of $u_{i}$ in $\mathcal{G}^{\prime}$ is the same. The input of the token mixer
phase on $\mathcal{G}$ is $\mathcal{V}$ as all of them are connected by a
hyperedge. Then we have:

$(\mathbf{H}_{\text{token}})_{i,j}=\mathbf{v}_{i,j}+\sigma\left(\frac{\exp(\mathbf{v}_{i,j})}{\sum_{k=1}^{M(L+1)+1}\exp(\mathbf{v}_{k,j})}\right).$
(16)

On the other hand, when applied to hypergraph $\mathcal{G}^{\prime}$ and
$(u_{k_{1}},u_{k_{2}})$. We have:

$(\mathbf{H}^{\prime}_{\text{token}})_{i,j}=\mathbf{v}_{i,j}+\sigma\left(\frac{\exp(\mathbf{v}_{i,j})}{\exp(\mathbf{v}_{k_{1},j})+\exp(\mathbf{v}_{k_{2},j})}\right),\>\>\>\>i\in\\{k_{1},k_{2}\\}.$
(17)

Since we use zero padding, for any $i\geq 3$,
$(\mathbf{H}_{\text{token}})_{i,j}\neq 0$ and
$(\mathbf{H}^{\prime}_{\text{token}})_{i,j}=0$. These zero rows, which capture
the size of the hyperedge, result in different encodings for each connection.

######  Remark 5.

To the best of our knowledge, the only anonymization process that is used on
hypergraphs is by Liu et al. [78], which uses simple walks on the CE and is
the same as Wang et al. [33]. Accordingly, it also suffers from the above
limitation. Also, note that this theorem shows the limitation of these
anonymization procedures when simply adopted to hypergraphs.

####  E.5 Proof of Proposition 2

######  Proposition 2.

Edge-independent random walks on hypergraphs [37], edge-dependent random walks
on hypergraphs [40], and simple random walks on the CE of hypergraphs are all
special cases of $r$-SetWalk, when applied to the $2$-projected graph, UP
graph, and $2$-projected graph, respectively. Furthermore, all the above
methods are less expressive than $r$-SetWalks.

###### Proof.

For the first part, we discuss each walk separately:

1

Simple random walks on the CE of the hypergraphs: We perform $2$-SetWalks on
the (weighted) 2-projected hypergraph with $\Gamma(.)=1$. Accordingly, for
every two adjacent edges in the 2-Projected graph like $e$ and $e^{\prime}$,
we have $\varphi(e,e^{\prime})=1$. Therefore, it is equivalent to a simple
random walk on the CE (2-projected graph).

2

Edge-independent random walks on hypergraphs: As is shown by Chitra and
Raphael [40], each edge-independent random walk on hypergraphs is equivalent
to a simple random walk on the (weighted) CE of the hypergraph. Therefore, as
discussed in  1, these walks are a special case of $r$-SetWalks, when $r=2$
and applied to (weighted) $2$-Projected hypergraph.

3

Edge-dependent random walks on hypergraphs: Let $\Gamma^{\prime}(e,u)$ be an
edge-dependent weight function used in the hypergraph random walk sampling.
For each node $u$ in the UP hypergraph, we store the set of
$\Gamma^{\prime}(e,u)$ that $e$ is a maximal hyperedge that $u$ belongs to.
Note that there might be several maximal hyperedges that $u$ belongs to. Now,
we perform 2-SetWalk sampling on the UP hypergraph with these weights and in
each step, we sample each hyperedge with weight
$\Gamma(u,e)=\Gamma^{\prime}(e,u)$. It is straightforward to show that given
this procedure, the sampling probability of a hyperedge is the same in both
cases. Therefore, edge-dependent random walks on hypergraphs are equivalent to
$2$-SetWalks when applied to the UP hypergraph.

As discussed above, all these walks are special cases of $r$-SetWalks and
cannot be more expressive than $r$-SetWalks. Also, as discussed in Theorem 1,
all these walks are less expressive than SetWalks, which are also special
cases of $r$-SetWalks, when $r=\infty$. Accordingly, all these methods are
less expressive than $r$-SetWalks. ∎

###  Appendix F Experimental Setup Details

####  F.1 Datasets

We use 10 publicly available222https://www.cs.cornell.edu/~arb/data/ benchmark
datasets, whose descriptions are as follows:

  * •

NDC Class [6]: The NDC Class dataset is a temporal higher-order network, in
which each hyperedge corresponds to an individual drug, and the nodes
contained within the hyperedges represent class labels assigned to these
drugs. The timestamps, measured in days, indicate the initial market entry of
each drug. Here, hyperedge prediction aims to predict future drugs.

  * •

NDC Substances [6]: The NDC Substances is a temporal higher-order network,
where each hyperedge represents an NDC code associated with a specific drug,
while the nodes represent the constituent substances of the drug. The
timestamps, measured in days, indicate the initial market entry of each drug.
The hyperedge prediction task is the same as NDC Classes dataset.

  * •

High School [6, 92]: The High School is a temporal higher-order network
dataset constructed from interactions recorded by wearable sensors in a high
school setting. The dataset captures a high school contact network, where each
student/teacher is represented as a node and each hyperedge shows Face-to-face
contact among individuals. Interactions were recorded at a resolution of 20
seconds, capturing all interactions that occurred within the previous 20
seconds. Node labels in this data are the class of students, and we focus on
the node class "PSI" in our classification tasks.

  * •

Primary School [6, 93]: The primary school dataset resembles the high school
dataset, differing only in terms of the school level from which the data is
collected. Node labels in this data are the class of students, and we focus on
the node class "Teachers" in our classification tasks.

  * •

Congress Bill [6, 94, 95]: Each node in this dataset represents a US
Congressperson. Each hyperedge is a legislative bill in both the House of
Representatives and the Senate, connecting the sponsors and co-sponsors of
each respective bill. The timestamps, measured in days, indicate the date when
each bill was introduced.

  * •

Email Enron [6]: In this dataset nodes are email addresses at Enron and
hyperedges are formed by emails, connecting the sender and recipients of each
email. The timestamps have a resolution of milliseconds.

  * •

Email Eu [6, 96]: In this dataset, the nodes represent email addresses
associated with a European research institution. Each hyperedge consists of
the sender and all recipients of the email. The timestamps in this dataset are
measured with a resolution of 1 second.

  * •

Question Tags M (Math sx) [6]: This dataset consists of nodes representing
tags and hyperedges representing sets of tags applied to questions on
math.stackexchange.com. The timestamps in the dataset are recorded at
millisecond resolution and have been normalized to start at 0.

  * •

Question Tags U (Ask Ubuntu) [6]: In this dataset, the nodes represent tags,
and the hyperedges represent sets of tags applied to questions on
askubuntu.com. The timestamps in the dataset are recorded with millisecond
resolution and have been normalized to start at 0.

  * •

Users-Threads [6]: In this dataset, the nodes represent users on
askubuntu.com, and a hyperedge is formed by users participating in a thread
that lasts for a maximum duration of 24 hours. The timestamps in the dataset
denote the time of each post, measured in milliseconds but normalized such
that the earliest post begins at 0.

The statistics of these datasets can be found in Table 3.

Table 3: Dataset statistics. HeP: Hyperedge Prediction, NC: Node
Classification

Dataset | NDC Class | High School | Primary School | Congress Bill | Email Enron | Email Eu | Question Tags M | Users-Threads | NDC Substances | Question Tags U
---|---|---|---|---|---|---|---|---|---|---
$|\mathcal{V}|$ | 1,161 | 327 | 242 | 1,718 | 143 | 998 | 1,629 | 125,602 | 5,311 | 3,029
$|\mathcal{E}|$ | 49,724 | 172,035 | 106,879 | 260,851 | 10,883 | 234,760 | 822,059 | 192,947 | 112,405 | 271,233
#Timestamps | 5,891 | 7,375 | 3,100 | 5,936 | 10,788 | 232,816 | 822,054 | 189,917 | 7,734 | 271,233
Task | HeP | HeP& NC | HeP & NC | HeP | HeP | HeP | HeP | HeP | HeP | HeP

####  F.2 Baselines

We compare our method to eight previous state-of-the-art methods and baselines
on the hyperedge prediction task:

  * •

CHESHIRE [11]: Chebyshev spectral hyperlink predictor (CHESHIRE), is a
hyperedge prediction methods that initializes node embeddings by directly
passing the incidence matrix through a one-layer neural network. CHESHIRE
treats a hyperedge as a fully connected graph (clique) and uses a Chebyshev
spectral GCN to refine the embeddings of the nodes within the hyperedge. The
Chebyshev spectral GCN leverages Chebyshev polynomial expansion and spectral
graph theory to learn localized spectral filters. These filters enable the
extraction of local and composite features from graphs that capture complex
geometric structures. The model with code provided is here.

  * •

HyperSAGCN [26]: Self-attention-based graph convolutional network for
hypergraphs (HyperSAGCN) utilizes a Spectral Aggregated Graph Convolutional
Network (SAGCN) to refine the embeddings of nodes within each hyperedge.
HyperSAGCN generates initial node embeddings by hypergraph random walks and
combines node embeddings by Mean (.) pooling to compute the embedding of
hyperedge. The model with code provided is here.

  * •

NHP [87]: Neural Hyperlink Predictor (NHP), is an enhanced version of
HyperSAGCN. NHP initializes node embeddings using Node2Vec on the CE graph and
then uses a novel maximum minimum-based pooling function that enables adaptive
weight learning in a task-specific manner, incorporating additional prior
knowledge about the nodes. The model with code provided is here.

  * •

HPLSF [89]: Hyperlink Prediction using Latent Social Features (HPLSF) is a
probabilistic method. It leverages the homophily property of the networks and
introduces a latent feature learning approach, incorporating the use of
entropy in computing hyperedge embedding. The model with code provided is
here.

  * •

HPRA [88]: Hyperlink Prediction Using Resource Allocation (HPRA) is a
hyperedge prediction method based on the resource allocation process. HPRA
calculates a hypergraph resource allocation (HRA) index between two nodes,
taking into account direct connections and shared neighbors. The HRA index of
a candidate hyperedge is determined by averaging all pairwise HRA indices
between the nodes within the hyperedge. The model with code provided is here.

  * •

CE-CAW: This model is a baseline that we apply CAW [33] on the CE of the
hypergraph. CAW is a temporal edge prediction method that uses causal
anonymous random walks to capture the dynamic laws of the network in an
inductive manner. The model with code provided is here.

  * •

CE-EvolveGCN: This is a snapshot-based temporal graph learning method that we
apply EvolveGCN [90], which uses Rnns to estimate the GCN parameters for the
future snapshots, on the CE of the hypergraph. The model with code provided is
here.

  * •

CE-GCN: We apply Graph Convolutional Networks [91] to the CE of the hypergraph
to obtain node embeddings. Next, we use MLP to predict edges. The
implementation is provided in the Pytorch Geometric library.

For node classification, we use additional five state-of-the-art deep
hypergraph learning methods and a CE-based baseline:

  * •

HyperGCN [19]: This is a generalization of GCNs to hypergraphs, where it uses
hypergraph Laplacian to define convolution.

  * •

AllDeepSets and AllSetTransformer [25]: These two methods are two variants of
the general message passing framework, Allset, on hypergraphs, which are based
on the aggregation of messages from nodes to hyperedges and from hyperedges to
nodes.

  * •

UniGCNII [28]: Is an advanced variant of UniGnn, a general framework for
message passing on hypergraphs.

  * •

ED-HNN [124]: Inspired by hypergraph diffusion algorithms, this method uses
star expansions of hypergraphs with standard message passing neural networks.

  * •

CE-GCN: We apply Graph Convolutional Networks [91] to the CE of the hypergraph
to obtain node embeddings. Next, we use MLP to predict the labels of nodes.
The implementation is provided in the Pytorch Geometric library. [91]

For all the baselines, we set all sensitive hyperparameters (e.g., learning
rate, dropout rate, batch size, etc.) to the values given in the paper that
describes the technique. Following [60], for deep learning methods, we tune
their hidden dimensions via grid search to be consistent with what we did for
CAt-Walk. We exclude HPLSF [89] and HPRA [88] from inductive hyperedge
prediction as it does not apply to them.

####  F.3 Implementation and Training Details

In addition to hyperparameters and modules (activation functions) mentioned in
the main paper, here, we report the training hyperparameters of CAt-Walk: On
all datasets, we use a batch size of 64 and set learning rate = $10^{-4}$. We
also use an early stopping strategy to stop training if the validation
performance does not increase for more than 5 epochs. We use the maximum
training epoch number of 30 and dropout layers with rate = $0.1$. Other
hyperparameters used in the implementation can be found in the README file in
the supplement.

Also, for tuning the model’s hyperparameters, we systematically tune them
using grid search. The search domains of each hyperparameter are reported in
Table 4. Note that, the last column in Table 4 reports the search domain for
hidden dimensions of modules in CAt-Walk, including SetMixer, MLP-Mixer, and
MLPs. Also, we tune the last layer pooling strategy with two options: SetMixer
or Mean(.) whichever leads to a better performance.

We implemented our method in Python 3.7 with PyTorch and run the experiments
on a Linux machine with nvidia RTX A4000 GPU with 16GB of RAM.

Table 4: Hyperparameters used in the grid search.

Datasets | Sampling Number $M$ | Sampling Time Bias $\alpha$ |  SetWalk Length $m$ | Hidden dimensions
---|---|---|---|---
NDC Class | 4, 8, 16, 32, 64, 128 | $\\{0.5,2.0,20,200\\}\times 10^{-7}$ | 2, 3, 4, 5 | 32, 64, 128
High School | 4, 8, 16, 32, 64, 128 | $\\{0.5,2.0,20,200\\}\times 10^{-7}$ | 2, 3, 4, 5 | 32, 64, 128
Primary School | 4, 8, 16, 32, 64, 128 | $\\{0.5,2.0,20,200\\}\times 10^{-7}$ | 2, 3, 4, 5 | 32, 64, 128
Congress Bill | 8, 16, 32, 64 | $\\{0.5,2.0,20,200\\}\times 10^{-7}$ | 2, 3, 4, 5 | 32, 64, 128
Email Enron | 8, 16, 32, 64 | $\\{0.5,2.0,20,200\\}\times 10^{-7}$ | 2, 3, 4, 5 | 32, 64, 128
Email Eu | 8, 16, 32, 64 | $\\{0.5,2.0,20,200\\}\times 10^{-7}$ | 2, 3, 4 | 32, 64, 128
Question Tags M | 8, 16, 32, 64 | $\\{0.5,2.0,20,200\\}\times 10^{-7}$ | 2, 3, 4 | 32, 64, 128
Users-Threads | 8, 16, 32, 64 | $\\{0.5,2.0,20,200\\}\times 10^{-7}$ | 2, 3, 4 | 32, 64, 128
NDC Substances | 8, 16, 32, 64 | $\\{0.5,2.0,20,200\\}\times 10^{-7}$ | 2, 3, 4 | 32, 64, 128
Question Tags U | 8, 16, 32, 64 | $\\{0.5,2.0,20,200\\}\times 10^{-7}$ | 2, 3, 4 | 32, 64, 128

###  Appendix G Additional Experimental Results

####  G.1 Results on More Datasets

Due to the space limit, we report the AUC results on only eight datasets in
Section 4. Table 6 reports both AUC and average precision (AP) results on all
10 datasets in both inductive and transductive hyperedge prediction tasks.

####  G.2 Node Classification

In the main text, we focus on the hyperedge prediction task. Here we describe
how CAt-Walk can be used for node classification tasks.

For each node $u_{0}$ in the training set, we sample
$\max\\{\text{deg}(u_{0}),10\\}$ hyperedges such as
$e_{0}=\\{u_{0},u_{1},\dots,u_{k}\\}$. Next, for each sampled hyperedge we
sample $M$ SetWalks with length $m$ starting from each $u_{i}\in e_{0}$ to
construct $\mathcal{S}(u_{i})$. Next, we anonymize each hyperedge that appears
in at least one SetWalk in $\bigcup_{i=0}^{k}\mathcal{S}(u_{i})$ by Equation 3
and then use the MLP-Mixer module to encode each
$\textsc{Sw}\in\bigcup_{i=0}^{k}\mathcal{S}(u_{i})$. To encode each node
$u_{i}\in e_{0}$, we use $\textsc{Mean}(.)$ pooling over SetWalks in
$\mathcal{S}(u_{i})$. Finally, for node classification task, we use a 2-layer
perceptron over the node encodings to make the final prediction.

Table 5 reports the results of dynamic node classification tasks on High
School and Primary School datasets. CAt-Walk achieves the best or on-par
performance on dynamic node classification tasks. While all baselines are
specifically designed for node classification tasks, CAt-Walk achieves
superior results due to 1 its ability to incorporate temporal properties (both
from SetWalks and our time encoding module), which helps to learn underlying
dynamic laws of the network, and 2 its two-step set-based anonymization
process that hides node identities from the model. Accordingly, CAt-Walk can
learn underlying patterns needed for the node classification task, instead of
using node identities, which might cause memorizing vertices.

Table 5: Performance on node classification: Mean ACC (%) $\pm$ standard
deviation. Boldfaced letters shaded blue indicate the best result, while gray
shaded boxes indicate results within one standard deviation of the best
result.

| Methods | High School | Primary School | Average Performance
---|---|---|---|---
Inductive | CE-GCN | 76.24 $\pm$ 2.99 | 79.03 $\pm$ 3.16 | 77.63 $\pm$ 3.07
HyperGCN | 83.91 $\pm$ 3.05 | 86.17 $\pm$ 3.40 | 85.04 $\pm$ 3.23
HyperSAGCN | 84.89 $\pm$ 3.80 | 82.13 $\pm$ 3.69 | 83.51 $\pm$ 3.75
AllDeepSets | 85.67 $\pm$ 4.17 | 81.43 $\pm$ 6.77 | 83.55 $\pm$ 5.47
UniGCNII | 88.36 $\pm$ 3.78 | 88.27 $\pm$ 3.52 | 88.31 $\pm$ 3.63
AllSetTransformer | 91.19 $\pm$ 2.85 | 90.00 $\pm$ 4.35 | 90.59 $\pm$ 3.60
ED-HNN | 89.23 $\pm$ 2.98 | 90.83 $\pm$ 3.02 | 90.03 $\pm$ 3.00
CAt-Walk | 88.99 $\pm$ 4.76 | 93.28 $\pm$ 2.41 | 91.13 $\pm$ 3.58
Transductive | CE-GCN | 78.93 $\pm$ 3.11 | 77.46 $\pm$ 2.97 | 78.20 $\pm$ 3.04
HyperGCN | 84.90 $\pm$ 3.59 | 85.23 $\pm$ 3.06 | 85.07 $\pm$ 3.33
HyperSAGCN | 84.52 $\pm$ 3.18 | 83.27 $\pm$ 2.94 | 83.90 $\pm$ 3.06
AllDeepSets | 85.97 $\pm$ 4.05 | 80.20 $\pm$ 10.18 | 83.09 $\pm$ 7.12
UniGCNII | 89.16 $\pm$ 4.37 | 90.29 $\pm$ 4.01 | 89.73 $\pm$ 4.19
AllSetTransformer | 90.75 $\pm$ 3.13 | 89.80 $\pm$ 2.55 | 90.27 $\pm$ 2.84
ED-HNN | 91.41 $\pm$ 2.36 | 91.74 $\pm$ 2.62 | 91.56 $\pm$ 2.49
CAt-Walk | 90.66 $\pm$ 4.96 | 93.20 $\pm$ 2.45 | 91.93 $\pm$ 3.71

Table 6: Performance on hyperedge prediction: AUC and Average Precision (%)
$\pm$ standard deviation. Boldfaced letters shaded blue indicate the best
result, while gray shaded boxes indicate results within one standard deviation
of the best result. N/A: the method has computational issues.

| Datasets | NDC Class | High School | Primary School | Congress Bill | Email Enron
---|---|---|---|---|---|---
| Metric | AUC | AP | AUC | AP | AUC | AP | AUC | AP | AUC | AP
Inductive | Strongly Inductive
CE-GCN | $52.31\pm 2.99$ | $54.33\pm 2.48$ | $60.54\pm 2.06$ | $59.92\pm 2.25$ | $52.34\pm 2.75$ | $56.41\pm 2.06$ | $49.18\pm 3.61$ | $53.85\pm 3.92$ | $63.04\pm 1.80$ | $57.70\pm 2.27$
CE-EvolveGCN | $49.78\pm 3.13$ | $55.24\pm 3.56$ | $46.12\pm 3.83$ | $52.87\pm 3.48$ | $58.01\pm 2.56$ | $55.68\pm 2.41$ | $54.00\pm 1.84$ | $50.27\pm 1.76$ | $57.31\pm 4.19$ | $54.52\pm 3.79$
CE-CAW | $76.45\pm 0.29$ | $78.58\pm 1.32$ | $83.73\pm 1.42$ | $82.96\pm 1.04$ | $80.31\pm 1.46$ | $82.84\pm 1.71$ | $75.38\pm 1.25$ | $77.19\pm 1.38$ | $70.81\pm 1.13$ | $72.07\pm 1.52$
NHP | $70.53\pm 4.95$ | $68.18\pm 4.31$ | $65.29\pm 3.80$ | $62.86\pm 3.74$ | $70.86\pm 3.42$ | $71.31\pm 3.51$ | $69.82\pm 2.19$ | $64.09\pm 2.87$ | $49.71\pm 6.09$ | $50.01\pm 4.87$
HyperSAGCN | $79.05\pm 2.48$ | $77.24\pm 2.05$ | $88.12\pm 3.01$ | $82.72\pm 2.93$ | $80.13\pm 1.38$ | $76.32\pm 2.96$ | $79.51\pm 1.27$ | $80.58\pm 2.61$ | $73.09\pm 2.60$ | $72.29\pm 3.69$
CHESHIRE | $72.24\pm 2.63$ | $70.31\pm 2.26$ | $82.54\pm 0.88$ | $80.34\pm 1.19$ | $77.26\pm 1.01$ | $77.72\pm 0.76$ | $79.43\pm 1.58$ | $78.63\pm 1.25$ | $70.03\pm 2.55$ | $72.97\pm 1.81$
CAt-Walk | 98.89 $\pm$ 1.82 | 98.97 $\pm$ 1.69 | 96.03 $\pm$ 1.50 | 96.41 $\pm$ 0.70 | 95.32 $\pm$ 0.89 | 96.03 $\pm$ 0.84 | 93.54 $\pm$ 0.56 | 93.93 $\pm$ 0.36 | 73.45 $\pm$ 2.92 | 74.66 $\pm$ 3.87
Weakly Inductive
CE-GCN | $51.80\pm 3.29$ | $50.94\pm 3.77$ | $50.33\pm 3.40$ | $48.54\pm 3.92$ | $52.19\pm 2.54$ | $53.21\pm 3.59$ | $52.38\pm 2.75$ | $50.81\pm 2.68$ | $50.81\pm 2.87$ | $55.38\pm 2.79$
CE-EvolveGCN | $55.39\pm 5.16$ | $57.24\pm 4.98$ | $57.85\pm 3.51$ | $63.26\pm 4.01$ | $51.50\pm 4.07$ | $52.59\pm 4.53$ | $55.63\pm 3.41$ | $5.19\pm 3.56$ | $45.66\pm 2.10$ | $50.93\pm 2.57$
CE-CAW | $77.61\pm 1.05$ | $80.03\pm 1.65$ | $83.77\pm 1.41$ | $83.41\pm 1.19$ | $82.98\pm 1.06$ | $80.84\pm 1.57$ | $79.51\pm 0.94$ | $80.39\pm 1.07$ | $\textbf{80.54}\pm\textbf{1.02}$ | $77.41\pm 1.28$
NHP | $75.17\pm 2.02$ | $77.23\pm 3.11$ | $67.25\pm 5.19$ | $66.73\pm 4.94$ | $71.92\pm 1.83$ | $72.30\pm 1.89$ | $69.58\pm 4.07$ | $72.48\pm 4.83$ | $60.38\pm 4.45$ | $55.62\pm 4.67$
HyperSAGCN | $79.45\pm 2.18$ | $80.32\pm 2.23$ | $88.53\pm 1.26$ | $87.26\pm 1.49$ | $85.08\pm 1.45$ | $86.84\pm 1.60$ | $80.12\pm 2.00$ | $73.48\pm 2.77$ | $78.86\pm 0.63$ | $\textbf{79.14}\pm\textbf{1.51}$
CHESHIRE | $79.03\pm 1.24$ | $78.98\pm 1.17$ | $88.40\pm 1.06$ | $86.53\pm 1.82$ | $83.55\pm 1.27$ | $79.42\pm 2.03$ | $79.67\pm 0.83$ | $80.03\pm 1.38$ | $74.53\pm 0.91$ | $75.88\pm 1.14$
CAt-Walk | 99.16 $\pm$ 1.08 | 99.33 $\pm$ 0.89 | 94.68 $\pm$ 2.37 | 96.54 $\pm$ 0.82 | 96.53 $\pm$ 1.39 | 96.83 $\pm$ 1.16 | 98.38 $\pm$ 0.21 | 98.48 $\pm$ 0.18 | 64.11 $\pm$ 7.96 | 67.68 $\pm$ 6.93
Transductive | HPRA | $70.83\pm 0.01$ | $67.40\pm 0.00$ | $94.91\pm 0.00$ | $89.17\pm 0.00$ | $89.86\pm 0.06$ | $88.11\pm 0.02$ | $79.48\pm 0.03$ | $77.16\pm 0.03$ | $78.62\pm 0.00$ | $76.74\pm 0.00$
HPLSF | $76.19\pm 0.82$ | $77.62\pm 1.42$ | $92.14\pm 0.29$ | $92.79\pm 0.15$ | $88.57\pm 1.09$ | $87.69\pm 1.61$ | $79.31\pm 0.52$ | $75.88\pm 0.43$ | $75.73\pm 0.05$ | $75.32\pm 0.08$
CE-GCN | $66.83\pm 3.74$ | $65.83\pm 3.61$ | $62.99\pm 3.02$ | $59.76\pm 3.78$ | $59.14\pm 3.87$ | $55.59\pm 3.46$ | $64.42\pm 3.11$ | $63.19\pm 3.34$ | $58.06\pm 3.80$ | $55.27\pm 3.12$
CE-EvolveGCN | $67.08\pm 3.51$ | $66.51\pm 3.80$ | $65.19\pm 2.26$ | $59.27\pm 2.19$ | $63.15\pm 1.32$ | $65.18\pm 1.89$ | $69.30\pm 2.27$ | $64.38\pm 2.66$ | $69.98\pm 5.38$ | $67.76\pm 5.16$
CE-CAW | $76.30\pm 0.84$ | $77.73\pm 1.42$ | $81.63\pm 0.97$ | $79.37\pm 0.53$ | $86.53\pm 084$ | $87.03\pm 1.15$ | $76.99\pm 1.02$ | $77.05\pm 1.14$ | $79.57\pm 0.14$ | $78.37\pm 1.15$
NHP | $82.39\pm 2.81$ | $80.72\pm 2.04$ | $76.85\pm 3.08$ | $75.37\pm 3.12$ | $80.04\pm 3.42$ | $80.24\pm 3.49$ | $80.27\pm 2.53$ | $77.82\pm 1.91$ | $63.17\pm 3.79$ | $66.87\pm 3.19$
HyperSAGCN | $80.76\pm 2.64$ | $80.50\pm 2.73$ | $94.98\pm 1.30$ | $89.73\pm 1.21$ | $90.77\pm 2.05$ | $88.64\pm 2.09$ | $82.84\pm 1.61$ | $81.12\pm 1.79$ | $\textbf{83.59}\pm\textbf{0.98}$ | $80.54\pm 1.66$
CHESHIRE | $84.91\pm 1.05$ | $82.24\pm 1.49$ | $95.11\pm 0.94$ | $94.29\pm 1.23$ | $91.62\pm 1.18$ | $92.72\pm 1.07$ | $86.81\pm 1.24$ | $83.66\pm 1.90$ | $82.27\pm 0.86$ | $81.39\pm 0.81$
| CAt-Walk | 98.72 $\pm$ 1.38 | 98.71 $\pm$ 1.36 | 95.30 $\pm$ 0.43 | 95.90 $\pm$ 0.44 | 97.91 $\pm$ 3.30 | 97.92 $\pm$ 2.95 | 88.15 $\pm$ 1.46 | 88.66 $\pm$ 1.57 | 80.47 $\pm$ 5.30 | 82.87 $\pm$ 3.50
|  |  |  |  |  |  |  |  |  |  |
| Datasets | Email Eu | Question Tags M | Users-Threads | NDC Substances | Question Tags U
| Metric | AUC | AP | AUC | AP | AUC | AP | AUC | AP | AUC | AP
Inductive | Strongly Inductive
CE-GCN | $52.76\pm 2.41$ | $50.37\pm 2.59$ | $56.10\pm 1.88$ | $54.15\pm 1.94$ | $57.91\pm 1.56$ | $59.45\pm 1.21$ | $55.70\pm 2.91$ | $54.29\pm 2.78$ | $51.97\pm 2.91$ | $55.03\pm 2.72$
CE-EvolveGCN | $44.16\pm 1.27$ | $49.15\pm 1.23$ | $64.08\pm 2.75$ | $60.64\pm 2.78$ | $52.00\pm 2.32$ | $52.69\pm 2.15$ | $58.17\pm 2.24$ | $57.35\pm 2.13$ | $54.57\pm 2.25$ | $57.16\pm 2.55$
CE-CAW | $72.99\pm 0.20$ | $73.45\pm 0.68$ | $70.14\pm 1.89$ | $70.26\pm 1.77$ | $73.12\pm 1.06$ | $72.64\pm 1.18$ | $75.87\pm 0.77$ | $73.19\pm 0.86$ | $74.21\pm 2.04$ | $76.52\pm 2.06$
NHP | $65.35\pm 2.07$ | $64.24\pm 1.61$ | $68.23\pm 3.34$ | $69.82\pm 3.41$ | $71.83\pm 2.64$ | $71.09\pm 2.83$ | $70.43\pm 3.64$ | $73.22\pm 3.03$ | $72.52\pm 2.90$ | $71.56\pm 2.26$
HyperSAGCN | $78.01\pm 1.24$ | $80.04\pm 1.87$ | $73.66\pm 1.95$ | $73.98\pm 1.35$ | $73.94\pm 2.57$ | $72.97\pm 2.45$ | $75.85\pm 2.21$ | $73.24\pm 2.75$ | $78.88\pm 2.69$ | $77.53\pm 2.28$
CHESHIRE | $69.98\pm 2.71$ | $70.10\pm 3.05$ | $N/A$ | $N/A$ | $76.99\pm 2.82$ | $74.03\pm 2.78$ | $76.60\pm 2.19$ | $74.91\pm 2.71$ | $75.04\pm 3.39$ | $75.46\pm 2.90$
CAt-Walk | 91.68 $\pm$ 2.78 | 91.75 $\pm$ 2.82 | 88.03 $\pm$ 3.38 | 88.46 $\pm$ 3.09 | 89.84 $\pm$ 6.02 | 91.58 $\pm$ 4.37 | 93.29 $\pm$ 1.55 | 94.26 $\pm$ 1.21 | 97.59 $\pm$ 2.21 | 97.71 $\pm$ 2.07
Weakly Inductive
CE-GCN | $49.60\pm 3.96$ | $55.01\pm 3.25$ | $55.13\pm 2.76$ | $51.48\pm 2.66$ | $57.06\pm 3.16$ | $58.37\pm 2.86$ | $60.92\pm 2.81$ | $55.93\pm 2.03$ | $56.85\pm 2.73$ | $57.19\pm 2.52$
CE-EvolveGCN | $52.44\pm 2.38$ | $50.61\pm 2.32$ | $61.79\pm 1.63$ | $59.61\pm 1.12$ | $55.81\pm 2.54$ | $50.63\pm 2.46$ | $58.48\pm 2.49$ | $55.90\pm 2.51$ | $54.10\pm 1.21$ | $56.13\pm 2.32$
CE-CAW | $73.54\pm 1.19$ | $74.10\pm 1.41$ | $77.29\pm 0.86$ | $77.67\pm 1.94$ | $80.79\pm 0.82$ | $81.88\pm 0.63$ | $77.28\pm 1.30$ | $79.24\pm 1.19$ | $76.51\pm 1.26$ | $77.17\pm 1.39$
NHP | $67.19\pm 4.33$ | $66.53\pm 4.21$ | $70.46\pm 3.52$ | $65.66\pm 3.94$ | $76.44\pm 1.90$ | $75.23\pm 3.96$ | $73.37\pm 3.51$ | $70.62\pm 3.71$ | $78.15\pm 4.41$ | $79.64\pm 4.32$
HyperSAGCN | $77.26\pm 2.09$ | $74.05\pm 2.12$ | $78.15\pm 1.41$ | $76.19\pm 1.53$ | $75.38\pm 1.43$ | $70.35\pm 1.63$ | $80.82\pm 2.18$ | $76.67\pm 2.06$ | $74.22\pm 1.91$ | $70.57\pm 1.02$
CHESHIRE | $77.31\pm 0.95$ | $76.01\pm 0.98$ | $N/A$ | $N/A$ | $81.27\pm 0.85$ | $82.96\pm 1.41$ | $80.68\pm 1.31$ | $80.78\pm 1.13$ | $77.60\pm 1.57$ | $79.48\pm 1.79$
CAt-Walk | 91.98 $\pm$ 2.41 | 92.22 $\pm$ 2.40 | 90.28 $\pm$ 2.81 | 90.56 $\pm$ 2.62 | 97.15 $\pm$ 1.81 | 97.55 $\pm$ 1.49 | 95.65 $\pm$ 1.82 | 96.18 $\pm$ 1.52 | 98.11 $\pm$ 1.31 | 98.25 $\pm$ 1.13
Transductive | HPRA | $72.51\pm 0.00$ | $71.08\pm 0.00$ | $83.18\pm 0.00$ | $80.12\pm 0.00$ | $70.49\pm 0.02$ | $72.83\pm 0.00$ | $77.94\pm 0.01$ | $75.78\pm 0.01$ | $81.05\pm 0.00$ | $81.71\pm 0.00$
HPLSF | $75.27\pm 0.31$ | $77.95\pm 0.14$ | $83.45\pm 0.93$ | $82.29\pm 1.06$ | $74.38\pm 1.11$ | $73.81\pm 1.45$ | $82.12\pm 0.71$ | $84.51\pm 0.62$ | $80.89\pm 1.51$ | $75.62\pm 1.38$
CE-GCN | $64.19\pm 2.79$ | $65.93\pm 2.52$ | $55.18\pm 5.12$ | $55.84\pm 4.53$ | $62.78\pm 2.69$ | $59.71\pm 2.25$ | $63.08\pm 2.19$ | $65.37\pm 2.48$ | $66.79\pm 2.88$ | $60.51\pm 2.26$
CE-EvolveGCN | $64.36\pm 4.17$ | $66.98\pm 3.72$ | $72.56\pm 1.72$ | $69.38\pm 1.51$ | $68.55\pm 2.26$ | $67.86\pm 2.61$ | $70.09\pm 3.42$ | $66.37\pm 3.17$ | $71.31\pm 2.92$ | $70.36\pm 2.72$
CE-CAW | $78.19\pm 1.10$ | $77.95\pm 0.98$ | $81.73\pm 2.48$ | $83.27\pm 2.34$ | $80.86\pm 0.45$ | $80.57\pm 1.08$ | $84.72\pm 1.65$ | $84.93\pm 1.26$ | $80.37\pm 1.77$ | $83.14\pm 0.97$
NHP | $78.90\pm 4.39$ | $76.95\pm 5.08$ | $79.14\pm 3.36$ | $78.79\pm 3.15$ | $82.33\pm 1.02$ | $81.44\pm 1.53$ | $81.38\pm 1.42$ | $82.17\pm 1.38$ | $78.99\pm 4.16$ | $80.06\pm 4.33$
HyperSAGCN | $79.61\pm 2.35$ | $75.99\pm 2.23$ | $84.07\pm 2.50$ | $84.22\pm 2.43$ | $79.62\pm 2.04$ | $79.38\pm 2.55$ | $85.07\pm 2.46$ | $85.32\pm 2.20$ | $85.18\pm 2.64$ | $80.99\pm 3.04$
CHESHIRE | $86.38\pm 1.23$ | $87.39\pm 1.07$ | $N/A$ | $N/A$ | $82.75\pm 1.99$ | $81.96\pm 1.75$ | $86.30\pm 1.57$ | $83.18\pm 1.92$ | $87.83\pm 2.15$ | $88.62\pm 1.76$
| CAt-Walk | 96.74 $\pm$ 1.28 | 97.08 $\pm$ 1.20 | 91.63 $\pm$ 1.41 | 92.28 $\pm$ 1.26 | 93.51 $\pm$ 1.27 | 94.98 $\pm$ 0.98 | 90.64 $\pm$ 0.44 | 91.96 $\pm$ 0.41 | 96.59 $\pm$ 4.39 | 97.06 $\pm$ 3.72

####  G.3 Performance in Average Precision

In addition to the AUC, we also compare our model with baselines with respect
to Average Precision (AP). Table 6 reports both AUC and AP results on all 10
datasets in inductive and transductive hyperedge prediction tasks. As
discussed in Section 4, CAt-Walk due to its ability to capture both temporal
and higher-order properties of the hypergraphs, achieves superior performance
and outperforms all baselines in both transductive and inductive settings with
a significant margin.

####  G.4 More Results on Rnn v.s. MLP-Mixer in Walk Encoding

Most existing methods on (temporal) random walk encoding see a walk as a
sequence of vertices and uses sequence encoders like Rnns or Transformers to
encode each walk. The main drawback of these methods is that they fail to
directly process temporal walks with irregular gaps between timestamps. That
is, sequential encoders can be seen as discrete approximations of dynamic
systems; however, this discretization often fails if we have irregularly
observed data [125].nper This is the main motivation of recent studies to
develop methods on continuous-time temporal networks [34, 126]. Most of these
methods are too complicated and sometimes fail to generalize [127]. In CAt-
Walk, we suggest a simple architecture to encode temporal walks by a time-
encoding module along with a Mixer module (see Section 3.4 for the details).
In this part, we evaluate the power of our Mixer module and compare its
performance when we replace it with Rnns [82]. Figure 7 reports the results on
all datasets. We observe that using MLP-Mixer with the time-encoding module in
CAt-Walk can always outperform CAt-Walk when we replace MLP-Mixer with a Rnn,
and mostly this improvement is more on datasets with high variance in their
timestamps. We relate this superiority to the importance of using continuous-
time encoding instead of sequential encoders.

Figure 7: The importance of using MLP-Mixer in CAt-Walk. Using an Rnn instead
of MLP-Mixer can damage the performance in all datasets. Rnns are sequential
encoders and are not able to encode continuous time in the data.

###  Appendix H Broader Impacts

Temporal hypergraph learning methods, such as CAt-Walk, benefit a wide array
of real-world applications, including but not limited to social network
analysis, recommender systems, brain network analysis, drug discovery, stock
price prediction, and anomaly detection (e.g. bot detection in social media or
abnormal human brain activity). However, there might be some potentially
negative impacts, which we list as:  1 Learning underlying biased patterns in
the training data, which may result in stereotyped predictions. Since CAt-Walk
learns underlying dynamic laws in the training data, given biased training
data, the predictions of CAt-Walk can be biased.  2 Also, powerful dynamic
hypergraph models can be used for manipulation in the abovementioned
applications (e.g., stock price manipulation). Accordingly, to prevent the
potential risks in sensitive tasks, e.g., decision-making from graph-
structured data in health care, interpretability and explainability of machine
learning models on hypergraphs is a critical area for future work.

Furthermore, this work does not perform research on human subjects as part of
the study and all used datasets are anonymized and publicly available.


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[1,2]Filippo Caruso

[1]Department of Physics and Astronomy, University of Florence, Via Sansone 1,
Sesto Fiorentino, 50019, Florence, Italy

2]LENS - European Laboratory for Non-Linear Spectroscopy, University of
Florence, Via Nello Carrara 1, Sesto Fiorentino, 50019, Florence, Italy

# Quantum-Noise-driven Generative Diffusion Models

Marco Parigi<EMAIL_ADDRESS>Stefano Martina<EMAIL_ADDRESS><EMAIL_ADDRESS>*  [

###### Abstract

Generative models realized with machine learning techniques are powerful tools
to infer complex and unknown data distributions from a finite number of
training samples in order to produce new synthetic data. Diffusion models are
an emerging framework that have recently overcome the performance of the
generative adversarial networks in creating synthetic text and high-quality
images. Here, we propose and discuss the quantum generalization of diffusion
models, i.e., three quantum-noise-driven generative diffusion models that
could be experimentally tested on real quantum systems. The idea is to harness
unique quantum features, in particular the non-trivial interplay among
coherence, entanglement and noise that the currently available noisy quantum
processors do unavoidably suffer from, in order to overcome the main
computational burdens of classical diffusion models during inference. Hence,
we suggest to exploit quantum noise not as an issue to be detected and solved
but instead as a very remarkably beneficial key ingredient to generate much
more complex probability distributions that would be difficult or even
impossible to express classically, and from which a quantum processor might
sample more efficiently than a classical one. Therefore, our results are
expected to pave the way for new quantum-inspired or quantum-based generative
diffusion algorithms addressing more powerfully classical tasks as data
generation/prediction with widespread real-world applications ranging from
climate forecasting to neuroscience, from traffic flow analysis to financial
forecasting.

###### keywords:

Generative Models, Diffusion Models, Quantum Machine Learning, Quantum Noise,
Quantum Computing

Figure 1: Depiction of the diffusion (from left to right) and denoising (from
right to left) processes within a diffusion probabilistic model framework. The
original image $\mathbf{x}_{0}$ sampled from the unknown data distribution
$p(\mathbf{x}_{0})$ is progressively perturbed ($t\rightarrow t+1$) by adding
noise to obtain a latent variable $\mathbf{x}_{T}$ from a known and tractable
distribution where the information is completely destroyed. In our framework
the diffusion process can be implemented with a classical or a quantum
stochastic dynamics. The denoising process is trained to approximate the
structure of the data distribution in order to generate new samples. The
latter is implemented step by step, using a classical (on the left in blue) or
quantum (on the right in orange) parameterized model $\hat{U}(\theta)$ in
order to approximate the backward mapping. The standard diffusion models
implement both the diffusion and the denoising processes in a classical
framework. We propose three different new approaches for the other cases: i)
classical diffusion and quantum denoising (CQGDM); ii) quantum diffusion and
classical denoising (QCGDM); iii) quantum diffusion and quantum denoising
(QQGDM). A similar picture can be applied to time series.

In Machine Learning (ML), diffusion probabilistic models, or briefly Diffusion
Models, are an emerging class of generative models used to learn an unknown
data distribution in order to produce new data samples. They has been proposed
for the first time by Sohl-Dickstein et al. [1] and take inspiration from
diffusion phenomena of non-equilibrium statistical physics. The underlying
core idea of the DMs is to gradually and slowly destroy the information
encoded into the data distribution until it became fully noisy, and then learn
how to restore the corrupted information in order to generate new synthetic
data. More precisely, the generic structure of diffusion models consists of
two stages: (i) a _diffusion_ (or _forward_) process and (ii) a _denoising_
(or _reverse_) process. In the former phase, a training data is progressively
perturbed by adding noise, typically Gaussian, until all data information is
destroyed. The increasing perturbation of information due to the
systematically and progressive injections of noise can be physically
understood as if the noise propagated inside the data structure, as shown in
Fig. 1 from left to right. Let us highlight the fact that in this first stage
the training of any ML model is not required. In the second phase, the
previous diffusive dynamics is slowly reversed in order to restore the initial
data information. The goal of this phase is to learn how to remove noise
correctly and produce new data starting from uninformative noise samples as in
Fig. 1 from right to left. In contrast to the forward diffusion process, the
noise extraction—and as a result the data information retrieval—is implemented
training a ML model typically based on a so-called U-Net neural network (NN)
architecture [2]. In detail, U-Net models are structured in a succession of
convolutional layers followed by an equal number of deconvolutional layers
where each deconvolution takes as input the output of the previous
deconvolution and also the copy of the output of the corresponding
convolutional layer in reverse order. The procedure described above allows DMs
to succesfully address the main complication in the design of probabilistic
models, i.e., being _tractable_ and _flexible_ at the same time [1, 3]. In
fact, alternatively to DMs there are other generative probabilistic models,
for instance, Autoregressive Models that are generally tractable but not
flexible, or Variational Auto Encoders [4] and Generative Adversarial Networks
[5] that are flexible but not tractable.

Diffusion models find use in computer vision for several image processing
tasks [6], such as, inpainting [7], super-resolution [8], image-to-image
translation [9], and image generation [10, 11, 12]. They are also successfully
adopted in several applications, for instance: Stable diffusion [13] that is
an open source model for high resolution image syntesis [10]; DALL-E 2 that is
a platform implemented by OpenAI [14] to generate photorealistic images from
text prompts [11]; Google Imagen [15] that combines _transformer_ language
models with diffusion models also in the context of text-to-image generation
[12]. Moreover, it has recently been shown that diffusion models perform
better than GANs on image synthesis [16].

Furthermore, diffusion models can also be applied to other contexts, for
instance in text generation [17, 18] and time-series related tasks [19, 20,
21]. For instance, time series forecasting is the task of predicting future
values from past history and diffusion models are employed to generate new
samples from the forecasting distribution [22, 23]. Moreover, diffusion models
can also be used in time series generation, which is a more complex task
involving the complete generation of new time-series samples from a certain
distribution [24, 25].

On the other side, very recently, we are witnessing an increasing interest in
quantum technologies. Near-term quantum processors are called Noisy
Intermediate-Scale Quantum (NISQ) devices [26] and they represent the-state-
of-the-art in this context. NISQ computers are engineered with quantum
physical systems using different strategies. For instance, a commonly used
technology employs superconductive-circuits-based platforms [27, 28] realized
with transmon qubits [29, 30]. This technology is exploited, for instance, by
IBM [31], Rigetti Computing [32], Google [33]. Moreover D-Wave [34] exploits
superconducting integrated circuits mainly as quantum annealers [35]. Xanadu
[36] is instead a company employing photons as information units within the
linear optical quantum computing paradigm [37] to realize their devices.
Finally, the quantum computation can be realized directly manipulating the
properties of single atoms. For instance, IonQ [38] realizes quantum devices
with trapped ions [39, 40], while Pasqal [41] and QuEra [42] realize analog
quantum computers with Rubidium Rydberg neutral atoms held in optical tweezers
[43]. All the mentioned devices can be in principle integrated in
computational pipelines that can involve also classical computation. In this
context they can be referred with the term quantum processing unit (QPU) that
can make some computational task much faster than its classical counterpart
(CPU) harnessing the quantum properties of particles at the atomic scale. The
main reason for building a quantum processor is the possibility of exploiting
inherent and peculiar resources of quantum mechanical systems such as
_superposition_ and _entanglement_ that, in some cases, allow to perform
computational tasks that are impossible or much more difficult via a classic
supercomputer [44, 45].

One of the most promising applications of NISQ devices is represented by
Quantum Machine Learning (QML) that is a recent interdisciplinary field
merging ML and quantum computing, i.e. data to be processed and/or learning
algorithms are quantum [46, 47, 48, 44]. Indeed, it involves the integration
of ML techniques and quantum computers in order to process and subsequently
analyze/learn the underlying data structure. QML can involve the adoption of
classical ML methods with quantum data or environments, for instance to
analyze noise in quantum devices [49, 50, 51, 52] or to control quantum
systems [53]. Alternatively, QML can consider the implementation of novel ML
techniques using quantum devices, for instance to implement visual tasks [54]
or generative models like Quantum Generative Adversarial Network (QGAN) [55,
56, 57, 58, 59] that are the quantum implementation of classical GAN or in
Natural Language Processing (NLP) context to generate text [60]. In fact,
quantum devices are capable of processing information in ways that are
different from the classical computation. Thus, the implementation of QML
models can offer an advantage over the corresponding classical ML models. The
latter is expected to arise in the form of _quantum speedup_ or a much smaller
number of training parameters, by harness the peculiar properties of quantum
systems, for instance, superposition, coherence and entaglement. However, NISQ
devices are indeed still very noisy and thus they do not perform the ideal
(pure) dynamics. Therefore, the system evolution is affected and driven by
_quantum noise_ due to the undesired interactions with external environment
and has to be described by the more general open quantum system formalism
[61].

In order to generalize DMs with quantum computing ideas, a crucial role is
played by noise. In classical information theory, noise is usually modeled by
the framework of probability theory, and in general via Markovian processes.
Accordingly, the main features of classical noise are a linear relationship
among successive steps of the dynamics whose evolution depends only on the
current state. Formally, noise is represented by a transition matrix that has
the properties of _positivity_ (non-negative entries) and _completeness_
(columns summing to one). In particular, Gaussian noise is a type of random
noise that is very often added to the input data of a DM in order to help its
learning to generate new data that is as similar as possible to the training
data, also in the case when the input is not perfect.

In the quantum domain, noise can be generated also by quantum fluctuations
that are typical of quantum systems, hence going much beyond the classical
noise sources. Mathematically, quantum noise is described by the more general
formalism of _quantum operations_ or _quantum maps_ [61], where, for instance,
the decoherence is the typical noise affecting the phase coherence among the
quantum states and, in fact, is the main enemy to fight with in order to build
up more powerful quantum processors. But what about if such noise is not only
detrimental for the quantum computation but it is instead actually very
beneficial for some ML tasks as we have observed in the past in other
different contexts [62, 63, 53]? Quantum noise might allow, for instance, to
generate much more complex (due to the presence of entanglement) probability
distributions that would be difficult (or even impossible) to express
classically and from which one can sample more efficiently via a quantum
processor than via a classical supercomputer.

In this article we therefore introduce and formalize the quantum versions of
DMs, particularly based on Denoising Diffusion Probabilistic Models and Score
Stochastic Differential Equations in the context of QML. More precisely, we
propose three potential quantum-noise-driven generative diffusion models that
can be both computationally simulated in the NISQ devices and implemented
_experimentally_ due to the naturally occurring noise effects in open quantum
systems. The three algorithms are: i) Classical-Quantum Generative Diffusion
Model (CQGDM) in which the forward diffusion process can be implemented in the
classical way, while the backward denoising with a Quantum Neural Network
(QNN) (that can be either a Parametrized Quantum Circuit (PQC) or an hybrid
quantum-classical NN); (ii) Quantum-Classical Generative Diffusion Model
(QCGDM) in which the noise diffusion process can be implemented in a quantum
way, while in the denoising process classical NNs are used; (iii) Quantum-
Quantum Generative Diffusion Model (QQGDM) where both the diffusion and the
denoising dynamics can be implemented in a quantum domain.

##  1  Classical-Quantum Generative Diffusion Model (CQGDM)

In this section we propose a model where the diffusion process is classical
while the denoising phase is implemented with a quantum dynamics. Moreover, as
a result of this setting, the training dataset is necessarily classical, for
instance, images, videos, time series, etc.

Formally, given an initial training data $\mathbf{x}_{0}$ sampled from a
generic and unknown probability distribution $p(\mathbf{x}_{0})$, the
procedure consists in a progressive destruction of the information encoded in
the initial data via a diffusive stochastic process. At the end, the data is
degraded to a fully noisy state $\mathbf{x}_{T}$ sampled from a classical
closed form and tractable _prior_ distribution $p(\mathbf{x}_{T})$ that
represents the latent space of the model. Here, tractable stands for the fact
that the distribution can be computationally calculated. The implementation of
this process can be obtained with different ways. For instance, in DDPM the
dynamics of forward diffusive process is implemented by classical Markov chain
[1, 3], while in Score SDE the stochastic evolution is determined by a
differential equation [64]. In detail, the former approach considers a
discrete-time stochastic process whose evolution, at every step, depends only
on the previous state and the transition relies on hand-designed kernels
$p(\mathbf{x}_{t}|\mathbf{x}_{t-1})$, $t=1,2,\dots,T$ (see Methods for more
details). Alternatively, in Score SDE the evolution is a continuous-time
process within a close time interval $t\in[0,T]$ and determined by the
stochastic differential equation:
$\operatorname{\mathrm{d}}\mathbf{x}=\mathbf{f}(\mathbf{x},t)\operatorname{\mathrm{d}}t+g(t)\operatorname{\mathrm{d}}\mathbf{w}$,
where $\mathbf{f}(\mathbf{x},t)$ is the drift coefficient, $g(t)$ is the
diffusion term, and $\mathbf{w}$ is the Wiener process (also known as standard
Brownian motion) that models the stochastic process [65]. The solutions of
this equation lead to the tractable prior distribution $p(\mathbf{x}(T))$.

Afterwards, in order to generate new data samples, the objective is to learn
how to reverse the diffusion process starting from the prior latent
distribution. In case of DDPM, calculating
$p(\mathbf{x}_{t-1}|\mathbf{x}_{t})$ is not computationally tractable and it
is classically approximated by a model parameterized with $\theta$ (e.g., a
NN): $p_{\theta}(\mathbf{x}_{t-1}|\mathbf{x}_{t})$ (see Methods). In the case
of Score SDE models, the quantity to be estimated is $\nabla_{\mathbf{x}}\log
p_{t}(\mathbf{x})$, where $p_{t}(\mathbf{x})$ is the density probability of
$\mathbf{x}(t)$ [64]. Here, for either DDPM and Score SDE diffusion processes,
we propose to implement the denoising process with a QNN that can be fully
quantum via a PQC or even a classical-quantum hybrid NN model.

In this context, the main advantage of using the quantum denoising process
instead of the classical one can be the possibility of using the trained
quantum model to efficiently generate highly dimensional data (e.g., images)
taking advantage of the peculiar quantum mechanical properties, such as
quantum superposition and entanglement, to speed up data processing [66, 67,
68]. Indeed, QPU devices could be very effective to overcome the main
computational burdens of classical diffusion model during this inference
process. As shown in Fig. 2, the denoising process for CQGDM crosses the
border between classical and quantum distribution spaces, this could take
advantage of the quantum speedup in order to accelerate the training of the
model.

##  2  Quantum-Classical Generative Diffusion Model (QCGDM)

In real experiments quantum systems are never perfectly isolated, but they are
easily subjected to noise, e.g., interactions with the environment and
imperfect implementations. Accordingly, we propose to physically implement the
diffusion process via a noisy quantum dynamics exploiting such quantum noise
as a positive boost.

In this setting a quantum dataset is considered, i.e., a collection of
_quantum_ data. Classical information can be embedded into the initial state
of a quantum system, allowing to treat classical data as quantum [48, 69, 70].
Even better, we could avoid the encoding of the classical data if we consider
quantum data as any result arising from a quantum experiment [71] or produced
directly by a quantum sensing technology [72]. Formally, quantum data is
identified with the density operator $\rho(t)$ living in
$\mathfrak{S}(\mathcal{H})$ being the set of non-negative operators with unit
trace acting on the elements of the Hilbert space $\mathcal{H}$ where the
quantum states live.

We here propose two approaches to implement the diffusion process: (i)
_quantum_ Markov chains generalizing their classical counterparts [73], and
(ii) Stochastic Schrödinger Equation (SSE) [74, 75, 76, 77] modelling the
dynamics of an _open quantum system_ subjected to an external noise source.

In the former approach (i), a quantum Markov chain can be described with a
composition of transition operation matrices mapping a density operator $\rho$
to another density operator $\rho^{\prime}$. TOMs are matrices whose elements
are _completely positive maps_ and whose column sums form a _quantum
operation_ (for more details refer to Section 5). A special case of TOMs are
the transition effect matrices whose columns are discrete positive operator
valued measures. A quantum Markov chains can be, thereby, implemented by a
sequence of quantum measurements [73].

The second approach (ii) employs SSEs to describe the physical quantum
diffusion process. Given a system in the state $\rho(t)$, its stochastic
evolution is determined by a SSE that takes the form
$\dot{\rho}(t)=-i[H(t),\rho(t)],$ with $\hbar=1$, and where the Hamiltonian
$H(t)=H_{s}(t)+H_{p}(t)$ consists of the sum of the Hamiltonian of the system
$H_{s}(t)$ and the stochastic term $H_{p}(t)$ representing the stochastic
dynamics to which the quantum system is subjected. Arbitrary sources of noise
applied to optimally controlled quantum systems were very recently
investigated with the SSEs formalism by our group [78].

The implementation of diffusion dynamics on quantum systems during the forward
stage can allow the processing of the data information not only by classically
simulated noise but also with quantum physical noise. Here, as previously
mentioned, let us remind that quantum noise is more general with respect to
its classical counterpart. In particular, the noise distributions used in
QCGDMs can be expressed (and more naturally arise by quantum dynamics) in more
general and powerful forms respect to the typical Gaussian distributions that
are commonly employed in classical DMs. In this set up, at the end of the
diffusion process, it is possible to obtain non-classical prior distributions
related to entangled state that do not exist in the classical information
scenario. In other terms there are probability density distributions that are
purely quantum. This can be used to implement diffusion processes that are not
possible to be implemented classically. At the end, during the denoising
phase, classical NNs can be used in order to remove noise and thus finally
generate new samples. Moreover, if the obtained prior distribution is not
classical, it is possible to consider the adoption of the denoising NN as a
discriminator to identify probability distributions that are purely quantum.
This could also be framed in a _security_ context. One can imagine a channel
where the communication of data takes place with the application of a quantum
diffusion process that maps to a purely quantum probability distribution. In
that case, the receiver can restore and so obtain the initial information only
with the training of a QNN and thus only with a quantum device. This might be
also exploited for quantum attacks/defence in cyber-security applications.

##  3  Quantum-Quantum Generative Diffusion Model (QQGDM)

In this last section we describe diffusion models within a fully quantum
physical framework. Precisely, the training data, the diffusion process and
the denoising process have all a quantum mechanical nature. This scenario can
be obtain by exploiting the quantum tools described above, namely, quantum
Markov chain or SSE for the forward diffusion phase, and a PQC for the
backward denoising phase.

Accordingly, all the advantages described in Sections 1 and 2 hold. The
adoption of a fully quantum pipeline for both the diffusion and denoising
phases would allow the possibility to obtain purely quantum prior
distributions that can be processed during the denoising phase with PQCs
obtaining a generation process that is not feasible classically. Moreover,
this approach could lead to an _exponential_ advantage in sample and time
complexity as shown in [72, 79]. As shown in Fig. 2, the diffusion and
denoising processes for QQGDM are entirely located in the space of quantum
distributions. This might lead to the speedup already described previously for
CQGDM and in addition to the possibility of exponentially reducing the
computational resources for storing and processing of data information [68].
Finally, it is also possible to access to complex quantum probability
distributions that are impossible or much more difficult to treat classically.

##  4 Discussion

The entanglement is a crucial quantum mechanical phenomenon occurring only in
the quantum domain (not classical analogue) when two or more quantum systems
interact. It is detected by measurement correlations between the quantum
systems that cannot be described with classical physics. Accordingly, quantum
systems are capable of representing distributions that are impossible to be
produced efficiently with classical computers [80, 46]. For this reason, a
quantum diffusion process is capable to explore probability density functions
that are not classically tractable.

Figure 2:  Relationship between the space of the probability distributions
that are tractable with classical computation ($\mathbf{C}$) and instead only
with quantum computation ($\mathbf{Q}$). We show the trajectories arising from
the mappings between probability distributions (colored and white shapes)
during the diffusion (blue wavy arrows) and denoising (red arrows) processes
for the four different combination: $\mathbf{CC}$, $\mathbf{CQ}$,
$\mathbf{QC}$ and $\mathbf{CC}$ indicating whether the diffusion (first
letter) and the denoising (second letter) are classical or quantum. The
initial data distribution (squares) is progressively transformed during
diffusion (changing color and shape) to an uninformative distribution
represented by the white circles, and vice versa during denoising. Completely
classical models are limited to operate within the space of classically-
tractable probability distributions, while completely quantum models can
manipulate quantum-tractable probabilities. Models that have classical
diffusion and quantum denoising are forced to work only with classical
probabilities, but during the denoising phase they can exploit quantum
properties within each step. Finally, models that have quantum diffusion and
classical denoising can manipulate quantum probabilities during the forward,
but in that case, it is not possible to train the classical backward to map
those probability distributions.

In Fig. 2 we highlight the relationship between the space of the probability
distributions that are tractable with classic computers, which we denote with
_classical distributions_ to be more concise, and the space of the probability
distributions that are tractable with quantum devices, which we denote with
_quantum distributions_ hereinafter. Moreover, we can observe several possible
trajectories that map probability distributions to other probability
distributions during the diffusion and denoising of the classical DM and of
the three proposed quantum approaches: CQGDM, QCGDM and QQGDM.

The classic DM realizes maps from classical distributions to other classical
distributions and the NN that implements the denoising are trained to realize
the inverse maps, i.e., to match the distributions crossed during the
diffusion.

In the CQGDM approach, the diffusion process is implemented classically. Thus,
all the probability distributions are necessarily classical. However, during
the denoising process, the quantum dynamics is free to explore also the
quantum probability space whithin each one of the steps hence exploiting
potential (noise-assisted and/or quantum-enhanced) shortcuts. This may give
huge advantage for the training of the denoising model. Let us stress that
when we evolve quantum systems within a QPU it is possible to process and
manipulate exponentially more information as compared to the classical case.

When we consider the fully quantum framework QQGDM we gain the advantage of
exploring quantum distributions also during the diffusion phase. For this
reason we could explore more complex noisy dynamics compared to the ones that
can be simulated in classical computers. Moreover, the two processes can be
experimentally implemented on real quantum processors. Furthermore, compared
to the CQGDM approach, and provided that the initial distribution of the
dataset is quantum, it is possible to design a QQGDM generative models that is
capable of generating complex quantum data that are not analytically
computable.

Besides, we would like point out that the QCGDM approach can be challenging to
be implemented. In detail, if the diffusion process leads to an entagled
quantum distribution it is impossible, for the previously mentioned reasons,
to efficiently train a classical NN to perform the denoising. This context
could be adopted as a proof of concept for the realization of a discriminator
for the quantum distributions from the classical ones. In other words, if it
is possible to train a model to perform the denoising, then the distribution
is classical.

As a future outlook, we would like to realize the implementation of the
quantum generative diffusion models either computationally via NISQ and/or
physically by using quantum sensing technologies. In particular, regarding
QCGDMs and QQGDMs, we propose to implement the diffusion process exploiting
naturally noisy quantum dynamics in order to take advantage of the possible
benefits of the quantum noise. Instead, regarding CQGDMs and QQGDMs, we
propose to use quantum implemented QMLs models, for instance QNNs and PQCs, to
learn the denoising process.

Finally, the design and realization of quantum generative diffusion models,
with respect to classical DMs, could alleviate and reduce the computational
resources (e.g. space of memory, time and energy) to successfully address ML
applications such as generation of high-resolution images, the analysis and
the prediction of rare events in time-series, and the learning of underlying
patterns in experimental data coming also from very different fields as, among
others, life and earth science, physics, quantum chemistry, medicine, material
science, smart technology engineering, and finance.

##  5 Methods

In this section we include some mathematical details on the classical and
quantum tools for the diffusion and denoising processed discussed in the main
text.

###  5.1 Classical methods

Here we formalize the classical methods used in the standard generative
diffusion models and used also for the relevant part of the proposed CQGDM and
QCGDM. In particular, we consider classical Markov chains for a Gaussian
pertubation and the NNs used for the classical denoising.

The classical diffusion process [1, 3] starts from an initial data sample
$\mathbf{x}_{0}$ drawn from an unknown generic distribution
$p(\mathbf{x}_{0})$. Gaussian noise is then iteratively injected for a number
$T$ of time steps to degrade the data to $\mathbf{x}_{T}$ sampled from a prior
Gaussian distribution $\mathcal{N}(0,\mathbf{I})$. In detail, the used
Gaussian transition kernel is in the form:

$p(\mathbf{x}_{t}|\mathbf{x}_{t-1})=\mathcal{N}(\mathbf{x}_{t};\sqrt{1-\beta_{t}}\mathbf{x}_{t-1},\beta_{t}\mathbf{I}),$
(1)

where $\beta_{t}\in(0,1)$ is an hyperparameter (fixed or scheduled over the
time) for the model at the time step $t$ that describes the level of the
injected noise, $\beta_{t}\mathbf{I}$ is the identity matrix, and
$\mathbf{x}_{t}$ and $\mathbf{x}_{t-1}$ are the random variables at the time
steps $t$ and $t-1$, respectively. In this way it is possible to calculate a
tractable closed form for the trajectory:

$p(\mathbf{x}_{1:T}|\mathbf{x}_{0})=p(\mathbf{x}_{0})\prod_{t=1}^{T}p(\mathbf{x}_{t}|\mathbf{x}_{t-1}).$
(2)

By obing so, for $T$ sufficiently high, $p(\mathbf{x}_{1:T}|\mathbf{x}_{0})$
converges to an isotropic Gaussian
$p(\mathbf{x}_{T})=\mathcal{N}(0,\mathbf{I})$.

The denoising phase starts from the Gaussian prior distribution and the
transition kernel that is implemented is in the form:

$p_{\theta}(\mathbf{x}_{t}|\mathbf{x}_{t-1})=\mathcal{N}(\mathbf{x}_{t-1};\mathbf{\mu}_{\theta}(\mathbf{x}_{t},t),\mathbf{\Sigma}_{\theta}(\mathbf{x}_{t},t)),$
(3)

and the closed form for the trajectory is:

$p_{\theta}(\mathbf{x}_{0:T}|\mathbf{x}_{T})=p(\mathbf{x}_{T})\prod_{t=1}^{T}p_{\theta}(\mathbf{x}_{t-1}|\mathbf{x}_{t}).$
(4)

Usually, a NN, specifically a U-Net architecture [2], is used to estimate the
mean $\mathbf{\mu}_{\theta}(\mathbf{x}_{t},t)$ and the covariance
$\mathbf{\Sigma}_{\theta}(\mathbf{x}_{t},t)$ in Eq. 3. In principle, the
approach to train the NN would be to find the parameters $\theta$ such that
$p_{\theta}(\mathbf{x}_{0})$ would be maximized for each training sample
$\mathbf{x}_{0}$. However $p_{\theta}(\mathbf{x}_{0})$ is intractable because
it is impossible to marginalize over all the possible trajectories. For this
reason, the common approach [3] is to fix the covariance and minimize the
loss:

$\mathcal{L}=\mathbb{E}_{t\sim[1,T]}\mathbb{E}_{\mathbf{x}_{0}\sim
p(\mathbf{x}_{0})}\mathbb{E}_{\mathbf{z}_{t}\sim\mathcal{N}(0,\mathbf{I})}||\mathbf{z}_{t}-\mathbf{z}_{\theta}(\mathbf{x}_{t},t)||^{2},$

where $\mathbf{z}_{t}$ is the real Gaussian noise, and
$\mathbf{z}_{\theta}(\mathbf{x}_{t},t)$ is the noise estimation of the model
at time $t$.

###  5.2 Quantum methods

Here we formalize the use of the quantum Markov chain introduced for the
diffusion processes of QCGDM and QQGDM in Sections 2 and 3 and the QNNs used
for the denoising of CQGDM and QQGDM in Sections 1 and 3.

Formally, a quantum Markov chain can described by two elements: i) a directed
graph $G$ whose sites represent the possible state that the quantum system can
occupy, ii) a TOM $\mathcal{E}=\mathcal{E}_{ij}$ whose elements are
_completely positive maps_ [81, 61] and whose column sums form a _quantum
operation_ [73, 44]. Formally, a positive a map is a linear transformation of
one positive bounded operator into another. A completely positive map is a
linear map $\phi:\mathcal{B(\mathcal{H})}\rightarrow\mathcal{B(\mathcal{H})}$,
where $\mathcal{B(\mathcal{H})}$ is the set of bounded linear operators acting
on the Hilbert space $\mathcal{H}$, such that the map $\phi\otimes I$ is
positive on the space
$\mathcal{B(\mathcal{H})}\otimes\mathcal{B(\mathcal{H^{\prime}})}$ for any
Hilbert space $\mathcal{H^{\prime}}$. A quantum operation is a completely
positive map $\phi$ preserving the trace, i.e., $\tr(\rho)=\tr(\phi(\rho))$,
with $\rho\in\mathcal{B(\mathcal{H})}$. Physically, the elements $E_{ij}$
describe the passage operation of the quantum system from site $j$ to site $i$
in one time step. Given a density operator $\rho$, representing the state of
system, the quantity $\mathcal{E}(\rho)$ is again a density operator.
Moreover, if $\mathcal{E}$ and $\mathcal{F}$ are two TOMs with the same size
and acting on the same Hilbert space, then the $\mathcal{EF}$ is again a TOM
by matrix multiplication. Accordingly, the dynamics of the quantum system
after a discrete number of time steps $n$ is described by the map
$\mathcal{E}^{n}=\mathcal{E}\mathcal{E}^{n-1}$, with $n=2,3...$, and the
initial state $\rho$ is transformed in the final state
$\mathcal{E}^{n}(\rho)$.

Let us now introduce the concepts of QNN [48] in the QML framework and how
they are trained. Formally, a QNN can be written as a product of layers of
unitary operations:

$\hat{U}(\mathbf{\theta})=\prod_{\ell=1}^{L}\hat{V}_{\ell}\hat{U}_{\ell}({\mathbf{\theta}_{\ell}}),$
(5)

where $\hat{V}_{\ell}$ and $\hat{U}_{\ell}({\mathbf{\theta}_{\ell}})$ are
fixed and parameterized unitary operations, respectively, for $\ell^{th}$
layer of QNN. The output of the QNN is:

$f(\mathbf{\theta})=\tr(\mathcal{M}\rho_{\theta})$ (6)

where $\mathcal{M}$ is an Hermitian operator representing the physical
observable,
$\rho_{\theta}=\hat{U}(\mathbf{\theta})^{\dagger}\rho_{0}\hat{U}(\mathbf{\theta})$
and $\rho_{0}$ is the initial state, which is the input of the QNN. The QNN is
optimized minimizing the difference between its output and the desired value.
Generally, the latter is performed with the gradient descent method with the
adoption of the parameters shift rule [82].

## Acknowledgements

M.P. and S.M. acknowledge financial support from PNRR MUR project
PE0000023-NQSTI. F.C. also acknowledges the European Union’s Horizon 2020
research and innovation programme under FET-OPEN GA n. 828946-PATHOS.

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  *[CQGDM]: Classical-Quantum Generative Diffusion Model
  *[QCGDM]: Quantum-Classical Generative Diffusion Model
  *[QQGDM]: Quantum-Quantum Generative Diffusion Model
  *[ML]: Machine Learning
  *[DMs]: Diffusion Model
  *[NN]: neural network
  *[GANs]: Generative Adversarial Network
  *[NISQ]: Noisy Intermediate-Scale Quantum
  *[QPU]: quantum processing unit
  *[QML]: Quantum Machine Learning
  *[QGAN]: Quantum Generative Adversarial Network
  *[GAN]: Generative Adversarial Network
  *[NLP]: Natural Language Processing
  *[QNN]: Quantum Neural Network
  *[PQC]: Parametrized Quantum Circuit
  *[DDPM]: Denoising Diffusion Probabilistic Model
  *[Score SDE]: Score Stochastic Differential Equation
  *[SSE]: Stochastic Schrödinger Equation
  *[TOMs]: transition operation matrix
  *[SSEs]: Stochastic Schrödinger Equation
  *[QCGDMs]: Quantum-Classical Generative Diffusion Model
  *[NNs]: neural network
  *[PQCs]: Parametrized Quantum Circuit
  *[DM]: Diffusion Model
  *[QQGDMs]: Quantum-Quantum Generative Diffusion Model
  *[CQGDMs]: Classical-Quantum Generative Diffusion Model
  *[QMLs]: Quantum Machine Learning
  *[QNNs]: Quantum Neural Network
  *[TOM]: transition operation matrix