# A Note on Machine Learning Approach for Computational Imaging

Bin Dong
Beijng International Center for Mathematical Research,
Peking University

###### Abstract

Computational imaging has been playing a vital role in the development of
natural sciences. Advances in sensory, information, and computer technologies
have further extended the scope of influence of imaging, making digital images
an essential component of our daily lives. For the past three decades, we have
witnessed phenomenal developments of mathematical and machine learning methods
in computational imaging. In this note, we will review some of the recent
developments of the machine learning approach for computational imaging and
discuss its differences and relations to the mathematical approach. We will
demonstrate how we may combine the wisdom from both approaches, discuss the
merits and potentials of such a combination and present some of the new
computational and theoretical challenges it brings about.

##  1 Introduction

The development of natural sciences has been heavily relying on visual
examinations. Therefore, images play a central role since they can accurately
record the phenomenon of interest and be further processed and analyzed by
algorithms to assist human decision-making. Advances in sensory, information,
and computer technologies have made it possible to apply some of the most
sophisticated developments in mathematics and machine learning to the design
and implementation of efficient algorithms to process and analyze image data.
As a result, the impact of images has now gone far beyond natural sciences.
Image processing and analysis techniques are now widely adopted in
engineering, medicine, technical disciplines, and social media, and digital
images have become an essential element of our daily lives.

The term “computational imaging” often refers to the process of forming images
from measurements via computations. It consists of two crucial steps, sensing
(or scanning) with a detector or electromagnetic beam and image reconstruction
with a numerical algorithm. In this note, we shall extend the definition of
computational imaging to include image analysis (or interpretation). Sensing
is the most fundamental step of computational imaging, where hardware design
is the central research topic assisted by sampling theory and algorithms
(e.g., compressed sensing [12, 22]). Its main objective is to acquire high-
quality measurements of the imaging subject efficiently. Image reconstruction
is a mid-level task in computational imaging that comes after the step of
sensing. Its main objective is to reconstruct high-quality images from
measurements. Mathematics has been the main driven force in the advancement of
image reconstruction for the past few decades [5, 13, 20]. Conversely, image
reconstruction also brings to mathematics new challenging problems and
fascinating applications that gave birth to many new mathematical tools, whose
application has even gone beyond the scope of image reconstruction. Image
analysis is a high-level task in computational imaging where both mathematical
and statistical models have played a significant role [52, 69]. The main
objective of image analysis is to extract meaningful information from images
to assist in human decision-making.

Although mathematical and statistical models have been successful in
computational imaging, they also face many challenges. In the sensing step,
although the theory of compressed sensing tells us random sampling is a good
option for some instances, only a handful of imaging modalities (e.g., MRI)
satisfy the theory’s assumptions. Furthermore, a better sensing mechanism
should be adaptive to each subject that the theory of compressed sensing is
not considered. A good adaptive sensing mechanism needs to decide for each
given imaging subject on which measurements to take to maximize a particular
quality metric. For image reconstruction, existing models and algorithms are
designed based on human knowledge. Although we know for each model which
general class of images is most suitable, e.g., we know TV model [60] is ideal
for piecewise constant images, for a given set of natural images, it is hard
to design a most suitable model entirely by hand (e.g., the design of the
regularization, the hyperparameters, etc.). For image analysis, the ultimate
objective is to extract relevant image features to facilitate decision-making.
Before deep learning, features are often handcrafted, which may not be well
adaptive to the data set or the underlying image analysis task. For example,
we know image edges, textures and contents are essential for image
segmentation. However, it is unclear what contents mean for different images
and whether there are other important features for image segmentation.

These challenges that limit the further development of computational imaging
can be rephrased in a mathematical term as the challenge of approximating
high-dimensional (HD) nonlinear functions of which we have limited or no
knowledge, primarily due to the well-known curse of dimensionality. For
example, making decisions on the set of measurements optimal for a given
imaging subject can be an intricate HD function that takes the current state
(e.g., the existing measurements) as the input and the next set of
measurements as the output; deciding on the best hyperparameters (e.g. the
sparsifying transformation, the sparsity promoting norm, the regularization
parameters, etc.) of image reconstruction models for a given data set is
another example; the embedding of images to the feature space (or latent
space) to facilitate different image analysis tasks can also be a complicated
HD function. This challenge is now being overcome by deep learning to various
extents [24]. Deep neural networks (DNNs), especially the convolutional neural
networks (CNNs), are very effective in approximating nonlinear HD functions.
Therefore, DNNs are now widely used in computational imaging, and the field
has advanced significantly over the past decade. We shall review some of these
exciting advancements, suggest some future opportunities, and raise some new
theoretical challenges.

##  2 Improving Image Reconstruction with Deep Learning

A typical learning-based image reconstruction can be summarized as the
following problem

$\begin{cases}\min_{\bm{\Theta}}\
\mathbb{E}_{({\bm{u}},{\bm{f}})\sim\mathcal{P}}\
\ell(\mathcal{F}_{\bm{\Theta}}({\bm{f}}),{\bm{u}})+r(\mathcal{F}_{\bm{\Theta}}),\qquad\mbox{supervised;}\cr\min_{\bm{\Theta}}\
\mathbb{E}_{{\bm{f}}\sim\mathcal{P}}\
\ell(\bm{A}\mathcal{F}_{\bm{\Theta}}({\bm{f}}),{\bm{f}})+r(\mathcal{F}_{\bm{\Theta}}),\qquad\mbox{unsupervised.}\end{cases}$
(2.1)

Here, $\mathcal{F}_{\bm{\Theta}}$ is an image reconstruction operator
parameterized by $\bm{\Theta}$ that takes ${\bm{f}}$ as input and the
reconstructed image as output, $\ell(\cdot,\cdot)$ is a certain loss function
measuring the distance between the two input arguments, $r(\cdot)$ is a
certain regularization on image reconstruction operator and $\mathcal{P}$ is
the distribution of the data. We shall call (2.1) the learning-based approach
(or deep learning approach when $\mathcal{F}_{\bm{\Theta}}$ is a DNN). Note
that a possible variant for (2.1) is to consider the dependence of
$\bm{\Theta}$ on $f$, i.e. $\mathcal{F}_{\bm{\Theta}(f;\bm{\Gamma})}$. Such
formulation makes the model $\mathcal{F}_{\bm{\Theta}(f;\bm{\Gamma})}$
adaptive to each data ${\bm{f}}$ rather than having a fixed model
$\mathcal{F}_{\bm{\Theta}}$ for all data in $\mathcal{P}$. This is closely
related to meta-learning [29].

The main difference between handcrafted modeling and learning-based approach
(especially deep learning approach) is twofold:

  1. 1.

the image reconstruction operator $\mathcal{F}_{\bm{\Theta}}$ is fully or
mostly human-designed for handcraft modeling, while
$\mathcal{F}_{\bm{\Theta}}$ is a complex nonlinear composite function with
less human-designed structures but millions of trainable parameters
${\bm{\Theta}}$ for deep learning approach;

  2. 2.

the objective function in the optimization problem contains an expectation
over a data distribution $\mathcal{P}$ for a learning-based approach, while
the expectation is absent for typical handcraft models.

The first difference leads to the well-known argument that DNNs and the
training dynamics of (2.1) are generally harder to interpret than handcrafted
modeling. The second difference is why learning-based models are more data-
adaptive than handcrafted models, while it also raises the new issue of
generalization since the expectation can only be approximated using samples.

Both handcrafted models and deep learning models have their advantages and
drawbacks depending on the applications. There has been an increasing effort
in the community to combine handcrafted modeling and deep modeling to enjoy
benefits from both approaches. One of the popular ways of such a combination
is the so-called unrolled dynamics approach. It started with the seminal work
of [26] where the authors showed that one could unroll the iterative soft-
thresholding algorithm (ISTA) to create a feed-forward network. Then, one can
train ISTA end-to-end to determine the parameters in ISTA so that they are
best suitable for the training data. The trained ISTA can achieve much smaller
errors than the original ISTA or even FISTA [8]. This work initiated an
approach to unroll a discrete dynamic system (an optimization algorithm or a
differential equation) to form a network for end-to-end training. Recently,
more and more examples showed that the unrolling dynamics approach well
balances model interpretability and efficacy. This includes unrolling discrete
forms of nonlinear diffusion equations for image restoration [43, 18] and
unrolling optimization algorithms for image reconstruction and inverse
problems [71, 87, 3, 70, 16, 39, 82, 85, 21, 40, 86]. The unrolling dynamics
approach can often result in deep models that have better interpretability
inherited from the original dynamics. Furthermore, these deep models normally
have much fewer trainable parameters than black-box deep neural networks
(e.g., using a plain CNN, U-Net, ResNet, etc.), which are more suitable for
learning on relatively small data sets. We refer the interested readers to
[48] for a comprehensive review on the unrolled dynamics approach.

The unrolled dynamics models often start with an iterative algorithm which can
be an optimization algorithm or a discretization of a certain evolution PDE.
For example, we can use the following iterative algorithm as the backbone
dynamics

$\begin{cases}\bm{u}^{k+1}&=\arg\min_{\bm{u}}E_{\lambda}(\bm{u},\bm{z}^{k};\bm{\beta}^{k}),\cr\bm{z}^{k+1}&=\arg\min_{\bm{z}}E_{\lambda}(\bm{u}^{k+1},\bm{z};\bm{\beta}^{k}),\cr\bm{\beta}^{k+1}&=\bm{\beta}^{k}+\gamma(\bm{H}\bm{u}^{k+1}-\bm{z}^{k+1}),\qquad
k=0,1,\ldots,K-1.\end{cases}$ (2.2)

Here, the function $E_{\lambda,\gamma}(\bm{u},\bm{z};\bm{\beta})$ is defined
as

$E_{\lambda,\gamma}(\bm{u},\bm{z};\bm{\beta})=\mathcal{D}(\bm{u},{\bm{f}})+\bm{\gamma}\mathcal{C}_{\bm{H}}(\bm{u},\bm{z};\bm{\beta})+\lambda\mathcal{R}(\bm{z}),$

where $\mathcal{D}(\bm{u},{\bm{f}})$ is a data fidelity term (e.g.,
$\mathcal{D}(\bm{u},{\bm{f}})=\frac{1}{2}\|\bm{A}{\bm{u}}-{\bm{f}}\|_{2}^{2}$),
$\mathcal{C}_{\bm{H}}(\bm{u},\bm{z};\bm{\beta})$ is a constraint term that
links the primal variable $\bm{u}$ with a certain sparsifying transformation
$\bm{H}$ (e.g.,
$\mathcal{C}_{\bm{H}}(\bm{u},\bm{z};\bm{\beta})=\frac{1}{2}\|\bm{H}{\bm{u}}-\bm{z}+\bm{\beta}\|$),
the auxiliary variable $\bm{z}$ and the dual variable $\bm{\beta}$, and
$\mathcal{R}(\bm{z})$ is the regularization term (e.g.,
$\mathcal{R}(\bm{z})=\|\bm{z}\|_{1}$). The iterative algorithm (2.2) defines a
discrete dynamics. If it is unrolled, it forms a feed-forward network that
takes ${\bm{u}}^{0}$ as input and ${\bm{u}}^{K}$ as output. It is common
practice to choose ${\bm{u}}^{0}$ as a coarse approximation of the image to be
reconstructed computed from ${\bm{f}}$. Thus, this feed-forward network forms
the nonlinear operator $\mathcal{F}_{\bm{\Theta}}$ in (2.1). The parameter
$\bm{\Theta}$ may include the hyperparameters $\lambda$ and $\gamma$ in (2.2),
and any trainable parameters introduced in the $\bm{u}$\- and
$\bm{z}$-subproblem.

Most existing unrolled dynamics models have the tendency to define as many
learnable components in the dynamics as possible, which makes the ultimate
model further deviate from the dynamics it started with. Having a lot of
learnable components increases the express power of the model to enable it to
approximate more complex mappings. At the same time, however, its training can
be more challenging and requires more data. In addition, an unrolled dynamic
model with too many learnable components may also be susceptible to
distribution shift during inference which is inevitable in practice.
Therefore, we suggest an opposite direction in [86] that keeps the learnable
components to the minimum and only converts the components to learnable units
when they are most critical to the reconstruction quality and too intricate to
be handcrafted.

Now, we briefly recall the model and main empirical findings of [86]. Consider
the following optimization model for image reconstruction

$\min_{\bm{u},\bm{z}}\frac{1}{2}\|\bm{A}\bm{u}-\bm{f}\|^{2}+\sum_{\ell=1}^{L}\left(\lambda_{i}\|\bm{z}_{\ell}\|_{1}+\frac{\gamma_{\ell}}{2}\|\bm{W}_{\ell}\bm{u}-\bm{z}_{\ell}\|^{2}\right),$
(2.3)

where $\bm{W}_{\ell}$ is a certain sparsifying transform such as wavelet frame
transform. The optimization problem (2.3) can be solved by the following
alternative optimization strategy

$\begin{cases}\bm{u}^{k+1}&=\arg\min_{\bm{u}}\|\bm{A}\bm{u}-\bm{f}\|^{2}+\sum_{\ell=1}^{L}\gamma_{\ell}\|\bm{W}_{\ell}\bm{u}-\bm{z}^{k}_{\ell}\|^{2},\cr\bm{z}^{k+1}_{\ell}&=\arg\min_{\bm{z}}\lambda_{\ell}\|\bm{z}_{\ell}\|_{1}+\frac{\gamma_{\ell}}{2}\|\bm{W}_{\ell}\bm{u}^{k+1}-\bm{z}_{\ell}\|^{2},\quad\ell=1,\ldots,L,\end{cases}$

with proper initialization $\bm{u}_{0}$ and $\bm{z}_{0}$. Solution to each of
the two subproblem takes the form

$\begin{cases}\bm{u}^{k+1}&=\left(\bm{A}^{\top}\bm{A}+\sum_{\ell=1}^{L}{\gamma}_{\ell}\bm{W}_{\ell}^{\top}\bm{W}_{\ell}\right)^{-1}\left[\bm{A}^{\top}\bm{f}+\sum_{\ell=1}^{L}{\gamma}_{\ell}\bm{W}_{\ell}^{\top}\bm{z}_{\ell}^{k}\right],\cr\bm{z}^{k+1}_{\ell}&=\mathcal{T}_{\lambda_{\ell}/\gamma_{\ell}}(\bm{W}_{\ell}\bm{u}^{k+1}),\quad\ell=1,\ldots,L,\end{cases}$
(2.4)

where $\mathcal{T}_{\beta}$ is the soft-thresholding operator with threshold
level $\beta$. Note that the $\bm{u}$-subproblem of (2.4) can be solved by the
conjugate gradient (CG) method. The design of the deep unrolled dynamics model
of [86] is to use a CNN as a hypernetwork that takes the current best
approximation of $\bm{u}$ as input and initialization for the CG method as
output at each step $k$ while keeping all other components manually selected.
Most existing unrolled dynamics methods tend to convert the sparsifying
transform, the soft-thresholding operator, or even the inversion of the linear
system of (2.4) to learnable units. However, [86] empirically demonstrated
that they are all unnecessary, and we only need to use a CNN to infer a good
initialization for the CG method that solves the $\bm{u}$-subproblem.
Comprehensive experiments of [86] show that such design of the unrolled
dynamics model is particularly beneficial in terms of robustness to
distribution shift that may be introduced by using a different image data set,
noise level/type, imaging conjuration, etc.

There are a few remaining questions on the unrolled dynamics approach that are
worth further exploration. Although empirical studies show that this approach
leads to a model $\mathcal{F}_{\bm{\Theta}}$ more interpretable and
generalizes better than some black-box deep learning models, there is still in
lack of theoretical support to these empirical studies. Some pioneering works
[81, 17] attempted to tackle this problem by analyzing the complexity and PAC-
learnability of these networks. Nonetheless, the unrolled dynamics approach
severely lacks theoretical guidance on model design and training. One
advantage of the unrolled dynamics approach is the incorporation of our
existing knowledge on the imaging modality and reconstruction algorithms.
However, this may as well limit the expressive power of the model leading to
only incremental improvements. Furthermore, the modeling of the underlying
imaging modality with an operator $\bm{A}$ is only an approximation which can
be a very rough one in practice. Thus, there are some works (e.g., [88]) that
suggest using black-box deep learning models instead, hoping the model itself
can automatically correct such inaccuracy. Of course, all of these are
speculations and require further empirical and theoretical studies.

While most of the models we talked about are within the scope of supervised
learning, unsupervised (and self-supervised) learning is another prevailing
approach for image reconstruction [10, 63, 74, 75, 31, 7, 37, 50, 57, 53, 54].
In comparison with the supervised learning-based approach, unsupervised
methods are more reliant on the regularization term (i.e., $r(\cdot)$ in
(2.1)) due to the lack of labels. Nonetheless, regularization is the key to
the success of both supervised and unsupervised learning-based approaches
since it can effectively reduce model complexity and improve generalization.
However, unlike mathematical approaches studied extensively for decades, the
regularization of the learning-based approach is still a mystery in general.
Many effective regularizations are implicitly induced by the stochasticity of
the data, model, and training algorithm [27, 32, 76, 19, 4, 58]. Therefore, a
collective understanding of these regularization effects and guidance on
exploiting them in practice is another important line of research.

##  3 Improving Sensing with Reinforcement Learning

In compressed sensing, the sensing strategy is random and not adaptive to the
imaging subject [12, 22], i.e., the measurements are randomly selected
independently from different imaging subjects. In theory, such a strategy is
proven for exact recovery using a convex model for only a handful of imaging
modalities such as MRI. However, the theory of compressed sensing fails to
cover many other imaging modalities, among which CT imaging is an important
example due to the coherence structure of Radon transform. In practice,
uniform sampling is often adopted due to its simplicity. However, for each
subject, a uniform or random sensing strategy may not be ideal. It is more
desirable to design a personalized sensing strategy for each subject to
achieve better reconstruction results.

In this section, we discuss how we may use reinforcement learning (RL) [72], a
powerful tool for sequential decision making, to improve sensing in
computational imaging. In particular, we shall review the work of [64] for CT
image reconstruction, where the we used RL to train a CT scanning policy that
is adaptive to each imaging subject. Note that RL has also been used in other
imaging modalities to improve sensing. For example, in Scanning Transmission
Electron Microscopy (STEM), recent work by [23] proposes to use RL to guide
the movement of the detector and uses a generator to generate reconstructed
images. In [56], RL is used to learn acquisition trajectories in MRI k-space
for a fixed image reconstruction model. RL is also applied to improve
autonomous exploration [46].

In [64], the scanning procedure is formulated as a Markov Decision Process
(MDP), where the state includes currently collected measurements, the action
determines the next measurement angle and the dose allocation, and the reward
depends on the reconstruction quality measured by the peak signal-to-noise
ratio (PSNR). Then, the scanning policy is trained using the proximal policy
optimization (PPO) algorithm [62]. The key to the application of RL in sensing
is the design of an appropriate MDP. Here, we review the MDP designed by [64]
as follows.

### MDP Formulation for Adaptive CT Scanning

The CT scanning process can be viewed as a sequential decision process, where
at each time step, we need to decide on the measurement angle and the
corresponding X-ray dose. Given an Image $I$ and the number of all possible
angles $N$ (e.g., $N=360$ if we can choose all the integer angles from $0$°to
$359$°), we now elaborate how the CT scanning process on $I$ can be formulated
as an MDP:

  * 1)

The state is a sequence $\vec{s_{t}}=(s_{1},s_{2},...,s_{t})$, where
$s_{t}=(p_{t},d_{t}^{\text{ac}},d_{t}^{\text{rest}})$. $p_{t}$ is the
collected measurement at time step $t$. $d_{t}^{\text{ac}}$ records the used
dose distribution up to time step $t$. The scalar $d_{t}^{\text{rest}}$
represents the amount of the remaining dose.

  * 2)

The action is $a_{t}=(a_{t}^{\text{angle}},a_{t}^{\text{dose}})$.
$a_{t}^{\text{angle}}$ is a one-hot vector recording the angle we choose at
time step $t$. $a_{t}^{\text{dose}}\in[0,1]$ is the fraction of dose that we
apply at the corresponding angle. We terminate the MDP when the total used
dose exceeds the total allowed dose.

  * 3)

The reward is computed as
$r(s_{t},a_{t})=\text{PSNR}(I_{t},I)-\text{PSNR}(I_{t-1},I)$, where $I$ is the
groundtruth image, $I_{t}$ is the reconstructed image at time step $t$, and
$\text{PSNR}(\hat{I},I)$ represents the PSNR value of the reconstructed image
$\hat{I}$. The reconstructed image $I_{t}$ is obtained by SART [51].

  * 4)

The transition model represents the scanning process of CT. At time step $t$,
given the state $\vec{s}_{t}$ and action $a_{t}$, the next state
$\vec{s}_{t+1}$ is simply the concatenation of $\vec{s}_{t}$ and
$s_{t+1}=(p_{t+1},d_{t+1}^{\text{ac}},d_{t+1}^{\text{rest}})$. Details on how
each of the three elements in $s_{t+1}$ is computed can be found in [64].

The policy network is chosen as a Recurrent Neural Network (RNN) to combine
all the information from the past measurements. The RNN is also carefully
designed to handle continuous action and discrete action simultaneously. We
refer the interested readers to [64] for details. Numerical experiments on a
large CT image dataset show that RL’s scanning policy significantly
outperforms the standard scanning strategies in terms of reconstruction
quality. In contrast to [23], the RL’s scanning policy can be directly
combined with different image reconstruction algorithms without retraining.

Note that sensing for many imaging modalities (e.g., CT and MRI) can be
formulated as combinatorial optimizations. In recently years, there is an
emerging line of research that applies RL to solve combinatorial optimization
problems (COPs) with exciting results, such as Travelling Salesman Problem
[9], Vehicle Routing Problem [35], Influence Maximization [47], Autonomous
Exploration [45], Integer Programming [73], etc. We can view RL as a trainable
optimization algorithm that can continue improving itself by attempting to
solve sampled COPs. However, the success of RL in solving a COP heavily relies
on the design of the MDPs, the exploitation of the special structures of the
COP in hand, and the choice of training algorithm and tricks. Nonetheless, RL
is a promising approach for COPs and, in particular, can make sensing of
computational imaging more data-adaptive and even task-driven (see the next
section).

##  4 Task-Driven Computational Imaging

In this section, we discuss a possibility to unify the three steps, i.e.,
sensing, image reconstruction, and image analysis, of computational imaging.
Although this is not a new idea, we can make it more practical and
computationally tractable with the latest advancements in machine learning and
its integration with mathematical approaches.

The motivation for making such unification in computational imaging is
twofold. Firstly, it induces a task-specific quality metric for sensing and
image reconstruction. Here, a task-specific quality metric is a metric to
evaluate the quality of a reconstructed image in fulfilling a specific image
analysis task. A proof-of-concept study regarding this was given by [80],
where we showed that an image abnormality detection task may induce a
significantly different image quality metric from pixel-level metric. Works in
computer vision [42, 41] and radiology [34] also suggest potentials of task-
driven imaging, and a theoretical framework was introduced by [2]. Therefore,
the unification may lead to more effective and practically relevant ways to
evaluating imaging qualities and more economic sensing strategies for a given
task or a set of tasks. Secondly, image analysis can benefit from such
unification as well (see [80] for some empirical evidence). This is because
the measurements acquired from sensing contain more information than the
underlying reconstructed image due to inevitable information loss induced by
any image reconstruction algorithm.

On the technical side, the unification is conceptually straightforward while
computationally challenging. We illustrate how the unification can be realized
under the supervised learning regime. Let
$\mathcal{M}_{\bm{\Theta}_{1}}:{\bm{u}}\mapsto{\bm{f}}$ be a parameterized
sensing operator that maps an image ${\bm{u}}$ to its measurements ${\bm{f}}$;
$\mathcal{F}_{\bm{\Theta}_{2}}:{\bm{f}}\mapsto{\bm{u}}$ be a parameterized
image reconstruction operator (e.g. the one defined in (2.1));
$\mathcal{G}_{\bm{\Theta}_{3}}:{\bm{u}}\mapsto{\bm{z}}$ be a parameterized
image analysis operator that takes an image ${\bm{u}}$ to its task-dependent
value ${\bm{z}}$ (e.g. one-hot vectors for image classification). Then, the
unification can be achieved by solving the following optimization problem

$\min_{\bm{\Theta}_{1},\bm{\Theta}_{2},\bm{\Theta}_{3}}\
\mathbb{E}_{({\bm{u}},{\bm{z}})\sim\mathcal{P}}\
\ell\left(\mathcal{G}_{\bm{\Theta}_{3}}\circ\mathcal{F}_{\bm{\Theta}_{2}}\circ\mathcal{M}_{\bm{\Theta}_{1}}({\bm{u}})\right),{\bm{z}}).$
(4.1)

Note that, unlike (2.1), we have dropped the regularization terms on the three
nonlinear operators for simplicity.

The optimization problem (4.1) can be a very tough problem to work with. First
of all, it is not clear whether there exists good parameterized approximations
to sensing, image reconstruction and image analysis, each of which is a
complicated HD mapping. Second challenge comes from solving the optimization
of (4.1) which includes the sampling of $\mathcal{P}$ and the design of
optimization algorithms. Latest advances of machine learning, especially deep
learning, have provided us with new tools to tackle (4.1). Both empirical [36,
59, 28, 25, 38, 24, 68] and theoretical studies [67, 77, 15, 78, 44, 49, 65,
66, 83] showed that DNNs are very effective in approximating nonlinear
operators in HD spaces. The combinatorial nature of the sensing operator
$\mathcal{M}_{\bm{\Theta}_{1}}$ can be effective handled by RL (as discussed
in Section 3) or Monte Carlo methods (e.g. Monte Carlo tree search [11]).
Furthermore, the back-propagation (BP) [61] based stochastic optimization
algorithms and the development of deep learning platforms such as Tensorflow
[1] and Pytorch [55] have made it easier to build and efficiently optimize
complex models such as (4.1) on multiple processors in parallel.

As a result, special instances and different versions of (4.1) has already
been considered in the literature. For example, [64, 56] optimized only
$\mathcal{M}_{\bm{\Theta}_{1}}$ of (4.1) with $\mathcal{F}_{\bm{\Theta}_{2}}$
fixed and $\mathcal{G}_{\bm{\Theta}_{3}}$ being an identity operator (with
${\bm{z}}={\bm{u}}$); [33, 23, 84] optimized (4.1) for both
$\mathcal{M}_{\bm{\Theta}_{1}}$ and $\mathcal{F}_{\bm{\Theta}_{2}}$ while
$\mathcal{G}_{\bm{\Theta}_{3}}$ is taken as an identity operator; [42, 41, 80,
30] optimized $\mathcal{F}_{\bm{\Theta}_{2}}$ and
$\mathcal{G}_{\bm{\Theta}_{3}}$ of (4.1) with $\mathcal{M}_{\bm{\Theta}_{1}}$
fixed. In the rising field of intelligent sensor design [6], all three
operators are considered except that $\mathcal{M}_{\bm{\Theta}_{1}}$ is often
relatively simple. In another emerging field known as the “deep optics” [79],
the operator $\mathcal{M}_{\bm{\Theta}_{1}}$ represents an optical module (a
physical layer) while
$\mathcal{G}_{\bm{\Theta}_{3}}\circ\mathcal{F}_{\bm{\Theta}_{2}}$ is the
digital image processing module. The optical and image processing modules are
first end-to-end trained for a specific imaging task. Then, the optical module
can be fabricated from the trained operator $\mathcal{M}_{\bm{\Theta}_{1}}$
using 3D printing [14]. This provides a framework for end-to-end task-driven
camera design.

Although the aforementioned works suggested potentials of the unification
(4.1), its full power is yet to be uncovered. Furthermore, we know very little
on the theoretical aspects of the problem (4.1), e.g., sample complexity,
approximation, generalization, robustness to distribution shift, etc. This
leaves a tremendous room for both empirical and theoretical studies.

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# A first step towards quantum simulating jet evolution in a dense
medium††thanks: Presented at Quark Matter 2022

João Barata  Physics Department, Brookhaven National Laboratory, Upton, NY
11973, USA

###### Abstract

The fast development of quantum technologies over the last decades has offered
a glimpse to a future where the quantum properties of multi-particle systems
might be more fully understood. In particular, quantum computing might prove
crucial to explore many aspects of high energy physics unaccessible to
classical methods. In this talk, we will describe how one can use digital
quantum computers to study the evolution of QCD jets in quark gluons plasmas.
We construct a quantum circuit to study single particle evolution in a dense
QCD medium. Focusing on the jet quenching parameter $\hat{q}$, we present some
early numerical results for a small quantum circuit. Future extensions of this
strategy are also addressed.

##  1 Introduction

Heavy ion collisions at RHIC and the LHC have offered an unique opportunity to
explore the quark gluon plasma, leading the way towards a better understanding
of QCD. Experimentally, there are multiple types of probes used to indirectly
measure the properties of this state of matter, being jets one of the most
powerful ones. When traversing the plasma, jets’ constituents suffer multiple
interactions with the background plasma, leading to their trajectories being
modified and resulting in the production of extra soft radiation [4].

Understanding how such modifications come into place allows to infer the
properties of the underlying plasma. However, from a theoretical point of view
it is hard to study these type of effects once the particle number becomes
large. Conversely, many of the interesting medium induced effects only emerge
at higher particle multiplicities, and thus understanding how multiparticle
interference mechanisms come into play is of great importance.

Quantum computers are quantum machines which ideally can simulate the full
quantum dynamics of other target quantum systems. Thus, in principle, one
could use such devices to simulate jet evolution in the presence of a
background, capturing higher order interference effects. In this way, quantum
computers might allow to broaden our understanding of jet quenching physics.
However, in reality, current quantum computers are still relatively small and
imperfect machines, and thus they are not capable of tackling such complex
questions [8]. Moreover, it is still not even clear how to efficiently (i.e.
using less resources than a classical machine) formulate full jet evolution at
the level of quantum operations on the computer.

In this work we detail our current effort to implement jet in-medium evolution
in a dense stochastic background. At this point, we limit our discussion to
the evolution of a single quark state, neglecting radiation production. This
first step will be useful when including gluons, since many aspects of the
implementation are common.

##  2 Jet evolution in a dense medium and the quantum simulation algorithm

We consider the evolution of a single quark (the jet) near the light cone with
momentum $p^{\mu}=(p^{+},p^{-},{\boldsymbol{p}})$. The medium is described by
a classical background field $A_{\mu}^{a}$. In the usual picture, the jet is
highly energetic and thus, in the light-cone gauge $A^{+}=0$, it is only
sensitive to the $A^{-}$ component of the field and its spacetime dependence
can be simplified as $A(x^{\mu})\approx A(x^{+},{\boldsymbol{x}})$. One can
then show that the quark evolves according to an effective Hamiltonian [4, 6]

$\displaystyle\hat{H}=\hat{K}+\hat{V}=\frac{\hat{\boldsymbol{p}}^{2}}{2p^{+}}+g\hat{A}^{-}_{a}T^{a}\;,$
(1)

at constant $p^{+}$ (the jet energy). The first term corresponds to the
kinetic energy of the quark and it accounts for the quantum diffusion of the
probe in the medium. The potential terms takes into account the gluon
exchanges between the medium and the quark.

In this approach, the medium is described by a stochastic field. As a
consequence, when computing any observable, one must average over all possible
medium configurations. In practice, we simulate the evolution of the system
over an ensemble of field configurations, such that for each configuration the
final quantum state is given by

$\displaystyle\begin{split}\ket{\psi_{L}}=&U(L;0)\ket{\psi_{0}}\;,\end{split}$
(2)

with $U$ the time evolution operator. Each field is generated by solving the
classical Yang-Mills equations in the presence of a stochastic color source
$\rho^{a}$, which is assumed to satisfy a Gaussian error form, i.e. its only
non-trivial correlator is

$\displaystyle\langle\\!\langle\rho_{a}(x^{+},{\boldsymbol{x}})\rho_{b}(y^{+},{\boldsymbol{y}})\rangle\\!\rangle=g^{2}\mu^{2}({\boldsymbol{x}})\delta_{ab}\delta^{2}({\boldsymbol{x}}-{\boldsymbol{y}})\delta(x^{+}-y^{+})\;,$
(3)

where $g^{2}\mu^{2}$ should be understood as the medium density.

In this work we are interested in computing the squared transverse momentum
acquired by the quark due to evolution in the medium. The medium averaged
expectation value of this observable can be related to the jet quenching
parameter $\hat{q}$ via

$\displaystyle\hat{q}\equiv\frac{\langle\\!\langle\langle\hat{\boldsymbol{p}}^{2}(L)\rangle\rangle\\!\rangle-\langle\\!\langle\langle\hat{\boldsymbol{p}}^{2}(0)\rangle\rangle\\!\rangle}{L}\,,$
(4)

where $L$ is the longitudinal length of the medium. Here we denote the medium
average as

$\displaystyle\langle\\!\langle\langle{\hat{\boldsymbol{p}}^{2}}\rangle\rangle\\!\rangle=\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\langle{\hat{\boldsymbol{p}}^{2}}\rangle_{i}\;,$
(5)

where the sum runs over different mediums configurations generated according
to Eq. 3 and the single brackets denote quantum expectation values.

To implement the evolution of the jet in a quantum computer we use the quantum
simulation algorithm [7]. Its main features can be summarized as follows:

  1. 1.

Input: A description of the target quantum system in terms of the system
Hamiltonian $H$ and underlying Hilbert space.

  2. 2.

Encoding: Mapping the degrees of freedom of the problem to qubits on the
quantum computer. One needs a mapping between the physical operators and
quantum gates that act on the quantum circuit.

  3. 3.

Initial state preparation: Preparing the initial state of the system from a
fiducial state given by the computer. In the current study, these states are
identified.

  4. 4.

Time evolution: After having prepared the initial state, one applies the time
evolution operator on the computational state, in terms of a series of quantum
gate operations.

  5. 5.

Measurement: Measurement of the final state, usually in such a way that
relevant correlators can be extracted efficiently.

##  3 Quantum circuit and results

The quantum circuit implementing the evolution of the quark according to the
quantum simulation algorithm is depicted in Fig. 1. There, the double lines
indicate a classical chanel of information which feeds into the quantum gate
implementing the evolution operator. Through this chanel, for each simulation,
one feeds the classical field values to the proper quantum circuit.

Figure 1: Circuit implementing the evolution of the quark in the medium. More
details can be found in Ref. [3].

The time evolution operator is decomposed into multiple small time evolution
operators. For each one of these small step evolutions, we further decompose
the evolution into a term controlled by the kinetic operator and the potential
term; for more details see Refs. [1, 3]. Finally, since we are working with a
digital computer, one must provide a finite description of the problem. For
that, we introduce a transverse lattice with position space spacing
$a_{\perp}$. Using a binary mapping, we can then match the momentum/spatial
modes of the probe to the qubits in the computer [1].

From the simulations, we extract the underlying transverse momentum
distribution of the quark by directly measuring the full quantum state. Then
from the constructed distribution one can compute $\hat{q}$ according to Eq.
4. Such an approach is inefficient and more optimized forms of extracting
correlators can be devised, see e.g. Refs. [1, 3] for further discussion.

Figure 2: Left: Jet quenching parameter as a function of the saturation scale
for two different lattices in a U(1) medium at infinite jet energy. Right: The
same observable for a SU(2) medium, using the coarser lattice and implementing
different values for $N_{\eta}$ at finite jet energy. Figures taken from Ref.
[1].

In Fig. 2 (left), we show the results obtained for the evolution of the jet
quenching coefficient $\hat{q}$ as a function of the saturation scale for a
U(1) background with $p^{+}=\infty$. We perform the simulation using two
transverse lattices: one with lattice spacing in position space
$a_{\perp}=0.30\,{\rm GeV}^{-1}$ and a finer one with $a_{\perp}=0.15\,{\rm
GeV}^{-1}$. In practice, the difference between these two lattices is related
to their sensitivity to discretization effects, with the finer lattice being
less sensitive. Indeed, we observe this in Fig. 2 (left), with the coarser
lattice showing a non-linear evolution at large values of
$Q_{s}^{2}=C_{F}g^{4}\mu^{2}L/(2\pi)$, which are not expected when compared to
the analytical calculation of $\hat{q}$ [1, 4]. When moving to the finer
lattice, we see that indeed these effects disappear and one recovers the
expected behavior.

Figure 3: Jet quenching parameter as a function of the saturation scale for an
anisotropic medium. Figure from Ref. [1].

The same exercise can be performed for a SU(2) medium at finite jet energy
($p^{+}=200$ GeV). The results are shown in Fig. 2 (right) for the same coarse
lattice. For these simulations, one needs to do an extra subdivision of the
evolution operator in order to ensure that Eq. 3 is satisfied. For that we
introduce $N_{\eta}$ layers, within each we generate a novel field
configuration. Indeed, we observe in the numerical result that as the number
of layers increases, the the curves tend to converge towards the analytical
result (modulus lattice effects).

Finally, the current approach can be used beyond the simplest models of the
medium. An example of an interesting scenario, which has been recently
explored [2, 5, 9], is to consider a medium with anisotropic transverse
profile. In Fig. 3 we show the results for $\hat{q}$ in the presence of a
medium where the Debye mass $m_{g}({\boldsymbol{x}})=m_{g}(1+\nabla
m_{g}\cdot{\boldsymbol{x}})$. Then, varying the value of $|\nabla m_{g}|$, we
observe that $\hat{q}$ gets corrected with respect to the homogenous case. In
particular, these modifications are not energy suppressed, unlike the ones
considered in Refs. [2, 5]. In conclusion, our approach offers a way to
explore this type of novel effects and gauge their importance for jet
quenching phenomenology.

##  4 Conclusion

In this proceeding, we have shown that the evolution of a single particle in a
QCD background can be studied using a quantum computer. Our results reproduce
known analytical results, but also point towards applications of this approach
to more realistic scenarios hard to access analytically. In the future, we
plan to extend this program to include the production of soft gluon radiation.

##  5 Acknowledgements

We are grateful to X. Du, M. Li, C. A. Salgado and W. Qian, who have made
important contributions to Ref. [1]. J.B.’s work is supported by the U.S.
Department of Energy, Office of Science, National Quantum Information Science
Research Centers under the “Co-design Center for Quantum Advantage” award and
by the U.S. Department of Energy, Office of Science, Office of Nuclear
Physics, under contract No. DE-SC0012704.

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