data consistent with the expected columnar grain growth with respect to the
build orientation. The micro-roughness data demonstrated a noticeable
improvement between the $None$ and $None$ & $None$ orientations and, with
lower confidence, this was confirmed in the UTS data. Linking microstructure
with micro-roughness, would provide a useful design metric in AM mirror
manufacture, where if the priority is optimum micro-roughness after HIP, then
a mirror design designed for a $None$ orientation would be preferred. However,
there is likely to be a trade-off between optimum micro-roughness and the
practicalities of AM designs rules and the lightweighting requirements of the
AM mirror.

##  4 Conclusion & future work

The goal of this paper was to add fundamental knowledge towards a strategy for
using AM to create low mass mirrors and, specifically, this paper adds
knowledge for the wavelength applications from visible (VIS) $\longrightarrow$
infrared (IR), where micro-roughness requirements are less challenging.
Aluminium continues to be a versatile material for low mass mirror fabrication
and AM will allow new opportunities to enhance design for function; however,
AM defects need to be considered. From the results presented, and supported
via the summary of AM Al mirrors in the Introduction (Section 1), HIP is a
quick, low cost, and effective heat treatment to deliver a uniform micro-
roughness across the optical surface, but due to the increase in grain size,
and consequently Sq value, applications are currently limited to the near-IR
($750\text{\,}\mathrm{nm}1400\text{\,}\mathrm{nm}$) - assuming $<$5% scatter
is a requirement of the mirror or system. To adopt AM Al mirrors in VIS
applications, alternative approaches are required to minimise the impact of AM
defects in the reflective surface.

Minimising AM defects in aluminium without HIP can be split into two
approaches: AM print parameters, and post-processing. AM print parameters can
be adapted for specific attributes, for example, the laser path and power can
be adjusted to create an optimum print to approach zero porosity [16]; or the
parameters can be adjusted to reduce porosity in a given location, for
example, near the optical surface [17]. However, although this approach has
been proven on sample cubes and cylinders, a low mass mirror substrate creates
a different thermal environment and an optimised setting on a cube does not
necessarily translate into an optimised setting on a low mass mirror geometry.
The AM print parameters approach is desirable as it removes porosity from the
source, but it should be noted that commercial AM bureaux are often reluctant
to adjust AM print parameters, which limits broad availability. The second
approach, post-processing, accepts porosity as an attribute of AM and adds a
further post-processing step. There is evidence that shot peening, or bead
blasting, is an effective process in reducing porosity between
$0\text{\,}\mathrm{\SIUnitSymbolMicro m}500\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ from the surface [29]. To implement this process effectively in mirrors,
shot peening should occur after rough machining and before SPDT.

In addition, there are two approaches that focus on minimising the
microstructure: underbuild for HIP, and grain refiners. The first approach
actively under-builds a sample with high porosity but fine microstructure, and
then uses HIP to densify the material. The aim in this approach is to target a
post-HIP microstructure comparable to a commercially printed microstructure.
The final approach is the addition of grain refiners to the AlSi10Mg powder to
limit grain growth. Previous research has demonstrated the potential of TiB2
and LaB6 to limit grain growth in AlSi10Mg [20]; however, how these grain
refiners would affect SPDT and the reflective surface is unknown.

Future work will focus on completing the microstructure analysis of the $None$
and $None$ orientations to provide clarity on the role of build orientation to
affect micro-roughness. With a complete experimental data package for the
Quintus-HIP, a second HIP iteration is planned with bespoke settings with the
aim to limit grain growth. Proof-of-concept studies exploring the potential of
shot peening to limit porosity in the vicinity of the SPDT layer are
anticipated in 2024. Research is on-going to adapt the AM method specifically
for mirror development, by optimising the print parameters and, in the future,
the application of grain refiners. In the long-term, the research presented in
this study needs to feedback into a practical application and to be applied in
low mass mirrors intended for space-based astronomy and Earth observation. In
this feedback, space qualification and optical requirements need to be
considered, particularly in how the different approaches might impact optical
performance in a thermally varying, harsh environment over time.

###### Acknowledgements.

The authors acknowledge the UKRI Future Leaders Fellowship ‘Printing the
future of space telescopes’ under grant # MR/T042230/1 and the Henry Royce
Institute for Advanced Materials for access the Aconity3D AconityLAB
facilities at The Royce Discovery Centre at the University of Sheffield,
funded through EPSRC grants EP/R00661X/1, EP/S019367/1, EP/P02470X/1 and
EP/P025285/1. Collaborators from the University of Sheffield acknowledge the
EPSRC Future Manufacturing Hub in Manufacture using Advanced Powder Processes
(MAPP) EP/P006566/1.

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# Resource Reduction in Multiplexed High-Dimensional Quantum Reed-Solomon
Codes

Shin Nishio<EMAIL_ADDRESS>Department of Informatics, School of
Multidisciplinary Sciences, SOKENDAI (The Graduate University for Advanced
Studies), 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo, 101-8430, Japan  Quantum
Information Science and Technology Unit, Okinawa Institute of Science and
Technology Graduate University, Onna-son, Kunigami-gun, Okinawa 904-0495,
Japan  National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku,
Tokyo, 101-8430, Japan  Nicolò Lo Piparo  Quantum Information Science and
Technology Unit, Okinawa Institute of Science and Technology Graduate
University, Onna-son, Kunigami-gun, Okinawa 904-0495, Japan  National
Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo, 101-8430,
Japan  Michael Hanks  QOLS, Blackett Laboratory, Imperial College London,
London SW7 2AZ, United Kingdom  William John Munro  NTT Basic Research
Laboratories & NTT Research Center for Theoretical Quantum Physics, NTT
Corporation, 3-1 Morinosato-Wakamiya, Atsugi, Kanagawa, 243-0198, Japan.
National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo,
101-8430, Japan  Kae Nemoto<EMAIL_ADDRESS>Quantum Information Science and
Technology Unit, Okinawa Institute of Science and Technology Graduate
University, Onna-son, Kunigami-gun, Okinawa 904-0495, Japan  National
Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo, 101-8430,
Japan  Department of Informatics, School of Multidisciplinary Sciences,
SOKENDAI (The Graduate University for Advanced Studies), 2-1-2 Hitotsubashi,
Chiyoda-ku, Tokyo, 101-8430, Japan

###### Abstract

Quantum communication technologies will play an important role in quantum
information processing in the near future as we network devices together.
However, their implementation is still a challenging task due to both loss and
gate errors. Quantum error correction codes are one important technique to
address this issue. In particular, the Quantum Reed-Solomon codes are known to
be quite efficient for quantum communication tasks. The high degree of
physical resources required, however, makes such a code difficult to use in
practice. A recent technique called quantum multiplexing has been shown to
reduce resources by using multiple degrees of freedom of a photon. In this
work, we propose a method to decompose multi-controlled gates using fewer
$\rm{CX}$ gates via this quantum multiplexing technique. We show that our
method can significantly reduce the required number of $\rm{CX}$ gates needed
in the encoding circuits for the quantum Reed-Solomon code. Our approach is
also applicable to many other quantum error correction codes and quantum
algorithms, including Grovers and quantum walks.

Quantum Communication, Quantum Error Correction, Quantum Multiplexing, Quantum
Networking

##  I Introduction

Quantum communication systems utilizing the principles of superposition and
entanglement will provide new capabilities that we can not achieve in our
current telecommunication counterparts [1, 2, 3, 4, 5]. Such communication is
expected to provide the basis for distributed quantum information processing
systems, including quantum key distribution [6, 7], blind quantum computation
[8, 9], distributed quantum computation [10], quantum remote sensing [11], and
the quantum internet [12]. However, the fragile nature of quantum states as
they propagate through such channels will severely limit the performance of
the systems. Quantum error correction codes (QECCs) are a fundamental tool to
address these issues as they can correct both channel loss and general errors
(gate errors, for instance).

Since the first QECC was found by Shor [13], quite a number of codes have been
proposed, including the stabilizer codes [14], quantum low-density parity-
check codes [15], and GKP codes [16]. Further, the Calderbank-Shor-Steane
(CSS) codes [17] are widely studied because they are quantum codes that can be
constructed using a variety of classical codes and inherit the good properties
of the existing classical codes [17]. For quantum computation, topological CSS
codes, including the surface codes [18] have known useful properties such as
high thresholds and capability of practical transversal gates due to the
locality of stabilizers. For communication and memory, the delay introduced in
the decoding is not as important as it is in computation [19], so complex
decoding circuits are acceptable compared to those typically used in the
computation. Therefore, for quantum communication and quantum memory-related
tasks, code rate and minimum distance are more important than the locality of
the stabilizers. It is known that the Quantum Reed-Solomon code [20] is one of
the most efficient codes for quantum communication from that point of view.

The classical Reed-Solomon (RS) code [21] has been used in various
communication systems [22, 23] due to it being a maximum distance separable
(MDS) code [24]. MDS codes are important because they can detect and correct
the greatest number of errors for a fixed codeword length $n$ and message
length $k$. In the RS code, multiple consecutive bits correspond to an element
of a Galois Field (GF) [21]. This allows this code to be particularly
resilient to a class of errors called burst errors (errors occurring in
consecutive bits) [22], where multiple bits are used to represent a single
element of the GF. Now the Quantum Reed-Solomon (QRS) codes [20] are known to
be one of the most efficient CSS codes because of their excellent ability to
correct qudit loss errors, which are the most critical issue in long-distance
quantum communication in optics [25]. However, QECCs require significant
physical sources (both in terms of the numbers of photons and qubits) for
their realization, making their implementation quite challenging. Recently
[26] it was shown that quantum multiplexing allows one to reduce the number of
resources when applied to the redundancy code [27] and small scale QRS codes.
This was achieved by using multiple degrees of freedom of the photon. Here we
show that by using such quantum multiplexing techniques, we can drastically
reduce the number of controlled-X ($\rm{CX}$) gates required to implement the
encoding circuit of QRS codes that could be used in quantum communication
tasks.

Now our paper is organized as follows. In Section II, we will present the
construction of QRS codes and their coding circuits. We then briefly review in
Section III the quantum multiplexing technique and propose a method to
implement multiple controlled gates ($C_{k}X$ gates) using fewer $\rm{CX}$
gates by utilizing quantum multiplexing. Section IV then shows the degree of
gate reduction that can be achieved when quantum multiplexing is applied to
the encoding circuit of the QRS code. The results and applications of the
proposed method to other quantum tasks are discussed in Section V.

##  II The Quantum Reed-Solomon Code

Figure 1: The encoding circuit for the $[[5,1,3]]_{5}$ Quantum Reed-Solomon
code using $5$ dimensional qudits. The two qudit gates are defined by the
black dot representing the control and the target with a square box containing
the “+” symbol represent the SUM gates. The discrete Fourier transform gate is
represented by the DFT labeled box.

Let us begin with a brief overview of the Quantum Reed-Solomon code. The
$[[d,2K-d,d-K+1]]_{d}$ QRS code [20, 28, 29, 30] is defined by the CSS
construction [17] of the classical RS code in which $d$ (a prime number) is
the dimension of the qudits used to encode the logic states. Further, $2K-d$
is the number of logical qudits, while $d-K+1$ is the minimum distance of the
code. In our work, we will consider the $2K-d=1$ case, which has the ability
to retrieve the original encoded quantum information when $(d-1)/2$ or less
qudits are lost. The encoding quantum circuit for the QRS code shown in Fig.1
can be implemented using a series of quantum SUM gates [16] and a discrete
Fourier transform (DFT) gate [31]. The SUM gate is a generalization of the
$\rm{CX}$ gate for $d$-dimensional qudit and is given by:

$\textrm{SUM}(\ket{A}\ket{B})=\ket{A}\ket{(A+B)\,[\textrm{mod}\,d]}$

where $A$ and $B$ are integers $\leq d-1$. Next the DFT gate is a
generalization of the Hadamard gate, which when applied to a single qudit,
creates a superposition of states given by:

$\textrm{DFT}(\ket{0})=\frac{\ket{0}+\ket{1}+\cdots+\ket{d-1}}{\sqrt{d}}$

To illustrate how these operations work, Fig.1 shows an example of the
encoding circuit for the $[[5,1,3]]_{5}$ code.

The number of SUM gates required to create the $[[d,1,d+1/2]]_{d}$ code using
a $d$-dimensional qudit is $(d^{2}+d-4)/2$, and is plotted in Fig.2. This
number increases quadratically with the qudit dimension. Further, the number
of $\rm{CX}$ gates used in each SUM gate also increases with the qudit
dimension (as we will show later), making the implementation of higher-
dimensional QRS codes more difficult. For our purposes, it is more convenient
to encode each logical $d$-dimensional qudit of the QRS code with $k$ qubits
where $2^{k-1}<d<2^{k}$. The SUM gate is essentially a modulo adder and can be
decomposed into two parts. The first part is the Ripple carry adder (RCA) [32]
that performs the following transformation:

$\textrm{RCA}(\ket{A},\ket{B})=\ket{A}\ket{(A+B)\,[\textrm{mod}\,2^{k}]}.$

The second part is a modulo operation that performs the following
transformation:

$\textrm{Mod}(\ket{A},\ket{(A+B)\,[\textrm{mod}\,2^{k}]})=\ket{A}\ket{(A+B)\,[\textrm{mod}\,d]}.$

In such a case, the RCA part adds two binary numbers having $k$ bits. This
part utilizes auxiliary qubits called “carry” to store the outcome of the
addition of two qubits having both values $1$. Further, the modulo operation
can also be split into two distinct elements. In the first element, a series
of logic gates check whether the outcome of the RCA exceeds $d$. In that case,
the result will be stored in an auxiliary qubit we label “check if”. A
specific example of how these auxiliary qubits have been used is presented in
Appendix A. The second element, which we label as a conversion element,
transforms the output of the RCA part in $[\textrm{mod}\,2^{k}]$
representation to the desired output of the SUM gate with $[\textrm{mod}\,d]$
representation.

Figure 2: Required number of SUM gates for encoding the $[[d,1,(d+1)/2]]_{d}$
QRS codes versus qudit dimension $d$. The dots represent prime $d$ values.

We need to consider this in a little more detail and so let us look at the
various outcomes of the RCA, which correspond to our three different cases. In
the first case, where the outcome of the RCA is less than $d$, the conversion
element will not change this outcome; hence no further action is needed. In
the second case, where the outcome is greater than $d-1$ but less than
$2^{k}$, “check if” auxiliary qubits are used for storing each value. Those
values are then used to convert the outcome to the desired state in the
conversion element. In the third case, where the outcome is greater than or
equal to $2^{k}$, “check if” auxiliary qubits will be used to store each
value, and the biggest “carry” qubit is also used to distinguish the value and
the value minus $2^{k}$ in the conversion element. However, the “check if”
qubit is not required when the outcome is $2^{k}$ if and only if
$2(d-1)=2^{k}$. This is because the biggest “carry” can be used as the “check
if” value since $2^{k}$ is the only one value that is bigger than $2^{k}$.
This SUM gate construction requires $k+d-2$ auxiliary qubits where $k$-qubits
are for the “carry” and $d-2$ qubits are for the “check if”.

The total number $N_{\rm{SUM}}$ of $\rm{CX}$ gates for each $\rm{SUM}$ gate in
a $d$-qudit system with $2^{k-1}<d\leq 2^{k}$ is given by:
$N_{\rm{RCA}}+N_{\rm{M}}$, where $N_{\rm{RCA}}(N_{\rm{M}})$ are the number of
gates for the RCA (modulo) parts, respectively. The number of gates required
by those are presented in the TABLE 1 where $C_{k}X$ is the $X$ gate
controlled by $k$-qubits ($C_{2}X$ is controlled by $2$ and is the well known
Toffoli gate), $C_{k+1}X$ gate is the $X$ gate controlled by $k$ qubits + 1
“carry” auxiliary qubit.

| $C_{k+1}X$ | $C_{k}X$ | $C_{2}X$ | $C_{1}X$ | Condition
---|---|---|---|---|---
$N_{\textrm{RCA}}$ |  |  | $3k-2$ | $2k-1$ |
$N_{M}$ |  | $d-1$ |  | $\displaystyle\sum_{i=d}^{2(d-1)}H_{D}(i,i\,[\textrm{mod}\,d])$ | $\rm{if}\,2(d-1)\leq 2^{k}$
| $2d-2^{k}-1$ | $2^{k}-d$ |  | $\displaystyle\sum_{i=d}^{2(d-1)}H_{D}(i,i\,[\textrm{mod}\,d])$ | $\rm{if}\,2(d-1)>2^{k}$
Table 1: The number of gates $N_{\textrm{RCA}}(N_{M})$ required for the RCA
(Modulo) operations in the $\rm{SUM}$ gate, respectively. $H_{D}(a,b)$ is a
function that returns the Hamming distance between the binary representations
of the input vector $a$ and $b$.

Now in Fig.3 we plot $N_{\textrm{RCA}}$ versus $d$. Since the RCA part is a
simple $k$ bit ripple carry adder, $N_{\textrm{RCA}}$ depends only on integer
$k$ for $2^{k-1}<d<2^{k}$ which is shown as the gray line. When $d$ is
sufficiently large, $N_{\textrm{RCA}}$ can be negligibly small compared to
$N_{M}$, and so $N_{\textrm{SUM}}$ is dominated by $N_{M}$.

In the modulo conversion part, the $C_{k}X$ gate can be decomposed into
$4(k-2)$ $\rm{C_{2}X}$ gates and the $C_{k+1}X$ gate can be decomposed into
$4(k-1)$ $\rm{C_{2}X}$ gates as shown in [33]. Further the $\rm{C_{2}X}$ gate
can be decomposed into $6\,\rm{CX}$, $2\,H$, $3\,T^{\dagger}$ and $5\,T$
gates. We will refer to this decomposition as a “general decomposition” and
compare it to our more efficient multiplexing decomposition later on. In the
next section, we will show that applying quantum multiplexing can drastically
reduce the number of $\rm{CX}$ gates required to implement these circuits. The
details of the modulo adder implementation are described in Appendix A.

Figure 3: The number $N_{\rm{RCA}}$ of $\rm{CX}$ gates required for performing
the RCA part of the SUM gate for $d$-dimensional qudits. The vertical gray
lines correspond to the $2^{m}$ integer values, while the dotted lines between
data points are used as a guide for the eye.

##  III Quantum Multiplexing

The quantum multiplexing technique [34] aims at reducing the number of
physical resources needed for a quantum communication system by using multiple
degrees of freedom (DOF) of the single-photon to carry and process
information. For instance, applying quantum multiplexing to error correction
codes permits one to reduce drastically the number of single photons and
qubits required to encode the information shared by Alice and Bob [34].
Another remarkable feature of quantum multiplexing is that it enables us to
substitute a Toffoli gate [35], which normally requires $6\,\rm{CX}$ gates,
with a single $\rm{CX}$ gate between two multiplexed photons (or no $\rm{CX}$
gates on a single photon). This is done by splitting one DOF of a photon into
two separate spatial modes and applying a $\rm{CX}$ gate directly on one mode,
as shown in Fig.LABEL:multiplexing_and_circuit(a). So the natural question is
how we apply it to our situation.

Figure 4: (a) Quantum circuit showing the $\rm{C_{2}X}$ gate between two
photons (left) and a multiplexing circuit in which the $\rm{C_{2}X}$ gate is
realized by an optical switch (OS) and a $\rm{CX}$ gate (right). (b) The
$\rm{C_{2}X}$ gate a photon having as control two timebins and another photon
having as target the polarization DOF (left) and its optical implementation
(right). (c) The $C_{k}X$ gate between two photons (left) and its optical
implementation (right).

In our work, we first generalize this method to a $\rm{C_{2}X}$ gate between a
photon having as control two timebins and another photon having as target the
polarization DOF as shown in Fig.LABEL:multiplexing_and_circuit(b).
Furthermore, we generalize the result of [35] by assuming that we can also
substitute the $C_{k}X$ gate and the $C_{k+1}X$ gate by a single $\rm{CX}$ and
a $\rm{C_{2}X}$ gate respectively when quantum multiplexing is applied. To
this aim, we require multiple optical switches OS (if the time-bin DOF is
being used) to split each component belonging to the photon having multiple
controls into single spatial modes. Then a single $\rm{CX}$ is applied between
the relevant component and the target photon. Finally, a series of optical
switches will restore the components into a single spatial mode, as shown in
Fig.LABEL:multiplexing_and_circuit(c). Our assumption can be proved by using
the induction principle shown in Appendix C.

We use this result then to substitute each $C_{k}X$ and $C_{k+1}X$ gates
required in the modulo conversion part of our encoding scheme with a single
$\rm{CX}$ gate and $\rm{C_{2}X}$ gate, respectively. Note that $C_{k+1}X$
requires a $\rm{C_{2}X}$ gate instead of a $\rm{CX}$ gate since $C_{k+1}X$ has
a control qubit in the auxiliary photon. We will refer to this decomposition
as a “multiplexing decomposition”. This will considerably decrease the number
of gates used, as we show in the next section.

##  IV Performance

Figure 5: (a) Plot of the required number of $\rm{CX}$ gates $N_{SUM}$ for the
construction of a single SUM gate versus prime number $d$, the size of the
qudit. The blue curve is based on the general decomposition [33] while the
magenta curve is based on Ralph et al. [36] and the red curve uses the
multiplexed decomposition shown in Fig.LABEL:multiplexing_and_circuit. (b)
Plot of the total number of $\rm{CX}$ gates $N_{tot}$ for constructing the
whole encoder of the $[[d,1,(d+1)/2]]_{d}$ QRS code versus $d$. (c) The ratio
$R$ between the blue curve and the magenta (red) curve plotted in (a),
respectively. The vertical gray lines correspond to $2^{m}$ for integer values
of $m$. (d) Plot of the required number of $\rm{CX}$ gates $N_{check-if}$ for
the “check if” part versus $d$ using quantum multiplexing.

Our focus now will be to compare the number of gates required to encode a QRS
code versus its dimension when single-mode photons and multiplexed photons are
used. The blue (red) curve of Fig.5(a) shows the total number of $\rm{CX}$
gates required to implement a single SUM gate for a $d$-dimensional QRS code
when single-mode (multiplexed) photons are used. The blue curve increases
rapidly with the dimension of the code reaching $21182$ $\rm{CX}$ gates
required for $d=139$ whereas the red curve increases modestly with the code
dimension (using only $1049$ at $d=139$). Thus, applying quantum multiplexing
has resulted in a significant reduction in CX gates. Moreover, for the
multiplexing case, at $d=67$ and $d=137$ the number of $\rm{CX}$ gates is
slightly less than $d=61$ and $d=131$, respectively. Although it seems an
inconsistent result, it can be explained by looking at the second case of
TABLE 1. The number of $C_{k}X$ gates and the number of $C_{k+1}X$ gates are
present in the “check if” part of the modulo conversion scheme. They reduce to
$CX$ and $C_{2}X$ gates when multiplexing is applied. Therefore, the total
number of $\rm{CX}$ gates will be much more affected by $N_{C_{k+1}X}$ than by
$N_{C_{k}X}.$ These numbers depend on the number of qubits $k$ used to encode
the states as well as on the dimension of the code, $d$. When $d\lesssim
2^{k}$, we require several “check if” auxiliary qubits to implement the modulo
addition meaning $N_{C_{k+1}X}$ will be high. On the other hand, when $d\ll
2^{k},$ $N_{C_{k}X}$ will be the preponderant term as we can now e use more
qubits for the sum and less “check if” qubits. Although it is not clear from
Fig.5(a) we also have that $N_{C_{k+1}X}(d=31)>N_{C_{k+1}X}(d=37).$ This
behavior is more evident in Fig.5(b), which shows the total number of
$\rm{CX}$ gates required to construct the whole encoder. The gate reduction by
quantum multiplexing is also significant in the overall cost of the encoding
circuit. Regardless of the value of $d$ we always require a number of
conversion gates proportional to the code dimension. Thus, the number of
conversion gates represented by the Hamming distance in TABLE 1 is not a
relevant term for determining the behavior.

Next Fig.5(c) shows the ratio of $N_{SUM}$ of the non-multiplexed case (blue)
over the multiplexed case (red). It allows us to quantify the improvement we
have. Here we can see that the curve “jumps” to a much higher ratio value when
it crosses the gray lines, which correpond to $2^{k},$ with integer $k=2..7.$
We label the prime numbers in which this happens by $d_{\textrm{cross}}$. At
$d>d_{\textrm{cross}}$ the ratio decreases as shown in regions ($d=32$ to
$63$, $64$ to $127$, and $128$ to $255$) of Fig.5(c). This is due to the
higher number of $\rm{C_{2}X}$ required as $d$ increases as already explained
in the previous paragraph. However, at $d>19$ the ratio increases slightly
until $d=31$ due to the fact that for this range of $d$ the number of “check
if” increases only slightly as shows in Fig.5(d). We also compare our results
with the ones obtained by [36] shown by the magenta curves in Fig.5. In [36],
the authors realize $C_{k}X$ gates by introducing one k-dimensional qudit,
$2k-1$ two qubit gates and single qudit gates they refer to $X_{a}$ and
$X_{b}$. Our results still show an advantage compared to the ones proposed in
[36] in terms of gate reductions. Furthermore, the system proposed in [36]
require special gates $X_{a}$ and $X_{b}$ which might require more two-qubit
gates when qubits representation is in use. We also determined that the
encoding circuit for $[[d-1,d-2t,2t+1]]_{d}$ QRS codes where $d=2^{m}$ for
integers $m$ does not require Toffoli or $C_{k}X$ gates, hence quantum
multiplexing does not give any advantage in terms of gates reduction for these
special cases. We show in the Appendix B an example of an implementation of a
QRS encoding circuit over GF($2^{m}$).

There is a well-known method [37] for constructing efficient encoding circuits
for stabilizer codes, but this method requires multiple-target gates but not
multiple-controlled gates. Therefore, the proposed method cannot be directly
applied to such encoding circuits. The proposed method can be applied to the
encoding circuit of quantum error correcting codes using multiple controlled
gates.

##  V Discussion

In this work, we have applied quantum multiplexing to the encoder of the
Quantum Reed-Solomon and have shown that it is possible to dramatically reduce
the number of controlled X gates utilized. Further, this method can be used to
implement the encoder of QRS with fewer $\rm{CX}$ gates when the dimension of
qudit $d$ is close to $2^{k}$ where $d$ is a prime number, and $k$ is greater
than 6. This reduction of the number of gates should make the implementation
of QRS codes more feasible - especially in the quantum communication setting.
Another method of realizing $C_{k}X$ gates is to decompose the $C_{k}X$ gates
directly into optical elements instead of $CX$ or $C_{2}X$ gates, which has
been proposed in [38]. This method requires beam splitters with transmittance,
mirrors, and phase shift for realizing multiple controlled-phase gates. For
realizing $C_{k}X$ gates in real physical systems, it is necessary to
quantitatively evaluate various resources and compare multiple implementation
methods.

Next, our approach can be applied to other error correction codes and quantum
algorithms that require a large number of gates. For example, most of the cost
of a quantum circuit implementation of a discrete-time quantum walk algorithm
[39] is the unitary operator that walks a “quantum walker” in a certain space,
called a Shift operator [40]. The circuit construction of the Shift operator
depends on the space in which the quantum walk is performed (for instance, in
a graph [41], one-dimensional space [42], or a hypercube [43]), but in many
cases, they consist of multiple controlled quantum gates [44]. This is because
the Shift operator depends on the state of the “quantum walker” at a certain
time and transitions to the quantum state of the “quantum walker” at the next
time step, and such dependence on the previous time is realized by control
gates. A second example is Grover’s search algorithm [45]. The practical cost
of Grover’s algorithm depends on the complexity of the unitary, called an
“oracle”, which depends on the diffusion operator (inversion about the
average). When the size of the search space is $N$-qubits, it is common to use
$C_{N-1}X$ gates in the diffusion part [46].

To summarize, our paper shows that when quantum multiplexing is used, it is
possible to significantly reduce the number of CX gates for the encoding
circuits of QRS codes and important quantum algorithms. Future work includes
optimization of circuits for correction and decoding of QRS codes and other
QECCs.

## Acknowledgements

SN acknowledge Min-Hsiu Hsieh from Foxconn Quantum Computing Center for useful
discusssions throughout this project. This work was supported by JSPS KAKENHI
Grant Number 21H04880, the MEXT Quantum Leap Flagship Program (MEXT Q-LEAP)
Grant Number JPMXS0118069605, and the JST Moonshot R&D Grant Number
JPMJMS2061.

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##  Appendix A Circuit implementation of the SUM gates (the Modulo adder)

This appendix describes the construction of a quantum circuit for a
$d$-dimensional SUM gate. The inputs of this gate are the pair of $k$ qubits
$A$ and $B$ corresponding to the pair of $d$-dimensional qudits, and the
output is the modulo with respect to $d$ of the sum of the two inputs.

As described in Section II, the circuit consists of an RCA part and a modulo
part. The RCA part adds the numbers of the same place in inputs $A$ and $B$
and stores the overflow of the place in the “carry” qubits for use in
calculating the next place. Fig.6 shows what information is stored in which
qubit during the calculation of the RCA part for $d=5$.

Figure 6: Input and output of the RCA operation. Carry qubits are used to
store the overflow of each places for $4$, $2$, and $1$. Check-if qubits are
not used here.

Next the modulo circuit converts the $\ket{(a+b)\,[\textrm{mod}\,2^{k}]}$
state stored in $B$ to the $\ket{(a+b)\,[\textrm{mod}\,d]}$ state. Such a
conversion is necessary for the case in which the sum of the inputs is between
$d$ to $2(d-1)$. We prepare a “check if” qubit for each of the numbers from
$d$ to $2(d-1)$, and then the “check if” qubit is flagged (flipped from
$\ket{0}$ to $\ket{1}$) when the sum of the inputs in each corresponding
values $d$ to $2(d-1)$. In order to perform such an operation, we perform
$C_{k}X$ gate with the data qubits $B$ as the control qubits with the
corresponding “check if” qubit as the target qubit. At this time, we choose
control and $0$-control based on the binary representation of each number. For
normal controls, the gate is applied to the target when the control qubit is
$\ket{1}$, but for $0$-control, the gate is applied when the control is
$\ket{1}$. If one wants to invert the “check if” qubit for $5\,(101)$, use the
control, $0$-control, and control for $C_{k}X$. Next, when the sum cannot be a
number greater than $2^{k}$, the most significant carry can be used as the
“check if” qubit of $2^{k}$. In this case, the $C_{k}X$ gate for “check if”
qubit inversion is also unnecessary.

Consider the following example. If $d=5$, then a conversion is required when
the sum of the inputs is between $5$ and $8$. Since
$0\,[\textrm{mod}\,8]=0\,[\textrm{mod}\,5]$,…
$4\,[\textrm{mod}\,8]=4\,[\textrm{mod}\,5]$, no conversion is required for
$d=0$ to $4$. You can also substitute $Carry_{8}$ as a “check if” qubit of $8$
in this case. After the flip of the “check if” qubits, converts the data
qubits $B$ based on the “check if” qubit. As an example, we execute controlled
$XIX,XXX,XIX,IXX$ gates with “check if” qubit as control qubit for the
conversion from $5(101)$, $6(110)$, $7(111)$, $8(000)$ to $0(000)$, $1(001)$,
$2(010)$, $3(011)$ for $d=5$.

It is important to note that when the maximum value that the sum can take
$2(d-1)$ is greater than $2^{k}$, the largest “carry” qubit must be used as a
control qubit at the same time. For example, in the $d=7$ case, the
$mod\,2^{k}\rightarrow mod\,d$ transformation is needed when the sum is
between $7$ and $14$. However, to determine that the sum is $8,9,10,11,12$,
not only must the data qubit be $000,001,010,011,100$, but $Carry_{8}$ must be
$1$. This makes it necessary to use the $C_{k+1}X$ gate instead of $C_{k}X$.
Such $C_{k+1}X$ gates can only be decomposed up to $\rm{C_{2}X}$ gates when
multiplexing decomposition is used since the control qubits are in two qudits
(data and carry). For the decomposition of $\rm{C_{2}X}$ gates to $\rm{CX}$
gates, we need to use the general method.

The case of constructing a modulo adder for the $5$ and $7$ dimensional qudit
using $2^{3}$ multiplexed photon is shown in the Fig.7 and 8.

Figure 7: Circuit layout for the SUM gate for qudit of dimension-$5$ in the
qubit system. It consists of a RCA and a modulo part. The black dots represent
control qubits, and the white dots represent $0$-control qubits. Figure 8:
Circuit layout for the SUM gate for qudit of dimension-$7$ in the qubit
system. This circuit includes the $C_{K+1}X=C_{4}X$ gates which has a control
qubit in $\textrm{Carry}_{8}$.

##  Appendix B The encoding circuit for the $GF(2^{m})$ QRS codes

Let us consider the $[[d-1,1,D\geqq\frac{N+1}{2}]]_{d}$ QRS code on
$GF(2^{m})$ with $d=2^{m}$. This code construction is based on
$[[kN,k(N-2K),d\geqq K+1]]$ (written in binary form) for $N-2K=1$. To
illustrate how this works let us take the $GF(2^{2})$ code as a simple
example. The Galois Field for the code is represented as

$GF(2^{2})=\\{0,1,\alpha,\alpha^{2}\\}$ (1)

where we choose $x^{2}+x+1=0$ as the primitive polynomial. Then the elements
of the field be written in polynomial and vector representation:

$\displaystyle GF(4)$ $\displaystyle=$
$\displaystyle\\{0,1,\alpha,\alpha^{2}\\}\,\,\textrm{exponential
representation}$ $\displaystyle=$
$\displaystyle\\{0,1,\alpha,\alpha+1\\}\,\textrm{polynomial representation}$
$\displaystyle=$ $\displaystyle\\{00,01,10,11\\}\,\textrm{vector
representation}$

We use the classical code $\mathcal{C}=[3,2,2]_{4}$ and it’s dual
$\mathcal{C}^{\perp}$ for our CSS construction. The generator polynomial for
the code is given as

$g^{\perp}(x)=(x-1)(x-\alpha)=x^{2}+(1+\alpha)x+\alpha$ (2)

from which we get the generator and parity check matrices:

$G=\begin{pmatrix}1&0&\alpha\\\ 0&1&\alpha^{2}\end{pmatrix}$ (3)
$H=\begin{pmatrix}\alpha&\alpha^{2}&1\end{pmatrix}.$ (4)

The quantum circuit for the encoder is shown in Fig.9

Now we denote the qudit gates required for our implementation as $C1$,
$C\alpha$, and $C\alpha^{2}$ defining them as follows:

$\displaystyle C1(\ket{a}\ket{b})$ $\displaystyle=$
$\displaystyle\ket{a}\ket{a+b}$ (5) $\displaystyle C\alpha(\ket{a}\ket{b})$
$\displaystyle=$ $\displaystyle\ket{a}\ket{\alpha a+b}$ (6) $\displaystyle
C\alpha^{2}(\ket{a}\ket{b})$ $\displaystyle=$
$\displaystyle\ket{a}\ket{\alpha^{2}a+b}$ (7)

These gates correspond to the addition and multiplication of Galois elements
for the calculation of coefficients of polynomials in Quantum Reed-Solomon
codes. They can be implemented in a multiplexing system as shown in the
Fig.10. As such, the gates of $\rm{GF}(2^{m})$ can be constructed using only
$\rm{CX}$ gates.

By generalizing the above method, we can calculate the cost of the gates we
will need for implementing the $[[d-1,1,D\geqq\frac{N+1}{2}]]_{d}$ QRS code.
First the $C1$ gate can be achieved with $k$ $\rm{CX}$ gates while the
$C\alpha^{n}$ gates can be implemented with
$\displaystyle\sum_{p=0}^{k-1}\rm{H_{w}}(\alpha^{n+p})$ $\rm{CX}$ gates for
the $GF(2^{k})$ system where $\rm{H_{w}}()$ is a function whose input is an
exponential representation of a element of the Galois Field and whose output
is a Hamming weight of the vector representation of the element. In this case,
we do not need the $C_{k}X$ gate to implement the encoding circuit of the
code. Therefore, the application of quantum multiplexing does not lead to any
advantage in terms of the reduction of the number of gates.

Figure 9: The encoding circuit of $[[3,1,D\geqq 2]]_{4}$ Quantum Reed-Solomon
code. Each horizontal line represents a 4-dimensional qudit (ququart). The two
qu-dit gates defined by the black dot representing the control and the target
with a square box containing the “$1$” ($\alpha,\alpha^{2}$) symbol represent
the $C1$ ($C\alpha,C\alpha^{2}$) gates defined in (B5-7). Figure 10: Qubit
implementation of the $C1$, $C\alpha$, and $C\alpha^{2}$ gates.

##  Appendix C Proof for the multiplexed decomposition

######  Theorem.

A $C_{k}X$ gate, which has $k\in\ \mathbb{Z}^{+}$ control time-bin qubits in a
photon and a target qubit in another photon, can be decomposed into a single
$\rm{CX}$ gate alongside several optical switches.

###### Proof.

Let $S(k)$ be the statement that the $C_{k}X$ gate in the theorem can be
decomposed into one $\rm{CX}$ gate and several optical switches.

We will now give a proof by induction on $k$ beginning with $k=1$. Since
$C_{1}X$ gate is a $\rm{CX}$ gate from a control photon to a target photon, it
holds by definition. Now for any integer $k\geqq 1$, if $S(k)$ holds, then
$S(k+1)$ also holds. Assume the induction hypothesis $S(k)$ is true. Since the
$C_{k+1}X$ gate can be realized by controlling the $C_{k}X$ gate with the
($k+1$)-th timebin qubit, it can be decomposed as shown in Fig.11. From
$S(k)$, the $C_{k}X$ gate can be implemented with one $\rm{CX}$ and several
OSs, therefore $S(k+1)$ holds.

Figure 11: (a) $C_{k+1}X$ between two photons. (b) The circuit implementation
for (a). Optical switches divides the (k+1)-th timebin qubit.

∎

Note that using the implementation of $\rm{CX}$ gate in the Appendix C of
[35], the target qubit can be either polarization or timebin.