# Resource-efficient photonic quantum computation with high-dimensional
cluster states

Ohad Lib,1∗ Yaron Bromberg,1

1 Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem
91904, Israel

∗To whom correspondence should be addressed; E-mail<EMAIL_ADDRESS>
> Quantum computers can revolutionize science and technology, but their
> realization remains challenging across all platforms. A promising route to
> scalability is photonic measurement-based quantum computation, where single-
> qubit measurements on large cluster states, together with feedforward,
> enable fault-tolerant quantum computation. However, generating large cluster
> states at high rates is notoriously difficult, as detection probabilities
> drop exponentially with the number of photons comprising the state. We
> tackle this challenge by encoding multiple qubits on each photon through
> high-dimensional spatial encoding, generating cluster states with over nine
> qubits at a rate of 100Hz. Additionally, we demonstrate that high-
> dimensional encoding substantially reduces the computation duration by
> enabling instantaneous feedforward between qubits encoded in the same
> photon. Our findings pave the way for resource-efficient measurement-based
> quantum computation using high-dimensional entanglement.

## Introduction

From simulating quantum systems (?) to factoring large numbers (?), quantum
computers promise substantial speedups over their classical counterparts.
Their realization, however, has proven to be extremely challenging across all
physical platforms. In matter-based systems, decoherence and noise limit the
achievable circuit depth (?), whereas, in photonic systems, weak photon-photon
interaction renders two-qubit gates difficult (?, ?). A potential remedy for
the lack of efficient two-photon gates is the measurement-based quantum
computing (MBQC) paradigm (?). In MBQC, a large initial entangled state called
a cluster state (?) is first generated, and universal quantum computation is
then performed through simple single-qubit measurements and classical
feedforward without the need for two-qubit gates (?).

However, while MBQC circumvents the need to perform photonic two-qubit gates
during the computation, it does so by shifting the difficulty to the
generation of large cluster states at high rates. Indeed, despite significant
theoretical (?, ?, ?, ?), experimental (?, ?, ?, ?), and technological (?)
advances since the first demonstration of entanglement in the 1970s (?, ?),
state-of-the-art realizations of photonic cluster states are still limited to
roughly ten qubits at sub-Hz detection rates (?, ?). This poor scalability is
mainly due to the exponential decrease in the generation and detection
probabilities when the number of photons comprising the state is increased,
which stems from both fundamental difficulties in entangling non-interacting
photons (?), and technical issues, such as finite collection and detection
efficiencies (?, ?). The exponential scaling with the number of photons sets a
fundamental trade-off in current experiments between the number of qubits in
the cluster state and its detection rate, limiting the scale of MBQC
demonstrations.

A promising approach for generating large cluster states without increasing
the number of photons is to encode the quantum information using hyper- or
high-dimensional entanglement (?, ?, ?, ?). In this case, the large Hilbert
space associated with each photon can encode multiple qubits or high-
dimensional quantum bits (qudits), thus reducing the number of required
photons (?). In the context of MBQC, early demonstrations based on
polarization-spatial hyper-entanglement have shown increased generation rates
of small cluster states (?, ?, ?) and reduced feedforward overhead (?). More
recently, the generation of noise-resilient $d=3$ qudit cluster states using
temporal-spectral encoding (?) and the encoding of error-protected qubits in
integrated photonic chips (?) were demonstrated.

Increasing the Hilbert space dimension of each photon brings new challenges to
the generation, manipulation, and certification of cluster states, due to the
large number of optical modes that need to be precisely controlled. In
addition, since the measurement of a single photon now collapses multiple
qubits simultaneously, performing complete measurement-based gates that
require feedforward on a single photon becomes nontrivial and could naively
only be done probabilistically. As a result, a clear advantage of high-
dimensional MBQC over current state-of-the-art two-dimensional systems has not
been observed so far.

Here, we experimentally generate large cluster states with up to $9.28$
qubits, that simultaneously exhibit a high detection rate of 100Hz. We
overcome the size-rate trade-off of binary encoding by entangling multiple
qudits within the Hilbert space of each photon, thus enlarging the state
without sacrificing its generation efficiency. We achieve this by utilizing
the spatial degree of freedom of entangled photons, which we precisely control
through ten bounces off a spatial light modulator. This demonstrates the
advantage of high-dimensional spatial encoding for photonic quantum
computation, joining recent works in quantum communication (?), quantum
networks (?, ?, ?), and entanglement certification (?, ?). In addition, we
show that the simultaneous collapse of multiple qubits upon the measurement of
their encoding photon is, in fact, beneficial for implementing feedforward
between them. We perform such feedforward instantaneously using tailored
linear optic circuits and demonstrate a potential four-fold reduction in the
duration of single-qubit rotations in MBQC. Our approach therefore reduces the
required experimental resources in terms of both the number of photons and
active feedforward hardware, highlighting the potential of high-dimensional
encoding for resource-efficient MBQC on different photonic platforms (?, ?, ?,
?).

## Spatially encoded cluster states

Our scheme for generating high-dimensional cluster states for MBQC is based on
the ability to first entangle different photons in high dimensions, then
establish entanglement between the multiple qudits encoded within each photon,
and finally measure the generated state in different bases. Figure 1
illustrates our implementation of these three tasks on the level of the
experimental realization (fig.1A), equivalent quantum circuit (fig.1B, top),
and resulting photonic cluster state (fig.1B, bottom). While this scheme is
compatible with high-dimensionally and hyper-entangled states encoded in any
degree of freedom, our experimental demonstration is in the spatial domain,
where tools to precisely control the quantum state, such as Mega-pixel spatial
light modulators, are readily available.

First, photon pairs are generated via spontaneous parametric down-conversion
(SPDC) (?), where high-dimensional spatial entanglement in multiple transverse
modes is achieved through careful adjustment of the momentum conservation
conditions in the nonlinear process (?, ?) (fig.1A, and materials and
methods). A binary amplitude mask defines $M$ spatial modes per photon, which
we can use for encoding $N=\log_{d}(M)$ d-dimensional qudits. The SPDC process
is equivalent to the quantum circuit presented on the left panel of fig.1B,
which generates pair-wise entanglement between qudits encoded in the two
photons (fig.1B, bottom left). In contrast to entanglement between different
photons, which often relies on probabilistic gates, qudits encoded within the
same photons can be arbitrarily entangled through deterministic unitary
transformations acting on the high-dimensional Hilbert space of each photon
(?, ?, ?, ?) (fig.1B, center). For example, generalized d-dimensional
controlled-Z (CZ) and controlled-NOT (CNOT) gates are implemented in the
experiment via mode-dependent phase shifts and mode-relabeling, respectively
(fig.1A, and supplementary information for additional details).

After generating the desired cluster state, performing measurements in
different bases is essential for both MBQC (?) and entanglement certification
(?). However, while high-dimensional encoding simplifies the generation of
large cluster states, it complicates their measurement in different bases.
This is due to the increased size of the required unitary transformations that
act in the high-dimensional Hilbert space of each photon (?).

To overcome this challenge, we use multi-plane light conversion (MPLC) - a
technique for performing arbitrary transformations on photonic spatial modes
by propagating them through a series of optimized phase masks (?). While
initially developed for transforming spatial modes of classical light (?, ?,
?, ?), recent small-scale demonstrations have highlighted the potential of
this approach for controlling single and entangled photons (?, ?, ?). We build
upon these works to realize a large-scale multi-plane light converter
consisting of ten programmable phase masks in a multi-pass configuration
(fig.1A), allowing us to measure all qudits at the required bases (fig.1B,
right). Importantly, while the number of phase masks required for realizing
arbitrary $M$x$M$ unitary transformations scales linearly with $M$, our
change-of-basis transformations acting on $N$ qudits consist of a tensor
product of $N$, $d$x$d$, unitary matrices acting on each qudit. Therefore, the
number of required phase masks scales linearly with $N=\log_{d}(M)$ and only
logarithmically with the number of modes $M$, making this approach more
scalable.

As a first demonstration of cluster state generation using high-dimensional
spatial entanglement, we create an eight-qubit cluster state encoded using
only two photons (fig.2A). In this case, each photon encodes information in
$M=16$ spatial modes, that are mapped onto $N=4$ qubits (see supplementary
information). By performing two sets of measurements in mutually unbiased
bases consisting of different combinations of Pauli X and Z operators, we
certify genuine eight-qubit entanglement (fig.2B). We observe strong
correlations on the diagonal with minor contributions from correlations
between other modes interfered via MPLC, obtaining a negative entanglement
witness of $\langle W_{C_{8,2}}\rangle=-0.39\pm 0.01$ (?) (see supplementary
information). We achieve a total detection rate of over 100Hz in both
measurement bases, which is well beyond current capabilities in equivalent
eight-photon experiments with two-dimensional encoding (?, ?).

Figure 1:  High-dimensional spatially encoded cluster states. (A) Illustration
of the experimental setup. Spatially entangled photon pairs are generated via
spontaneous parametric down-conversion (SPDC) and are transmitted through a
binary mask with a grid of up to 50 apertures. Due to momentum conservation
(inset), when photon A passes through one of the top 25 apertures, photon B
passes through the anti-symmetric bottom aperture. Mode-dependent phases and
mode relabeling are equivalent to entangling CZ and CNOT operations between
the qudits encoded in the Hilbert space of each photon, respectively. A multi-
plane light converter consisting of a spatial light modulator and a mirror in
a multi-pass configuration implements programmable high-dimensional unitary
transformations, enabling coincidence measurements in arbitrary bases. (B)
Quantum circuit with $N$ qudits per photon (top) and qudit graph
representations (bottom) of the above experimental setup. The SPDC source
generates a state of high-dimensional entangled photons, that is equivalent,
up to local Hadamard gates ($H_{d}$), to a multi-qudit state initialized in
$|+_{d}\rangle^{\otimes 2N}=[1/\sqrt{d}(|1\rangle+\cdots+|d\rangle)]^{\otimes
2N}$, followed by CZ gates creating pair-wise entanglement between the $2N$
qudits encoded by the two photons (left). Unitary transformations on each
photon are equivalent to entangling operations between its encoded qudits
(center), allowing for deterministic connectivity settings within the state.
Measurement-based quantum computing can then be achieved by measuring the
different qudits at arbitrary bases (right). Figure 2:  Generation and
certification of an eight-qubit cluster state. (A) An eight-qubit cluster
state $|C_{8,2}\rangle$ is encoded using two photons (A and B) entangled in
$M=16$ spatial modes per photon. Vertices and edges represent qubits and CZ
operations, respectively. (B) To certify the generation of the desired cluster
state, we measure correlations in two mutually unbiased bases consisting of
different combinations of Pauli X and Z operators. Based on these
measurements, we obtain a negative entanglement witness of $\langle
W_{C_{8,2}}\rangle=-0.39\pm 0.01$, certifying genuine eight-qubit
entanglement.

## Efficient MBQC via instantaneous intra-feedforward

After generating a large cluster state, MBQC is realized via single-qubit
measurements and classical feedforward (?, ?). Feedforward is a critical
component of the MBQC paradigm, as it counteracts the inherent randomness of
each single-qubit measurement by accordingly updating the measurement basis of
the following qubits (?). Importantly, feedforward also sets the temporal
order of the computation and, ultimately, its duration (?). This is because,
in contrast to non-commuting gates in the circuit model, single-qubit
measurements could otherwise be performed simultaneously.

When each photon in the cluster state encodes multiple qubits, two different
types of feedforward steps can be identified: between qubits encoded on
different photons (coined inter-feedforward), and between qubits encoded on
the same photon (coined intra-feedforward). While inter-feedforward steps can
be realized through active classical hardware as in standard MBQC, the
implementation of intra-feedforward steps warrants a different approach. This
is because the measurement of a single photon now collapses multiple qubits
simultaneously, preventing a naive implementation of feedforward between them.
Nevertheless, as we will show, a judicious design of linear optic circuits can
take advantage of this collapse for performing multiple feedforward steps
instantaneously and passively.

Inspired by early ideas for performing a single feedforward step by measuring
the polarization of a photon conditioned on its momentum (?), we show that
complete measurement-based gates with multiple intra-feedforward steps can be
realized via passive linear optic circuits. We replace active feedforward
steps with equivalent multi-qubit gates that reflect the feedforward
structure, followed by simultaneous measurement of all qubits encoded in the
photon. Such instantaneous feedforward is achieved deterministically through
tailored high-dimensional linear optic circuits that implement the multi-qubit
gates within the Hilbert space of a single photon. This approach substantially
reduces the computation time, as measurements originally made at different
time steps with slow, active, feedforward between them, can now be performed
simultaneously on qubits encoded in the same photon.

We experimentally demonstrate the advantages of instantaneous intra-
feedforward for MBQC by implementing measurement-based single-qubit rotations
on a five-qubit linear cluster state. While in most quantum computing
architectures single-qubit rotations are the simplest operations to implement,
in MBQC, they are the most time-consuming gates because they require four
clock cycles and feedforward steps (?) (fig.3A). However, using intra-
feedforward between four qubits encoded on a single photon, we realize the
same gate, in a single time step (fig.3B). To this end, we map each slow
feedforward step in the original scheme (?) (fig.3C, top) with an equivalent
linear optical circuit (fig.3C, bottom). From these 4x4 circuits, we construct
a 16x16 optical circuit that performs all intra-feedforward steps
simultaneously, which are then implemented by the MPLC (see fig.S4 in the
supplementary information). Following the intra-feedforward steps, the final
inter-feedforward step is realized by updating the Pauli frame of the last
qubit in the data analysis (see supplementary information). To demonstrate the
ability to perform single qubit rotations, we focus on two input states that
are rotated by angles $\alpha$ and $\beta$ around the X and Z axes,
respectively (fig.3D-F, and supplementary information). We observe good
agreement of the output expectation values as a function of rotation angle, up
to lower visibility resulting mainly from imperfect fidelity, confirming the
applicability of instantaneous intra-feedforward for MBQC.

Figure 3: Single-qubit rotation using instantaneous intra-feedforward. (A) In
MBQC, arbitrary single-qubit rotations parameterized by rotation angles
$(\alpha,\beta,\gamma)$ around the $(X,Z,X)$ axes are performed using a five-
qubit chain. Classical feedforward compensates for the inherent randomness in
the measurement outcomes $m_{i}=\pm 1$ by changing the measurement basis of
the following qubits. This is implemented by a unitary $U(m_{i}\theta)$ that
adds a phase $m_{i}\theta$ to the $|1\rangle$ state of the qubit. The duration
of the total gate is thus dictated by the slow feedforward process that
imposes temporal order on the measurements. (B) When four qubits are encoded
using a single high-dimensional photon, all qubits are measured simultaneously
when the photon is detected. Intra-feedforward between them is achieved
instantaneously using tailored transformation on the 16-dimensional photon
Hilbert space, which can be realized using a 16x16 linear optic circuit. A
single intra-feedforward step (top) and its equivalent linear optic circuit
(bottom) are shown in panel (C). Black and blue lines represent qubits and
optical modes, respectively. We experimentally demonstrate single-qubit
rotations on a chain of five qubits, four of which are encoded by a single
photon (D). The expectation values of the output rotated states $|\phi\rangle$
as a function of angle are presented in panels (E, F). In (E), an input state
$|\psi\rangle=1/\sqrt{2}(|0\rangle+i|1\rangle)$ is rotated around the X axis
by an angle $\alpha$, while in (F), an input state
$|\psi\rangle=1/\sqrt{2}(|0\rangle+|1\rangle)$ is rotated around the Z axis by
an angle $\beta$.

## Realization of a $d=5$ qudit cluster state

Until this point, we have focused on the encoding of multiple _qubits_ within
each photon in the cluster state. However, high-dimensional encoding naturally
provides the flexibility to encode multiple _qudits_ as well (?). While MBQC
was predominately studied in the context of qubits, a few notable works have
recently suggested potential benefits in implementing gates (?, ?), algorithms
(?, ?, ?), and error correction (?) using qudits, especially in prime
dimensions (?, ?).

To this end, we demonstrate a two-photon, $d=5$, four-qudit cluster state
encoded using a total of 50 spatial modes (fig.4A). To certify genuine
entanglement, we use an entanglement witness, adapted for $d=5$ qudit cluster
states, that requires measurements in two mutually-unbiased bases (?) (fig.4B,
see supplementary information). We obtain a negative entanglement witness of
$\langle W_{C_{4,5}}\rangle=-0.51\pm 0.01$, clearly certifying genuine four-
qudit entanglement.

To put our result in perspective, we compare our high-dimensional cluster
state with previous demonstrations in terms of both size and detection rate.
In terms of size, the Hilbert space dimension of our cluster state
($dim(\mathcal{H})=625$) is equivalent to that of $9.28$ qubits, making it the
largest cluster state generated via SPDC (?). This marks the first time high-
dimensional encoding outperforms state-of-the-art SPDC experiments with binary
encoding. In terms of rate, our total detection rate of 100Hz is over two
orders of magnitude higher than any demonstration of cluster states at this
scale, including recent demonstrations with deterministic sources (?).

We further quantitatively compare recent two- and high-dimensional
demonstrations of cluster states (?, ?, ?, ?, ?, ?) by looking at the
effective quantum resource rate as a figure of merit (?). The effective
quantum resource rate takes into account both cluster size and detection rate
and is defined as the multiplication of the rate by the dimension of the
state’s Hilbert space. We plot the effective quantum resource rate of
different experiments as a function of the number of qubits encoded per photon
in figure 4C, showing that beyond the advantage of instantaneous intra-
feedforward, high dimensional encoding is beneficial for high-rate generation
of large cluster states. By encoding over four qubits per photon, we are able
to achieve a record-high effective quantum resource rate with our $d=5$
cluster state.

Figure 4: $\mathbf{d=5}$ qudit cluster state. We certify the generation of a
four-partite five-dimensional qudit cluster state (A) via correlation
measurements in two bases (B). Here, X and Z are the generalized five-
dimensional Pauli operators. We obtain a negative entanglement witness of
$\langle W_{C_{4,5}}\rangle=-0.51\pm 0.01$, certifying genuine four-qudit
entanglement. (C) The effective quantum resource rate, defined as the product
of the state’s Hilbert space dimension and detection rate (?), is presented as
a function of the number of qubits encoded per photon, for different
realizations of photonic cluster states. Our $d=5$ cluster state has a Hilbert
space dimension of 625 and a detection rate of approximately 100Hz, yielding a
large effective quantum resource rate of dim($\mathcal{H}$)$\cdot$rate
$\approx 6.25\cdot 10^{4}$Hz. Data for other experiments presented in (C) is
taken from fig.2 (eight qubits at approximately $130$Hz) and the following
references (?, ?, ?, ?, ?, ?).

## Discussion

We demonstrate the generation of the largest cluster states ever produced via
SPDC, with up to $9.28$ qubits, at high rates. This achievement is made
possible by encoding cluster states using high-dimensional spatial encoding,
reducing the number of required photons and thus circumventing the size-rate
trade-off. We control the large number of photonic modes by implementing a
large-scale programmable multi-plane light converter, which allows us to
switch between the measurements required for performing measurement-based
gates and certifying genuine entanglement. Additionally, we have shown that
high-dimensional encoding enables instantaneous feedforward between qubits
encoded within the same photon, which substantially boosts the computation
speed in MBQC.

This work highlights an important question: What is the optimal encoding
dimension for scalable, fault-tolerant, MBQC? Answering this question amounts
to finding a sweet spot between the benefits of high-dimensional encoding,
such as increased information capacity per photon, and drawbacks, such as
complex state generation and manipulation. In our experiment, at least up to
25 modes per photon, the benefit of reducing the number of photons outweighs
the added complexity of controlling multiple spatial modes, resulting in
substantially higher detection rates compared with two-dimensional encoding.

Our results also emphasize the importance of further research into high-
dimensional encoding in deterministic platforms (?, ?, ?, ?, ?, ?, ?), as an
alternative for our SPDC source. Specifically, on the theoretical side,
answering open questions on the optimal success probability of heralded
generation and fusion of high-dimensional entangled states (?, ?, ?, ?, ?)
would complement our work and enable a more complete discussion on high-
dimensional architectures for fault-tolerant MBQC. On the experimental side,
scaling up our demonstration to even larger cluster states would further test
the limits of high-dimensional encoding for MBQC. This could be achieved in
the near future through the combination of high-dimensionally entangled
photons generated in thin films (?) and resonant metasurfaces (?) together
with large-scale MPLC (?, ?) or integrated photonic circuits (?, ?, ?).
Additionally, incorporating state-of-the-art superconducting nanowires or
single-photon avalanche diode (SPAD) arrays (?, ?) with better detection
efficiencies will further increase the high detection rate achieved in our
experiment.

In a broader context, we believe that our first demonstration of a high-
dimensional cluster state generation scheme that outperforms equivalent qubit
experiments will pave the way to new applications in quantum computation and
communications in different degrees of freedom of light (?, ?, ?, ?). In
addition to exploring higher rates and larger cluster sizes in MBQC, our
scheme holds promise for enhancing long-range quantum communication with
cluster-state-based all-photonic quantum repeaters (?, ?). To this end, the
cluster states generated in our experiment can be made compatible with long-
range free-space or fiber links by leveraging the capability of MPLC to
convert hundreds of spatial modes into orbital-angular-momentum or fiber modes
(?, ?). We thus envision that high-dimensional quantum computing and
communications could be combined with recent demonstrations of high-
dimensional quantum memories (?) to demonstrate important building blocks for
a future quantum internet based on high-dimensional encoding (?, ?).

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## Acknowledgments

We thank O. Katz and B. Dayan for helpful discussions. Funding: This project
was supported by the Zuckerman STEM Leadership Program. O.L. acknowledges the
support of the Clore Scholars Programme of the Clore Israel Foundation. This
joint research project was financially supported by the State of Lower Saxony,
Hannover, Germany. Author contributions: O.L. and Y.B. conceived and
conceptualized the project. O.L. designed and built the experimental setup,
carried out the experiment, and performed the data analysis under the
supervision of Y.B. All authors contributed to the writing of the manuscript.
Competing interests: The authors declare no competing interests. Data and
materials availability: All data is available in the main text and
supplementary information or at https://doi.org/10.5281/zenodo.8358012.

## Supplementary Materials for resource-efficient photonic quantum computation
with high-dimensional cluster states

## Materials and methods

Experimental setup. The experimental setup is presented in fig.S1A. Spatially
entangled photons are generated via type-I spontaneous parametric down-
conversion (SPDC) by pumping a Barium Borate (BBO) crystal with a $30mW$,
$405nm$ continuous-wave laser beam. The waist of the pump beam at the crystal
plane determines the transverse momentum conservation conditions and is
inversely proportional to the spatial correlation width between the entangled
photons at the far field. The crystal thickness, on the other hand, determines
the total width of the far-field intensity distribution. Choosing a pump waist
of $w\approx 600\mu m$ and crystal thickness of $8mm$, we obtain a correlation
width of $\approx 100\mu m$ in a $\approx 6mm$ wide cone at the Fourier plane
of a $f=150mm$ lens. We place at the Fourier plane a binary amplitude mask,
made of a Chromium-covered glass slide carving 50 circular spatial modes with
a radius of $100\mu m$ and a spacing of $300\mu m$ on a $5X10$ etched grid.
The entangled photons are then imaged onto the first plane of the multi-plane
light converter (MPLC) using lenses $L2-L5$, having focal lengths
$f=50,50,150,150mm$ respectively. Two mirrors, $M1$, and $M2$, are placed in
the image and Fourier planes of the first plane of the MPLC to allow for
precise and independent control over the entrance position and angle of the
light into the MPLC, respectively.

The MPLC consists of a spatial light modulator (SLM, Hamamatsu X13138-02), a
parallel dielectric mirror placed $43.5mm$ from the SLM, and a right-angle
prism placed $69mm$ from the SLM (fig.S1B). In total, the entangled photons
impinge onto the SLM ten times at different locations before propagating
another $43.5mm$ to two transversely motorized $100\mu m$ fibers coupled to
avalanche photo-diode single-photon detectors. A polarizer and two $810\pm
10nm$ filters ensure the detection of photons at the desired polarization and
wavelength. Coincidence counts are recorded using a time tagger (Swabian
instruments, Time Tagger 20) with a $400ps$ coincidence window. For the
alignment of the MPLC, a flip mirror redirects the light at the output to a
camera on a moving rail using an imaging system (not shown). The alignment
procedure is described in greater detail below.

Figure S1:  Experimental setup. (A) A sketch of the experimental setup. $L$\-
lens, M- mirror, P- polarizer, MPLC- multi-plane light converter, P-
polarizer, FM- flip mirror, BS- beam splitter. (B) Detailed view of the MPLC.
The photons bounce back and forth between a spatial light modulator and a
mirror. A prism is used after the fifth plane to redirect the beam onto the
lower part of the SLM.

Calculating phase masks. For any given transformation, the required phase
masks are calculated using the wavefront matching algorithm with 30
iterations, based on the code described in ref. (46). The size of each phase
mask is 140X360 pixels. Crosstalk between pixels and diffraction losses are
minimized by restricting the diffraction angles from the phase masks to $15$
percent of the entire angular range set by the $12.5\mu m$ pixels. Once the
phase masks are calculated, they are displayed on the SLM without
modification. Mode-dependent phases are then added on the first plane of the
MPLC to encode the desired cluster state and compensate for small optical
aberrations in the preceding optical setup.

## Supplementary text

## Multi-plane light conversion

### Alignment

Properly aligning MPLCs is known to be experimentally challenging, and is
sometimes performed with the aid of optimization algorithms with experimental
feedback, even for a relatively small number of planes. In our case, due to
the large number of planes and weak SPDC signal, we decided to avoid blind
optimization and developed an optical alignment procedure that takes advantage
of the large number of planes in our MPLC.

Once the beam is aligned reasonably well and bounces off the SLM ten times,
the goal of the alignment procedure is to find the ten
$(x_{j},y_{j}),j=1,...,10$ coordinates where the center of the beam hits the
SLM. These locations then determine where the phase masks are placed on the
SLM. Since an average misalignment of a few pixels is already detrimental to
the transformation, we position the phase mask with an accuracy of $\pm 1$
pixels.

Our alignment procedure relies on sequentially imaging the different planes of
the SLM onto an sCMOS camera and looking at the intensity distribution of the
SPDC light when applying $\pi$ phase steps at different positions on the SLM.
The camera is placed on a rail after a set of imaging lenses and can directly
image planes nine, ten, and eleven (the effective plane of the detectors) of
the MPLC (fig.S1A). Planes one to eight are imaged onto the camera as well by
adding appropriate quadratic phases to other MPLC planes, which effectively
act as programmable imaging lenses.

The first step of the alignment procedure is to find the position of the beam
at the first plane. We image the first plane to the camera and scan the
position of a $\pi$ phase step on the SLM. After the imaging system, the $\pi$
phase step is manifested as a dark line on the camera due to high diffraction
loss occurring at the phase step. Since each mode at the first plane has a
radius of only $100\mu m$, we were able to find the center of the beam with an
accuracy of approximately a single pixel ($12.5\mu m$). This procedure is
repeated for the second plane, albeit with slightly lower accuracy due to the
diffraction of the modes.

For planes three to ten, the diffracted SPDC intensity distribution is wide,
and it is hard to define the center of the beam. We, therefore, again use
other planes of the MPLC as programmable lenses and image the modes from the
first plane to the desired plane, and then onto the camera. For example, to
find the position of phase mask three, we first image plane one onto plane
three by setting a quadratic phase on plane two, and then image plane three
onto the camera using a quadratic phase on plane seven. We repeat this process
for all planes, including self-consistency checks where the position of
certain planes is determined in multiple ways. For example, plane one can be
imaged onto plane five using a quadratic phase on either plane three, planes
two and four, plane two, or plane four. In addition, planes two and three can
be imaged onto plane five as well to check that their measured centers are
consistent. The entire alignment procedure takes less than an hour, and the
MPLC typically remains aligned and stable for over a month.

### Transformation characterization

To characterize the different transformations implemented via the MPLC, we co-
align a classical $808nm$ laser beam with the SPDC light, through the binary
amplitude mask and the MPLC. We reconstruct the matrix $A$ that represents the
MPLC transformation using two sets of input modes. First, we inject the laser
beam into the MPLC through a single aperture in the binary mask and record the
intensities at all output modes. By repeating this measurement for all the
apertures, we obtain the absolute values of the matrix elements $|A_{ij}|$,
i.e. the square root of the recorded intensity at output mode $i$ for input
mode $j$. As, in general, the transformation of the MPLC has finite efficiency
(see next section), we do not assume the unitarity of the matrix $A$ at any
point. Next, we obtain the relative phases between the matrix elements. We
coherently inject light into all the apertures of the binary masks,
simultaneously, and look at the output intensities for different relative
phases at the input plane (set by the SLM). The relative phases between the
matrix elements within each row of the matrix can then be obtained through a
simple implementation of the Gerchberg-Saxton phase retrieval algorithm.
Obtaining the relative phases within each row is sufficient since the relative
phases between rows, or equivalently between output modes, are irrelevant to
our intensity and correlation measurements at the output.

We experimentally reconstructed the 18 transformations used for evaluating the
entanglement witnesses in figures 2,4. For the eight-qubit cluster states, the
desired transformations on each photon are four-dimensional Hadamard
operations $U=H_{4}$ between 8 different sets of four modes, while for the
four-qudit cluster state, five-dimensional discrete Fourier transformation
$U=DFT_{5}$ between 10 sets of modes are required. We compare the
reconstructed matrices $A$ with the desired ones $U$ to obtain their fidelity
according to the Frobenius norm,
$F=\frac{|Trace(AU^{\dagger})|}{\sqrt{Trace(AA^{\dagger})Trace(UU^{\dagger})}}$,
achieving $F_{H_{4}}=0.95\pm 0.02$ and $F_{DFT_{5}}=0.93\pm 0.01$.

### Loss estimation

To estimate the optical loss in the system, we measured the total coincidence
rate before the MPLC and compared it with that from the entanglement
certification measurements of the eight-qubit and four-qudit cluster states
after the MPLC. The same detectors, fibers, and filters are used for all
measurements. As the two-photon coincidence rate drops quadratically with
loss, the estimated loss is given by the square root of the ratio between the
output and input coincidence rates. The measured average losses for the eight-
qubit and four-qudit cluster states are $-9.1\pm 0.1dB$ and $-10.4\pm 0.2dB$,
respectively. The use of high-dimensional encoding enables us to obtain high
detection rates of large cluster states in the presence of such loss. The
detection rate could be boosted even further by using fixed phase plates
instead of an SLM, which can substantially reduce loss.

## Encoding cluster states with high-dimensional entanglement

In this section, we describe the mapping between two-photon high-dimensional
entangled states generated via SPDC and multi-qubit/qudit graph states. Due to
transverse momentum conservation, for $M$ modes per photon, the output state
of the entangled photons is
$|\psi\rangle=\frac{1}{\sqrt{M}}\sum_{i=1}^{M}|ii\rangle$, where modes that
are anti-symmetric with respect to the center of the beam have the same label
$i=1,...,M$. For any given total dimension $M$, the state of each photon can
be represented using $N$, $d$-dimensional, qudits with $d^{N}=M$. The two-
photon quantum state can thus be written as a sum over all possible strings
$s_{i}$ of $N$ numbers from 1 to $d$,
$|\psi\rangle=\frac{1}{\sqrt{d^{N}}}\sum_{s_{i}=\\{1,...,d\\}^{N}}|s_{i}s_{i}\rangle$.
Note that all possible strings for each photon are represented in the sum and
that the value of qudit $n\in\\{1,...,N\\}$ in both photons is always
identical due to their entanglement. Therefore, the quantum state is
equivalent to a tensor product of $N$, $d$-dimensional, entangled states and
takes the form
$|\psi\rangle=(\frac{1}{\sqrt{d}}\sum_{i=1}^{d}|ii\rangle)^{\otimes N}$. This
state is equivalent to the graph state presented on the left side of fig.1B,
up to the local operation of the conjugate transpose of the $d$-dimensional
Hadamard gate ($H_{d}^{\dagger}$) on each of the qudits on one of the two
photons (figure 1B, left).

After the SPDC process creates pair-wise entanglement between the qudits of
the two photons (fig.1B, left), the connectivity of the cluster state is set
by entangling qudits encoded within the same photon using local unitary
transformations on the optical modes. CZ gates can be directly applied to
qudits that did not experience the extra $H_{d}^{\dagger}$ gate. The CZ gate
between two qudits encoded on the same photon is equivalent to a mode-
dependent phase shift of $(i-1)(j-1)2\pi/d$ where $i=1,...,d$ and $j=1,...,d$
are the values of the two qudits at this mode. For example, in the qubit case,
this amounts to a $\pi$ phase shift of the $11$ mode (considering standard 0/1
encoding of the qubits), as can be seen in fig.S2A. To apply a CZ gate between
two qudits where one of them has an additional $H_{d}^{\dagger}$ gate, the CZ
gate is sandwiched between two $H_{d}$ gates, making it equivalent to a CNOT
gate. The first $H_{d}$ gate compensates for the extra $H_{d}^{\dagger}$ gate
resulting from the SPDC process while the second one is absorbed in the
measurement basis. The CNOT gate between two qudits encoded on the same photon
is equivalent to relabeling. For example, in the qubit case, a CNOT gate
between two qubits is equivalent to the label swapping of the $11$ and $10$
modes (fig.S2B).

For the eight-qubit cluster state, each photon encodes $N=4$ qubits ($d=2$) in
$M=16$ optical modes. We choose the order of the labels to be
$q_{1}q_{2}q_{3}q_{4}$ for photon A and $q_{8}q_{7}q_{6}q_{5}$ for photon B
(fig.S2C). When choosing identical labels for anti-symmetric modes with
respect to the center of the 32 apertures, the SPDC process ensures
entanglement between the qubit pairs 1-8, 2-7, 3-6, and 4-5. Following the
SPDC process, a CZ gate is applied to qubits two and three, and a CNOT gate
between qubits one and two, and between qubits three and four (see fig.S2C).
The CZ gate is equivalent to a $\pi$ phase shift on modes where
$q_{2}q_{3}=11$, and the CNOT gates are equivalent to mode relabeling, as
shown in fig.S2B. Specifically, the CNOT gate between qubits one and two
(marked with red) is equivalent to the label swapping of modes where
$q_{2}=1$, by changing $q_{1}$ from 0 to 1 or vice versa. Similarly, the CNOT
gate between qubits three and four (marked with green) is implemented by
changing the label of $q_{4}$ whenever $q_{3}=1$. The effect of the CZ and
CNOT gates is illustrated in fig.S2C.

Similarly, for the four-qudit cluster state, each photon encodes $N=2$, $d=5$,
qudits in $M=25$ spatial optical modes. A CZ operation between qudits two and
three is performed by applying mode-dependent phase shifts to form the four-
qudit cluster (up to Hadamard operations on qudits one and four, fig.S2D). In
this case, a phase shift of $(i-1)(j-1)2\pi/d$ is applied for a mode labeled
with $q_{2}q_{3}=ij$.

Figure S2:  Cluster state encoding. (A) A CZ gate between two qubits is
equivalent to a $\pi$ phase shift on the spatial mode encoding the
$|11\rangle$ state. (B) Similarly, a CNOT gate between two qubits is
equivalent to the label swapping of the modes encoding the $|10\rangle$ and
$|11\rangle$ states (outlined in green). (C) The eight-qubit cluster state is
encoded in 16 spatial modes per photon. The modes of the two photons are
positioned in anti-symmetric locations with respect to the center of the 32
modes, marked with a white X sign. Edges in the graph representation of the
state are realized via mode-dependent phase shifts (CZ) and mode-relabeling
(CNOT). The blue edge is equivalent to a $\pi$ phase shift added to modes
where both the second and third qubits have the value $1$. The red and green
edges are equivalent to mode relabeling (see text above). Modes that have been
relabeled in the addition of the red and green edges are marked with the
corresponding color. (D) The four-qudit $d=5$ cluster state is encoded using
25 spatial modes per photon. For clarity, only the labels 11,15,51, and 55 are
explicitly written in the figure. The edge between qudits two and three is
equivalent to the addition of a phase $(i-1)(j-1)2\pi/5$ to each mode $ij$ of
photon A.

## Entanglement certification

To experimentally certify the entanglement in the generated cluster states, we
use entanglement witnesses $W_{2N,d}$, which are evaluated by measuring the
correlations between the two photons (each encoding $N$ qudits) in two
mutually unbiased bases. Measurements in two bases are sufficient in our case,
as the graphs associated with the quantum state are two-colorable (51). For
the eight-qubit case, we use a standard entanglement witness of the form
$W_{2N,2}=3-2(\prod_{odd\,k}\frac{(S_{k}+1)}{2}+\prod_{even\,k}\frac{(S_{k}+1)}{2})$,
where $S_{k}$ is the stabilizer associated with the kth qubit, given by the
Pauli $X$ operator acting on the kth qubit and Pauli Z operator acting on its
neighbors (51). Similarly, for the four-qudit cluster state, we use a recent
generalization of this entanglement witness that takes the form
$W_{2N,d}=\frac{d+1}{d-1}-\frac{d}{d-1}(\prod_{odd\,k}(\frac{1}{d}\sum_{p=1}^{d}S_{k}^{p})+\prod_{even\,k}(\frac{1}{d}\sum_{p=1}^{d}S_{k}^{p}))$,
where $d$ is the dimension of the qudit and $S_{k}^{p}$ is the stabilizer
$S_{k}$ in power $p$ (61).

In the main text, for clarity, the measurement bases are written according to
the standard cluster state stabilizers, absorbing into the measurement the
additional local Hadamard gates that act on some of the qubits. Here, we write
the stabilizers and entanglement witnesses for the quantum state, including
the local Hadamard gates, to make the actual measurement bases realized via
the MPLC in the experiment more transparent. The entanglement witnesses for
the eight-qubit and four-qudit cluster states are given by

$\begin{array}[]{l}W_{C_{8,2}}=3-\frac{1}{8}(X_{2}Z_{3}Z_{4}Z_{5}Z_{6}X_{7}X_{8}+X_{2}Z_{6}X_{7}X_{8}+X_{2}Z_{4}Z_{5}X_{7}X_{8}+X_{2}Z_{3}X_{7}X_{8}+X_{1}Z_{4}Z_{5}Z_{6}X_{8}\\\
+X_{1}Z_{3}Z_{6}X_{8}+X_{1}Z_{3}Z_{4}Z_{5}X_{8}+X_{1}X_{8}+X_{1}X_{2}Z_{3}Z_{4}Z_{5}Z_{6}X_{7}+X_{1}X_{2}Z_{6}X_{7}+X_{1}X_{2}Z_{4}Z_{5}X_{7}\\\
+X_{1}X_{2}Z_{3}X_{7}+Z_{4}Z_{5}Z_{6}+Z_{3}Z_{6}+Z_{3}Z_{4}Z_{5}+Z_{1}Z_{2}X_{3}X_{5}X_{6}Z_{7}Z_{8}+Z_{1}Z_{2}X_{3}X_{4}X_{6}Z_{7}Z_{8}+Z_{1}X_{4}X_{5}Z_{7}Z_{8}\\\
+Z_{1}Z_{7}Z_{8}+Z_{1}X_{3}X_{5}X_{6}Z_{8}+Z_{1}X_{3}X_{4}X_{6}Z_{8}+Z_{1}Z_{2}X_{4}X_{5}Z_{8}+Z_{1}Z_{2}Z_{8}+X_{3}X_{5}X_{6}Z_{7}+X_{3}X_{4}X_{6}Z_{7}\\\
+Z_{2}X_{4}X_{5}Z_{7}+Z_{2}Z_{7}\left.+Z_{2}X_{3}X_{5}X_{6}+Z_{2}X_{3}X_{4}X_{6}+X_{4}X_{5}+2\right)\par\end{array}$
(1)

$\begin{array}[]{l}W_{C_{4,5}}=\frac{14}{10}-\frac{1}{10}Re(Z_{1}^{4}Z_{2}^{2}X_{3}X_{4}+Z_{1}^{4}Z_{2}^{3}X_{3}^{2}X_{4}^{2}+Z_{1}^{4}Z_{2}^{4}X_{3}^{3}X_{4}^{3}+Z_{1}^{4}I_{2}X_{3}^{4}X_{4}^{4}\\\
+Z_{1}^{3}Z_{2}^{3}X_{3}X_{4}+Z_{1}^{3}Z_{2}^{4}X_{3}^{2}X_{4}^{2}+Z_{1}^{3}I_{2}X_{3}^{3}X_{4}^{3}+Z_{1}^{3}Z_{2}X_{3}^{4}X_{4}^{4}+I_{1}Z_{2}X_{3}X_{4}\\\
+I_{1}Z_{2}^{2}X_{3}^{2}X_{4}^{2}+Z_{1}^{4}Z_{2}I_{3}I_{4}+Z_{1}^{3}Z_{2}^{2}I_{3}I_{4}+I_{1}I_{2}Z_{3}Z_{4}^{4}+I_{1}I_{2}Z_{3}^{2}Z_{4}^{3}+X_{1}X_{2}Z_{3}^{2}Z_{4}^{4}\\\
+X_{1}X_{2}Z_{3}^{3}Z_{4}^{3}+X_{1}X_{2}Z_{3}^{4}Z_{4}^{2}+X_{1}X_{2}I_{3}Z_{4}+X_{1}X_{2}Z_{3}I_{4}+X_{1}^{2}X_{2}^{2}Z_{3}^{3}Z_{4}^{4}+X_{1}^{2}X_{2}^{2}Z_{3}^{4}Z_{4}^{3}\\\
\left.+X_{1}^{2}X_{2}^{2}I_{3}Z_{4}^{2}+X_{1}^{2}X_{2}^{2}Z_{3}Z_{4}+X_{1}^{2}X_{2}^{2}Z_{3}^{2}I_{4}\right).\par\end{array}$
(2)

For the eight-qubit cluster state, each measurement basis includes two Pauli
$X$ operators for each photon. This is because there are at most two Pauli $X$
operators per photon in each term of the entanglement witness, and when there
are fewer, they are replaced with Identity operators, which can be evaluated
in the $X$ basis as well. The MPLC is thus used to perform the tensor product
of two Hadamard operations on the modes of each photon, interfering different
sets of four spatial modes. Similarly, for the four-qudit cluster state, each
measurement basis consists of at most one single five-dimensional Pauli $X$
operator per photon, and the MPLC performs the five-dimensional discrete
Fourier transform, interfering different sets of five spatial modes. In this
case, one measurement basis amounts to interfering the modes over the rows of
the 5x5 mode grid and the other over the columns. The expectation values of
all terms in the expressions for the entanglement witnesses are presented in
fig.S3, clearly showing genuine entanglement in the generated states.

Figure S3:  Entanglement certification. Expectation values for all terms in
the entanglement witness of the eight-qubit (A) and four-qudit (B) cluster
states. All values are above the required average to obtain a negative
entanglement witness (black dashed line). In the qudit case, the individual
expectation values can have imaginary components, which are expected to be
zero and are shown to indeed be negligible.

## Implementation of instantaneous feedforward

### Obtaining the five-qubit linear cluster state

For the demonstration of single-qubit rotation, we use a linear cluster state
with four qubits encoded on photon A and a fifth qubit encoded on photon B
(fig.3D). This state is obtained from the eight-qubit cluster state by
appropriately measuring qubits six to eight. Qubits six and seven are removed
from the cluster by projecting them onto the $+1$ eigenstate of the $Z$ basis,
which is equivalent to the $X$ basis in the experiment due to the additional
Hadamard gates acting on these qubits. Qubit eight is measured in either the
$Z$ or $Y$ basis, to both remove it from the cluster and to change the input
state $|\psi\rangle$ on qubit $1$.

### Instantaneous feedforward with linear optics

As described in the main text, single qubit rotations in MBQC are implemented
using a chain of five qubits. In standard binary encoded MBQC, the input state
on the first qubit is rotated and teleported to the fifth qubit by measuring
qubits one to four. The rotation is parameterized by three rotation angles,
$(\alpha,\beta,\gamma)$, around the $(X,Z,X)$ axes. To counteract the
randomness in the measurement results of the four qubits, their measurement is
performed sequentially in bases determined by previous measurement results
(fig.S4A). Using this measurement structure with classical feedforward, the
output state equals the desired state up to known Pauli correction dependent
on the results of the previous measurements.

In our experiment, the first four qubits are encoded using sixteen spatial
modes of a single photon. Since the qubits are encoded using a single photon,
multi-qubit gates that mimic the structure of the standard quantum circuit
with classical feedforward can be implemented using a linear optics circuit
encoding the desired unitary transformation on the optical modes. To this end,
we designed a linear optic circuit that performs intra-feedforward between
four qubits encoded on a single photon for measurement-based single-qubit
rotations, as described in fig.S4B. This approach circumvents the need for the
slow feedforward steps between the measurement of different qubits and enables
fast single-qubit rotations without the use of active feedforward between
qubits encoded on a single photon. The known Pauli corrections on the last
qubit are taken into account in the data analysis.

Figure S4:  Instantaneous feedforward with linear optics. In MBQC, a general
single-qubit rotation is performed by measuring four qubits out of a five-
qubit chain (A). Feedforward between measurements is used for counteracting
the inherent randomness of the measurement results. When the four qubits are
encoded within the same photon, as is the case in our experiment, an
equivalent linear optic circuit acting on sixteen optical modes can be used
instead (B). In (B), red rectangles represent phase shifters, and black
vertical lines with end dots represent beam splitters.

Specifically, in our implementation, we set $\gamma=0$ and start from an
eight-qubit cluster state, as described in the previous section. The first and
last layers of eight beam splitters in fig.S4B are automatically realized by
the extra Hadamard gates on qubits one and four in the generation of the
state. The rest of the linear optic circuit is realized via the MPLC. We add
the mode-dependent phases $\pm\alpha$ on the first MPLC plane, followed by the
relevant beam splitter operations simultaneously realized via planes one to
six of the MPLC. The phases $\pm\beta$ are then added on plane six, followed
again by eight simultaneous beam splitter operations realized via planes six
to ten of the MPLC. For the second photon, qubits six and seven are measured
in the $X$ basis as described before. Qubits five and eight are measured at
different bases depending on the prepared input state and the output
measurement basis.

### Extended results

We demonstrate measurement-based single-qubit rotations by looking at the
oscillations of output expectation values of the rotated state as a function
of the rotation angles $\alpha$ and $\beta$ around the $X$ and $Z$ axes,
respectively. We choose two different sets of an input state and output
observable to demonstrate dependence on both $\alpha$ and $\beta$ (fig.S5).
Despite reduced visibility resulting from noise and imperfect transformations,
good agreement with the expected rotations is observed in both cases. The
results in fig.3E,F of the main text are one-dimensional averages over the
data in fig.S5A,B, respectively. To reduce the measurement time in the single-
qubit rotation experiment, in this set of measurements, we raised the power of
the pump power to 125mW.

Figure S5:  Single qubit rotation. Expectation values for the Pauli $Z$ (A)
and $Y$ (B) operators as a function of the rotation angles $\alpha$ and
$\beta$, for input states
$|\psi\rangle=\frac{1}{\sqrt{2}}(|0\rangle+i|1\rangle)$ and
$|\psi\rangle=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$, respectively.


# Anomalous magnetoentropic response of skrymion crystals

Ahmed R. Saikia  Department of Physics, Indian Institute of Technology
Roorkee, Roorkee 247667, India  Narayan Mohanta<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology Roorkee, Roorkee 247667,
India

###### Abstract

We investigate theoretically magnetoentropic signatures of the crystal phase
of magnetic skyrmions of various kinds, commonly appearing in two dimensions,
viz., Néel, Bloch and anti skyrmions. Using Monte Carlo calculations based on
spin Hamiltonians, we obtain magnetic entropy change $\Delta S_{m}$ in the
presence of three different types of Dzyaloshinskii-Moriya interactions
responsible for these skyrmions. The phase mapping of $\Delta S_{m}$ using
skyrmion counting number $N_{sk}$ in temperature-magnetic field plane reveals
fluctuation-dominated weak first-order transition in the precursor phase of
the skyrmions, and a sign change in $\Delta S_{m}$ when the system enters into
the skyrmion crystal phase—in agreement with recent experimental findings. We
also find that the fractional entropy change in going from a ferromagnetic
phase to the skyrmion crystal phase is much larger compared to the
conventional route of paramagnetic phase to ferromagnetic phase, used for the
purpose of magnetic cooling. The magnetoentropic signatures of the different
types of skyrmion crystals are found to be similar. Our results indicate that
the skyrmion crystals exhibit enhanced cooling efficiency and have the
potential to upgrade the existing magnetic cooling methods.

##  I Introduction

Chiral spin fluctuations in magnets with antisymmetric Dzyaloshinskii-Moriya
interaction (DMI) near the onset of magnetically ordered phases give rise to
anomalies in various physical observables including magnetization, specific
heat, electric polarization and magnetic susceptibility Pappas  _et al._
(2009); Ruff  _et al._ (2015); Pardeep  _et al._ (2023). These chiral spin
fluctuations, driven primarily by the frustration introduced by the DMI and
aided by intrinsic inhomogeneities, also generate strong signatures in the
magnetic entropy near the temperature-driven transition to the ordered phases.
In chiral magnets and spinel compounds, which support skyrmion crystal (SkX)
phase at high temperatures, the magnetic entropy response was found to clearly
distinguish the SkX phase from other topologically trivial phases such as
conical and ferromagnetic phases Ge  _et al._ (2015); Han  _et al._ (2017);
Bocarsly  _et al._ (2018); Jamaluddin  _et al._ (2019); Dhital and DiTusa
(2020); Zuo  _et al._ (2021). The SkX phase appears also robustly in two
dimensions, at the interface between two oxide compounds and in van der Waals
magnets, stabilized by a magnetic anisotropy which originates from interfacial
strain and favors the long-range magnetic order Li  _et al._ (2014); Yokouchi
_et al._ (2015); Matsuno  _et al._ (2016); Rowland  _et al._ (2016); Hoffmann
_et al._ (2017); Nakamura  _et al._ (2018); Vistoli  _et al._ (2019); Mohanta
_et al._ (2019); Park  _et al._ (2021). Within a range of temperatures above
the critical temperature for the SkX phase, the average scalar spin chirality
$\chi\\!=\\!\langle{\bf S}_{i}\cdot({\bf S}_{j}\times{\bf S}_{k})\rangle$ can
be finite even when $\langle{\bf S}_{i}\rangle\approx 0$. The nature of the
phase transition from the paramagnetic phase to the SkX phase is, therefore,
dictated primarily by the spin chirality fluctuation.

In this paper, we investigate the behavior of the magnetic entropy as the
transition to the SkX phase takes place in two dimensional magnets, and
explore the classification strategy of the SkX phase using the magnetoentropic
response. The central object of our investigation is the magnetic entropy
change, which can be obtained using the following Maxwell’s thermodynamics
relation

$\displaystyle\Big{(}\frac{\partial S}{\partial
H}\Big{)}_{T}=\Big{(}\frac{\partial M}{\partial T}\Big{)}_{H}$ (1)

The above relation provides the magnetic entropy change $\Delta S_{m}$ in
terms of the temperature dependence of magnetization. We perform Monte Carlo
calculations and obtain $\Delta S_{m}$ as a function of temperature and
magnetic field, and obtain the magnetoentropic response of various ordered
phases. Magnetic skyrmions are topologically stable spin texture which can be
characterized by an invariant known as the skyrmion number, defined as

$\displaystyle N_{sk}=\frac{1}{4\pi}\int{\bf S}\cdot\Big{(}\frac{\partial{\bf
S}}{\partial x}\times\frac{\partial{\bf S}}{\partial y}\Big{)}dx~{}dy.$ (2)

The SkX phase exhibits a large value of $N_{sk}$; hence it provides an useful
tool to map out the SkX phase by comparing $N_{sk}$ with $\Delta S_{m}$ in the
phase plane spanned by magnetic field and temperature. We perform this
magnetoentropic mapping for three different types of DMI in two dimensions
which give rise to three common types of skyrmions, viz., Néel, Bloch and anti
skyrmions. Besides, $\Delta S_{m}$ can also shed light on the puzzle in
understanding the role of critical spin fluctuations as the system transitions
to the SkX phase when the temperature is lowered. The puzzle arises because
specific heat measurements reveal a first-order transition at zero magnetic
field to the helical state Bauer  _et al._ (2013), while neutron probes reveal
Landau soft-mode mechanism of weak crystallization to the SkX phase at finite
magnetic fields Wilhelm  _et al._ (2011); Samatham and Ganesan (2013);
Kindervater  _et al._ (2019). These experimental observations suggest that
phase transitions in Dzyaloshinskii-Moriya magnets occur via a precursor phase
in which chiral spin textures emerge with an abundance of spin fluctuations.
The magnetic entropy change $\Delta S_{m}$ can provide a great amount of
information about the fluctuation-induced gradual first-order transition to
the SkX phase Caron  _et al._ (2009a). Valleys and peaks in $\Delta S_{m}$ can
reveal phase transitions which can be detected in experiments and give
information complementary to the conventional signatures in the heat capacity
$C\\!=\\!T(\partial S/\partial T)$ Dhital and DiTusa (2020). We find that for
all three types of skyrmions, there is a change in the slope in the
temperature variation of $\Delta S_{m}$ as the system enters into the
precursor phase from the spin-disordered paramagnetic phase. Also, $\Delta
S_{m}$ undergoes a change in sign as the system enters the fully-ordered SkX
phase. These features enable us to identify the critical temperatures for the
precursor phase and the fully-ordered SkX phase. The magnetic entropy change
can also provide valuable insights into entropy-limited topological protection
and lifetime of skyrmions Wild  _et al._ (2017).

Conventional magnetic alloy compounds exhibiting a large magnetic entropy
change also suffer from considerable thermal hysteresis effect which makes
these materials unsuitable for applications in refrigerators due to their low
performance in the cyclic operation at high cycle frequencies Caron  _et al._
(2009b). The irreversibility of the magnetization also causes serious problems
in estimating the magnetocaloric effect from the magnetization measurements
Amaral and Amaral (2009); Balli  _et al._ (2009); Carvalho  _et al._ (2011).
We here also discuss about magnetic cooling efficiency of a cooling cycle
involving a SkX phase, and compare it qualitatively with the conventional
cycle involving the paramagnetic and ferromagnetic phases. Based on our
estimation of a large fractional entropy change, we propose that the cyclic
route between the ferromagnetic phase and the SkX phase can enable a more
efficient magnetic cooling method. Compounds hosting topological spin textures
such as the SkX can, therefore, be a promising alternative for the
refrigeration applications with high efficiency.

The remainder of this paper is organized as follows: in section II, we discuss
our theoretical model and provide details of our Monte Carlo annealing
simulation. In section III, we present the numerical results of the magnetic
entropy change $\Delta S_{m}$ as the temperature is reduced while approaching
various phases viz. spin spiral (SS), SkX and ferromagnetic (FM) phases, all
stabilized at low temperatures in our considered two-dimensional spin systems.
In section III, we discuss the experimental relevance of our results, enhanced
magnetic cooling efficiency of the SkX phase, and summarize our results.

##  II Model and method

We consider two-dimensional spin systems with competing Heisenberg exchange
interaction and DMI, in the presence of an external magnetic field applied
perpendicular to the two-dimensional plane (considered to be the $x$-$y$
plane) and an easy-plane magnetic anisotropy. The model Hamiltonian for such
systems is given by

$\displaystyle\mathcal{H}=-J\sum_{\langle ij\rangle}{\bf S}_{i}\cdot{\bf
S}_{j}+\mathcal{H}_{\rm DMI}-H_{z}\sum_{i}S_{zi}-A\sum_{i}|S_{zi}|^{2}$ (3)

where ${\bf S_{i}}\\!\equiv\\!(S_{xi},S_{yi},S_{zi})$ represents the spin
vector of magnitude $S\\!=\\!1$ at site $i$, $J$ is the strength of nearest-
neighbor Heisenberg exchange interaction, $\mathcal{H}_{\rm DMI}$ is the
Hamiltonian describing the DMI which is mentioned below, $H_{z}$ is the
strength of the perpendicular magnetic field, and $A$ is the easy-plane
magnetic anisotropy which helps to stabilize the magnetic skyrmions and the
long-range order in the SkX Bogdanov and Hubert (1999); Banerjee  _et al._
(2014); Gungordu  _et al._ (2016); Leonov and Kezsmarki (2017). We used
$A\\!=\\!0.001J$ throughout this paper.

As discussed widely in literature, the DMI term is relevant in materials with
strong spin-orbit coupling and broken inversion symmetry. Magnets with
sizeable Rashba spin-orbit coupling, such as those interfacing another
compound in a two-dimensional plane, are candidate systems where the DMI
amplitude is typically large. Here we consider three different types of DMI,
commonly appearing in two dimensions and responsible for the Néel, Bloch and
anti skyrmions. The vectors representing these three types of DMIs are shown
in Fig. 1, and the explicit forms of $\mathcal{H}_{\rm DMI}$ in these three
cases are expressed as

$\displaystyle\mathcal{H}_{\rm DMI}=-D\sum_{\langle
ij\rangle}(\hat{z}\times\hat{r}_{ij})\cdot({\bf S}_{i}\times{\bf S}_{j}),$ (4)
$\displaystyle\mathcal{H}_{\rm DMI}=-D\sum_{\langle
ij\rangle,\hat{e}=\hat{x},\hat{y}}(\pm\hat{e})\cdot({\bf S}_{i}\times{\bf
S}_{j=i\pm\hat{e}}),$ (5) $\displaystyle\mathcal{H}_{\rm DMI}=-D\sum_{\langle
ij\rangle}\eta(\hat{z}\times\hat{r}_{ij})\cdot({\bf S}_{i}\times{\bf S}_{j}),$
(6)

where, in the last expression $\eta\\!=\\!+1$ for $j\\!=\\!i\pm\hat{x}$ and
$\eta\\!=\\!-1$ for $j\\!=\\!i\pm\hat{y}$. The DMI in Eq.(4), Eq.(5) and
Eq.(6) stabilizes, respectively, the Néel, Bloch and anti skyrmions in two
dimensional lattices. We consider $J\\!=\\!1$, and express all other energies
in units of $J$.

The SkX phase can also originate without the explicit need of the DMI or
external magnetic field, from spin frustration or interfacial phase
frustration Ozawa  _et al._ (2017); Hayami and Motome (2021); Mohanta and
Dagotto (2022). We anticipate our discussions below and the magnetoentropic
signatures to be relevant in those unconventional SkXs as well.

Figure 1: Schematic representation of the three types of Dzyaloshinskii-Moriya
interaction (DMI) considered in the two-dimensional geometry. Blue spheres
represent nearest-neighbor lattice sites, with indices $i$ and $j$. Red arrows
denote the vectors of the DMI in the three cases, which give rise to (a) Néel,
(b) Bloch and (c) anti skyrmions, respectively. Figure 2: Spin configurations
in the spin spiral (SS) and the skyrmion crystal (SkX) phases, obtained in
Monte Carlo annealing process at a low temperature $T\\!=\\!0.001J$ on a
24$\times$24 square lattice with periodic boundary conditions. (a), (b), (c)
SS texture realized in the absence of any magnetic field (i.e. at
$H_{z}\\!=\\!0$) for the three types of DMI discussed in Sec. II. (d),(e),(f)
Triangular crystal structure of Néel, Bloch and anti skyrmions, respectively,
obtained at a magnetic field $H_{z}\\!=\\!1.8J$. The colors in the arrows
represent the $z$ component of the spin vector, blue (red) being the down (up)
spin arrangement. Other parameters used are $J\\!=\\!1$, $D\\!=\\!0.5J$, and
$A\\!=\\!0.001J$. Figure 3: (a),(b),(c) Temperature variation of magnetic
entropy change $\Delta S_{m}$ for the three types of DMI discussed in Sec. II,
at a magnetic field $H_{z}\\!=\\!4.4J$ which stabilizes the FM phase (blue
region) at low temperatures. (d),(e),(f) Temperature variation of $\Delta
S_{m}$ for the three types of DMI, at $H_{z}\\!=\\!1.8J$ which stabilizes the
SkX phase (red region) at low temperatures. Other parameters used are
$J\\!=\\!1$, $D\\!=\\!0.5J$, and $A\\!=\\!0.001J$. The upward turn in $\Delta
S_{m}$ indicates the entrance to the ordered phase via the fluctuation-
dominated regime. The change in the sign of $\Delta S_{m}$ occurs when the
skyrmions stabilize and form a triangular crystal at low temperatures. Figure
4: Magnetic entropic change $\Delta S_{m}$ (top row in (a),(b),(c)) and the
skyrmion number $N_{\rm sk}$ (bottom row in (d),(e),(f)) in the temperature
$T$ vs. magnetic field $H_{z}$ plane, for the three types of DMI
configurations. The top and bottom rows provide a magnetoentropic mapping of
the skyrmion crystal phase—shown by the red regions in top panels and bright
yellow regions in below panels. Parameters used are $J\\!=\\!1$,
$D\\!=\\!0.5J$, and $A\\!=\\!0.001J$.

Monte Carlo annealing: Spin configurations are obtained using Monte Carlo (MC)
annealing procedure on a square lattice of size 24$\times$24 with open
boundary conditions. The annealing process was started at a high temperature
$T\\!=\\!10J$ with an initial random spin configuration, and the temperature
was gradually lowered down to $T\\!=\\!0.001J$ in 500 steps. At each
temperature step, $10^{8}$ MC spin update steps are performed. At each spin
update step, a new direction of the spin vector at a randomly-selected site is
chosen randomly within a small cone of angle $\Delta\theta=2^{\circ}$ around
the initial spin direction. The new trial spin configuration is accepted or
rejected using the standard metropolis algorithm: accepted if the energy
difference $\Delta E$ between the two spin configurations is negative, and
accepted with the Boltzmann probability $e^{-\Delta E/(k_{B}T)}$ when $\Delta
E$ is positive; the latter is to incorporate the thermal spin fluctuation
which increases with temperature.

Calculation of magnetic entropy change: The change in the magnetic entropy at
a given temperature $T$ and magnetic field $H$ is obtained, using Eq. (1), as

$\displaystyle\Delta
S_{m}(H,T)=\frac{\mu_{0}}{T_{2}-T_{1}}\int_{0}^{H}M(T_{2},H)-M(T_{1},H)dH$ (7)

where, $T_{1}\\!=\\!T-\Delta T$, $T_{2}=T+\Delta T$ are two considered
temperatures around $T$, $\Delta T$ is incremental temperature gradient,
$\mu_{0}$ is the free-space permeability, and $M$ is the average
magnetization. The above integral is evaluated numerically using the Simpson’s
method from the magnetization data obtained in MC simulations. To enhance the
smoothness of the magnetic entropy variation with $T$, a Savitzky-Golay filter
was applied. This filtering scheme uses a $4^{\rm th}$ order polynomial for
local fitting, and reduces high-frequency noise from the entropy signal,
revealing a clearer representation of the overall trend in the entropy change.

Calculation of skyrmion number: The skyrmion number in Eq. (2) can be
expressed in discretized form as

$\displaystyle N_{sk}=\frac{1}{4\pi}\sum_{i,j}{\bf S}_{ij}.[({\bf
S}_{i+1,j}-{\bf S}_{i,j})\times({\bf S}_{i,j+1}-{\bf S}_{i,j})],$ (8)

in which the integration in Eq. (2) has been replaced by a summation and the
partial derivatives are evaluated using a central-difference scheme. Magnetic
skyrmions can be classified by the topological index $N_{sk}$, since under
transformation of the underlying lattice from a torus to a sphere a skyrmion
gives a full coverage of the sphere—a unique feature of the topological
structure of the skyrmion. The number $N_{sk}$, therefore, helps in the
characterization of the skyrmions, especially the SkX phase, in the phase
plane spanned by the magnetic field and temperature.

##  III Results

###  III.1 Three types of skyrmion crystals

In our two-dimensional spin systems, the SkX phase appears at low temperatures
within a range of external magnetic fields, for all three types of DMI
described above. The MC-generated spin configurations representing the chiral
ordered phases, viz. the spin spiral (SS) which appear in the absence of any
magnetic field and the SkX, for the three types of DMIs at a low temperature
$T\\!=\\!0.001J$ are shown in Fig. 2. Fig. 2(a),(b),(c) show one of the two
natural solutions of diagonal arrangements for the SS configurations. On the
other hand, Fig. 2(d),(e),(f) show triangular crystals of Néel, Bloch and anti
skyrmions, respectively. Promising platforms in which such two-dimensional
skyrmions can appear are interfaces between compounds with competing magnetic
interactions and broken inversion symmetry at the interface Li  _et al._
(2014); Yokouchi  _et al._ (2015); Matsuno  _et al._ (2016); Hoffmann  _et
al._ (2017); Nakamura  _et al._ (2018); Vistoli  _et al._ (2019); Mohanta  _et
al._ (2019); Park  _et al._ (2021). At finite temperatures, the phase
boundaries between the SS, the SkX and the fully-polarized FM phases are not
sharp, but rather exhibit a region with metastable spin configurations with
characteristics of the neighboring phases and governed primarily by thermal
fluctuations Mohanta  _et al._ (2019). One such metastable phase which is
important for the below discussion is the precursor phase of the skyrmions
which prevails at temperatures above the critical temperature for the SkX
phase.

###  III.2 Temperature dependence of entropy change

Fig. 3 shows the temperature dependence of the magnetic entropy change $\Delta
S_{m}$ for the three types of DMI discussed in Sec. II and at two values of
the magnetic field $H_{z}\\!=\\!4.4J$ and $H_{z}\\!=\\!1.8J$, at which the
system is stabilized in the FM and the SkX phases at low temperatures,
respectively, upon cooling from the spin-disordered paramagnetic phase. At
$H_{z}\\!=\\!4.4J$ (Fig. 3(a),(b),(c)), for which the FM phase is stabilized
at low temperatures, the magnitude of $\Delta S_{m}$ is peaked at a critical
temperature $T\\!\approx\\!3J$, at the onset of the FM phase, below which it
starts to decrease. A similar behavior is observed at $H_{z}\\!=\\!1.8J$, for
which the SkX phase is stabilized at low temperatures—$\Delta S_{m}$ shows
upturn below a temperature $T\\!\approx\\!3J$. In addition, in this case
$\Delta S_{m}$ changes sign at $T\\!\approx\\!1.8J$. This positive entropy
change is revealed in all three types of the DMI settings.

###  III.3 Entropy response in the skyrmion crystals

To understand the connection between the features in the magnetic entropy
change $\Delta S_{m}$ and various temperature dependent phase transitions, we
compare $\Delta S_{m}$ with the skyrmion number $N_{sk}$ in the parameter
plane spanned by temperature $T$ and the external magnetic field $H_{z}$, as
shown in Fig. 4. At larger values of the magnetic field
($H_{z}\\!\gtrsim\\!3J$), for which the fully-polarized FM phase is stabilized
at low temperatures, $\Delta S_{m}$ varies continuously with temperature and
exhibits a maximum in its absolute value near the critical temperature for the
transition to the FM phase, as shown by the dark blue regions in Fig.
4(a)-(c). At smaller values of the magnetic field ($H_{z}\\!\lesssim\\!0.5J$),
the transition takes place from the spin-disoriented paramagnetic to the SS
phase upon lowing the temperature. In this case, the change in the magnetic
entropy is nearly zero, as shown by the white regions in Fig. 4(a)-(c). At
intermediate values of the magnetic field
($1.5\\!\lesssim\\!H_{z}\\!\lesssim\\!2.5J$), the SkX phase is stabilized at
low temperatures, as evident from Fig. 4(d)-(f). Remarkably, $\Delta S_{m}$
undergoes a sign change as the system enters into the SkX phase either from
the paramagnetic phase or from the fully-polarized FM phase. The positive
entropy change, shown by the red regions in Fig. 4(a)-(c), is found for all
three types of skyrmions realized in our two-dimensional systems, and also
observed in three-dimensional materials in experiments Bocarsly  _et al._
(2018); Dhital and DiTusa (2020); Zuo  _et al._ (2021).

Spin fluctuations play an important role in the temperature-driven phase
transitions in Dyaloshinskii-Moriya magnets Janoschek  _et al._ (2013).
Magnetic entropy change, obtained via measurements of magnetization and
specific heat, can provide a great amount of information about the nature of
the fluctuation-dominated phase transitions Bocarsly  _et al._ (2018);
Kautzsch  _et al._ (2020); Schueller  _et al._ (2020). Both experimental and
theoretical studies suggest that the temperature-driven transition from the
paramagnetic to the SkX phase occurs via a precursor phase in which non-
trivial topological spin textures of skyrmionic character are developed within
a paramagnetic background with abundance of fluctuations Wilhelm  _et al._
(2011); Kindervater  _et al._ (2019); Mohanta  _et al._ (2020a). This is
evident from the plots of the skyrmion number $N_{\rm sk}$ in Fig. 4(d)-(f),
as this number fluctuates and its non-zero value is extended up to much higher
temperature above the SkX phase, which prevails at temperatures
$T\\!\lesssim\\!1.5J$. This fluctuation-dominated weak first order phase
transition, also known as the Brazovskii transition, exists also at higher
fields, as it can be seen from the magnetic entropy change in going from the
paramagnetic to the ferromagnetic phase. The results presented in Fig. 3 and
Fig. 4 enable us to conclude that the precursor phase involving skyrmions
exists at temperatures $T\\!\lesssim\\!3J$, below which $\Delta S_{m}$ changes
its slope. The essential features of $\Delta S_{m}$ are qualitatively the same
for all three types of DMI considered in our models.

##  IV Discussion and summary

Because of the sign reversal in the magnetic entropy change $\Delta S_{m}$ in
the SkX phase, the fractional change in the magnetic entropy in transitioning
from the fully-polarized FM phase to the SkX phase i.e. $|\Delta S_{m}^{\rm
SkX}-\Delta S_{m}^{\rm FM}|/\Delta S_{m}^{\rm FM}$ is always larger than the
fractional change in the magnetic entropy in transitioning from the PM phase
to the FM phase i.e. $|\Delta S_{m}^{\rm FM}-\Delta S_{m}^{\rm PM}|/\Delta
S_{m}^{\rm PM}$. In Fig. 5(a), the two routes for magnetic cooling process are
shown; the PM to FM path is used in conventional magnetic cooling methods
based on the magnetocaloric effect (MCE). The phase boundaries can be drawn
explicitly by using various observables such as magnetization, spin-spin
correlation function, syrmion number and topological Hall conductivity, as
done in previous studies Mohanta  _et al._ (2019, 2020b).

Figure 5: (a) Schematic phase diagram, showing two routes of phase transition
(green arrows) for magnetic cooling cycle: one between the paramagnetic (PM)
and ferromagnetic (FM) phases, and another between the skyrmion crystal (SkX)
and the FM phases. (b) Proposed cooling cycle between the SkX and the FM
phases. Increasing magnetic field causes FM ordering, which results in
increase in temperature of the material. Once excess heat is expelled out, the
material is again demagnetized, and the temperature is decreased.

The MCE also depends significantly on the nature of the variation in
magnetization near the phase transitions. Based on the nature of the magnetic
transition, a variety of magnetic materials have been proposed to enhance the
efficiency of the MCE Franco  _et al._ (2012). Materials with a first-order
phase transition to the magnetically-ordered phase exhibit a large change in
the magnetization over a small temperature range. It results in an abrupt
change in the magnetic entropy, which leads to losses by hysteresis and
irreversibilities of the magnetic state during demagnetization Mellari (2023).
The energy lost during the cycle involving the magnetization and
demagnetization, is dissipated as heat via the magnetic circuits. Since the
Maxwell’s relation in Eq. (1) is valid for systems in equilibrium, the
dissipative kinetics essentially reduces the efficiency of the MCE of these
materials. Additional structural transitions have also been reported in the
vicinity of the first-order magnetic phase transition Murtaza  _et al._
(2020). Materials with a smooth phase transition to the magnetically-ordered
phase are, on the other hand, free from these penalizing phenomena such as
hysteresis loss, slow kinetics and structural transition. Because of the above
advantages, we propose materials hosting the SkX phase to be good candidates
for the magnetocaloric applications. A typical cycle of the magnetic cooling
method involving a SkX phase and a FM phase is shown in Fig. 5(b). The
proposed refrigeration cycle is performed by adiabatically magnetizing the
system to FM phase from SkX phase, causing magnetic heating. This heat is then
expelled through the system which again undergoes magnetic cooling through
adiabtic demagnetization. Finally, the system absorbs heat from the
environment and reaches its thermal equilibrium, from which it can restart the
magnetic refrigeration cycle.

To summarize, we investigated the magnetic entropy change in two-dimensional
magnets hosting various kinds of skyrmion crystals and the nature of
fluctuation-dominated phase transitions. Our results confirm several
experimental reports and suggest that the magnetic entropy change can be used
as a general probe, as an useful alternative to neutron probes in momentum
space Muhlbauer  _et al._ (2009); Pappas  _et al._ (2009); Kindervater  _et
al._ (2019), transmission-electron microscopy imaging in real space Yu  _et
al._ (2010, 2011); Tokunaga  _et al._ (2015a) or topological Hall effect-based
probes Yokouchi  _et al._ (2015); Matsuno  _et al._ (2016); Nakamura  _et al._
(2018); Vistoli  _et al._ (2019), to analyze the nature of phase transitions
in Dzyaloshinskii-Moriya magnets in two and three dimensions. Our study also
provides interesting perspectives about magnetocaloric effect in the SkX
phase, and suggest that magnetic materials which host the SkX phase can be
used in efficient magnetic cooling applications. More stable topological
magnetic phases, such as hopfions which appear in three-dimensional materials,
open up a new path towards the exploration of magnetocaloric applications with
enhanced efficiency at elevated temperatures Tokunaga  _et al._ (2015b); Kent
_et al._ (2021); Rybakov  _et al._ (2022).

## Acknowledgements

NM acknowledges support of an initiation grant (No. IlTR/SRIC/2116/FIG) from
IIT Roorkee and SRG grant (No. SRG/2023/001188) from SERB. Numerical
calculations were performed at the computing resources of PARAM Ganga at IIT
Roorkee, provided by National Supercomputing Mission, implemented by C-DAC,
and supported by the Ministry of Electronics and Information Technology and
Department of Science and Technology, Government of India.

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