# QFNN-FFD: Quantum Federated Neural Network for Financial Fraud Detection

Nouhaila Innan1, Alberto Marchisio2,3, Muhammad Shafique2,3, and Mohamed
Bennai1 1Quantum Physics and Magnetism Team, LPMC, Faculty of Sciences Ben
M’sick,
Hassan II University of Casablanca, Morocco
2eBRAIN Lab, Division of Engineering, New York University Abu Dhabi (NYUAD),
Abu Dhabi, UAE
3Center for Quantum and Topological Systems (CQTS), NYUAD Research Institute,
NYUAD, Abu Dhabi, UAE
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###### Abstract

This study introduces the Quantum Federated Neural Network for Financial Fraud
Detection (QFNN-FFD), a cutting-edge framework merging Quantum Machine
Learning (QML) and quantum computing with Federated Learning (FL) for
financial fraud detection. Using quantum technologies’ computational power and
the robust data privacy protections offered by FL, QFNN-FFD emerges as a
secure and efficient method for identifying fraudulent transactions within the
financial sector. Implementing a dual-phase training model across distributed
clients enhances data integrity and enables superior performance metrics,
achieving precision rates consistently above 95%. Additionally, QFNN-FFD
demonstrates exceptional resilience by maintaining an impressive 80% accuracy,
highlighting its robustness and readiness for real-world applications. This
combination of high performance, security, and robustness against noise
positions QFNN-FFD as a transformative advancement in financial technology
solutions and establishes it as a new benchmark for privacy-focused fraud
detection systems. This framework facilitates the broader adoption of secure,
quantum-enhanced financial services and inspires future innovations that could
use QML to tackle complex challenges in other areas requiring high
confidentiality and accuracy.

###### Index Terms:

Quantum Neural Network, Quantum Federated Learning, Quantum Machine Learning,
Fraud Detection, Finance

##  I Introduction

In the rapidly evolving financial technology landscape, privacy is a
fundamental pillar, crucial for upholding the trust and integrity of financial
transactions and services [1]. As digital transactions become more prevalent,
the volume of sensitive data handled by financial institutions grows
exponentially, making robust privacy measures indispensable [2]. The emergence
of Quantum Machine Learning (QML) marks a transformative era [3, 4, 5],
promising unprecedented computational capabilities by exploiting quantum
physics [6], while simultaneously raising pivotal concerns about privacy and
data security. This paper introduces the Quantum Federated Neural Network for
Financial Fraud Detection (QFNN-FFD), a pioneering framework that influences
the quantum-enhanced processing power of Quantum Computing (QC) with the
privacy-preserving attributes of Federated Learning (FL). The synergy of QML
with FL jointly improves the efficiency and accuracy of detecting fraudulent
activities, while safeguarding sensitive financial data against the ever-
looming threats of breaches and unauthorized access.

QFNN-FFD represents a significant leap forward in applying quantum
technologies to real-world economic challenges and sets a new benchmark for
privacy-centric approaches in the fintech domain. By deploying this framework,
financial institutions can potentially harness the unique advantages of
QC—such as rapid processing of large datasets—while also benefiting from the
decentralized nature of FL, which keeps sensitive data localized and reduces
the risk of central points of failure. As shown in Fig. 1, Quantum Federated
Learning (QFL) has shown superior performance in various fields [7, 8, 9],
prompting our decision to implement it in finance. Our framework has
demonstrated its capability to enhance both accuracy and privacy protection
through comparative analysis with existing models works [10, 11, 12]. This
approach meets and often surpasses current industry standards, providing a
scalable, secure framework that adapts seamlessly to diverse operational
environments while maintaining high accuracy in fraud detection under various
conditions.

(a)(b) Figure 1: Comparison of ML and FL accuracies in classical and QC
contexts across various fields and experiments. Panel (a)illustrates the
performance of different experiments within the finance sector. Panel (b)
compares QML with QFL across four domains: healthcare, IoT, computer v, and
finance. In classical computing contexts, FL generally demonstrates superior
performance compared to ML [13, 14]. In QC contexts, QFL exhibits slight
improvements over QML [15, 16, 17]. These findings highlight the potential of
QFL and provide a compelling rationale for its adoption, particularly in the
finance sector.

Our contributions significantly impact the fintech sector by providing a
scalable, secure framework that adapts to various operational environments
while maintaining high accuracy in fraud detection under different conditions.

Figure 2: The QFNN-FFD process flow. The diagram outlines the end-to-end
workflow from input through to output. Datasets are processed and fed into the
QFNN-FFD, built upon the PennyLane library. The model undergoes training and
testing for 100 iterations, incorporating a variety of noise models using
noise simulators from IBM’s Qiskit. The quantum simulator within PennyLane is
utilized to emulate a quantum environment. The output is evaluated based on
performance metrics, including accuracy, precision, recall, F1 score, and mean
squared error loss, providing a comprehensive assessment of the model’s
capability to detect fraudulent transactions.

Our novel contributions, encapsulating a comprehensive workflow that
significantly enhances fintech security measures, are listed as follows and
shown in Fig. 2:

  * •

Introducing a novel QFNN-FFD that uniquely combines QML algorithms with FL
architecture to enhance both the computational capabilities and the privacy
aspects of fraud detection systems, ensuring that sensitive financial data
remains within its local environment.

  * •

Demonstrating superior analytical capabilities by analyzing complex
transaction patterns more effectively than traditional models, comparative
experimental results reveal that QFNN-FFD consistently outperforms existing
fraud detection systems in terms of accuracy, thereby establishing a new
benchmark for the industry.

  * •

Recognizing the challenges posed by quantum decoherence and noise by testing
our QFNN-FFD across six different quantum noise models to validate its
robustness ensures that our framework is not only theoretically but also
practically viable in real-world QC environments, maintaining high performance
under various noise conditions.

##  II Background and Related Works

FL is a Machine Learning (ML) paradigm in which multiple parties [18, 19, 20],
termed clients, collaborate under the oversight of a central server to address
an ML task without exchanging their raw data.

Figure 3: Schematic representation of the FL architecture. The diagram shows
multiple users (clients), each with their local dataset, independently
training local models. These models are then transmitted as model updates to a
central server. The server aggregates these updates to improve the global
model, which is then distributed back to the users for further refinement.
This cycle ensures data privacy and security, as raw data never leaves the
local premises of each user.

As illustrated in Fig. 3, clients contribute model updates, computed from
their local datasets, to the server. Mathematically, each client $i$ computes
an update $\Delta\theta_{i}$ based on its local data $D_{i}$:

$\Delta\theta_{i}=-\eta\nabla L(\theta;D_{i}),$ (1)

where $\eta$ is the learning rate and $L$ is the loss function evaluated with
the ML model parameters $\theta$. These updates $\Delta\theta_{i}$ are then
sent to the central server, which aggregates them to update the global model
using a weighted average:

$\theta\leftarrow\theta+\sum_{i=1}^{n}\frac{|D_{i}|}{D}\Delta\theta_{i},$ (2)

where $D=\sum_{i=1}^{n}|D_{i}|$ represents the total size of data across all
clients, and $|D_{i}|$ is the size of the local dataset of client $i$. This
aggregation method effectively mitigates concerns related to privacy, data
security, and data access rights, which are particularly pertinent when
dealing with sensitive information scattered across disparate locations.

The progression of FL into the QC domain has precipitated the inception of QFL
[17, 21]. This methodology exploits quantum mechanics’ distinctive properties
to augment privacy and computational efficiency. In [22], the study delineated
the first fully operational QFL framework capable of processing exclusively
quantum data. This innovation indicated the establishment of the inaugural
quantum federated dataset, facilitating the collaborative learning of quantum
circuit parameters by quantum clients in a decentralized manner—a cornerstone
in adapting quantum technologies to federated contexts.

Subsequently, the notion of dynamic QFL frameworks was advanced in [8], which
introduced the Slimmable QFL. This framework was designed to adapt to varying
network conditions and constraints on computing resources by dynamically
modulating the training parameters of Quantum Neural Networks (QNNs).
Empirical studies demonstrated that SlimQFL sustained superior classification
accuracy under fluctuating conditions compared to conventional static QFL
methods. Moreover, integrating secure protocols, such as blind quantum
computation, into distributed learning environments enhanced the
confidentiality aspect of QFL.

The research outlined in [23] proposed a quantum protocol that leveraged the
computational capacities of remote quantum servers while safeguarding the
privacy of the underlying data. This protocol proved its robustness against
prevalent security threats, including gradient attacks, rendering it
especially beneficial in domains that demand boosted security for distributed
learning tasks.

It is essential to recognize the expansive applications of QFL across various
industries and how these applications introduce specialized implementations in
sectors requiring high data privacy and computational precision. Particularly
in the financial industry, where the confidentiality and integrity of data are
paramount, the transition from general data protection to targeted fraud
detection represents a critical evolution of QFL capabilities.

The effectiveness of QFL in securely managing and processing data within
healthcare and genomics, as explored in [24], serves as a foundation for its
application in the more complex and sensitive realm of financial transactions.
This broad applicability underscores the potential of QFL to enhance privacy
and computational efficiency in highly effective scenarios.

Advancing into financial fraud, significant research has been conducted to
apply QC and QML in detecting financial fraud. In [12], they developed quantum
protocols for anomaly detection, applying them to credit card fraud. They
compared quantum kernel methods to classical ML benchmarks, revealing the
potential for quantum approaches to achieve superior precision significantly
as the number of qubits increased. This demonstrated that quantum methods can
become increasingly advantageous with the scaling of quantum systems.

Furthermore, in [25], they explored using a Quantum Support Vector Machine
(QSVM) for real-world financial data, presenting one of the first end-to-end
applications of QML in the financial sector. Their findings highlighted the
complementary nature of QSVM to classical techniques, offering new methods for
feature exploration that enhanced fraud detection accuracy.

As the application of QML in fraud detection advances, several innovative
approaches have emerged. For instance, in [26], they explored using QML
models, including the Variational Quantum Classifier (VQC) and different QNNs.
These models showed promising results in classifying fraud and non-fraud
transactions, demonstrating QML’s potential in financial applications. Despite
their success, these models faced challenges such as needing more efficient
quantum algorithms and the ability to handle larger and more complex datasets.

In [27], the study addressed the latency in traditional fraud detection
systems by implementing a QML approach using a Support Vector Machine (SVM)
enhanced with quantum annealing solvers. This approach significantly
outperformed traditional models’ speed and accuracy, particularly on time-
series data from bank loans, which are typically highly imbalanced.

In [28], they discussed a hybrid model that combines QNNs with classical
neural networks to enhance fraud detection capabilities. This study
implemented two distinct methods: a hybrid quantum-classical neural network
and topological data analysis for noise reduction and improved classification
accuracy. Such hybrid models leverage the computational power of quantum
devices while maintaining the versatility and robustness of classical ML
frameworks.

Generative adversarial networks (GANs) have also been adapted to quantum
settings to tackle the instability and inefficiency of classical sampling
methods. [29] introduced variational quantum-classical Wasserstein GANs
(WGANs), which incorporated a hybrid quantum-classical generator with a
classical discriminator. This model was effective on a credit card fraud
dataset, providing competitive performance with classical counterparts in
terms of F1 score.

Further advancing the field, in [30], they presented an approach using data
re-uploading techniques to train single-qubit classifiers that perform
comparably to classical models under similar training conditions. This study
highlights the potential for QML to achieve significant results with minimal
quantum resources, opening new avenues for efficient quantum computations.

Moreover, in [31] and [32], they highlighted the real-time challenges in fraud
detection. They utilized quantum annealing to develop frameworks that enhance
the speed of fraud detection, addressing the critical issue of timely response
in fraudulent transaction identification.

These studies collectively demonstrate the growing capability of QML to
enhance fraud detection but often neglect the aspect of data privacy in their
computational frameworks. Most QML models focus primarily on computational
advantages without integrating robust privacy measures. Our QFNN-FFD framework
addresses this gap by combining the privacy-preserving features of FL with the
power of QC. By ensuring that data remains local and only aggregate updates
are shared, our framework enhances the security and privacy of the distributed
learning process, setting a new standard in applying quantum technologies to
sensitive financial operations.

##  III QFNN-FFD Framework Design

In this section, we introduce a novel QFNN-FFD framework that integrates the
quantum computational capabilities of QML with the distributed, privacy-
preserving nature of FL, as described in Algorithm 1.

Data: QNN circuit, dataset split among $N$ clients, learning rate $\eta=0.1$,
maximum local iterations $T$.

Result: Accuracy, precision, recall, F1 score, and loss

Initialization: Parameters $\theta$ randomly initialized in $[0,1]$;

for  _each client $i=1$ to $N$_ do

Initialize local model parameters $\theta_{i}\leftarrow\theta$;

for  _each local iteration $t=1$ to $T$_ do

foreach  _batch in local dataset_ do

Encode data into quantum states;

Apply QNN circuit with current parameters $\theta_{i}$;

Perform quantum measurements to obtain classical outputs;

Calculate loss using MSE;

Optimize $\theta_{i}$ using Adam optimizer with learning rate $\eta$;

Evaluate local model on validation set and adjust $\theta_{i}$;

If convergence criteria are met, exit loop early;

Synchronize and send optimized local parameters $\theta_{i}$ to central
server;

On central server:;

Aggregate local parameters to update global model; Broadcast updated global
parameters $\theta$ back to each client;

for  _each client $i=1$ to $N$_ do

Update local model parameters $\theta_{i}\leftarrow\theta$;

Evaluate model performance on a global validation set to ensure
generalization;

Algorithm 1 QFNN-FFD Framework

###  III-A QNN Circuit Design and QFL Integration

Central to this approach is a QNN circuit, shown in Fig. 5. The QNN model has
demonstrated its powerful capabilities in various applications, particularly
fraud detection. Like typical QML models, as shown in Fig. 4, it begins with
data encoding, followed by a sequence of quantum operations that form the core
of the processing circuit, and concludes with measurement to extract
actionable insights [33, 34, 35, 36].

Figure 4: General schematic of a QML model workflow. The process begins with
qubits in the zero state $(|0\rangle)$. The qubits undergo data encoding to
represent the input data in quantum form. Then, a parametrized quantum
circuit, $U(\theta)$, transforms the qubit states, where $\theta$ represents
tunable parameters. The transformed quantum states are measured, converting
quantum information into classical output. This output is evaluated using a
predefined loss function, and a classical optimization algorithm iteratively
adjusts $\theta$ to minimize the loss, thereby refining the QML model’s
performance.

The QFNN-FFD framework operates on data distributed across $N$ clients, each
possessing a subset of the overall dataset, thereby preventing the need for
central data aggregation and enhancing data privacy. Training of the QFNN-FFD
is directed in a federated manner, where local models on each client are
independently trained using their data subsets.

In the local model, the first step is to encode classical data into quantum
states through angle encoding. Each data feature $x_{i,j}$ from the vector
$\mathbf{x}_{i}$ of client $i$ is mapped onto two rotation angles,
$\theta_{i,j}$ for the $R_{y}$ rotation and $\phi_{i,j}$ for the $R_{z}$
rotation. These rotations are then applied to the qubits sequentially to
modify both their phase and orientation:

$R(\theta_{i,j},\phi_{i,j})=R_{y}(\theta_{i,j})R_{z}(\phi_{i,j}),$ (3)

where $R_{y}(\theta_{i,j})=e^{-i\theta_{i,j}Y/2}$ and
$R_{z}(\phi_{i,j})=e^{-i\phi_{i,j}Z/2}$, with $Y$ and $Z$ representing the
Pauli-Y and Pauli-Z matrices, respectively.

We apply a series of controlled operations to achieve an entangled quantum
state that captures correlations between different features. One effective
method is using a sequence of CNOT gates, which create entanglements between
successive qubits:

$U_{\text{ent}}=\prod_{k=1}^{n-1}\text{CNOT}_{k,k+1},$ (4)

where $\text{CNOT}_{k,k+1}$ applies a CNOT gate between the $k$-th and
$(k+1)$-th qubits. This sequence creates a chain of entanglements across the
qubit register, which is crucial for leveraging quantum correlations.

This setup ensures that the quantum states are intricately linked, which is
crucial for capturing complex correlations in the dataset. The full quantum
state preparation for client $i$ is thus represented by:

$\ket{\psi_{i}}=\left(\bigotimes_{j=1}^{n}R_{y}(\theta_{i,j})R_{z}(\phi_{i,j})\right)\cdot\text{CNOT}\ket{0}^{\otimes
n}.$ (5)

Figure 5:  An overview of the QFNN-FFD framework. This flowchart presents the
multi-stage process, beginning with data preprocessing and distribution to
various users. Each user independently conducts a local training phase on a
QNN circuit, followed by an optimization stage. The optimized local models are
then transmitted to a central cloud server for global aggregation, culminating
in an enhanced federated model. The lower part of the figure illustrates the
quantum circuit’s structure, showcasing the intricate interplay of qubits and
quantum gates (rotations and CNOT gates) during the computation process.

###  III-B Optimization and Training Process

The Adam optimizer is integral to the training process of our QFNN-FFD
framework due to its adaptive learning rate capabilities, which significantly
enhance convergence speed and efficiency. The Adam optimizer’s update rule is
particularly well-suited for the demands of quantum circuit training and is
defined as follows:

$\theta_{t+1}=\theta_{t}-\frac{\eta}{\sqrt{\hat{v}_{t}}+\epsilon}\hat{m}_{t},$
(6)

where $\eta$ represents the learning rate, $\hat{m}_{t}$ and $\hat{v}_{t}$ are
the estimates of the first and second moments of the gradients, respectively,
and $\epsilon$ is a small constant to avoid division by zero. This
configuration allows each parameter update to be adjusted dynamically based on
the individual gradients’ variability, providing a tailored approach to
parameter optimization.

In the context of our QFNN-FFD, the Adam optimizer’s role extends to
effectively minimizing the Mean Squared Error (MSE) loss function during the
training process. The MSE loss function is crucial for calibrating the model’s
predictive accuracy and is expressed as:

$L(\theta)=\frac{1}{m}\sum_{j=1}^{m}(y_{j}-\hat{y}_{j}(\theta))^{2},$ (7)

where $m$ is the batch size, $y_{j}$ are the actual labels of transactions,
and $\hat{y}_{j}(\theta)$ represents the predicted labels output by the model.
This loss function quantifies the error between the model’s predictions and
the true labels, guiding the optimizer to focus on reducing these
discrepancies. The optimization process iterates through a maximum of $T$
local iterations, refining the model’s ability to discern fraudulent
transactions accurately.

###  III-C Parameter Aggregation and Model Evaluation

Following local optimization, each client’s parameters $\theta_{i}$ are
transmitted to a central server. They are aggregated through a simple
averaging process to update the global model parameters $\theta$. This cyclic
process of local optimization and global aggregation iteratively enhances the
QFNN-FFD’s performance, evaluated on a global validation set for
generalizability and efficacy. The mathematical foundation of parameter
optimization within the QFNN-FFD employs the Adam optimizer, adjusting
$\theta$ as

$\theta_{t+1}=\theta_{t}-\eta\cdot\text{Adam}(\nabla_{\theta}L(\theta_{t}))$
(8)

where $\text{Adam}(\nabla_{\theta}L(\theta_{t}))$ calculates the adjustment
based on the gradient of the loss function with respect to the parameters
$\theta$ at iteration $t$. This optimization ensures a gradual refinement of
the model’s parameters.

After the local training phases, the optimized parameters $\theta_{i}$ from
each client are securely aggregated at a central server using a federated
averaging algorithm:

$\theta_{\text{global}}=\frac{1}{N}\sum_{i=1}^{N}\theta_{i},$ (9)

This aggregation step effectively combines insights from all the distributed
models, enhancing the global model’s generalizability and robustness, steering
the QFNN-FFD towards higher accuracy in fraud detection (see Algorithm 1).

The globally updated parameters are redistributed to all clients for further
training, cycling through local optimization and global aggregation to
progressively improve the QFNN-FFD’s performance. This iterative process
enhances computational efficiency and maintains strict privacy standards.

Integrating QML with FL in our QFNN-FFD framework fosters the high-efficiency
processing of complex financial data and upholds stringent data privacy
standards. This dual advantage, coupled with the model’s mathematical rigor
and strategic parameter optimization, positions the QFNN-FFD as an effective
tool in the fight against financial fraud, marking a significant leap forward
in applying QC to real-world challenges in the financial sector.

##  IV Results and Discussion

###  IV-A Experimental Setup

In our study, we utilize the IEEE-CIS Fraud Detection dataset [37]. It is
divided into two primary files: identity and transaction, linked by
TransactionID. It encompasses both numerical and categorical features
essential for identifying fraudulent activities. The preprocessing steps begin
with optimizing the dataset’s memory usage by refining data types,
significantly reducing its memory footprint. This is followed by a detailed
analysis of missing values, which helps identify and quantify missing data to
inform our approach to managing these instances. Subsequently, features are
categorized and processed: categorical variables undergo one-hot encoding,
while numerical variables are standardized. To counteract the class imbalance
between fraud and non-fraud instances, an up-sampling technique is employed to
ensure equitable representation of both classes.

Our QFNN-FFD is implemented using PennyLane for model architecture and Qiskit
for simulating quantum noise [38, 39], enabling a realistic QC environment.
The framework, structured around four qubits, employs the Adam optimizer
($\eta$=0.1) across dual training phases—local and global—with up to 100
iterations for each across 15 clients. This setup is characterized by 32
initially random parameters, which are optimized through evaluations on a
training set comprising 115,386 instances (80% of the total dataset of 144,233
instances) and a validation set comprising 28,847 instances, which is 20% of
the total dataset. We focus on binary classification accuracy and MSE as key
metrics. Operational deployment occurs within an environment characterized by
a configuration consisting of 4 virtual CPUs (vCPUs), 25 gigabytes (GB) of
RAM, and a single NVIDIA Tesla V100 virtual GPU (vGPU). This setup offers a
balance of processing power, memory capacity, and advanced GPU
acceleration—crucial factors for efficiently handling the intensive
computations required by our QFNN-FFD framework.

###  IV-B Accuracy and Loss Analysis

The validation accuracy and loss trends for the QFNN-FFD, as shown in Fig. 6,
provide valuable insights into the model’s performance over iterations, which
is the average outcome of 10 trials of QFNN, which ensures the reliability of
the results by accounting for variability in the model’s performance.
Initially, the model’s accuracy begins at 0.735 and demonstrates a steady
upward trend, culminating in a plateau of 0.95 at \raisebox{-0.9pt}1⃝,
consistently maintained from iteration 35 onwards. This performance plateau
signifies that the framework not only swiftly attains a high confidence level
in its fraud detection capabilities but also sustains this efficacy over time.
Alongside, the validation loss diminishes from an initial 0.275 to 0.02,
reflecting the model’s enhanced precision in identifying fraudulent
transactions.

This reduction in validation loss is significant as it suggests a substantial
enhancement in the model’s ability to differentiate between fraudulent and
legitimate transactions with minimal error, thereby reducing the likelihood of
costly false positives. The pronounced improvement in accuracy and reduction
in loss observed between iterations 10 and 20 at \raisebox{-0.9pt}2⃝ marks a
critical learning phase for the model. By iteration 35, the model achieves and
upholds a state of high accuracy and minimal loss at \raisebox{-0.9pt}3⃝,
indicative of its robust learning mechanism and stability. This phase
showcases the effective convergence of the quantum and FL components,
optimizing the model’s parameters for high-stakes decision-making
environments. The sustained model performance beyond the 35th iteration
underscores the QFNN-FFD’s ability for dependable and steady fraud prediction
within QC environments.

Moreover, the robust validation performance of QFNN-FFD highlights its
practical applicability. The high validation accuracy suggests effective
pattern recognition is crucial for fraud detection, while the low and stable
loss indicates minimized rates of false positives and negatives—essential for
the operational deployment of any fraud detection system. This balance is
particularly important in financial contexts where the cost of false negatives
can be extraordinarily high. Given the observed performance plateau,
implementing an early exit strategy in training could economize on
computational resources without compromising effectiveness, optimizing overall
efficiency. This strategy underscores the framework’s capability to deliver
high performance while efficiently managing computational demands, setting a
new standard for privacy-focused, quantum-enhanced financial services.

123Stable plateau indicatinghigh performance3Rapid improvement phase2Initial
spike inloss reduction Figure 6:  Evolution of Validation Metrics as a
Function of Iteration Count. The plot illustrates the optimization trajectory
over 100 iterations, with validation accuracy demonstrating an upward trend
towards convergence, and loss exhibiting a reciprocal decrease, indicative of
the model’s improving generalization on unseen data. 321 Figure 7:
Comparative Impact of quantum noise models on QFNN-FFD Framework Accuracy. The
graph systematically evaluates the framework’s accuracy under the influence of
six quantum noise models: depolarizing, phase damping, amplitude damping,
bitflip, phaseflip, and bitphaseflip. The noise parameters are adjusted from 0
(indicating no noise) to 1 (signifying maximum noise interference), providing
insights into the relative performance stability of the QFNN-FFD framework
across a spectrum of quantum noise intensities.

###  IV-C Quantum Noise Analysis

In our experiments, we expose the QFNN-FFD framework to a spectrum of quantum
noise models [40], aiming to simulate the challenging conditions of near-term
QC devices. As presented in Fig. 7, under the depolarizing noise model,
accuracy remains high at 0.97 but plummets to 0 when the noise parameter
reaches 1, indicating the model’s noise tolerance limit. In the bitflip noise,
the QFNN-FFD shows resilience, maintaining a 0.97 accuracy until the noise
parameter hits 0.3 at \raisebox{-0.9pt}1⃝, after which it significantly drops
to 0.2 at a noise level of 1, marking the model’s performance threshold. This
illustrates how bitflip errors, which flip the state of a qubit, begin to
degrade the system’s performance only at higher noise levels, demonstrating a
strong error tolerance up to a critical point. The amplitude damping noise
leads to a less severe decrease in accuracy, from 0.97 to 0.4 at
\raisebox{-0.9pt}2⃝ as noise increases, while phase damping impacts it more,
reducing accuracy to 0.09, highlighting sensitivity to phase perturbations.
These results underscore the QFNN-FFD’s varying sensitivity to different types
of quantum noise, with phase damping proving particularly detrimental. This
sensitivity is crucial for understanding which quantum error correction
techniques might be most effective in enhancing the robustness of the model.
Remarkably, against phaseflip and bitphaseflip noises, the QFNN-FFD maintains
over 0.9 accuracy up to a noise parameter of 0.7 at \raisebox{-0.9pt}3⃝, only
dropping to 0.74, demonstrating significant robustness and potential
compatibility with existing quantum technologies. This resilience against
phaseflip and bitphaseflip noises suggests that the model’s quantum circuitry
may be naturally more protected against these types of errors, possibly due to
the nature of the quantum gates used or the initial state preparation.

Such robustness implies the QFNN-FFD’s potential compatibility with current
quantum technology, where such noise is prevalent. The robust performance of
the QFNN-FFD across these diverse noise profiles strongly indicates its
applicability in quantum-enhanced fraud detection systems. The data clearly
illustrates how the QFNN-FFD could provide reliable performance, guiding
future enhancements in quantum error correction to fortify the model against
the most vulnerable types of noise. These findings are pivotal, as they
demonstrate the framework’s current efficacy and its potential for adaptation
and improvement with the maturation of quantum technologies.

###  IV-D Comparison with Existing Works

TABLE I: Comparison of QML frameworks on different financial fraud datasets. Reference | Precision | Recall | F1-Score | Accuracy
---|---|---|---|---
[10] | 84 | 84.44 | 75.68 | 83.92
[11] | 96.1 | 79.5 | 86 | 94.5
[12] | 90 | – | – | –
Our QFNN-FFD | 95 | 96 | 95 | 95

Compared to the results in Table I, our QFNN-FFD outperforms other QML models
applied to similar datasets, achieving superior performance metrics. These
metrics include precision, recall, F1-score, and accuracy, where QFNN-FFD
demonstrates comprehensive superiority across all fronts. This performance is
a testament to the model’s efficacy and highlights its ability to effectively
integrate complex quantum computations within an FL framework. Unlike the
existing models [10, 11, 12], which focus solely on performance, QFNN-FFD
additionally integrates a privacy-preserving FL approach. This ensures high
detection accuracy of 95% and enhanced data privacy, establishing QFNN-FFD as
a leading solution for secure and efficient fraud detection in fintech.

###  IV-E Discussion

Our results show that the framework achieves high validation accuracy,
maintains low loss across various operational conditions, and exhibits
resilience against diverse quantum noise models. Such robustness underlines
the framework’s suitability for real-world QC environments known for their
integral noise issues. In direct comparison with existing quantum and
classical models, QFNN-FFD surpasses typical performance metrics, making it a
superior choice for fraud detection. This performance is particularly notable
given the framework’s integration of privacy-preserving FL, which safeguards
sensitive financial data during detection. This dual benefit of enhanced
accuracy and increased data privacy sets QFNN-FFD apart as a leading solution
for secure and effective fraud detection in the fintech industry. Furthermore,
the framework’s ability to maintain high performance under various noise
conditions suggests its potential for broader applications beyond financial
services, including sectors where data sensitivity and security are paramount.
Integrating advanced quantum computational capabilities with robust privacy
features positions QFNN-FFD as a scalable solution for future challenges in
secure data processing and analysis.

##  V Conclusion

Our research successfully demonstrates the potential of QFNN-FFD in enhancing
fraud detection within the financial sector. By integrating advanced QC
techniques with FL, we present a novel approach that significantly improves
accuracy and efficiency compared to conventional methods. Our findings reveal
that the QFNN-FFD framework, supported by a robust computational
infrastructure and optimized through sophisticated preprocessing techniques,
can effectively identify fraudulent transactions with high precision. Its
resilience against various quantum noise models is particularly noteworthy,
indicating its suitability for real-world application in the near-term QC
landscape. This resilience, coupled with the model’s ability to maintain high
performance under different noise conditions, underscores the practical value
of our approach. Furthermore, the QFNN-FFD’s adaptability to quantum noise
suggests a promising direction for future research in quantum error correction
and noise mitigation strategies. Our study contributes to the emerging field
of QC by providing an efficient framework for applying QML while ensuring
privacy to solve complex problems in finance. Expanding beyond finance, this
framework has the potential to revolutionize fields such as healthcare and
cybersecurity, where privacy and data sensitivity are paramount, thus marking
a significant milestone in the interdisciplinary application of QML. In
conclusion, the QFNN-FFD framework addresses key challenges in the fintech
sector and also sets a precedent for the deployment of quantum technologies in
privacy-critical applications, offering substantial implications for both
academic research and industry practices. It encourages further exploration
and development within the QC, QML, and FL communities, aiming to unlock new
possibilities for handling complex, large-scale data analysis tasks in an
increasingly digital and interconnected world.

## Acknowledgment

This work was supported in part by the NYUAD Center for Quantum and
Topological Systems (CQTS), funded by Tamkeen under the NYUAD Research
Institute grant CG008.

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# A silicon source of heralded single photons at 2 $\mu$m.

S. Signorini<EMAIL_ADDRESS>Nanoscience Laboratory, Department
of Physics, University of Trento, Via Sommarive 14, 38123, Trento, Italy  M.
Sanna  Nanoscience Laboratory, Department of Physics, University of Trento,
Via Sommarive 14, 38123, Trento, Italy  S. Piccione  Nanoscience Laboratory,
Department of Physics, University of Trento, Via Sommarive 14, 38123, Trento,
Italy  M. Ghulinyan  Centre for Sensors and Devices, Fondazione Bruno Kessler,
38123, Trento, Italy  P. Tidemand-Lichtenberg  Department of Photonics
Engineering, DTU Fotonik, Technical University of Denmark, Roskilde, 4000,
Denmark  C. Pedersen  Department of Photonics Engineering, DTU Fotonik,
Technical University of Denmark, Roskilde, 4000, Denmark  L. Pavesi
Nanoscience Laboratory, Department of Physics, University of Trento, Via
Sommarive 14, 38123, Trento, Italy

###### Abstract

Mid infrared integrated quantum photonics is a promising platform for
applications in sensing and metrology. However, there are only few examples of
on-chip single photon sources at these wavelengths. These have limited
performances with respect to their C-band counterparts. In this work, we
demonstrate a new approach to generate heralded single photons in the mid
infrared on a silicon chip. By using a standard C-band pump, the inter-modal
spontaneous four wave mixing enables the generation of the herald idler at
1259.7 nm and the heralded signal at 2015 nm. The idler photon is easily
detected with a common infrared single photon detector while the signal photon
is upconverted to the visible before its detection. In this way, we are able
to operate a mid infrared source without the need of mid infrared detectors
and laser sources. By measuring a heralded $g^{(2)}$ of $0.23\,\pm\,0.08$ we
demonstrate the single photon behaviour of the source as well as the
feasibility of multi-photon coincidence measurements beyond 2 $\mu$m with our
setup. The source exhibits a high intrinsic heralding efficiency of
$(59\,\pm\,5)\%$, a maximum coincidence to accidental ratio of
$40.4\,\pm\,0.9$ and a generation probability of
$\left(0.72\,\pm\,0.10\right)$ W-2.

††preprint: AIP/123-QED

##  I Introduction

Mid infrared (MIR) light (2 - 15 $\upmu$m) is of importance in a wide range of
technological applications. Free space telecommunication Su  _et al._ (2018),
LIDAR Weibring, Edner, and Svanberg (2003), environmental monitoring Fix  _et
al._ (2016), medicine and biology Evans  _et al._ (2007); Bellisola and Sorio
(2012); Potter  _et al._ (2001); Miller, Bourassa, and Smith (2013) are only
few of the several fields where MIR optics plays a role. In particular, gas
sensing exploits the strong absorption bands in the MIR Popa and Udrea (2019)
to enhance remarkably the sensitivity of absorption spectroscopy measurements
Petersen  _et al._ (2014); Vainio and Halonen (2016); Ghorbani and Schmidt
(2017). Despite the great interest in developing MIR applications, these are
still hindered by immature optical MIR devices. Quantum optics offers new
solutions to mitigate such limitations. Sub-poissonian light can be used to
beat the shot noise limit Brida, Genovese, and Berchera (2010); Whittaker  _et
al._ (2017). Entangled photons have been used to demonstrate new imaging and
spectroscopy techniques able to get rid of detection technology limitations,
namely ghost imaging Pittman  _et al._ (1995); Morris  _et al._ (2015) or
undetected photon measurement Lemos  _et al._ (2014); Kalashnikov  _et al._
(2016); Vergyris  _et al._ (2020). To enable quantum enhanced MIR metrology
leveraging these quantum based measurement strategies, a source of single or
entangled photons beyond 2 $\upmu$m is required. Up to now, these techniques
have been investigated only with bulky, alignment tolerant and expensive
instrumentation, based on free space nonlinear crystals Kalashnikov  _et al._
(2016); Prabhakar  _et al._ (2020). To develop feasible, robust and affordable
quantum technologies, miniaturization and cost effectiveness are crucial. Such
requirements can be met by means of integrated photonics. In particular,
silicon photonics integrated circuits are characterized by mature CMOS
(complementary metal oxide semiconductor) fabrication technology, which allows
for robust, stable, low power consuming and efficient light manipulation at
the chip scale Lockwood and Pavesi (2010). On-chip MIR quantum measurements
would enable efficient and cost effective sensors, boosting the development of
MIR and quantum technologies. Recently, an on-chip silicon-on-insulator (SOI)
source of MIR pairs has been reported Rosenfeld  _et al._ (2020). However, in
this work a pump in the MIR is used, and both the paired photons are beyond 2
$\upmu$m, thus requiring specific MIR technologies for both the pump and the
detection. Recently, we demonstrated that inter-modal spontaneous four wave
mixing (SFWM) can be used in silicon waveguides to generate correlated pairs
with one photon in the near infrared (NIR) and the other in the MIR by using a
standard C-band pump Signorini  _et al._ (2018, 2019). However, we never
detected the MIR correlated photon. Instead we inferred its existence by
measuring the high energy photon in the pair.
In this work, we demonstrate a SOI waveguide source of heralded MIR single
photons based on inter-modal SFWM, peforming the MIR detection by means of an
upconversion system Mancinelli  _et al._ (2017). The herald photon lays in the
NIR, where it can be efficiently detected with traditional InGaAs single
photon avalanche photodiodes (SPADs). Moreover, the photons are generated in
discrete bands, thus removing the need for narrow band filters to select the
operating wavelengths of signal and idler. As a result, the heralding
efficiency is increased with respect to traditional intra-modal SFWM, as
witnessed by the measured intrinsic heralding efficiency $\eta_{I}=59(5)\,\%$.
The large detuning of the generated photons is also beneficial for the pump
and Raman noise rejection, that can be easily removed with broadband filters.
The pump is a standard 1550 nm pulsed laser. Therefore, we do not require MIR
technologies to operate a source beyond 2 $\mu$m. We assessed the single
photon behaviour of the source by measuring a heralded $g^{(2)}_{h}(0)$ of
0.23(8). We monitored the idler-signal coincidences, reporting a maximum
coincidence to accidental ratio of 40.4(9), exceeding the performance of
current integrated sources of MIR heralded photons Rosenfeld  _et al._ (2020).
The paper is organized in the following way: in section II we describe the
chip design and the experimental setup. In section III our approach to data
analysis is extensively described. In section IV the results relative to the
source characterization are reported. Section V concludes the paper.

Figure 1:  a) Simulated intensity profiles of the TE0 and TE1 spatial modes in
the multimode waveguide. b) Experimental setup used in the experiment. For the
pump (green) we used a pulsed laser at 1550.3 nm (40 ps pulse width, 80 MHz
repetition rate), which, after passing through a band pass filter (F) and a
polarization controller (PC), is coupled to the chip via a tapered lensed
fiber. The chip schematics is shown in the bottom part. On the chip, after a
3-dB directional coupler (DC), half of the pump remains on the TE0, while the
other half is converted to the TE1 via an asymmetric directional coupler
(ADC1) (92$\%$ efficiency). In this way, the pump reaches the multimode
waveguide (MMWG) equally splitted on the TE0 and TE1 modes. In the MMWG, the
inter-modal SFWM process generates the idler (blue) and signal (red) photons
in the TE0 and TE1 modes respectively. The signal is then converted to the TE0
via another asymmetric directional coupler (ADC2). In this way, idler and
signal can be easily separated on chip. Idler and signal are then out-coupled
from the chip via two tapered lensed fibers. Pump residual and Raman noise are
rejected from the idler beam by means of a short pass filter (SP) with a cut-
off wavelength of 1335 nm. The idler is then detected via an InGaAs SPAD (ID
Quantique IDQ210), triggered by the pump, with a gate width of 1.90 ns. The
signal, after being out-coupled from the chip, is polarization rotated through
a free space half-wave plate $\left(\lambda/2\right)$ and upconverted to the
visible through an upconverter system (UC). The UC includes a long pass filter
with a cut-on wavelength of 1900 nm, which rejects the C-band pump. To be
noticed that the UC introduces noise photons collinear to the upconverted
signal and centered at the same wavelength. A bandpass filter (BP) is used to
filter away part of this noise, without filtering the upconverted signal
(purple). Then, the signal photons are analyzed by means of a Hanbury Brown
and Twiss (HBT) interferometer. The HBT interferometer is composed by a 50/50
beam splitter (BS) with two visible silicon SPADs (Excelitas SPCM-AQRH-12)
monitoring the BS reflection and transmission ports. The visible SPADs are
used in free-running mode. A time tagging unit (Swabian Time Tagger 20) is
used to monitor individual singles and coincidences between the three
detectors.

##  II Chip design and experimental setup

Conventional intra-modal SFWM involves only one waveguide mode in the
conversion of two input pump photons into an idler photon and a signal photon.
On the contrary, inter-modal SFWM leverages the different chromatic
dispersions of different optical spatial modes of a photonic waveguide to
achieve phase matching Signorini  _et al._ (2018). Different modal
combinations are possible, depending on the waveguide cross-section, which
also determines the generated signal and idler wavelengths. In this work, we
use the transverse electric (TE) fundamental (TE0) and first (TE1) waveguide
modes in a rib SOI waveguide. The waveguide has a width of 1.95 $\upmu$m and a
height of 0.190 $\upmu$m over a 0.3 $\upmu$m thick slab. The waveguide length
is 1.5 cm. The waveguide and the slab are in silicon, while the top and bottom
claddings are in silica. The simulated intensity profiles of the TE0 and TE1
modes are shown in Fig. 1a.

The inter-modal combination used in our work involves the pump on both the TE0
and TE1, the idler on the TE0 and the signal on the TE1. A peculiar advantage
of intermodal SFWM is the generation of the signal and idler photons on
different waveguide modes. In this way, idler and signal can be easily
separated with high efficiency through an on-chip mode converter. The
experimental setup is detailed in Fig. 1b. The upconverter (UC) is constituted
by a continuous wave (CW) laser cavity, where a Nd:YVO4 pumped intra-cavity
periodically poled lithium niobate (PPLN) allows for sum-frequency generation
(SFG) between the intra-cavity laser (1064 nm) and the input MIR photons. We
used a PPLN from HC Photonics with a length of 25 mm, tuned in temperature to
upconvert the MIR signal at 2015 nm to the visible at 696 nm. The UC used is
the same of Mancinelli et al. Mancinelli  _et al._ (2017), though tuned at the
wavelengths of interest here. The transfer function of the UC is reported in
Fig. 2b, showing a full width at half maximum (FWHM) of $1.15\,\pm\,0.12$ nm.
We used a pump pulsed laser centered at $1550.30\,\pm\,0.05$ nm with 40 ps
pulse width and 80 MHz repetition rate. The generated idler spectrum is
reported in Fig. 2. We measured a discrete band centered at 1259.7 $\pm$ 0.5
nm, with a FWHM of 2.0 $\pm$ 0.3 nm. The measured FWHM of the idler is
compatible with the simulated one of 1.81 nm, as shown in Fig. 2a. According
to the energy conservation, the signal is generated at 2015.2 $\pm$ 1.5 nm.
From the measured idler bandwidth we estimated a FWHM of 5.1 $\pm$ 0.8 nm for
the signal. Therefore, the UC filters the signal photons according to the
spectrum shown in Fig. 2b.

Figure 2:  a) Measured intensity spectrum of the idler beam. The fit has been
made with a gaussian function, showing a FWHM of $2.87\,\pm\,0.07$ nm. This
measurement is affected by the transfer function of the monochromator used to
perform the measurement, which enlarges the actual bandwidth of the
generation. We simulated the idler spectrum considering also the widening due
to the monochromator (orange dashed line). To evaluate the actual bandwidth of
the idler (2.0 $\pm$ 0.3 nm) we deconvolved the response function of the
monochromator. b) Measured spectral response of the upconverter. The response
has been fitted by a squared sinc function, as expected for a sum frequency
generation process. The FWHM is $1.15\,\pm\,0.12$ nm.

##  III Data analysis

With SFWM, the detection probabilities per pulse for the idler ($p_{i}$),
signal ($p_{s}$), coincidences ($p_{si}$) and accidentals ($p_{acc}$) are
quadratic with the pump power $P$. In the limit of low transmission
efficiencies for the signal and idler Harada  _et al._ (2009), they can be
written as

$p_{i}=\xi P^{2}\eta_{i}+d_{i},$ (1a) $p_{s}=\xi P^{2}\eta_{s}+d_{s},$ (1b)
$p_{si}=\xi P^{2}\eta_{i}\eta_{s},$ (1c) $p_{acc}=p_{i}p_{s},$ (1d)

where $\xi$ is the generation probability per pulse per squared unit power
Rosenfeld  _et al._ (2020), $\eta_{i},\,\eta_{s}$ are the total transmission
efficiencies for the idler and signal channels (from generation to detection),
$d_{i},\,d_{s}$ are the dark count probabilities per pulse for the idler and
signal respectively. Eq. (1c) refers to net coincidences, thus without
accidentals. In eqs. (1), noise photons coming from the pump residual and
Raman scattering, typically linear with the pump power, have not been
considered, being negligible in our experimental setup. Singles and
coincidence rates can be calculated by multiplying the probabilities in eqs.
(1) by the repetition rate $R_{p}$ of the pump laser. Together with SFWM other
nonlinear phenomena take place in the waveguide. Two photon absorption (TPA),
cross two photon absorption (XTPA) and free carrier absorption (FCA) have to
be modelled properly in order to recover the actual generation and
transmission efficiency of the pairs. TPA, XTPA and FCA play an important role
in increasing the losses in the waveguide for both the pump and the generated
photons. As a result, the detection probabilities are no longer quadratic with
the input pump power Boyd (2019). A further effect is the nonlinearity of the
idler detector. To model the linear and nonlinear losses affecting pump,
signal and idler photons, we solved the differential equations for the pulse
propagation involving TPA, FCA and propagation losses, assuming that the pump
power is equally split on the TE0 and TE1 modes Borghi  _et al._ (2017).
According to this modeling we can rewrite eqs. (1) as

$p_{si}\simeq\xi\bar{P}_{p}^{2}\bar{\eta}_{i}\bar{\eta}_{s}\eta_{ND}\equiv\bar{p}_{si},$
(2a)
$p_{i}\simeq\left(\xi\bar{P}_{p}^{2}\bar{\eta}_{i}+d_{i}\right)\eta_{ND}\equiv\bar{p}_{i},$
(2b) $p_{s}\simeq\xi\bar{P}_{p}^{2}\bar{\eta}_{s}+d_{s}\equiv\bar{p}_{s},$
(2c) $p_{acc}\simeq\bar{p}_{i}\bar{p}_{s}\equiv\bar{p}_{acc},$ (2d)

where

$\bar{P}_{p}=\sqrt{\frac{1}{L}\int_{0}^{L}P_{p}^{2}(z)dz},$ (3a)
$\bar{\eta}_{j}=\bar{\eta}_{j}^{on}\eta_{j}^{off},$ (3b)
$\bar{\eta}^{on}_{j}=\frac{1}{L}\int_{0}^{L}\eta^{on}_{j}(z)dz,$ (3c)

where $j=i,s$, $L$ is the waveguide length, $P_{p}(z)$ is the on-chip pump
power along the waveguide, $\eta^{on}_{j}(z)$ is the transmission efficiency
for a photon generated at $z$ along the waveguide accounting only for the
linear and nonlinear on-chip losses, $\eta^{off}_{j}$ is the transmission
efficiency accounting only for the losses occurring off chip (fiber-chip
coupling, filtering) and $\eta_{ND}$ models the nonlinear response of the
idler detector. Details about the derivation of eqs. (2) are reported in
Supplementary material.

##  IV Results

###  IV.1 Generation probability and heralding efficiency

To monitor the coincidences between signal and idler, we used a start-and-stop
detection system, using the idler as the start trigger and the signal as the
stop detection Signorini and Pavesi (2020). Coincidences are evaluated within
a coincidence window $\Delta t_{c}$. To be noticed that while for the idler
channel the detection rates (both signal and dark counts) are fixed by the
detection gate width of the idler detector (1.90 ns), for the signal the rates
depend on the coincidence window used in post processing. Therefore, given
$R_{dc,i}=620\,\textrm{cps}$ and $R_{dc,s}=2150\,\textrm{cps}$ the dark count
rates at the idler and signal detectors,

$d_{i}=R_{dc,i}/R_{p}=7.75\times 10^{-6},$ (4)

while

$d_{s}=1-\textrm{e}^{-R_{dc,s}\Delta t_{c}},$ (5)

considering a Poisson distribution for the signal noise (SPAD dark counts and
UC noise).

In order to fit the measured rates and retrieve the generation probability
$\xi$, we can reduce eqs. (2) to

$y_{i}=\frac{\bar{p}_{i}-\eta_{ND}\,d_{i}}{\bar{\eta}_{i}^{on}\eta_{ND}}=\xi\bar{P}_{p}^{2}\eta_{i}^{off}=a_{i}\bar{P}_{p}^{2},$
(6a)
$y_{s}=\frac{\bar{p}_{s}-d_{s}}{\bar{\eta}_{s}^{on}}=\xi\bar{P}_{p}^{2}\eta_{s}^{off}=a_{s}\bar{P}_{p}^{2},$
(6b)
$y_{si}=\frac{\bar{p}_{si}}{\bar{\eta}_{i}^{on}\eta_{ND}\bar{\eta}_{s}^{on}}=\xi\bar{P}_{p}^{2}\eta_{i}^{off}\eta_{s}^{off}=a_{si}\bar{P}_{p}^{2},$
(6c)

with $a_{i}=\xi\eta_{i}^{off}$, $a_{s}=\xi\eta_{s}^{off}$,
$a_{si}=\xi\eta_{i}^{off}\eta_{s}^{off}$. $y_{i}$, $y_{s}$, $y_{si}$ can be
calculated from the measured singles, coincidence and noise rates and from the
simulated $\bar{\eta}^{on}_{j}$ and the measured $\eta_{ND}$ (see
Supplementary material). Modeling exactly the nonlinear losses is a non
trivial task, being the nonlinear parameters highly variable with the
fabrication process and the geometry used. Therefore, we fit $y_{i}$, $y_{s}$,
$y_{si}$ for an input power $<$ 0.5 W (i.e. $\bar{P}_{p}<0.4$ W), where the
nonlinear losses are not the dominant ones. We use $f(x)=ax^{2}+b$ as the
fitting function, retrieving $a_{i}$, $a_{s}$ and $a_{si}$. In this way, we
can evaluate $\xi$ (in units of $W^{-2}$ of peak power) and the off-chip
transmissions, resulting in

$\xi=\frac{a_{i}\,a_{s}}{a_{si}}=\left(0.72\pm 0.10\right)W^{-2},$ (7a)
$\eta_{i}^{off}=\frac{a_{si}}{a_{s}}=\left(2.81\pm 0.17\right)\times 10^{-3},$
(7b) $\eta_{s}^{off}=\frac{a_{si}}{a_{i}}=\left(3.97\pm 0.20\right)\times
10^{-4},$ (7c)

where we used $\Delta t_{c}=1.1$ ns (3$\sigma$ bin width) and the
uncertainties are evaluated at 1 standard deviation of the fitting
coefficients. Details about the nonlinear parameters and propagation losses
used in the model are reported in Supplementary materials. From these results
we calculate the intrinsic heralding efficiency $\eta_{I}$ as Signorini and
Pavesi (2020)

$\eta_{I}=\frac{R^{net}_{si}}{\left(R_{i}-R_{dc,i}\right)\,\bar{\eta}_{s}^{off}}=59\pm
5\,\%,$ (8)

where $R^{net}_{si}$ is the measured net coincidence rate and $R_{i}$ is the
measured idler rate. By normalizing for the signal channel losses, the
$\eta_{I}$ allows to compare different sources only on the bases of their
intrinsic properties, getting rid of the setup used. Our high value comes from
the low on-chip signal losses and the moderate filtering losses to select the
signal wavelength. The heralding efficiency can be further improved by
optimizing the matching between the signal and UC bandwidths.

###  IV.2 Coincidence to accidental ratio

To quantify the efficiency of coincidence detection, the coincidence-to-
accidental ratio (CAR) is used. CAR is analogous to a signal-to-noise ratio
comparing the rate of true coincidences with the accidental ones. True
coincidences come from simultaneous detection of a signal and an idler
belonging to the same pair. Coincidences between signals and idlers belonging
to different pairs or coincidences with noise photons or dark counts
contribute to the accidentals Signorini and Pavesi (2020); Harada  _et al._
(2009). The measurement of CAR is carried out with the start-stop coincidence
detection described in sec. IV.1. We used the setup in Fig. 1b with a single
visible SPAD at the output of the UC after removing the beams splitter. In
fact, the CAR does not involve the intra-beam correlations. As shown in Fig.
3, the coincidences occur with a temporal delay $\delta t$ = 0 ns. The other
peaks, spaced with the laser repetition period, are due to accidentals. Please
notice that the zero-delay peak includes also accidental coincidences.
Therefore, the CAR is evaluated as

$\textrm{CAR}=\frac{\textrm{coincidence counts}}{\textrm{accidental
counts}}=\frac{N^{raw}_{si}-N_{acc}}{N_{acc}},$ (9)

with $N^{raw}_{si}$ the total coincidence counts falling in the zero delay bin
and $N_{acc}$ the accidental counts, evaluated as the average over all the
accidental peaks. The true coincidences, also called as net coincidences, are
calculated as $N_{si}^{net}=N_{si}^{raw}-N_{acc}$. Depending on the $\Delta
t_{c}$ used, the ratio between coincidence and accidentals in the individual
bin changes, changing the CAR. In Fig. 4 we report the measured CAR and the
corresponding net coincidences as a function of the on-chip peak pump power.
To be noticed that the peak power in the plot is the power at the input of the
multimode waveguide after fiber-chip coupling losses, it is not $\bar{P}_{p}$.
We report the results with a coincidence window of 1.1 ns and of 2 ns. With
the 1.1 ns window the CAR is higher, with a maximum of 40.4(9) at 115 mW. At
this power the rate of net coincidences is 0.316(3) cps. The net coincidences
are almost the same for the two windows, demonstrating that with the larger
coincidence window we are mainly introducing noise rather than signal. CAR and
net coincidences have been also simulated starting from the parameters
calculated in sec. IV.1 and sec. III. They are reported as solid lines in the
figure and are calculated as Harada  _et al._ (2009)

$\textrm{CAR}=\frac{\bar{p}_{si}}{\bar{p}_{i}\,\bar{p}_{s}}=\frac{\xi\bar{P}_{p}^{2}\bar{\eta_{i}}\bar{\eta_{s}}}{\left(\xi\bar{P}_{p}^{2}\bar{\eta_{i}}+d_{i}\right)\,\left(\xi\bar{P}_{p}^{2}\bar{\eta_{s}}+d_{s}\right)},$
(10a)
$N_{si}^{net}=\xi\bar{P}_{p}^{2}\bar{\eta}_{i}\bar{\eta}_{s}\eta_{ND}R_{p}.$
(10b)

Simulated and experimental values of CAR are in agreement in the whole range
of pump power used. This agreement demonstrates that the main effects and
phenomena involved in the generation process have been properly considered and
modelled. The net coincidence rates are in agreement at low power, while at
higher power the nonlinear losses have been overestimated. A perfect agreement
would require a precise knowledge of all the nonlinear parameters of the
material.
The larger CAR here measured with respect to other works Rosenfeld  _et al._
(2020) demonstrates that the overall system, considering both the generation
and detection stages, is competitive with respect to solutions already
demonstrated on the silicon platform.

Figure 3:  Two-fold coincidences as a function of the delay $\delta t$ between idler (start) and signal (stop) detections. We collect the events with a coincidence window of 0.05 ns (blue). In post processing, we use a larger coincidence window, here 1.1 ns (orange), in order to take into account the majority of the coincidence events. The coincidence peak is the highest one, placed at $\delta t=0$ ns. The laser repetition period is clearly visible from the accidental peaks. In the inset, we focus on the zero-delay bin, comparing the coincidence peak shape with the post processing coincidence window.  Figure 4:  Measured CAR (circles) and net coincidence rates (triangles) with $\Delta t_{c}=1.1$ ns (orange) and $\Delta t_{c}=2$ ns (blue). The data are reported versus the on-chip peak pump power. The experimental points are compared with the simulated values for both the CAR (solid lines) and the net coincidence rates (dashed lines). With $\Delta t_{c}=1.1$ ns the CAR is remarkably higher with respect to the 2 ns bin, with only a limited reduction in the coincidence rate. The better performance obtained with the smaller $\Delta t_{c}$ is due to the lower noise integrated within the coincidence bin.  Table 1: Comparison with state of the art MIR heralded sources. Platform | Process | Generation probability | CAR | CAR | $\mathbf{g^{(2)}_{h}(0)}$ | $\eta_{I}$ | Reference
---|---|---|---|---|---|---|---
|  | (W-2) | max | @ $N^{net}_{si}\sim$ 1 Hz |  | ($\%$) |
Mg:PPLN | SPDC | - | 180 $\pm$ 50 | - | - | - | Prabhakar  _et al._ (2020)
SOI | intra-modal SFWM | 0.28 | 25.7 $\pm$ 1.1 | 25.7 $\pm$ 1.1 | - | 5 | Rosenfeld  _et al._ (2020)
SOI | inter-modal SFWM | 0.72 $\pm$ 0.10 | 40.4 $\pm$ 0.9 | 27.9 $\pm$ 0.5 | 0.23 $\pm$ 0.08 | 59 $\pm$ 5 | This work

###  IV.3 Heralded g${}^{(2)}_{h}$

To asses the single photon nature of the emission, we measured the heralded
$g^{(2)}$, that we indicate as $g^{(2)}_{h}$. Using the setup in Fig. 1b, we
tuned the delays in order to have the signal detection on one visible SPAD
coincident with the idler detection on the InGaAs SPAD. The coincidence
between these two detectors, with a coincidence window $\Delta t_{c}=$ 2 ns,
was used as the start trigger, while the detection from the remaining visibile
SPAD, that we will call "delayed signal", was used as the stop trigger. In
this way, we monitored the three-fold coincidences as a function of the delay
$\delta t$ between the start and stop events. At the same time, we measured
the two-fold coincidences between the idler and the delayed signal. We used a
coincidence window of 2 ns to monitor the three-fold coincidences. The
$g^{(2)}_{h}$ can be given as Signorini and Pavesi (2020)

$g^{(2)}_{h}(\delta t)=\frac{N_{12i}(\delta t)}{N_{1i}(0)N_{2i}(\delta
t)}N_{i},$ (11)

where $1,2,i$ label respectively the first signal detector, the second signal
detector (that is the delayed signal) and the idler detector. $N_{12i}$
corresponds to the three-fold coincidence counts, $N_{1i}$ and $N_{2i}$ are
the two-fold coincidence counts between the idler and the signal detectors,
and $N_{i}$ corresponds to the idler counts. We can normalize eq. (11) by
$N_{i}$ and $N_{1i}(0)$, such that

$g^{(2)}_{h}(\delta t)=\frac{N_{12i}(\delta t)}{\langle N_{12i}(\delta t\neq
0)\rangle}\frac{\langle N_{2i}(\delta t\neq 0)\rangle}{N_{2i}(\delta t)},$
(12)

with $\langle N_{12i}(\delta t\neq 0)\rangle$ and $\langle N_{2i}(\delta t\neq
0)\rangle$ the average of the three-folds and two-folds coincidences for
$\delta t$ different from zero. If the emission is truly at the single photon
level, $g^{(2)}_{h}(0)$ should be lower than 0.5Signorini and Pavesi (2020).
The measured $g^{(2)}_{h}(0)$ as a function of the on-chip peak pump power is
reported in Fig. 5. For an input power of 0.33 W we measured
$g^{(2)}_{h}(0)=0.23(8)$, demonstrating the single photon regime of the
source. The corresponding $g^{(2)}_{h}(\delta t)$, calculated as in eq. (12),
is reported in the inset of Fig. 5. We discarded the neighbouring bins of the
zero delay bin, affected by spurious coincidences due to photon emissions from
triggered silicon SPADs Kurtsiefer  _et al._ (2001).

Figure 5:  Comparison between the measured (blue points) and simulated (light
blue area) $g_{h}^{(2)}(0)$ as a function of the on-chip peak power. In the
inset is reported the measurement for the $g^{(2)}_{h}(\delta t)$ at an on-
chip peak power of 0.33 W. The bins adjacent to the zero-delayed one have been
removed due to the SPADs emitted photons.

To verify the goodness of the modeling introduced in sec. III, we used the
calculated $\xi$, $\bar{P}_{p}$, $\bar{\eta}_{i}$ and $\bar{\eta}_{s}$ in sec.
IV.1 to simulate the expected $g_{h}^{(2)}(0)$. Considering the general
formula for the heralded second order coherence, we can write

$g_{h}^{(2)}(0)=\frac{\bar{p}_{12i}\bar{p}_{i}}{\bar{p}_{1i}\bar{p}_{2i}},$
(13)

where $\bar{p}_{12i}$ is the probability per pulse of having a three-fold
coincidence. To model the experimental results, we have to consider all the
possible coincidence events that may involve signal and/or noise detections.
By considering all the possible events leading to a three-fold coincidence
(see Supplementary Material), we can rewrite $\bar{p}_{12i}$ as

$\displaystyle\bar{p}_{12i}=$
$\displaystyle\sum_{n=2}^{\infty}n^{2}(n-1)\wp(n)\,\bar{\eta}_{1}\bar{\eta}_{2}\bar{\eta}_{i}\eta_{ND}$
(14) $\displaystyle+$
$\displaystyle\sum_{n=1}^{\infty}n^{2}\wp(n)\,(\bar{\eta}_{1}d_{2}+d_{1}\bar{\eta}_{2})\bar{\eta}_{i}\eta_{ND}$
(15) $\displaystyle+$
$\displaystyle\frac{1}{2}\sum_{n=2}^{\infty}n(n-1)\wp(n)\,\bar{\eta}_{1}\bar{\eta}_{2}d_{i}\eta_{ND}$
(16) $\displaystyle+$
$\displaystyle\sum_{n=1}^{\infty}n\wp(n)\,\bar{\eta}_{1}d_{2}d_{i}\eta_{ND}$
(17) $\displaystyle+$
$\displaystyle\sum_{n=1}^{\infty}n\wp(n)\,d_{1}\bar{\eta}_{2}d_{i}\eta_{ND}$
(18) $\displaystyle+$
$\displaystyle\sum_{n=1}^{\infty}n\wp(n)d_{1}d_{2}\bar{\eta}_{i}\eta_{ND}$
(19) $\displaystyle+$ $\displaystyle\,d_{1}d_{2}d_{i}\eta_{ND},$ (20)

with $\wp(n)$ the photon number distribution. In eq. (14), $\bar{\eta}_{i}$ is
as in eq. (3c), while $\bar{\eta}_{1}$ and $\bar{\eta}_{2}$ have to take into
account also the effect of the beam splitter, thus, according to eq. (3c),
they can be written as

$\bar{\eta}_{1}=\bar{\eta}_{s}T^{2}_{BS}\eta_{BS},$ (21a)
$\bar{\eta}_{2}=\bar{\eta}_{s}R^{2}_{BS}\eta_{BS},$ (21b)

with $T_{BS}$ and $R_{BS}$ the transmission and reflection coefficients of the
beam splitter, $T^{2}_{BS}+R^{2}_{BS}=1$, and $\eta_{BS}$ modeling the losses
of the beam splitter. In our case, $T^{2}_{BS}=R^{2}_{BS}=0.5$ and
$\eta_{BS}=1$. In eqs. (21) we are assuming the same detection efficiency for
the two visible SPADs. Considering all the events leading to a two-fold
coincidence, we can rewrite $\bar{p}_{1i}$ and $\bar{p}_{2i}$ as

$\displaystyle\bar{p}_{ki}=$
$\displaystyle\sum_{n=1}^{\infty}n^{2}\wp(n)\bar{\eta}_{k}\bar{\eta}_{i}\eta_{ND}$
(22) $\displaystyle+$
$\displaystyle\sum^{\infty}_{n=1}n\wp(n)\left(\bar{\eta}_{k}d_{i}+d_{k}\bar{\eta}_{i}\right)\eta_{ND}$
(23) $\displaystyle+$ $\displaystyle\,d_{k}d_{i}\eta_{ND},$ (24)

with $k=1,2$. To be noticed that in eq. (14) and eq. (22) we are neglecting
events with more than one photon reaching the same detector, being unlikely
with the involved transmission efficiencies (i.e. $\bar{\eta}_{i}$,
$\bar{\eta}_{1}$ and $\bar{\eta}_{2}$ are all $\ll$1). We are also neglecting
events where photon detections and dark count detections occur simultaneously
on the same detector. The photon number distribution of a squeezed source
ranges between a poissonian (infinite modes emission) and a thermal (single
mode emission) distribution Takesue and Shimizu (2010); Signorini and Pavesi
(2020). We solved eq. 14 and eq. 22 for the poissonian emission,

$\wp(n)=\frac{\mu^{n}}{n!}\textrm{e}^{-\mu},$ (25)

and for the thermal emission,

$\wp(n)=\frac{\mu^{n}}{(1+\mu)^{n+1}},$ (26)

where $\mu$ is the average number of pair per pulse. Eqs. 25 and 26 define a
lower and an upper boundary for $g^{(2)}_{h,sim}$. In computing $g^{(2)}_{h}$
we calculated $\mu$ as $\mu=\xi\bar{P}_{p}^{2}$ and we measured the noise
affecting the three channels. $d_{i}$ is the same of the CAR measurements,
$d_{1}=2.30\times 10^{-6}$ and $d_{2}=2.32\times 10^{-6}$. We simulated an
area for the expected value of the $g^{(2)}_{h}(0)$, that is upper bounded by
the thermal case and lower bounded by the poissonian case. The simulation is
reported in Fig. 5. The measured $g^{(2)}_{h}$ is compatible with the
simulated values, confirming the reliability of the modeling. We want to
stress that in this case we are not performing a fit of the measured
$g^{(2)}_{h}$ and that the experiment and the simulation are completely
independent. The experimental points in Fig. 5 are closer to the upper bound
rather than to the lower one, suggesting an emission statistics closer to the
thermal one. This is compatible with the unheralded $g^{(2)}$ of the signal
beam Signorini and Pavesi (2020), measured in Fig. 6 as a function of the pump
power. The unheralded $g^{(2)}$ results to be 1.67(2) at a power of 1.08 W,
compatible with the simulated value of 1.66 (dashed line) calculated from the
simulated joint spectral intensity (JSI) Signorini and Pavesi (2020); Borghi
(2020). The measured $g^{(2)}$ demonstrates that the source is closer to a
thermal emission, justifying the experimental $g^{(2)}_{h}$. In Fig. 6 we also
report the simulated values for a source whose statistics is between the
thermal (upper bound) and the poissonian one (lower bound). At low powers, the
dark counts dominate, and in both cases the $g^{(2)}$ goes to 1. At high
powers, the $g^{(2)}$ asymptotically increases to its actual value. In this
way, we explain the power dependent behaviour of the experimental data.
Further details about the measurement and simulation of $g^{(2)}$ are reported
in Supplementary materials.

Figure 6:  The measured unheralded $g^{(2)}(0)$ (orange dots) is reported as a
function of the on-chip peak power. We report in the inset the simulated JSI,
from which we calculated the expected $g^{(2)}$ (dashed black line), that is
compatible with the experiment. The measured points fall within the simulated
values (light orange area), upper bounded by a source with thermal emission
statistics and lower bounded by a source with poissonian emission statistics
(constant $g^{(2)}=1$).

##  V Conclusions

In this work, we demonstrated a heralded single photon source beyond 2 $\mu$m
based on inter-modal SFWM on a silicon chip. This source has two main
peculiarities: the discrete band generation and the large detuning between the
signal and idler photons. The discrete band generation removes the need for
tight filtering to select idler and signal wavelengths, and the generated
photons experience a higher transmission with respect to standard continuous
band sources, witnessed by the high experimental $\eta_{I}=59(5)\,\%$. The
large detuning has two advantages: on one side, it enables an easier pump and
nonlinear noise rejection; on the other side, it allows to generate the herald
photon in the NIR, benefiting of an efficient detection technology. As a last
advantage, this heralded source based on inter-modal SFWM requires a common
C-band pump laser, easier to be integrated and operated on a silicon chip. We
performed a complete characterization of the source. We demonstrated the sub-
poissonian statistics of the source by measuring $g^{(2)}_{h}(0)=0.23(8)$. We
characterized the CAR, finding a maximum value of 40.4(9), and the generation
probability per pulse, with a measured value of 0.72(10) W-2. These
performances are competitive with other reported silicon sources of MIR
photons (Table 1) demonstrating the promising perspectives of inter-modal SFWM
for bright and efficient sources of correlated photons beyond 2 $\upmu$m. The
source can be significantly improved by reducing the propagation losses and
optimizing the matching between the signal and upconverter bandwidths. With
this work we demonstrate a new approach to MIR quantum photonics, providing a
high quality source of quantum light beyond 2 $\mu$m without the need of MIR
technologies. This result paves the way towards low cost, efficient and
integrated solutions for quantum photonics beyond 2 $\upmu$m, offering new
opportunities to the developing field of MIR photonics.

## SUPPLEMENTARY MATERIAL

See supplementary material for further details about the experimental setup,
the measurements and the theoretical calculations.

###### Acknowledgements.

This work was partially supported by grants from Q@TN provided by the
Provincia Autonoma di Trento. The authors acknowledge HC Photonics, which
fabricated the PPLN crystals used for the upconversion system. S.S. wants to
thank Dr. Massimo Borghi, for fruitful discussions and precious suggestions,
and Mr. Davide Rizzotti for its careful revision of the manuscript.

## DATA AVAILABILITY

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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