#  AI-Generated Network Design: A Diffusion Model-based Learning Approach

Yudong Huang, Minrui Xu, Xinyuan Zhang, Dusit Niyato, ,
Zehui Xiong, Shuo Wang, Tao Huang,
Y. Huang and X. Zhang are with the State Key Laboratory of Networking and
Switching Technology, BUPT, Beijing, 100876, P.R. China (e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>M. Xu and D. Niyato are with
the School of Computer Science and Engineering, Nanyang Technological
University, Singapore (e-mail<EMAIL_ADDRESS>dniyato@ntu.edu.sg). Z.
Xiong is with Information Systems Technology and Design (ISTD) Pillar,
Singapore University of Technology and Design, Singapore (email:
zehui_xiong@sutd.edu.sg). S. Wang, and T. Huang are with the State Key
Laboratory of Networking and Switching Technology, BUPT, Beijing, 100876, P.R.
China. They are also with the Purple Mountain Laboratories, Nanjing, 211111,
P.R. China (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The future networks pose intense demands for intelligent and customized
designs to cope with the surging network scale, dynamically time-varying
environments, diverse user requirements, and complicated manual configuration.
However, traditional rule-based solutions heavily rely on human efforts and
expertise, while data-driven intelligent algorithms still lack
interpretability and generalization. In this paper, we propose the AIGN (AI-
Generated Network), a novel intention-driven paradigm for network design,
which allows operators to quickly generate a variety of customized network
solutions and achieve expert-free problem optimization. Driven by the
diffusion model-based learning approach, AIGN has great potential to learn the
reward-maximizing trajectories, automatically satisfy multiple constraints,
adapt to different objectives and scenarios, or even intelligently create
novel designs and mechanisms unseen in existing network environments. Finally,
we conduct a use case to demonstrate that AIGN can effectively guide the
design of transmit power allocation in digital twin-based access networks.

###### Index Terms:

Future Networks, Generative Artificial Intelligence, Diffusion Model,
Reinforcement Learning.

##  I Introduction

The future networks have attracted much attention due to the unprecedented
capabilities of bearing the Metaverse, satellite-terrestrial integrated
networks, connected vehicles, and other emerging applications[1]. However,
with the rapid growth in network scale and user demands, recent years have
witnessed dramatic increases in the complexity and dynamics of network
designs. For instance, the cross-domain network planning requires a lot of
expert knowledge and experience, and layered protocol designs heavily rely on
human efforts. Moreover, in large-scale time-varying scenarios, it is
intractable to construct a complete system model with appropriate
constraints[2]. Faced with enormous operation and maintenance costs, the
future networks pose intense demands for intelligent, automated, and
customized network designs.

Recently, artificial intelligence-generated content (AIGC) provides an
exciting perspective for intention-driven network design. Based on generative
AI techniques and applications, such as diffusion models[3] and ChatGPT[4],
AIGC exhibits outstanding inference capabilities in text processing, code
writing, and text-to-image, where humans interact with intelligent machines
through natural language to efficiently complete design tasks. In particular,
diffusion models have been successfully leveraged to optimize the decision-
making in reinforcement learning (RL), where we can encode the design
intention into the conditioning information and explicitly guide the
generation of realistic trajectories (i.e., state-action-reward sequences that
represents the decision-making process) with flexible combination of multiple
constraints. Although datasets collected from network systems are not
normalized data like image pixels, many network optimization problems are
viable to model as the decision-making process of RL.

On the other hand, previous studies mainly focused on solving network
optimization problems by using intelligent algorithms, such as deep
reinforcement learning (DRL)[5] and federated learning, while the lack of
interpretability and generalization makes intelligence-based networking
systems prohibitive to deploy in practice. Digital twins are expected to
narrow this gap by verifying the algorithms in virtual worlds that map from
physical infrastructure. With the advances of generative AI, RL, and digital
twins, we envision a paradigm shift from rule-based manual design towards
intention-driven intelligent design for future networks.

In this paper, we propose the AIGN (AI-Generated Network), a novel network
design paradigm that can automatically generate customized solutions to adapt
to dynamically time-varying environments. Unlike traditional optimization
methods, AIGN does not require building a complete network system model and
knowing all constraints prior. Moreover, by merging the generative ability of
diffusion model with the RL, AIGN is promising to achieve intelligent
decision-making in many network design scenarios, such as satellite-
terrestrial network planning and mobile ubiquitous computing.

The main contributions of this article are as follows.

  * •

We present the novel AIGN framework and analyze its appealing properties,
including scalability for large-scale optimization, flexibility for multi-
constraint combination, and interpretability for intent-driven configuration.

  * •

To enable the AIGN, we propose a diffusion model-based learning approach,
where the critical techniques for generating novel network designs are
detailed.

  * •

We conduct a proof-of-concept simulation in digital twin-based access
networks. The results show that AIGN can effectively guide the design of
transmit power allocation by learning the reward-maximizing trajectories.

  * •

We discuss the challenges and potential research directions brought by the
AIGN, such as high-quality datasets, performance metrics, and interaction
platforms.

The rest of the article is organized as follows. We commence with the AIGC and
network design issues, clarify our motivation, and give the design principles.
Next, we present the AIGN framework. Then the key functional components of
diffusion model-based learning approach are detailed. Following that, we give
the case study and analyze the challenges. Finally, we draw the main
conclusions.

Figure 1: Comparing traditional network expert design with AIGN. (a) Rule-
based methods require manually building the system model, and pursue the
optimal results that heavily depend on expert knowledge. (b) AIGN prioritizes
the decision-making process with high rewards, and memorizes these
trajectories as experiences. This allows AIGN to automatically generate
various customized designs that are aligned with the intention.

##  II AI-Generated Network

This section overviews the generative AI techniques, and their applications in
the network design, emphasizing the motivation and design principles.

###  II-A AI-Generated Content

AIGC refers to the use of generative AI techniques (e.g., Autoregressive
Models, Variational Auto-Encoders, Generative Adversarial Networks, Flow-based
Models, and diffusion models[3]) to automatically produce high-quality
customized content. Recently, diffusion models have shown groundbreaking
success in the task of text-to-image and decision-making. Thanks to the great
potential in creating new patterns, diffusion models have also been applied to
generate the molecular conformer and protein structure, which unveils a new
revolution in exploiting the generative AI for science. This paper focuses on
the RL-based decision-making problems in network fields. Since the diffusion
models have been proven to combine with RL[6], we adopt the diffusion models
as the main representative of generative techniques, while other generative
models may also be utilized.

The technique of generative AI lies in that: Given observed samples $x$ from a
distribution of interest (e.g., an image), the generative AI learns to model
its true data distribution $p(x)$. Once learned, it can generate new samples
from the approximate model at will. Then, by explicitly controlling the data
generated through conditioning information $y$ (e.g., a text), we can learn
the conditional distribution $p(x\mid y)$ (e.g., a new image in response to
text prompts)[7].

###  II-B Network Design Issues

In the past decades, network protocols have laid the foundation of the
Internet, while optimization algorithms and configuration parameters greatly
affect network performance. As shown in Fig. 1(a), the current design approach
for large-scale networks mainly includes three aspects: (1) The layered
protocol is used to simplify the design. Each layer provides a set of
communication/networking functions. (2) The network domain is adopted by
partitioning networks into regions to improve scalability. (3) Human experts
empirically design large-scale optimization problems by specifying the system
models and objective functions with multiple constraints, especially across
layers and domains. Based on the rule and model, the experts can form a search
space for the problem, and then use combinatorial optimization (such as greedy
iterative strategy) or intelligent algorithms to find the optimal solution
(e.g., maximize throughput, minimize power consumption, and reduce hardware
device and its deployment expense).

Nevertheless, future networks not only require full-coverage connections in
the air, space, ground, and sea, but also need to provide real-time and high-
bandwidth services for ubiquitously connected users. As the network size and
number of users proliferate, it is difficult to manually formulate the network
problem from a global perspective and directly construct a complete system
model with all constraints. Moreover, the traditional rule-based optimal
solutions are prone to suffer from dynamic time-varying environments, where
customized solutions that quickly adapt to a variety of realistic situations
are more essential than a rigid optimal solution. To be specific, we present
four typical scenarios to analyze large-scale network design issues.

Use case 1: VR/AR video transmission. As virtual reality (VR)/augmented
reality (AR) and high-definition live stream are applied to education, medical
care, entertainment, and business, video traffic occupies most of the current
Internet bandwidth. To improve the Quality of Experience (QoE) metrics,
content providers struggle with constructing high-quality video transmission
networks by systematically considering bitrate selection, cache placement,
routing, and other optimization problems. However, peak demands (e.g.,
concerts and sport events) easily cause network congestion and video
interruption, challenging the flexibility and scalability of manual-configured
transmission systems, further leading to significant revenue losses for
content providers.

Use case 2: Satellite-terrestrial network planning. The satellite-terrestrial
backbone network interconnects hundreds of regions and serves hundreds of
terabits of traffic at all the time. With the rapidly growing bandwidth demand
of emerging applications, such as big data and Metaverse, the operators spend
billions of dollars every year to plan and upgrade backbone networks. Given
traffic demand forecast and requirements (e.g., failure redundancy for
reliability), network planning refers to continuously making cross-layer
decisions, including both the optical layer (e.g., fiber path to build, fibers
to turn on, spectrum to use) and the IP layer (e.g., link capacity to add and
routers to procure), to minimize the network infrastructure expenditure.
Moreover, satellite clusters have dynamic topologies and unstable time-varying
orbital connections, which further exacerbate the challenges of joint
satellite-terrestrial optimization.

Use case 3: MAC protocols design. The wireless medium access control (MAC)
protocols are mainly responsible for the channel access of user equipments.
The protocol design involves complex interaction processes and configuration
parameters, which must be tailed to specific purposes and scenarios, such as
improving throughput, reducing power consumption, and guaranteeing fairness.
Especially under the heterogeneous network (HetNet) deployment scenarios, each
wireless access network, e.g., 5G-NR, Wi-Fi, Bluetooth, Zigbee, and even
satellite access network, has its own protocols and properties, such as
capacity, delay, coverage, security, access technology, power consumption, and
cost. The HetNet poses great challenges for efficiently managing and
coordinating multiple types of wireless access technologies. Considering the
even more complex and diversified settings in future networks, AI-generated
protocol design is essential to relieving humans from these burdensome
undertakings[8].

Use case 4: Mobile ubiquitous computing. Computing resource allocation plays a
particularly important role in ubiquitous connection scenarios, such as
Industrial Internet of Things, vehicles, drones, and satellites. Due to the
limited on-chip computing resources of local devices, computing-intensive
tasks usually need to be offloaded to edge nodes or cloud nodes. The
offloading decisions are expected to be globally optimal under complicated
system models that include computing, communication, power, mobility,
distance, and delay constraints. As concepts such as fog computing, device-to-
device computing, in-network computing, and serverless computing continue to
be proposed, it is quite intractable for network experts to give a complete
system model or a feasible solution in practice.

Figure 2: AIGN framework with the diffusion model-based learning approach.
After the operator inputs the intentions (e.g., the network capacity
planning), AIGN samples from high-reward trajectories and infers the decision-
making process from the initial state to the target solution with customized
policy $\pi^{*}$.

###  II-C Motivation

AI techniques profoundly change the way of design and deployment for future
networks. On the one hand, recent studies on AI-assisted network design
leverage various Neural Network structures, such as Deep Neural Network,
Recurrent Neural Network, and Graph Neural Network, to learn the nonlinear
mapping between problem parameters and optimal solutions. Especially, DRL has
been widely used in various communication and network scenarios due to the
advantages of label-free training and long-term optimization. On the other
hand, there are three significant limitations. First, it still requires expert
knowledge to combine multiple network problems to a large-scale optimization
problem and empirically tune numerous learning parameters. Second, the
underlying Neural Networks remain blackboxes where the performance stays
uninterpretable for network operators. The lack of interpretability [9] makes
intelligence-based networking systems prohibitive to deploy in practice.
Third, a model well trained in one environment may perform much worse than
traditional rule-based schemes in another new environment. The lack of
generalization means endless model design and parameter tuning, which greatly
limits the development of intelligence in the network.

To deal with this dilemma, this paper targets to explore AI-generated methods
to assist network design tasks. Specifically, we propose a new paradigm for
the network design, named AIGN. As shown in Fig. 1(b), AIGN is a generative
network expert system that allows operators to specify customized properties
of the network solution and automatically create network design details
without requiring the exhaustive system model and constraints. Driven by the
holistic data (e.g., channel state, traffic matrix, operation logs, and
expertise) and design intentions (e.g., constraints and objectives), AIGN is
able to learn new problems from basic optimization problems, automatically
combine multiple network subsystems, or even create novel designs and
mechanisms unseen in existing network environments, which enhances the
interpretability and generalization capabilities of intelligent networks.

###  II-D Design Principles

Generally, large-scale optimization problems consist of multiple subproblems.
Specific subproblems with simple constraints are usually well-studied and RL
is viable to generate labeled trajectory datasets for particular network
environments. The authors in [9] decompose networking systems into local
systems and global systems, and use the decision tree method to interpret a
single RL agent’s decision-making process in the local system. In [10], the
curriculum learning is introduced to improve the generalization of RL. They
use rule-based algorithms to generate training curricula. By iteratively
training the RL agent in environments where it performs worse than rule-based
baselines, the RL agent achieves asymptotic performance in multiple
environments (problems). In [11], the wireless MAC protocol design is
decoupled into a set of building blocks, each of which is trained with an RL
agent. Multiple agents share their information with a global agent for
clustered decision-making. Since the future networks are composed of a large
number of heterogeneous and decentralized network entities, multi-agent RL
allows each network entity to learn its optimal policy by observing not only
the environments but also the policies of other entities, which can
significantly improve the learning efficiency. Thus, our first observation is
that large-scale network optimization problems can be transformed into a data-
driven paradigm of multi-agent RL[12].

Then, the critical challenge is to enable RL agents to intelligently
collaborate and learn from each other, so that they can work together to
generate new network designs. One of the most powerful solutions is generative
AI. The authors in [13] prove the feasibility of using Generative Adversarial
Networks (GANs) to automatically generate synthetic packet header traces for
networking tasks (e.g., telemetry and anomaly detection). However, these
generated traces are purely packet records (such as IP address, port number,
and the number of bytes) rather than collected from state-action-reward sets
of RL environments; thus they cannot be directly utilized for decision-based
problem optimization. Recently, a Decision Diffuser[6] framework based on the
diffusion model has been proposed in the robotic field. Given several RL
trajectories collected from simple constraint environments, the Decision
Diffuser generates not only typical behaviors satisfying individual
constraints but also new behaviors that maximize rewards by flexibly combining
multiple constraints at test time. Thus, our second observation is that by
introducing the inference ability of generative models to the collaboration
among multi-agent RL, it is expected to achieve expert-free problem
optimization and automatic generation of new network designs.

##  III AIGN Design

Motivated by the above issues and trends, we design the AIGN framework with
the diffusion model-based learning approach. Several appealing properties of
AIGN are derived, and the key technical components are detailed.

###  III-A AIGN Framework

As shown in Fig. 2, the workflow of AIGN is composed of the following steps.
(1) Collect expert datasets from all past online RL experiments. Data
collected from network systems is not normalized data like image pixels, which
makes it extremely intractable to unify large-scale network datasets.
Fortunately, many network optimization problems can be modeled as the
decision-making process of RL, where the trajectories are stored and utilized
in a standardized manner. In addition, operation logs and expertise can be
encoded into trajectory labels to further improve the quality of datasets. (2)
The operator inputs network design intentions (e.g., I want a forty-node IP
and optical topology with traffic matrix $M$ and reliability $\lambda$. To
minimize the costs, what will the network look like?). The Large Language
Model (LLM), e.g., ChatGPT, can be utilized to execute the intention
compilation and generate explanations. The intentions are abstracted into
optimization objectives, constraints, and specific requirements. Then, the
conditional datasets corresponding to intentions are sampled to train an
offline RL. With the ability of in-context learning, AIGN infers the design
details step by step through dialogue with operators. (3) Finally, the
operator obtains a variety of customized network solutions by just clicking a
button. The solutions are evaluated in the digital twin-based virtual
counterpart to reduce the cost and risk of deployment. As the generative
network models continuously evolve, AIGN has several appealing properties as
follows.

Scalability for large-scale optimization. The future networks must become
highly scalable to support a large number of heterogeneous networking
scenarios. In practice, a single local network with hundreds of nodes and
links may be translated into an ILP (Integer Linear Programming) problem with
millions of variables and constraints[5]. To achieve joint optimization of
heterogeneous networks, even highly-skilled experts require much effort to
find an actionable solution. By continuously learning from various network
subsystems, AIGN intelligently synthesizes large-scale optimization problems,
generalizes well to new environments, and pursues long-term reward
maximization, while traditional hand-tuned heuristics are prone to local
optima.

Flexibility for multi-constraint combination. In the process of network
design, we concern not only about the solution outcomes, but also about the
impact of each part on performance and costs. For instance, the wireless MAC
protocol design may require removing the retransmission mechanism to reduce
the power consumption, or adding an adaptive sending function to improve the
average channel throughput. AIGN provides the service to simultaneously
compare alternatives that are optimized for multiple objectives and
constraints, avoiding the tedious trial-and-error modeling process.
Consequently, AIGN will greatly accelerate the innovation of network protocols
and mechanisms.

Interpretability for intention-driven configuration. After the network has
been deployed, operators are often overwhelmed with failures and alarm events.
Due to the lack of interpretable intelligence, the existing network management
heavily relies on human analysis and manual configuration. By synchronously
sensing the physical network environments with digital twins and learning from
historical operating experience, AIGN may intelligently give appropriate
operational suggestions, such as increasing link capacity or migrating the
path of heavy-loaded applications when network congestion occurs. This
transparency of AIGN allows network operators to understand how the system
works and make informed adjustments with fewer potential errors. Further, AIGN
is expected to realize the intention-driven networks and automatic network
management.

###  III-B Diffusion Model-based Learning Approach

We make a concrete step towards the AIGN by proposing a diffusion model-based
learning approach. The approach is divided into two stages. The first stage is
to explore various environments and generate labeled trajectories by online RL
agents, and the second stage is to generate new designs by the offline RL with
the diffusion model. Note that AIGN is not limited to the diffusion model, and
other generative models could also be applied to the AIGN. Next, we present
the key technical details and analyze the theory behind AIGN.

Stage 1: Generate labeled trajectories by RL agents. Many communication and
network problems[14] can be formulated as a Markov Decision Process (MDP),
where an online RL agent is trained to make decisions based on trial-and-error
interactions with an environment. At each step, the agent observes the current
state $s_{t}$ of the environment and selects an action $a_{t}$ to take, which
may lead to a new state $s_{t+1}$ and a corresponding reward $r_{t}$. The goal
of the agent is to learn a policy $\pi$ that maximizes the expected cumulative
reward over a sequence of state-action-reward trajectory $\tau$. By encoding
operation logs and expertise (e.g., returns, specific constraints, and
optimization objectives) into trajectories, we can obtain the labeled
trajectories that are analogous to well-marked standard image datasets, where
generative AI techniques have already found success.

Stage 2: Generate new designs by diffusion model. The diffusion model is based
on a Markov chain that simulates a forward noising process $q(x_{k+1}|x_{k})$,
which gradually adds noise to the orginal data $x_{0}$ until the data $x_{k}$
becomes a Gaussian distribution $x_{k}\sim\mathcal{N}(0,I)$. Then, by sampling
the data $x_{k-1}$ with the conditioning information $y$, the reverse
denoising process $p_{\theta}(x_{k-1}|x_{k},y)$ iteratively removes the noise
and finally learns the new conditional data $\tilde{x_{0}}$[3], i.e., the
target data that satisfies user requirements. The temporal U-Net[6], a neural
network consisting of repeated convolutional residual blocks, can be trained
to predict the perturbed noise $\epsilon_{\theta}$ that needs to be removed at
each denoising step.

Treating labeled trajectories $\tau$ in online RL as the original data
distributions, i.e., $x_{0}({\tau})$, the offline RL with the diffusion model
has unprecedented abilities to handle complex high-dimensional state spaces
and learn new customized policy $\pi^{*}$ that closely match the expected
network solutions. For instance, we can explicitly define the conditioning
variables $\left\\{y^{i}(\tau)\right\\}_{i=1}^{n}$ as a one-hot vector that
represents the returns or satisfied constraints, where $i$ is the indice and
$n$ is the number of variables. As shown in Fig. 2, the high-reward
trajectories under multiple combinations of IP-layer, optical-layer, and
reliability constraints are sampled as experiences. After conditioning on the
IP ($y_{1}$), optical ($y_{2}$), and reliability ($y_{3}$), the offline RL
with the diffusion model infers the decision-making process from the initial
state to the target solution that satisfies multiple constraints at the same
time.

##  IV Case Study

In order to evaluate the performance of proposed diffusion model-based
learning approach, we create a proof-of-concept prototype under the wireless
access scenarios. Simulation results exhibit that AIGN can guide the power
allocation.

Figure 3: Simulation results for accessing two users. The initial power value is set to (0, 0), and the proposed approach quickly generates the optimal power allocation scheme of (28, 31). TABLE I: Multi-user access solutions. The reward values are calculated based on the sum-data utility function. Best reward values | 20 Users | 50 Users | 80 Users
---|---|---|---
Soft actor-critic | 254.04 | 488.90 | 632.89
Batch-constrained deep Q-learning | 254.16 | 489.08 | 632.59
Diffusion model-based learning | 254.15 | 489.02 | 633.02

Simulation Setup: The uplink transmission power allocation for sum-data
utility maximization with interference in multi-access base stations is a
classic NP-hard problem in wireless networks. We build a digital twin
simulation platform to design the optimal discrete power selection scheme
($p_{i,j}$) under time-varying environments with different channel gains
$h_{i,j}$, noise power $\sigma^{2}$, the indices of base stations $j$, and the
indices of users $i$. The optimization objective is set to maximize the
marginal utility, which is a logarithmic revenue of rate $R_{i,j}$ subtracting
the linear power cost $C_{i,j}$. The rate $R_{i,j}$ is computed according to
the signal-to-interference-plus-noise ratio (SINR) and Shannon formula. The
power cost $C_{i,j}$ is proportional to the power $p_{i,j}$.

In stage 1, we train online RL agents to explore various environments with the
soft actor-critic (SAC) algorithm. The SAC is a variant of the Actor-Critic
algorithm that uses the maximum entropy framework to encourage exploration
during training, where the actor learns a stochastic policy that maximizes the
expected sum of rewards over time and the critic estimates the Q-values of
state-action pairs. These training trajectories are stored in the experience
buffer and labeled with information such as returns, constraints, and
environmental parameters. In stage 2, we feed about one hundred thousand
labeled trajectories to the diffusion model and train the reverse denoising
process of U-Net[6] with maximum-likelihood estimation.

Simulation Results: To facilitate understanding and visualization, but without
loss of generality, we firstly set the number of base stations to 1 and the
number of users to 2. As shown in Fig. 3, after just inputting the channel
gain and noise parameters, the diffusion model quickly generates the reward-
maximizing trajectory and finds the best power allocation scheme $(28,31)$,
which proves that AIGN can achieve expert-free problem optimization. Then, by
further conditioning on the user numbers of 20, 50, and 80, AIGN samples from
the corresponding conditional datasets and give the transmit power allocation
solutions that relate to the required user numbers. As shown in Table I, the
scheme rewards of the diffusion model-based learning approach are 254.15,
489.02, and 633.02, which are very close to the online SAC and offline batch-
constrained deep Q-learning (BCQ) algorithms. As shown in Table II[6][15],
both the proposed approach and offline BCQ algorithms learn directly from the
datasets, which have faster convergence speed than online RL. Offline BCQ is
less time-consuming but the performance highly depends on the quality of data
distribution. If the dataset is biased or incomplete, the agent will learn
suboptimal policies or even fail to learn. Our approach also requires large
and diverse datasets to avoid overfitting, and the conditioning generation
ability of the diffusion model helps the agent to overcome the dataset bias
and adapt to conditional environments. Due to the limit of space, we leave
more simulations based on AIGN as future work.

TABLE II: Comparison of SAC, BCQ and the proposed diffusion model-based learning approach. Methods | SAC | BCQ | Proposed
---|---|---|---
Category | Online RL | Offline RL | Offline RL
Features |  Interact with environments |  Learn from datasets |  Learn from datasets
Convergence | Slow and unstable | Fast | Fast
Pros |  Suitable for changing environments |  Less time- consuming |  Adapt to conditional environments
Cons |  Time- consuming |  Dataset bias and overfitting |  Require more tuning

##  V Challenges and Future Prospects

AIGN paves the way for intelligent and automated network design, while leaving
some challenges to be discussed. In this section, we analyze the challenges
brought by the AIGN paradigm and highlight the potential research directions.

Realistic sources for high-quality datasets: Realistic data sources collected
from physical infrastructure (e.g., ground-truth channel states and traffic
matrix) are the best datasets to guarantee the fidelity and accuracy of
learning trajectories. Besides, synthetic datasets produced by simulators,
expertise, or other mixed methods are promising alternatives to overcome the
data access barrier and mitigate the privacy concerns. The dataset
characteristics, which heavily influence the performance of AIGN, still remain
to be studied. To evaluate the quality of different datasets, the Trajectory
Quality (TQ) defined by the average dataset return and the State-Action
Coverage (SACo) calculated by the number of unique state-action have been
proposed[15]. Moreover, reinforcement learning from human feedback (RLHF) is a
powerful method to further improve the quality of trajectory datasets.

Generic model with performance metrics: Although replacing network experts is
the ultimate goal of AIGN, training a generic AIGN model to cope with various
network design scenarios is quite challenging. Apart from the convergence
speed and average reward, metrics to measure the design performance of AIGN
are still lacking. For example, the degree of task completion and the
percentage of constraint satisfaction may be added to test the inference
ability of AIGN. Traditional solvers (e.g., CPLEX and Gurobi) can also be
leveraged to verify and further optimize AIGN-designed solutions. Moreover,
developing ChatGPT-based plug-and-play tools for AIGN to aid network design
and management is a promising direction.

Digital twins as interaction platforms: Since directly deploying the AIGN to
the real network environments can be expensive and very risky, combining AIGN
with the digital twin is a potential research direction. The digital twin
provides a real-time interaction platform between the physical infrastructure
and the virtual counterpart, where we can continuously verify the effect of
AIGN and synchronize the final designed solution with good performance to the
physical entity. Moreover, with advanced analytics and modeling techniques,
digital twin can also help reduce costs, improve efficiency, and accelerate
the network development process, such as predictive maintenance, performance
optimization, training, and testing. Additionally, unlike two-dimensional
images, the exploration space of network problems grows exponentially with the
number of users. Digital twin methods for visualizing network designs in high-
dimensional space still remain to be investigated.

##  VI Conclusions

This article has merged the generative AI techniques and RL to enable AIGN,
thereby earthing the potential of a data-driven paradigm for expert-free
network optimization and promoting the value of intelligent decision-making in
intent-driven network design. First, the AIGN framework with several appealing
properties has been presented, and a novel diffusion model-based learning
approach is proposed. Then, a proof-of-concept prototype has been conducted to
prove that AIGN can guide the network design of power allocation under
wireless access scenarios. Finally, the challenges of the proposed AIGN are
discussed. Since intelligence will play a defining role in the development of
future networks, we hope AIGN can inspire follow-up research.

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Yudong Huang received his B.S. degree from BUPT, China, in 2019. He is
currently working toward his Ph.D. degree in BUPT. He is a visiting Ph.D.
student with the School of Computer Science and Engineering, Nanyang
Technological University, Singapore, from 2022. His current research interests
include deterministic networks and network intelligence.
---
Minrui Xu [S’23] received the B.S. degree from Sun Yat-Sen University,
Guangzhou, China, in 2021. He is currently working toward the Ph.D. degree in
the School of Computer Science and Engineering, Nanyang Technological
University, Singapore. His research interests mainly focus on Metaverse,
quantum information technologies, and deep reinforcement learning.
---
Xinyuan Zhang received her B.S. degree from BUPT, China, in 2019. She is
currently working toward her Ph.D. degree in BUPT. She is a visiting Ph.D.
student with the ISTD Pillar, Singapore University of Technology and Design,
Singapore, from 2022. Her current research interests include satellite-
terrestrial integrated networks and edge intelligence.
---
Dusit Niyato [M’09, SM’15, F’17] is currently a professor in the School of
Computer Science and Engineering, Nanyang Technological University, Singapore.
He received the B.Eng. degree from KMITL, Thailand in 1999 and Ph.D. in
electrical and computer engineering from the University of Manitoba, Canada in
2008. His current research interests include generative AI and network
intelligence.
---
Zehui Xiong [M’20] is an Assistant Professor at Singapore University of
Technology and Design. Prior to that, he was a researcher with Alibaba-NTU
Joint Research Institute, Singapore. He received the Ph.D. degree in Computer
Science and Engineering at Nanyang Technological University, Singapore. His
research interests include wireless communications and edge intelligence.
---
Shuo Wang received his B.S. degree in communication engineering from Zhengzhou
University, China, in 2013 and his Ph.D. degree from Beijing University of
Posts and Telecommunications in 2018. He is currently an assistant professor
at BUPT. His current research interests include deterministic networks and
network intelligence.
---
Tao Huang received the B.S degree from Nankai University, Tianjin, China, in
2002, the M.S. and Ph.D. degrees in communication and information system from
the BUPT, China, in 2004 and 2007, respectively. He is currently a Professor
with the BUPT. His current research interests include network architecture and
network intelligence.
---


# Continuous Variable Quantum Algorithms: an Introduction

Samantha Buck Email<EMAIL_ADDRESS>University of Guelph, 50 Stone Rd E,
Guelph, ON. Canada  Robin Coleman Email<EMAIL_ADDRESS>University of
Guelph, 50 Stone Rd E, Guelph, ON. Canada  Hayk Sargsyan Email:
<EMAIL_ADDRESS>Department of Physics, Yerevan State University, 1 Alex
Manoogian, 0025 Yerevan, Armenia

Quantum computing is usually associated with discrete quantum states and
physical quantities possessing discrete eigenvalue spectrum. However, quantum
computing in general is any computation accomplished by the exploitation of
quantum properties of physical quantities, discrete or otherwise. It has been
shown that physical quantities with continuous eigenvalue spectrum can be used
for quantum computing as well. Currently, continuous variable quantum
computing is a rapidly developing field both theoretically and experimentally.
In this pedagogical introduction we present the basic theoretical concepts
behind it and the tools for algorithm development. The paper targets readers
with discrete quantum computing background, who are new to continuous variable
quantum computing.

###### Contents

  1. Introduction
  2. 1 Quantum Continuous Variables
    1. 1.1 General Concepts
    2. 1.2 Eigenvalues and eigenfunctions
    3. 1.3 Bra-ket notation
  3. 2 Hilbert Space
  4. 3 Quantum Harmonic Oscillator
  5. 4 Continuous Variable Quantum Algorithms
    1. 4.1 Grover’s algorithm.
    2. 4.2 The Deutsch–Jozsa algorithm
  6. 5 Summary
  7. A Uncertainty relations
  8. B Dirac’s $\delta$ function

## Introduction

Nowadays, one usually encounters quantum mechanics either because they are a
physics (or related) student, or they got interested in quantum computing. In
any case it’s extremely easy to be carried away by the beauty of discreteness
all-around quantum mechanics and forget about continuous quantum phenomena.
There is a clear modern trend in transforming the curricula of quantum
mechanics into discrete quantum computing biased ones, where discrete
phenomena and calculations in finite dimensional Hilbert spaces are
emphasized. The ”older” way of teaching based on coordinate representation
wave functions, de Broglie waves, Fourier transforms, etc. seems to be fading
away, however, this kind of knowledge is crucial for continuous variable
quantum computing (CVQC). In this pedagogical introduction we aim to bridge
this gap. Here we present the basic mathematical concepts and tools used when
dealing with continuous quantities in quantum mechanics and explain how to use
them for CVQC algorithm development.

Many physical quantities, such as position and momentum or the quadratures of
electromagnetic field, can accept values from a continuous spectrum in quantum
mechanics. Due to the nature of quantum mechanics and Heisenberg’s uncertainty
relations the precise manipulation of continuous quantum quantities is
fundamentally impossible. Moreover, the existence of noise in quantum systems
further worsens the situation and it seems that there is no perspective in
using continuous quantum quantities for computation. However, the developments
in relevant experimental realizations and quantum error correction codes have
motivated the investigation of CVQC as an independent computational paradigm.
The question of universality of CVQC is a subtle one, however it has been
addressed in the restricted case of polynomial hamiltonians [1]. Since then, a
number of algorithms have been developed for CVQC. Besides the algorithms
originally specific to continuous variables, most of the well-known quantum
algorithms in discrete quantum computing (Deutsch–Jozsa, Grover, Shor, etc.)
have been adapted to the continuous variable setting.

Quantum computing, in general, is any computation which is done by the
exploitation of quantum properties of physical quantities, be they discrete,
continuous or the combination of both111It is possible for a physical quantity
to have an eigenvalue spectrum which is discrete in some region(s) and
continuous in other(s).. Moreover, it is also possible to combine systems with
continuous variables with traditional discrete quantum computing and implement
the so called hybrid computing [2, 3]. Our focus in this paper is the CVQC as
an independent paradigm.

Section 1 will serve as a crash course on relevant concepts and definitions
known from quantum mechanics. Explanations go as deep as necessary for the
overall integrity and scope of the text. For more detailed explanations from
ground up the reader is invited to read quantum mechanics textbooks (e.g. [4,
5, 6]), however it is not necessary for complete understanding of this paper.

Sections 2 and 3 are incremental development on top of the previous section to
cover the notion of Hilbert spaces in relation to continuous variables, and
quantum harmonic oscillator. These are crucial topics for CVQC and algorithm
development.

In section 4 we discuss a few famous algorithms to demonstrate how all the
mathematical tools introduced in the previous section work out in real life.
We also briefly introduce strawberryfields [7], a Python library which can be
used for CVQC simulation.

Section 5 summarizes the paper and provides some future directions.

##  1 Quantum Continuous Variables

In the quantum computing literature the term continuous variable is loosely
used to refer to a physical quantity that can accept any value in an interval
- in contrast to discrete variables which can accept only distinct values. An
example of a continuous variable is the $x$ coordinate of a particle. In
quantum mechanics physical quantities are represented in terms of
operators222In this paper we have adopted the standard convention of denoting
an operator with a hat on top, e.g. $\hat{x}$ and their continuity properties
are encoded in the eigenvalue spectrum of these operators. If the eigenvalue
spectrum of some operator is continuous then the corresponding physical
quantity can serve as a continuous variable for computation. In this paper we
use the term continuous variable as a collective reference to the physical
quantity itself, the corresponding operator and the eigenvalue.

###  1.1 General Concepts

Any continuous variable has a conjugate which is yet another continuous
variable. In the case of the coordinate $\hat{x}$ the conjugate quantity is
the $x$ component of momentum $\hat{p}_{x}$. For simplicity we will drop the
subscript of the momentum operator. In this paper we will almost always denote
a continuous variable and its conjugate by $\hat{x}$ and $\hat{p}$, however
this does not necessarily mean they are coordinate and momentum. They can be
any pair of conjugate continuous variables. We just use this notation to
continuously hint that the mathematical properties of these variables are
exactly the same as for coordinate and momentum. The physical realizations can
be non-related to coordinate and momentum.

$\hat{x}$ and $\hat{p}$ satisfy the following commutation relation

$[\hat{x},\hat{p}]=i\hbar,$ (1.1)

where $[\hat{x},\hat{p}]=\hat{x}\hat{p}-\hat{p}\hat{x}$ is the commutator. As
you can see the two conjugate continuous variables do not commute. The result
of this non-commutativity is another important property, which is the
Heisenberg’s uncertainty principle

$\Delta x\Delta p\geq\frac{\hbar}{2},$ (1.2)

where $\Delta x=\sqrt{\overline{(\hat{x}-\bar{x})^{2}}}$ is the standard
deviation, and the overbar notation means ensemble average. If this is
completely new to you, appendix A provides more details and explanation about
(1.2).

As we know the state of a system in quantum mechanics is described by a wave
function333We consider only pure states, so we don’t talk about density
matrices, only wave functions.. A wave function is the amplitude of a
probability distribution, i.e. the absolute squared of a wave function gives
the probability distribution over the values of its argument. Consider a
quantum system described by a wave function $\psi(q)$ where $q$ is a
continuous variable. In this case it is said that the wave function is in the
$q$ representation. In particular the quantity $|\psi(q)|^{2}dq$ is the
probability for the system to have value of $q$ in the interval $q,q+dq$. $q$
is oftentimes referred to as the ”space” one is working in. In practise, $q$
can be any physical quantity, for example coordinate $x$, momentum $p$, energy
$E$, etc. The same state of a system can be described in different
representations and there should be some relation between the wave functions
in different representations. These relations give transformation laws, so
that having the wavefunction in one representation one can transform it into
the wavefunction in another representation. It is known from quantum mechanics
that the transformation law is particularly straightforward if one transforms
from a representation to its conjugate representation: it is a Fourier
Transform. Given a state $\phi(p)$ in the $p$ representation the corresponding
state in $x$ representation can be obtained via the Fourier transform

$\psi(x)=\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty}\phi(p)e^{ipx/\hbar}dp$
(1.3)

and vice versa

$\phi(p)=\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty}\psi(x)e^{-ipx/\hbar}dx$
(1.4)

When we choose a particular representation, we can also talk about the
explicit forms of the operators $\hat{x}$ and $\hat{p}$. In the
$x$-representation the operator $\hat{x}$ just coincides with its numerical
value $x$ and the operator $\hat{p}$ has the following explicit form

$\hat{p}=-i\hbar\frac{\partial}{\partial x}.$ (1.5)

Conversely, in the $p$-representation the operator $\hat{p}$ just coincides
with its numerical value $p$ and the operator $\hat{x}$ has the following
explicit form

$\hat{x}=i\hbar\frac{\partial}{\partial p}.$ (1.6)

The explicit forms of the operators $\hat{x}$ and $\hat{p}$ in any
representation $q$ can be derived based on the above formulas, but we will not
cover it here since it’s outside of the scope of the current paper. The
commutation relation (1.1) and the uncertainty principle (1.2) are always the
same, regardless of the representation.

###  1.2 Eigenvalues and eigenfunctions

Given a continuous variable $\hat{f}$, the eigenvalue equation in some
$q$-representation is

$\hat{f}\psi_{f}(q)=f\psi_{f}(q).$ (1.7)

In the discrete case we usually enumerate the eigenvalues and eigenvectors
with integer indices. In the continuous case we just use the letter without a
hat to denote the eigenvalues and the same letter as a subscript to denote the
corresponding eigenstate. The eigenstates $\psi_{f}(q)$ describe a state where
the physical quantity underlying the operator $\hat{f}$ has exactly determined
value equal to $f$.

It is known from quantum mechanics that an arbitrary wave function $\psi(q)$
can be decomposed (i.e. written as a superposition) in terms of eigenstates of
any continuous variable $\hat{f}$ as follows

$\psi(q)=\int_{-\infty}^{+\infty}c(f)\psi_{f}(q)df,$ (1.8)

where the function $c(f)$ represents the decomposition coefficients. Recall,
that in the case of discrete variables we have a summation over discrete
eigenstates, and a set of coefficients $c_{n}$. In the continuous case the
summation transforms into integration, and the set of coefficients becomes a
continuous function $c(f)$. The function $c(f)$ is the wavefunction of the
state under consideration in the $f$-representation and the equation (1.8)
gives transformation law between any two representations $f$ and $q$. This is
easily seen in the particular case of $\hat{q}=\hat{x}$ and $\hat{f}=\hat{p}$.
Given the formulas (1.7) and (1.5) the eigenfunctions of $\hat{p}$ in the
$x$-representation are determined from the following equation

$-i\hbar\frac{\partial\psi_{p}(x)}{\partial x}=p\psi_{p}(x)$ (1.9)

and they are equal to444The solution for the equation (1.9) alone is
$\psi_{p}(x)=\text{const}\times e^{\frac{i}{\hbar}px}$. The value of the
constant is determined from normalization condition which will be covered in
the next subsection.

$\psi_{p}(x)=\frac{1}{\sqrt{2\pi\hbar}}e^{\frac{i}{\hbar}px}.$

Substituting this into the equation (1.8) we instantly recover the Fourier
transformation law (1.3). Conversely, considering $\hat{q}=\hat{p}$ and
$\hat{f}=\hat{x}$ the inverse Fourier transform (1.4) is recovered.

In the theory of CVQC we are quite often interested in states with exactly
determined value of $\hat{x}$ or $\hat{p}$ (i.e. eigenstates of $\hat{x}$ or
$\hat{p}$). If in (1.8) we take $\hat{f}=\hat{p}$, we obtain the expansion of
any $q$-representation wave function in terms of eigenstates of $\hat{p}$
(i.e. in terms of states where $\hat{p}$ has exactly determined value)

$\psi(q)=\int_{-\infty}^{+\infty}c(p)\psi_{p}(q)dp$ (1.10)

Let’s consider the expansion of the wave function for a particular state - the
state $\psi_{x}(q)$ which is an eigenstates of $\hat{x}$ in the
$q$-representation (i.e. has exactly determined coordinate $x$). We get

$\psi_{x}(q)=\int_{-\infty}^{+\infty}c_{x}(p)\psi_{p}(q)dp,$ (1.11)

where $c_{x}(p)$ is now the wave function of the state with exact coordinate
$x$ in the $p$-representation. Given the definition (1.6), we know that the
eigenstates $c_{x}(p)$ are determined from the eigenvalue equation
$i\hbar\frac{\partial c_{x}(p)}{\partial p}=xc_{x}(p)$ and are equal to
$c_{x}(p)=\frac{1}{\sqrt{2\pi\hbar}}e^{-\frac{i}{\hbar}px}$. So we have the
following relation

$\psi_{x}(q)=\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{+\infty}e^{-\frac{i}{\hbar}px}\psi_{p}(q)dp$
(1.12)

With analogous considerations the reverse formula can be obtained as well

$\psi_{p}(q)=\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{+\infty}e^{\frac{i}{\hbar}px}\psi_{x}(q)dx$
(1.13)

###  1.3 Bra-ket notation

In discrete quantum computing we are used to Dirac’s bra-ket notation. In
CVQC, the bra-ket notation can be used as well, however there are a few
caveats and the bra-ket notation should be used with caution.

In the previous subsections we have mostly used $q$ as placeholder
representation whenever we didn’t want to concretize the representation and
derive general formulas regardless of the representation. In the bra-ket
notation a state with exact coordinate $x=x_{0}$, and not concretized
representation is denoted as $|x_{0}\rangle$. A state with exact momentum
$p=p_{0}$, and not concretized representation is denoted as $|p_{0}\rangle$.
Whenever we want to concretize the representation, it is specified as a
subscript to the bra-kets, (e.g. $|x_{0}\rangle_{p}$). With this notation, the
formulas (1.12) and (1.13) have the following form

$\displaystyle|x\rangle$
$\displaystyle=\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{+\infty}e^{-\frac{i}{\hbar}px}|p\rangle
dp$ (1.14) $\displaystyle|p\rangle$
$\displaystyle=\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{+\infty}e^{\frac{i}{\hbar}px}|x\rangle
dx$ (1.15)

The bra vectors are defined as complex conjugate of the ket vectors. The inner
product of a bra and a ket is defined as follows

$\langle
x_{1}|x_{2}\rangle=\int_{-\infty}^{+\infty}\psi_{x_{1}}^{\ast}(q)\psi_{x_{2}}(q)dq.$
(1.16)

An important caveat compared to the case of discrete variables is the
normalization of bra-kets. From discrete variable quantum computing we know
that two eigenkets corresponding to different eigenvalues of some operator are
orthogonal (i.e. their inner product is equal to zero), and each eigenket is
normalized to $1$ (i.e. the inner product of a ket with itself is equal to
$1$). If there were a discrete set of possible values for $x$, denoted as
$x_{i}$, we could simply adopt the Kronecker delta notation to summarize this
orthonormalization conditions in one equation $\left\langle x_{i}\mid
x_{j}\right\rangle=\delta_{x_{i}x_{j}}$. This is not true in the continuous
case, and it can be easily seen for example by calculating the quantity
$\langle p|p\rangle$ in the $x$-representation

$\displaystyle\langle p|p\rangle$
$\displaystyle=\int_{-\infty}^{+\infty}\psi_{p}^{\ast}(x)\psi_{p}(x)dx$
$\displaystyle=\frac{1}{2\pi\hbar}\int_{-\infty}^{+\infty}dx.$

It is readily seen that this expression is nowhere near being equal to $1$, it
is in fact infinite. In quantum mechanics this problem is solved by
introducing Dirac’s delta function instead of the Kronecker’s notation

$\left\langle x^{\prime}\mid x\right\rangle=\delta\left(x^{\prime}-x\right)$
(1.17)

where $\delta\left(x^{\prime}-x\right)$ is the Dirac delta function. If you
are not familiar with Dirac’s delta function, appendix B contains additional
explanations and useful formulas for you. The fact that a ket $|x\rangle$ is
not normalizable to $1$ is sometimes referred to as ”non square
integrability”, meaning that the integral of wave function absolute squared is
not a finite number. There are not physical states corresponding to these non-
normalizable kets - the kets of real states are always normalizeable to $1$.
From quantum mechanics we know that the normalization integral is an integral
over a probability distribution, and the integration over the entire range
should always be $1$. Nevertheless, in quantum mechanics it is allowed to use
these states as mathematical tools and they make a lot of calculations easier.
One just needs to keep in mind and always use the correct orthonormalization
relation (1.17).

##  2 Hilbert Space

In discrete quantum computing one uses the eigenstates of a discrete variable
(e.g. the hamiltonian, in which case the eigenstates are the discrete energy
levels) as computational basis. In the case of CVQC we would like to be able
to act analogously, (i.e. use the eigenstates of a continuous variable as a
computational basis). It is well known that for discrete quantum computing it
is convenient to describe computations in an abstract Hilbert space [8]. CVQC
is not any different, however the topic needs cautious handling. The concept
of a Hilbert space in the discrete case is pretty straightforward (see e.g.
[9] for a simple introduction). In layman’s terms, it can be thought of as a
space with a lot of perpendicular axes and each discrete eigenstate of a
quantum system is considered as a unit vector along one of the axes. Any state
of the system is represented as a vector and can be understood as a linear
combination (superposition) of such unit vectors. If one tries to do the same
for eigenstates corresponding to a continuous variable they will immediately
notice that the Hilbert space is infinite dimensional555This might not seem an
indicative of a problem at first glance, since infinite dimensional Hilbert
spaces are abundant in quantum mechanics even for discrete variables. However,
for discrete quantum computational purposes the Hilbert space is always finite
dimensional. and the labeling of axes is continuous. This means the axes are
not countable, i.e. one cannot actually index them.

To demonstrate the above on an example, let’s consider a
spin-$\nicefrac{{1}}{{2}}$ particle which is a typical representative of a
qubit. One could use a 2-dimensional Hilbert space for this system, with two
basis kets, $|up\rangle$ and $|down\rangle$. The wavefunction of an arbitrary
state would be a superposition of these two basis kets and we would talk about
probabilities $P(+\nicefrac{{1}}{{2}})$ and $P(-\nicefrac{{1}}{{2}})$ for the
system to be $|up\rangle$ and $|down\rangle$ respectively. We don’t talk about
probabilities such as $P(0.3)$ or $P(0.301)$, since there are no corresponding
states in our basis. However, an observable like position is continuous, where
we can ask the probability that the position is 0.3, or 0.301, etc. Thus the
kets corresponding to each possible value of the coordinate should be present
in the basis of the Hilbert space. Since there are infinitely many possible
values of position, we need infinitely many axes in the Hilbert space.
Moreover, since the possible values of coordinate are not countable (i.e. for
every two distinct values one can always find another one which is in
between), the axes in the Hilbert space are not countable either.

Yet another problem with this approach of constructing a Hilbert space is the
fact that the basis kets are not normalizable (see section 1 for more
details). However, the mathematical apparatus of quantum mechanics allows
having the construction of Hilbert spaces with non-physical basis kets [5], so
we use the Hilbert space construction described above for CVQC.

A continuous variable in quantum mechanics can accept values in the infinite
range $(-\infty,+\infty)$ or some finite range $[a,b]$. The formalism is the
same for both cases, so we will just consider the case of infinite range.
Similar to the case of discrete quantum computing we can talk about
superpositions of basis ket vectors and entanglement between variables. A
superposition state is characterized by a complex function $f(x)$ representing
the amplitude of each ket

$|\psi\rangle=\int_{-\infty}^{+\infty}f(x)|x\rangle dx.$ (2.1)

Recall that in the case of discrete states one has summation over the basis
and here we have integration. A simple example of an entanglement can be
represented by a function $g(x)$ and a real constant $c$

$|\psi\rangle=\int_{-\infty}^{+\infty}g(x)|x,x-c\rangle.$ (2.2)

In this example the first and the second registers are perfectly correlated,
so that if, for example, the first one is measured to be some value $x_{0}$,
the second one is immediately known to be $x_{0}-c$.

We can also talk about various operations (or gates) acting on states in this
Hilbert space. By far the most common one is the Fourier gate $\mathcal{F}$
which is the analog of the Hadamard gate. The Fourier transform takes a ket
$|x\rangle$ to a ket $|p\rangle$ of the conjugate variable:
$\mathcal{F}|x\rangle=|p\rangle$. Thus, its action can be explicitly written
based on eq. (1.14)

$\mathcal{F}|x\rangle=\frac{1}{\sqrt{2\pi\hbar}}\int
e^{\frac{i}{\hbar}xy}|y\rangle\mathrm{d}y$

where both $x$ and $y$ are position kets.

Another common and simple operator in CVQC is the displacement operator. In
its most general form it can be represented as an operator depending on two
parameters $\Delta x$ and $\Delta p$

$\hat{D}(\Delta x,\Delta p)=e^{i\Delta x\hat{p}-i\Delta p\hat{x}}.$ (2.3)

This operator in essence displaces the state in the position space by $\Delta
x$ and in momentum space by $\Delta p$. When multiple displacement operators
are applied it can be shown that it can be equivalently described by a single
displacement operator with a euclidean sum of the displacements and the
momenta of the constituent operators [10]. The displacement operator and other
operators will be discussed in greater detail within the context of the
quantum harmonic oscillator in section 3 .

Let’s also briefly discuss a projective measurement on value $x_{0}$. One
might naively think, that in analogy with the discrete case, this projection
operation can be represented as $|x_{0}\rangle\langle x_{0}|$, however this
does not work for continuous variables. There are two main reasons for this:

  1. 1.

projection operation defined this way is not idempotent666Idempotent means
that if you apply the operator multiple times one immediately after another,
you get the same effect as if you applied the operator only once..

  2. 2.

no measuring apparatus can have infinite precision to exactly distinguish the
value $x_{0}$ from its arbitrarily close neighbors.

Both of these problems are resolved if we consider a finite precision
projection operation

$\hat{P}_{x_{0}}=\int_{x_{0}-\Delta x/2}^{x_{0}+\Delta x/2}|x\rangle\langle
x|dx,$ (2.4)

where $\Delta x$ is the precision. With this operator, the second concern
resolved by definition. Let us convince ourselves that this operator is indeed
idempotent and the first concern is resolved as well. For this purpose it is
enough to show that the action of this operator on an arbitrary superposition
state $|\psi\rangle$ is the same as its double action on the same state, i.e.
$\hat{P}_{x_{0}}\hat{P}_{x_{0}}|\psi\rangle=\hat{P}_{x_{0}}|\psi\rangle$.
Indeed

$\displaystyle\hat{P}_{x_{0}}|\psi\rangle$ $\displaystyle=\int_{x_{0}-\Delta
x/2}^{x_{0}+\Delta x/2}|x\rangle\langle
x|dx\int_{-\infty}^{+\infty}f(x^{\prime})|x^{\prime}\rangle dx^{\prime}$
$\displaystyle=\int_{x_{0}-\Delta x/2}^{x_{0}+\Delta x/2}|x\rangle
dx\int_{-\infty}^{+\infty}f(x^{\prime})\langle x|x^{\prime}\rangle
dx^{\prime}$ $\displaystyle=\int_{x_{0}-\Delta x/2}^{x_{0}+\Delta
x/2}|x\rangle
dx\int_{-\infty}^{+\infty}f(x^{\prime})\delta(x-x^{\prime})dx^{\prime}$
$\displaystyle=\int_{x_{0}-\Delta x/2}^{x_{0}+\Delta x/2}f(x)|x\rangle dx$
(2.5)

and

$\displaystyle\hat{P}_{x_{0}}\hat{P}_{x_{0}}|\psi\rangle$
$\displaystyle=\int_{x_{0}-\Delta x/2}^{x_{0}+\Delta x/2}|x\rangle\langle
x|dx\int_{x_{0}-\Delta x/2}^{x_{0}+\Delta x/2}f(x^{\prime})|x^{\prime}\rangle
dx^{\prime}$ $\displaystyle=\int_{x_{0}-\Delta x/2}^{x_{0}+\Delta
x/2}f(x)|x\rangle dx$ (2.6)

##  3 Quantum Harmonic Oscillator

So far we have considered the properties of continuous variables in quantum
mechanics in general, without assuming a particular physical system. There are
several physical systems which can be used to accomplish CVQC, however the
most famous is the implementation with quantum optical systems. The continuous
variables in these systems are the quadratures of the electromagnetic field,
however one does not need knowledge in quantum electrodynamics in order to
study the continuous variable algorithms. The mathematical description of the
quadratures in a quantum optical mode is exactly the same as the mathematical
description of a quantum harmonic oscillator.

A one dimensional quantum harmonic oscillator is described by the following
Hamiltonian

$\hat{H}=\frac{\hat{p}^{2}}{2}+\frac{\hat{x}^{2}}{2}=-\hbar^{2}\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}+\frac{x^{2}}{2}.$
(3.1)

We have taken the mass and the frequency of the oscillator to be $1$. As a
result the reduced Planck constant $\hbar$ is dimensionless, however we do not
assign any particular numerical value to it in this paper and we keep it as a
letter throughout777In the literature there are several conventions about
assigning a value to $\hbar$, e.g. some people prefer to set $\hbar=1$, others
set it as $\hbar=2$ or even $\hbar=1/2$. We keep $\hbar$ as a symbol in this
paper to be consistent with everyone.. We don’t need to care about the
frequency either, since in usual realizations of CVQC frequencies of all
oscillators are the same.

Next, let’s define the following complex operator

$\hat{a}=\frac{1}{\sqrt{2\hbar}}(\hat{x}+i\hat{p}).$ (3.2)

The coordinate and the momentum operators are given in terms of $\hat{a}$ and
$\hat{a}^{\dagger}$ as follows

$\displaystyle\hat{x}=\sqrt{\frac{\hbar}{2}}(\hat{a}+\hat{a}^{\dagger})$ (3.3)
$\displaystyle\hat{p}=-i\sqrt{\frac{\hbar}{2}}(\hat{a}-\hat{a}^{\dagger}).$
(3.4)

If we then have the operator $\hat{a}^{\dagger}$ or $\hat{a}$ act on the
energy eigenstate of the system, $\ket{n}$, we see that it produces a higher
or lower eigenstate respectively

$\displaystyle\hat{a}^{\dagger}\ket{n}=\sqrt{n+1}\ket{n+1}$ (3.5)
$\displaystyle\hat{a}\ket{n}=\sqrt{n}\ket{n-1}\quad.$ (3.6)

This means that $\hat{a}^{\dagger}$ essentially adds a quantum of energy to
the system, hence this operator is generally referred to as the creation
operator. The opposite is true for $\hat{a}$, leading to the name,
annihilation operator. We can also combine these to get the quanta of energy
out of an eigenstate, this is called the number operator $\hat{N}$ given as

$\displaystyle\hat{N}=\hat{a}^{\dagger}\hat{a}$ (3.7)
$\displaystyle\hat{N}\ket{n}=n\ket{n}$ (3.8)

We can also use these operators to define some useful operations, namely the
phase shift and displacement operators. These are defined as $\hat{U}(\theta)$
and $\hat{D}(\alpha)$ respectively and represent the following:

$\displaystyle\hat{U}(\theta)=e^{-i\theta\hat{N}}$ (3.9)
$\displaystyle\hat{D}(\alpha)=e^{\alpha\hat{a}^{\dagger}-\alpha^{*}\hat{a}}.$
(3.10)

Before we start to construct states with these operators we need to consider
how to visualize these states. One useful measure that we use is the Wigner
quasi-probability distribution (or Wigner function) over the phase space (i.e.
the joint space of coordinate and momentum).

With the help of the aforementioned operators we can define coherent states,
which mathematically are eigenstates of the annihilation operator, herefore

$\hat{a}\ket{\alpha}=\alpha\ket{\alpha}$ (3.11)

These coherent states inherently have circular uncertainly in phase space due
to the uncertainty principle. These states can then be squeezed, in that a
squeezing operator can act on the coherent state to squeeze the circular
uncertainty into an ellipse in phase space. Mathematically this squeezing
operator is written as

$\hat{S}(\zeta)=e^{1/2(\zeta^{*}\hat{a}^{2}-\zeta\hat{a}^{\dagger 2})}$ (3.12)

We can now create coherent squeezed states starting from vacuum states by
simply applying the displacement operator and the squeezing operator. Xanadu’s
strawberryfields offers a comprehensive quantum computing toolkit that can be
used in python or on their website in an interactive applet. This program
comes with all of the aforementioned operations to prepare these states to be
used in algorithms that will be discussed in detail in section 4. We can
observe how these states change by plotting the Wigner function. As an
example, a vacuum state can be created and then displacement and squeezing
operators act on it to obtain a displaced squeezed state. With Dgate applying
the displacement operator and Sgate applying the squeezing operator, they can
be done together in one line with the DisplacedSqueezed function.

⬇

import strawberryfields as sf

from strawberryfields.ops import *

prog = sf.Program(1)

with prog.context as q:

# The initial states are

# prepared with q[0]

Vacuum() | q[0]

Dgate(1) | q[0]

Sgate(1) | q[0]

# Alternatively this can be

# Done with the following:

# DisplacedSqueezed(1,0,1,0) | q[0]

and we can see how the Wigner function changes with the three steps in Figure
1.

Figure 1: These contour plots show the Wigner function values in phase space
for a vaccuum state (top left), a state displaced with the parameter of $0.5$
(top right), a vacuum state squeezed with the parameter $0.5$ (bottom left),
and a vacuum state with both the aforementioned displacement and squeezing
applied (bottom right).

All of the operators introduced above act on a single harmonic oscillator, but
systems can have multiple coupled harmonic oscillators. These coupled
oscillators can interact through various operators. One such operator is the
mixing888In optics this is the beamsplitter, however since in this paper we
are avoiding optics as much as we can, we use the term ”mixing” instead which
is defined as

$\hat{B}(\theta,\phi)=\exp{\left(\theta\left(e^{i\phi}\hat{a}_{1}\hat{a}_{2}^{\dagger}-e^{i\phi}\hat{a}_{1}^{\dagger}\hat{a}_{2}\right)\right)}$
(3.13)

where $\theta$ represents the transmittivity angle between the and $\phi$ is
the phase angle. We can create a $50-50$ mixer by setting $\theta=\pi/4$, and
the mixer will be symmetric by setting $\theta=\pi/2$.

At this point we can now create harmonic oscillators in coherent states
representing laser light starting from the vacuum state and we can begin to
couple these systems. We have thus far, however, introduced continuous
variables in the context of a real optical system without discussing how these
continuous variables are handled in a computational sense. So let’s take a
moment to discuss these continuous variables used in computation.

##  4 Continuous Variable Quantum Algorithms

In this section we take a look at several quantum algorithms to give the
reader an idea of how all the introduced mathematical concepts work. There are
many algorithms for continuous variable quantum computing, and they can be
grouped into two categories: 1. algorithms originally specific to continuous
variables, 2. algorithms originally invented for discrete variable quantum
computing and then adapted for continuous variables. We will restrict
ourselves in looking at a few examples from the second category only.

As we have already mentioned in the sections 1 and 2 the exact eigenstates
$|x\rangle$ of a continuous variable $\hat{x}$ are not physical states, hence
they are not realizable in any real physical device. In reality you can create
highly localized states around some value $x$ but never an exactly localized
one. Nevertheless, on the theoretical front, the mathematical apparatus of
quantum mechanics provides tools to work with these states without creating
any theoretical artifacts. Being able to work in a Hilbert space defined by
these non-physical states makes things a lot easier. When developing CVQC
algorithms one usually goes through the following procedure:

  1. 1.

Develop the algorithm assuming that exact eigenstates are physically possible.

  2. 2.

If applicable, investigate the features and properties of the algorithm and
the speedup over the classical counterpart.

  3. 3.

Drop the assumption about exact eigenstates and investigate the algorithm by
replacing them with realistic states. A realistic state corresponding to an
exact eigenstate $|x_{0}\rangle$ is a state which is localized around $x_{0}$,
so can be described as a continuous superposition of exact eigenstates as
defined in (2.1). A common choice for realistic states (which is also easy to
achieve experimentally) are states with Gaussian distribution 3.

$|\psi\rangle=\frac{\sqrt{R}}{\pi^{1/4}}\int_{-\infty}^{+\infty}e^{-\frac{R^{2}}{2}x^{2}}|x\rangle
dx.$

For the conjugate variable $p$ this state is Gaussian as well

$|\psi\rangle=\frac{1}{\sqrt{R}\pi^{1/4}}\int_{-\infty}^{+\infty}e^{-\frac{1}{2R^{2}}p^{2}}|p.\rangle
dp$

These are exactly the squeezed states introduced in the section 3, and $R>1$
is the squeezing factor. Note that if $R>1$, the state is squeezed in the $x$
direction and stretched in the $p$ direction, and vice versa if $R<1$. The
fact that the states cannot be squeezed in both directions is the direct
consequence of Heisenberg’s uncertainty relations. A more complete
introduction to squeezed states from the perspective of continuous variables
can be found in [11].

###  4.1 Grover’s algorithm.

In Grover’s quantum search algorithm, an unsorted list of $N$ entries can be
searched to find an unmarked item with $O(\sqrt{N})$ steps, instead of the
best classical case which requires $O(N)$ steps. This quantum search
algorithm, originally proposed for discrete-variable quantum systems, can be
generalised to the continuous variable representation [12].

From the theory of discrete quantum computing we know that the basic elements
required for Grover’s search algorithm with discrete-variables are an oracle
which marks the solution to the searching problem, Hadamard gates to generate
state superposition, and finally the Grover diffusion operator, which is also
sometimes referred to as an amplitude amplification operation. The oracle is
given as a black box operation and we are not aware of its internal workings.
What we know is the way the oracle marks the special element: given a
superposition of all possible states the oracle inverts the sign of the
special state. This is usually done in two equivalent ways: 1. using an
ancillary qubit to handle the sign flip based on the oracle’s output, and 2.
in-place sign inversion without using a special ancillary qubit. When
considering the continuous variable version of Grover’s algorithm below, we
will concentrate on the in-place variant.

The statement of Grover’s search problem in the continuous variable setting
can be understood as follows. There is a continuous variable $x$ and an oracle
which highlights a special value $x_{f}$ from the interval $(-L/2,L/2)$ with
some precision $\Delta x$. Essentially, we have a collection of
$N=\frac{L}{\Delta x}$ intervals and our task is to find the one which
contains the value $x_{f}$. The continuous variable Grover algorithms achieves
this in $O(\sqrt{N})$ steps.

Consider a continuous variable in the initial state $|x_{0}\rangle$ and apply
the Fourier transform on it999The original paper on continuous variable Grover
search uses the notation $\hbar=\frac{1}{2}$ and considers $n$ continuous
variables. For the sake of simplicity we consider searching along one
continuous variable only. The Fourier transform in the case of $n$ variables
is  $\mathcal{F}|x\rangle=\frac{1}{\sqrt{(2\pi\hbar)^{n}}}\int
e^{\frac{i}{\hbar}xy}|y\rangle\mathrm{d}y$

$\mathcal{F}|x_{0}\rangle=\frac{1}{\sqrt{2\pi\hbar}}\int
e^{\frac{i}{\hbar}x_{0}x}|x\rangle\mathrm{d}x$

Then apply the oracle $\mathcal{O}$. Next, apply an inverse Fourier transform
followed by a selective inversion operator about the state $|x_{0}\rangle$

$I_{x_{0}}\mathcal{F}^{\dagger}\mathcal{O}$

where $I_{x_{0}}=2\hat{P}_{x_{0}}-\mathbb{I}$ is the inversion operation, with
$\mathbb{I}$ being the identity operator and $\hat{P}_{x_{0}}$ the projection
operator defined in (2.4).

Grover’s algorithm then proceeds by repeating the above steps $\sqrt{N}$ times

$\dots\underbrace{I_{x_{0}}\mathcal{F}^{\dagger}\mathcal{O}\mathcal{F}}_{\text{2}}\underbrace{I_{x_{0}}\mathcal{F}^{\dagger}\mathcal{O}\mathcal{F}}_{1}|x_{0}\rangle$

to gradually transform the initial state $|x_{0}\rangle$ into a state where
$|x_{f}\rangle$ and its immediate surroundings have high probability.

The operation $\mathcal{O}$ is given as a black box oracle, so from the
perspective of Grover’s algorithm the implementation details and the inner
workings of the oracle are not important to us. Nevertheless, if you are
interested in implementing oracles, then it’s worth to mention that the oracle
itself can be implemented simply as an inversion operation about the state
$|x_{f}\rangle$. The original paper on continuous variable Grover search
actually assumes that the oracle is given as the inversion operator
$I_{x_{f}}$.

The construction above assumes the existence the exact eigenstate
$|x_{0}\rangle$. To get rid of this assumption, we can substitute
$|x_{0}\rangle$ with a Gaussian distribution peaked at $x_{0}$. It has been
shown that with this substitution the algorithm described above still achieves
success in the same complexity [12].

###  4.2 The Deutsch–Jozsa algorithm

Assuming the existence of exact eigenstates, the continuous variable Deutsch-
Jozsa algorithm has been introduced in [13] and the version with realistic
Gaussian states has been developed in [14]. Here we give a brief overview of
the results.

Let’s consider a binary function $f(x)$ where $x$ is a real number101010A
function mapping real numbers to binary: $R\rightarrow\\{0,1\\}$. For
simplicity let’s also consider that $x\in(-\infty,+\infty)$. The function
$f(x)$ is guaranteed to be either constant or balanced. The case of a constant
function is straightforward to imagine. The challenging part is to come up
with a good definition of what it means for a continuous function to be
balanced. The strict mathematical definition can be found in [13], but the
basic idea is that if we combine all intervals where the function is equal to
$1$ the result should be of the same order as the combined interval where the
function is equal to $0$.

The function $f(x)$ is given as an oracle and the Deutsch–Jozsa problem asks
to find out whether a given oracle implements a constant function or a
balanced one. This problem which would classically require infinite
evaluations of the function can be solved with one evaluation using the
Deutsch-Jozsa algorithm in a perfect continuous variable quantum computer
(i.e. a thought computer where the exact eigenstates are physical) [13]. It
has been also shown that considering realistic Gaussian states instead of the
idealized eigenstates, the continuous variable Deutsch-Jozsa algorithm can
still be accomplished in a finite number of steps, thus still providing
infinite speedup over the classical counterparts [14].

In order to accomplish this two continuous variables are prepared in the
initial states and then a Hadamard gate is effectively applied through the use
of the Xgate and Rgate functions available in Strawberry Fields. This is then
passed to the oracle. Following the oracle another rotatition is applied to
the first qunat. The first variable can then be measured and if the output is
zero then the oracle function is constant and if it is non-zero then the
function is balanced. This can be done in python as follows using the Xgate as
a sample oracle for this situation.

⬇

import strawberryfields as sf

from strawberryfields.ops import *

prog = sf.Program(2)

with program.context as q:

# Prepare initial states

Squeezed(2) | q[0]

Squeezed(2) | q[1]

Xgate(3) | q[0]

Xgate(pi/2) | q[1]

Rgate(pi/2) | q[0]

Rgate(pi/2) | q[1]

# Example Oracle

# Constant zero in this case

Xgate(1) | q[1]

Rgate(-pi/2) | q[0]

# Check if the output is zero

MeasureX | q[0]

This creates the circuit shown in Figure 2 which produces the Wigner function
shown in Figure 3.

Figure 2: Continuous Variable Deutsch-Jozsa algorithm circuit visualized in
Strawberry Fields. Where an X gate is used as the example oracle.

Note that the example oracle used is constant displacement in the $x$
direction which results into the Wigner function being concentrated around
$x=0$, indicating that the oracle function is constant one.

Figure 3: The second mode of the Wigner function represented as a contour plot
with an initial state of $r=1.0$ and $\phi=0.5$

##  5 Summary

In this pedagogical paper we have introduced the mathematical tools used in
the theory of continuous variable quantum computing (CVQC), from basics of
quantum mechanics up to continuous variable quantum algorithms. The reader
should now be knowledgeable enough to explore the field starting from the
classic reviews [15, 16, 17, 18] and the algorithms [13, 14, 12, 19, 20].

There are several important topics which we have not covered in the current
version of this paper, however below we make sure to provide the critical
references required for getting started with these topics.

One of them is continuous variable error correction, which can be studied
along the same lines as in the case of discrete quantum computing [21, 22, 23,
24].

The most discussed experimental realizations of CVQC are based on optical
devices, where the two quadrature amplitudes of each mode are conjugate
continuous variables. Operations are accomplished via various optical elements
and homodyne measurement [7, 25]. Nevertheless, other experimental
realizations have also been recently discussed [26, 27].

A slight detour from the main course of CVQC, but still closely related field
of research is boson sampling (BS) [28]. BS is an active modern field of
research, especially on the experimental front [29, 25], and it has been
recently used to demonstrate quantum computational advantage [30].

Gaussian boson sampling (GBS) is a variation of BS which is easier to
implement experimentally [31, 32, 33]. Besides being yet another candidate of
a computational problem which can be used to demonstrate quantum computational
advantage, a bunch of applications have been found for GBS as well [34].

## Acknowledgments

This work has been done as a part of Quantum Computing Mentorship Program held
by Quantum Open Source Foundation (QOSF). S.B. and R.C. are the mentees, H.S.
is the mentor.

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##  Appendix A Uncertainty relations

Generally speaking, the relation (1.2) expresses ”uncertainty” in that it
necessarily places limits on what we can know about a particle from
simultaneous measurements of position and momentum. To conceptually illustrate
the Heisenberg’s uncertainty principle, consider $N$ (a very large integer)
non-interacting identical particles. This is called an ensemble of particles.
Alternatively, you can also consider a single particle in one universe and
imagine a lot of identical copies of the universe. Let’s measure the
coordinate of each particle. We will have $N$ numbers as a result. The average
of these numbers would be the ensemble average of the measured coordinate, and
this is what is denoted by $\bar{x}$. All the other quantities with bars on
top of them should be understood in the same manner. If one is looking into a
state where $\Delta x$ is very small, i.e. the coordinate measurement results
are narrowly distributed around the average value, we are dealing with a
highly localized particle. In this case, Heisenberg’s uncertainty principle
tells us that the distribution of momentum cannot be narrow, and the particles
in the ensemble contain widely distributed values of momentum.

Heisenberg’s uncertainty principle is not unique to conjugate continuous
variables. In general, for any non commuting continuous variables $\hat{A}$
and $\hat{B}$ the following uncertainty relation is true

$\Delta A\Delta
B\geq\frac{\left|\overline{\hat{A}\hat{B}-\hat{B}\hat{A}}\right|}{2},$

where $|\cdot|$ means absolute value. In the case if $\hat{A}$ and $\hat{B}$
are conjugate variables, their commutator is given by the equation (1.1) and
the Heisnberg’s original principle (1.2) is obtained.

##  Appendix B Dirac’s $\delta$ function

Dirac’s delta function $\delta(x)$ can be defined by

$\int dxf(x)\delta\left(x^{\prime}-x\right)=f\left(x^{\prime}\right).$

There is no actual function that does this, although one can think of
$\delta\left(x^{\prime}-x\right)$ as a sort of limit of ordinary functions
that vanish when $x^{\prime}-x$ is not very close to 0 but are very big when
$x^{\prime}-x$ is very close to 0, with the area under the graph of
$\delta\left(x^{\prime}-x\right)$ equal to $1.$ The precise way to think of it
is that $\delta\left(x^{\prime}-x\right)$ is a ”distribution”, where a
distribution $F$ maps nice well behaved functions $f$ to (complex) numbers
$F[f]$.

A commonly used identity for Dirac’s $\delta$ function is

$\frac{1}{2\pi}\int_{-\infty}^{+\infty}e^{ip(x-x^{\prime})}dp=\delta(x-x^{\prime})$