# Towards quantum-enabled cell-centric therapeutics

Saugata Basu  Purdue University, Department of Mathematics, West Lafayette,
IN, USA  Jannis Born  IBM Research, IBM Research Europe, Zurich, Switzerland
Aritra Bose  IBM Research, IBM Thomas J Watson Research Center, Yorktown
Heights, NY, USA  Sara Capponi  IBM Research, Almaden Research Center, San
Jose, CA, USA  Center for Cellular Construction, San Francisco, CA, USA
Dimitra Chalkia  Athos Therapeutics Inc., Los Angeles, CA, USA  Timothy A Chan
Center for Immunotherapy and Precision-Immuno-Oncology, Cleveland Clinic,
Cleveland, OH, USA  National Center for Regenerative Medicine, Cleveland
Clinic, Cleveland, OH, USA  Hakan Doga  IBM Quantum, Almaden Research Center,
San Jose, CA, USA  Frederik F. Flöther  QuantumBasel, uptownBasel Infinity
Corp., Arlesheim, Switzerland  Gad Getz  Massachusetts General Hospital Cancer
Center, Boston, MA, USA  Broad Institute of MIT and Harvard, Cambridge, MA,
USA  Department of Pathology, Massachusetts General Hospital, Boston, MA, USA
Department of Pathology, Harvard Medical School, Boston, MA, USA  Mark
Goldsmith  Algorithmiq Ltd, Helsinki, Finland  Tanvi Gujarati  IBM Quantum,
Almaden Research Center, San Jose, CA, USA  Aldo Guzmán-Sáenz  IBM Research,
IBM Thomas J Watson Research Center, Yorktown Heights, NY, USA  Dimitrios
Iliopoulos  Athos Therapeutics Inc., Los Angeles, CA, USA  Gavin O. Jones  IBM
Quantum, Almaden Research Center, San Jose, CA, USA  Stefan Knecht
Algorithmiq Ltd, Helsinki, Finland  Dhiraj Madan  IBM Research, IBM Research
India, India  Sabrina Maniscalco  Algorithmiq Ltd, Helsinki, Finland  Nicola
Mariella  IBM Quantum, IBM Research Europe, Dublin, Ireland  Joseph A. Morrone
IBM Research, IBM Thomas J Watson Research Center, Yorktown Heights, NY, USA
Khadijeh Najafi  IBM Quantum, IBM Thomas J Watson Research Center, Yorktown
Heights, NY, USA  Pushpak Pati  IBM Research, IBM Research Europe, Zurich,
Switzerland  Daniel Platt  IBM Research, IBM Thomas J Watson Research Center,
Yorktown Heights, NY, USA  Maria Anna Rapsomaniki  IBM Research, IBM Research
Europe, Zurich, Switzerland  Anupama Ray  IBM Research, IBM Research India,
India  Kahn Rhrissorrakrai  IBM Research, IBM Thomas J Watson Research Center,
Yorktown Heights, NY, USA  Omar Shehab  IBM Quantum, IBM Thomas J Watson
Research Center, Yorktown Heights, NY, USA  Ivano Tavernelli  IBM Quantum, IBM
Research Europe, Zurich, Switzerland  Meltem Tolunay  IBM Quantum, Almaden
Research Center, San Jose, CA, USA  Filippo Utro  IBM Research, IBM Thomas J
Watson Research Center, Yorktown Heights, NY, USA  Stefan Woerner  IBM
Quantum, IBM Research Europe, Zurich, Switzerland  Sergiy Zhuk  IBM Quantum,
IBM Research Europe, Dublin, Ireland  Jeannette M. Garcia Corresponding
author. {jmgarcia<EMAIL_ADDRESS>listed in alphabetical order with
exception of corresponding authors. IBM Quantum, Almaden Research Center, San
Jose, CA, USA  Laxmi Parida†‡ IBM Research, IBM Thomas J Watson Research
Center, Yorktown Heights, NY, USA

###### Abstract

In recent years, there has been tremendous progress in the development of
quantum computing hardware, algorithms and services leading to the expectation
that in the near future quantum computers will be capable of performing
simulations for natural science applications, operations research, and machine
learning at scales mostly inaccessible to classical computers. Whereas the
impact of quantum computing has already started to be recognized in fields
such as cryptanalysis, natural science simulations, and optimization among
others, very little is known about the full potential of quantum computing
simulations and machine learning in the realm of healthcare and life science
(HCLS). Herein, we discuss the transformational changes we expect from the use
of quantum computation for HCLS research, more specifically in the field of
cell-centric therapeutics. Moreover, we identify and elaborate open problems
in cell engineering, tissue modeling, perturbation modeling, and bio-topology
while discussing candidate quantum algorithms for research on these topics and
their potential advantages over classical computational approaches.

##  1 Introduction

The history of computing is a story of remarkable achievements that continue
to transform almost every aspect of our society. From the invention of the von
Neumann architecture to the emergence of Moore’s law to the recent rise of
artificial intelligence (AI), computing has enabled unprecedented advances in
natural science, engineering, and medicine [1]. However, as we approach the
physical limits of classical computing, we face new challenges and
opportunities that require a paradigm shift in how we process and manipulate
information. Quantum computing can provide such a paradigm shift.

By harnessing the power of quantum physics, quantum computers can potentially
perform some tasks exponentially faster than classical computers and solve
problems that are practically intractable for classical computers, such as
simulating quantum mechanical systems [2] or decrypting contemporary
cryptography [3]. With their exponential rise in computational power (each
additional qubit doubles the quantum state space), quantum computers may
enable unprecedented transformative potential in the next decades. However to
effectively steer quantum algorithm development and avoid ”reinventing the
wheel,” application domains with vast untapped potential for novel
computational approaches need to be identified. We believe that, similar to
high-energy physics [4], healthcare [5] and drug discovery [6] are prime
examples of areas that could see a tremendous impact from quantum-classical
computational workflows because they require accurate and reliable simulations
of complex systems (for example, molecules, proteins, and cells) or
necessitate learning complex behaviors from limited experimental data. In the
following, we identify areas in healthcare and life sciences (HCLS) that have
seen great advances in the recent years and in which we believe significant
benefits from quantum computing applications will be possible.

### Current technological shift in healthcare and life sciences

In the past decade, technological advancements have turned biological
discovery into an information-rich, quantitative science. From super-
resolution microscopy techniques that image macromolecules with nanometer
resolution [7] to organoid technologies that mimic human organs [8] to spatial
single-cell methods able to generate three-dimensional molecular maps of whole
tissues [9], new ways of interrogating biological systems across all scales of
organization have emerged. These technologies have fueled ambitious efforts,
such as creating a human cell atlas, i.e. a detailed map of all individual
cells in the human body [10], and have transformed how we explore fundamental
questions in health and disease. Although most outcomes are still descriptive,
studies that show promise in identifying patterns with clinical significance
are appearing across a variety of diseases, including cancer [11, 12, 13],
cardiovascular disease [14], and diabetes [15]. As a result of such
technological advancements, one of the emerging paradigms is the ability to
engineer cells to carry out therapeutic functions [16, 17]. Reprogramming
immune cells has been proven to be successful in treating hematological
cancers [18, 19, 20, 21, 22] and the effort has recently been extended to
treating solid tumors [23, 24, 25] and other diseases [26, 27, 28, 29], while
also taking advantage of the most recent technologies such as mRNA delivery
[30, 31].

On the computational front, AI (herein defined as intelligent software
automating routine labor, understanding and/or recognizing images, text
patterns, etc.) and machine learning (ML) (herein defined as the set of
algorithms and the mathematical and statistical methods allowing the computer
to learn from data) have accelerated discovery in HCLS. The use of AI and ML
have revolutionized several fields favoring the development and implementation
of novel methodologies often based on data-driven approaches. One prominent
example of the data-driven solutions provided by AI is in the field of
structural biology, where the longstanding problem of predicting the three-
dimensional (3D) structure of a protein given a sequence [32] has seen
significant improvement via transformer-based architectures [33]. This work
has had a profound effect on the field of synthetic biology by showcasing the
potential for using data-driven approaches based on ML methods to solve
scientific problems. For instance, in the last few years, novel ML
architectures have been developed to generate large protein complexes [34, 35,
36, 37] and design de novo proteins and enzymes [38, 39, 40, 41, 42].

AI/ML models have been used successfully to predict the effects of noncoding
variants [43] and reach human-level performance in automated whole-cell
segmentation, a task that traditionally involved hours of manual processing
[44]. Additionally, several AI/ML approaches have shown great promise in
improved disease diagnosis or prognosis. Deep convolutional neural networks
(CNNs) have matched the accuracy of radiologists for predictions of lung
cancer risk from CT images [45], have outperformed human dermatologists in
classification of skin lesions [46], and have exceeded the performance of
established models for breast cancer risk discrimination [47]. As AI is
rapidly moving towards foundation model-based learning [48], generalist multi-
task medical AI foundation models are emerging [49]. Although the progress in
applying AI models to biological data has been impressive, important
limitations that hinder their applications to the clinic still persist [50].
While some limitations are related to native properties of biological systems,
such as their innate complexity and scale [51], others are associated with
shortcomings of AI algorithms, e.g. their inability to learn in data-limited
contexts, model overfitting, or learning saturation [52].

### Quantum-enabled healthcare and life science trajectory

Quantum computing may soon provide researchers with quantum-enabled tools that
could expand the limits of computing to unprecedented capabilities, opening up
previously unimagined avenues for addressing some of these challenges. Quantum
algorithms make use of a radically different computing paradigm that may
potentially represent and learn from biological data more efficiently,
tackling classically difficult computing problems in healthcare and life
sciences. Here, we advocate for the adoption of quantum computing to open up
new frontiers for biological research that could enable biomedical
discoveries. There exist multiple areas for potential significant impact by
quantum computing in HCLS that each merit deeper discussions, including
biomarker discovery, clinical trial optimization, imaging analysis, and drug
protein design and discovery. While naturally there has been a great deal of
focus on chemical simulations in biomolecular systems using techniques from
quantum simulation [53, 54, 55], here we will focus on applications of quantum
machine learning and optimization that have newly realizable potential for
healthcare applications owing to recent advancements in quantum hardware and
software development. We will elaborate on some of these technological
advances in quantum computing from both a hardware and software perspective,
with an emphasis on quantum optimization and quantum machine learning.
Importantly, we will present our vision to reimagine healthcare and drug
discovery, summarized by Quantum Enabled Cell-Centric Therapeutics, which aims
to leverage advancements in single cell and spatial single-cell technologies
to create a holistic view of the cellular and metabolic activities in disease
tissue to better understand disease dynamics and improve therapeutic design.
Here we will highlight four areas of research explorations that address
various aspects of a cell-centric therapeutic design philosophy and that may
serve as a example of how bringing together quantum computing advancements,
AI/ML models, and cutting-edge developments in biological research can
transform therapeutic discovery and improve healthcare. There is a growing
body of work exploring the application of quantum technologies in healthcare,
medicine, and the life sciences [56]. Hence, we intend for this paper to also
serve as a call to both quantum and HCLS researchers to participate in and
help shape this vision, devising new biology-inspired quantum algorithms and
proof-of-concepts.

##  2 Quantum computing state-of-the-art

Challenges in healthcare and life sciences present opportunities to leverage
the unique features of quantum computing to derive novel biological insights
to improve patient care. The scale of today’s quantum devices are on the order
of tens to hundreds of qubits and remain susceptible to noise [57]. Moreover,
the development of error correction techniques used to protect quantum
information is still in an early stage [58, 59, 60]. However with recent
quantum developments, it may now be the time to begin addressing these HCLS
challenges in earnest. Qubit counts have increased [57], dynamic circuits with
mid-circuit measurements have been introduced [61], the fidelities of 1- and
2-qubit gates have improved [62], and the speed of execution of quantum
circuits has increased [63]. Advances in error mitigation and error
suppression techniques, when coupled with circuit cutting and knitting
techniques [64, 65, 66], have enabled researchers to scale up the size of
their quantum experiments. As an example, quantum chemistry simulations using
circuit cutting techniques, such as entanglement forging, have enabled users
to utilize double the number of qubits [67] while quantum-classical embedding
techniques enabled the scaling up to relevant system sizes [68, 69]. These
advancements have demonstrated significant progress towards reaching the stage
where quantum computers can solve certain meaningful problems faster, cheaper,
or more accurately than classical computers alone for selected applications
[70, 71, 72]. This progress has led to the emergence of quantum machine
learning (QML), which aims to use quantum algorithms to analyze large datasets
more efficiently than classical machine learning algorithms [73]. Depending on
the specific use case, quantum computing in general, and QML in particular,
may enable different types of benefits, in metrics such as accuracy, energy
efficiency, input data requirements, and speed. For instance, there is early
evidence in specific instances that quantum algorithms applied to electronic
health records are better at handling small, noisy data sets and producing
acceptable accuracies than classical approaches [74]. At the end of 2022, IBM
announced the production of a 433-qubit processor and that a 100-qubit device
capable of achieving a depth of 100 will be available in 2024, representing a
new testbed for quantum circuits [75]. It is now possible for quantum circuits
to be designed for simple drug discovery problems [6, 76, 77, 78] and executed
on real quantum hardware using these new capabilities, such as circuit cutting
and error mitigation, to complement research on advancing the classical and
quantum algorithms.

Figure 1:  Quantum computing state-of-the-art.
A: Reproduced with permission from [63]. Different levels of metrics to
express the quality of quantum hardware. Along with number of qubits, quantum
gate quality is an important quality metric. This is typically expressed in
terms of Quantum Volume first introduced in [79]. Another quality metric is
the speed of execution of quantum circuit layers which is expressed as CLOPS
first introduced in [63]. B: A mind-map of representative quantum algorithms.
C: Qiskit Runtime (accessed on May 10, 2023 [80]) allows a server-less quantum
program execution environment.

###  2.1 Quantum hardware

Like bits for classical computers, qubits are the basic unit of quantum
computation. While classical bits can only be in a state of $0$ or $1$, qubits
can exist in a superposition of both states, meaning that they can represent
multiple states simultaneously. Additionally, qubits can be entangled, which
means that their states are correlated in a way without classical equivalent.
Finally, qubits are measured probabilistically and one can measure the
interference of their probabilities. These three properties come from the
principles of quantum mechanics [81]. There are several approaches to building
qubits, including neutral atom qubits [82], spin qubits [83], topological
qubits [84], trapped-ion qubits, and superconducting qubits, each with its
advantages and challenges as assessed by different quantum computing metrics
(Fig. 1A).

While much of the quantum algorithms and applications that we will discuss
here are platform-agnostic, we will focus the majority of our discussion on
one of the most widely used qubit technologies, superconducting qubits. These
qubits rely on the employment of superconducting Josephson’s junctions, which
leverage the properties of superconducting materials to create and manipulate
the two level systems required for quantum computation. These superconducting
qubits are made using superconducting circuits [72, 85, 86] that are required
to operate at extremely low temperatures, typically just a few degrees above
absolute zero, in order to take advantage of the unique property of such
materials, which can conduct electricity with no resistance Superconducting
qubits are promising candidates for building quantum computers because they
can be easily fabricated using standard semiconductor fabrication techniques
and have demonstrated long coherence times.

###  2.2 Near-term vs. fault-tolerant quantum computing

Near-term quantum computing refers to the current state of quantum computing
technology, where small-scale quantum processors are available for research
and development purposes. These devices can perform simple quantum algorithms
with limited numbers of qubits. However, these devices are prone to errors,
and must be carefully calibrated and controlled to produce reliable results.
Fault-tolerant quantum computing, on the other hand, refers to the theoretical
possibility of building large-scale quantum processors that can operate
reliably in the presence of noise and errors. These devices would be able to
run complex quantum algorithms with many qubits, and would be a major
breakthrough in the development of practical quantum computing technology.

The key difference between near-term and fault-tolerant quantum computing
(FTQC) is how errors are handled. FTQC uses quantum error correction (QEC) and
near-term quantum computing uses error mitigation that trades circuit
executions for reduced impact of noise for certain measurements such as
observables. In earlier devices, error mitigation [87, 88] is limited, and the
devices are only capable of running small-scale quantum algorithms that can
tolerate some errors. Moreover, application of different circuit compilation
[89] and decomposition techniques [64] are relatively explicit in near-term
quantum computing. However, evidence of quantum utility through quantum error
mitigation have started emerging for nontrivial problems [90]. Both the
increase in the number of qubits and improved gate quality have contributed to
the progress. In fault-tolerant devices [91], error correction is much more
robust, and it requires error rates that are not yet readily achievable, much
less at the scale that is needed and hence QEC is just not practical soon.
Therefore, advances both in error correction and error mitigation are needed
to create a continuous path to quantum advantage. Another important difference
is the number of qubits that can be used. Near-term devices typically have on
the order of hundreds of qubits, numbers and qubit quality will continue to
scale in the near future, where error-corrected devices may require thousands
or even millions of qubits [92]. Thus, fault-tolerant quantum computing will
require the development of new technologies for qubit fabrication, control,
and error correction.

Despite these challenges, researchers are making progress towards fault-
tolerant quantum computing, with many promising new technologies and
algorithms being developed. However, it is still unclear when or if practical
fault-tolerant quantum computers will be built. In the meantime, near-term
quantum computing remains an active area of research and development, with
many companies and research groups working to build better quantum processors
and software tools. These devices are expected to have an impact on fields
such as chemistry, optimization, and machine learning.

###  2.3 Quantum toolkit and services

The history of quantum software toolkits in recent years has been
characterized by rapid growth and development, as researchers and companies
worked to create more advanced and user-friendly tools for quantum
programming. In 2017, IBM released Qiskit, an open-source quantum software
toolkit that quickly became one of the most widely used tools in the field
[93]. Qiskit runs on IBM Quantum Experience, the first ever cloud-based
quantum computing service. Qiskit provides users with a comprehensive set of
tools for designing and optimizing quantum circuits, as well as simulating and
running quantum programs on real quantum hardware. Other toolkits include
Forest [94], PennyLane [95], CirQ [96], Braket [97], and so on. Well-developed
quantum software toolkits, like Qiskit, tend to have complex and elaborate
modules for different domains of applications. Qiskit provides a server-less
model to off-load quantum computes to IBM Quantum cloud through Qiskit Runtime
as shown in Figure 1C. Toolkits, like Qiskit, are essential for enabling
researchers and developers to explore the potential of quantum computing, and
to build the next generation of quantum applications, such as Algorithmiq’s
Aurora [98], a package for advanced quantum chemistry simulations for drug
design and discovery. As these quantum software toolkits continue to evolve
and improve, we will see new tools, features, and applications brought online
to unlock new areas of HCLS research that are difficult to address using
classical computing technologies.

###  2.4 Quantum hardware roadmap

Quantum computing is expected to go through revolutionary changes in the
coming years, including expected near term advancements in architectural
primitives like modular chip design, classically- and quantumly-connected
modules, etc. A combination of larger number of qubits, higher quality quantum
gates, and faster clock cycles will allow users to solve significantly larger
problems that are common in healthcare and life science. At the kernel level,
threaded primitives, dynamics circuit, built-in error mitigation are also
expected to be available at similar timeframe. These will allow an algorithm
developer to take advantage of advanced algorithm development techniques from
classical computing and expand them into quantum algorithms. Quantum
serverless architecture along with advanced circuit decomposition and
compilation techniques will enable users to scale up their problem size
rapidly. In addition to IBM’s announced production of a 433-qubit processor in
2022 and a 100-qubit device capable of achieving a depth of 100 available in
2024 [75], IBM has targeted the development of a 100,000 qubit quantum system
by 2033 [99]. In summary, we are entering a new regime in which quantum
computers can be used to study and gain insights for important scientific
problems, where any non-trivial result will not be easily certifiable with
classical alternatives. Deeper collaboration with domain experts is needed to
identify the smallest practical problem at that scale. Overall, recent
developments in quantum computing hardware are driving progress towards the
realization of practical quantum computers and empowering quantum algorithms
to achieve heretofore unattainable results on actual quantum hardware, and we
will provide, as an example, how these quantum advancements can be applied
towards a cell-centric therapeutic discovery process.

###  2.5 Quantum algorithms

The history of quantum algorithm development began in the early 1980s when
Richard Feynman and Yuri Manin suggested that quantum computers could solve
problems that classical computers could not [100, 101]. In 1994, Peter Shor
developed a quantum algorithm for factoring large numbers, which is believed
to be intractable for classical computers [3]. This algorithm is the basis for
much of modern cryptography. In 1996, Lov Grover developed a quantum algorithm
for searching unsorted databases, which can provide a quadratic speedup
compared to classical algorithms [102]. In 1997, Seth Lloyd developed a
quantum algorithm for simulating quantum systems, which has potential
applications in fields such as chemistry and materials science [103]. Since
then, researchers developed a variety of other quantum algorithms (Fig. 1B)
for tasks such as solving linear equations [104], classification [105],
optimizing functions [106], and simulating physical systems [107, 108]. These
algorithms have been further specialized and optimized for different areas of
applications including HCLS [6, 74].

####  2.5.1 Quantum simulation

Quantum algorithms for chemistry simulation are a promising new approach to
studying the behavior of atoms and molecules [109]. These algorithms exploit
the power of quantum computers to simulate the quantum mechanical effects that
are essential for understanding the properties of matter. One of the most
important quantum algorithms for chemistry simulation is the variational
quantum eigensolver (VQE) algorithm [107]. VQE can be used to calculate the
ground state energy of a molecule, which is a fundamental property that
determines its stability and reactivity. The VQE algorithm has been used to
simulate the properties of a wide variety of molecules, including water [110],
methane [111], deuteron [112, 113], and large organic molecules such as
butyronitrile [69] and ferrocence [114].

Quantum algorithms are also being developed to simulate excited states of
quantum systems. Excited state simulations are needed for understanding the
behavior of many physical and chemical systems, including electronic and
magnetic properties of materials. While VQEs have been used to find these
excited state energies of a quantum system by optimizing a variational ansatz
wave function [115], quantum subspace expansion (QSE) and the quantum
equation-of-motion (qEOM) algorithms [116] have also been developed to perform
excited state simulations. [117] These methods simulate the excited states of
a quantum system by expanding the wave function in a basis of excited states
[118]. The quantum Lanczos algorithm has also been found useful for excited
state simulations [119] by efficiently computing the eigenvalues and
eigenvectors of a Hamiltonian matrix, which can be used to determine the
excited states of a quantum system. In addition to these specific algorithms,
there is growing interest in developing general-purpose quantum algorithms for
excited state simulations. These algorithms would be able to simulate a range
of quantum systems and phenomena, including the excited states of many-body
systems. The development of quantum algorithms for ground and excited state
simulations is still in its early stages, but researchers are optimistic about
the potential impact of these algorithms on the field.

####  2.5.2 Quantum operations research

Quantum algorithms for operations research is a rapidly evolving area of
research that focuses on applying quantum computing to solve problems in
operations research. Quantum Monte Carlo algorithms can simulate the behavior
of quantum systems, which is useful for solving problems in optimization,
finance, and other fields [66]. Quantum algorithms for mixed-integer
programming problems are able to exploit problem structure to maximize
fractional Grover speedup [120]. Quantum walk algorithms may have the
potential for quantum advantage over classical Monte Carlo or random walk
algorithms [121, 122, 123]. Additionally, a large number of optimization
problems may be represented as quadratic unconstrained binary optimization
(QUBO) problems which allows quantum approximate optimization algorithm (QAOA)
to create the potential of maximizing fractional Grover speedup [106, 124].

####  2.5.3 Quantum machine learning

Quantum machine learning (QML) is an evolving field that combines the power of
quantum computing with the techniques of machine learning [125, 126, 73]. QML
algorithms have the potential to significantly improve the accuracy and
efficiency of machine learning models, particularly for large datasets. One
key advantage of QML is the ability to perform certain computations
exponentially faster and more efficiently than classical computers [105, 127].
Another advantage is the potential for QML to solve problems that are not
easily solved with classical techniques. QML algorithms can be used for a
variety of tasks, including classification, regression, clustering, and
dimensionality reduction. Some popular QML algorithms include the quantum
support vector machine [128], quantum k-means clustering [129], and quantum
neural networks [130]. While QML is still in its early stages, researchers and
industry experts are optimistic about its potential impact on fields such as
drug discovery, finance, and materials science. However, one major challenge
of QML is the need for large-scale, fault-tolerant quantum computers, which
are not yet available. More importantly, the quantum amenability to practical
datasets for machine learning is still not well understood. Nevertheless,
quantum machine learning is an exciting area of research with promising
possibilities for advancing the field of machine learning.

Quantum Support Vector Machine The quantum support vector machine (QSVM) is an
emerging approach that combines principles from quantum computing and
classical machine learning to enhance the capabilities of support vector
machines (SVMs) [105]. SVMs are powerful algorithms used for classification
and regression tasks, but they face limitations when dealing with large and
complex datasets. QSVMs aim to overcome these limitations by leveraging
quantum feature maps that encodes information into a Hilbert space.
Fundamentally, the QSVM algorithm employs a quantum kernel function that
measures the similarity between quantum feature vectors as the inner product
between corresponding density matrices. These feature vectors encode
information about the input data and are represented as quantum states. The
kernel function can be computed by running the quantum circuit for feature
maps corresponding to one of the inputs followed by the inverse circuit for
the other quantum circuit and finally measuring in standard basis. One can
then optimize the dual on a classical machine based on the kernel function
computed above on quantum circuit. While the theoretical foundation of quantum
feature maps is still an area of active research, asymptotic quantum speedup
has been demonstrated for a certain quantum feature map [127], and empirical
quantum advantage has been demonstrated for electronic health record data in a
few very specific settings [74].

Quantum Neural Networks The convergence of quantum information science and
machine learning has given rise to a novel and promising paradigm known as
quantum neural networks (QNNs) [125, 73]. QNNs combine the principles of
quantum mechanics and classical neural networks. Like their classical
counterparts, QNNs consist of interconnected nodes, or ”neurons”, that process
and transmit information. However, unlike classical neural networks that rely
on classical bits, QNNs employ qubits as their basic units of information. The
architecture of a QNN comprises three key components: the input layer, hidden
layers, and output layer. Each layer consists of qubits and quantum gates. The
input layer encodes the input data, which is then processed through the hidden
layers via quantum gates. Finally, the output layer produces the desired
output based on the learned patterns. Training QNNs involves adjusting the
parameters of the quantum gates to optimize the network’s performance. Quantum
algorithms, such as quantum gradient descent and quantum variational
algorithms, play a crucial role in the training process. Quantum gradient
descent adapts the parameters of the gates by minimizing the loss function,
while variational algorithms optimize the parameters by leveraging quantum
optimization techniques. It has been shown that a class of quantum neural
networks is able to achieve a better effective dimension, which is a robust
capacity measure, than comparable feed-forward networks and can be trained
faster [130].

Quantum Topological Data Analysis Topological data analysis (TDA), introduced
in early 2000s [131, 132] is a novel data science method that combines tools
from algebraic topology and computational geometry to study and analyze the
shape of the data to reveal hidden structures and patterns while gaining
insights independent of noise. Some successful applications of TDA include
areas such as medicine, biology, image analysis, network analysis, and multi-
variate time series analysis [133, 134, 135, 136]. In particular, TDA can
efficiently extract higher dimensional features from a noisy data set. While
TDA found many applications in different research areas, including healthcare
and life sciences [137, 138, 139, 140, 141, 142], it is known that the
computational cost increases exponentially in the number of data points or in
the dimension of the topological features targeted. The first quantum
algorithm for TDA offered an exponential speed up targeting the regime where
classical TDA struggles, in particular higher dimensional features [143].
Under certain constraints, this algorithm utilizes some standard quantum
protocols such as multi-targeted Grover’s search algorithm and quantum phase
estimation (QPE). This created a surge of attention to develop improved
quantum algorithms for TDA [144, 145, 146, 147, 148, 149], providing both
theoretical and experimental framework for researchers to analyze higher-
dimensional data using quantum computing.

Quantum Cumulant Calculation Cumulant calculation is a mathematical technique
that can be used to analyze and understand complex, high-dimensional, and
noisy data sets including healthcare data, and has been used in conjunction
with TDA to better capture high-dimensional relationships. By using cumulant-
based analyses one can identify patterns and relationships within the data,
which are often challenging to extract using traditional statistical methods.
Cumulants provide a way to identify redescriptions [150], which can be used to
identify and generate logical relationships among variates that may indicate
underlying biological processes, and whose connectivities provide information
about distinct pathways to disease [151, 152]. The idea of redescriptions is
that patients who develop conditions (e.g. atherosclerosis) tend to share
clusters of other comorbidities (e.g. hypertension) and can be captured in
groups according to their diagnoses of these conditions. Often, healthcare
data are strongly correlated and therefore higher-order moments tend to
include the impact of such lower order correlations. These higher-order
associations may reveal specific biological pathways driving disease and can
lead to distinguishing multiple distinct pathways captured by edges in a
network formed by multi-omics variables [153, 154]. It is possible to exclude
potential spurious lower-order correlations by computing cumulants of higher-
order moments involving products of collections of variates since cumulants
vanish if the products of collections of variates partition into independent
subsets [155]. In order to study all the moments or the cumulants of a
sequence of random variables it is usual to consider the moment as well as the
cumulant generating functions. These are functions of $n$ indeterminates where
$n$ is the number of random variables in question. These generating functions,
which contain all the information about the moments and cumulants, can be
quite complicated and can be challenging to compute. An analogous situation
occurs in statistical physics. The partition function in various Ising models
is an exponential sized sum over all configurations where the indeterminates
correspond to magnetization variables. Computing the partition function is
equally challenging as computations of moments or cumulants because of the
exponential sized sums. Various approaches have been suggested for computing
good approximations. In particular, taking all the magentization variables to
be equal one obtains a polynomial in one variable. The study of the zeros of
this univariate polynomial (via the Lee-Young theorem [156]) gives important
information about the macroscopic behavior of the corresponding physical
system – such as phase transition. Efficient quantum circuits has been
developed for computing the complex zeros of the partition function. An
analogous quantum approach towards the computation of moments and cumulants
will be very interesting.

Quantum Network Medicine The emerging field of network medicine [157] applies
network science approaches to investigate the molecular complexity of a
particular disease, as well as the molecular relationships among apparently
distinct phenotypes, integrating information from relevant Omics databases.
The modern era has seen an exceptional growth in molecular interaction data
such as molecular networks, including protein interaction networks, whose
nodes are proteins that are linked to each other by physical interactions;
metabolic networks, whose nodes are metabolites that are linked if they
participate in the same biochemical reactions; and regulatory networks, whose
directed links typically represent regulatory relationships between a
transcription factor and a gene. However, such networks are known to be vastly
incomplete, with large proportions of the true interactions being yet unknown
[158]. Moreover, if we are to efficiently search for new drugs and drug
combinations or pathogenic interactions within and between cells, there is a
pressing need for computational methods that can access the immense molecular
space until now largely unexplored. Quantum computing may be a key ingredient
in enabling the full potential of network medicine. Recently, it has been
proposed to combine network medicine and quantum algorithms in a novel
research field, quantum network medicine, to lay the foundations of a new era
of disease prevention and drug design [159]. A successful example of the
potential of this field is the recent demonstration of a link prediction
algorithm based on continuous-time quantum walks [160]. This algorithm has
also been successfully adapted to identify disease modules and new disease
pathways.

##  3 Quantum-enabled cell-centric therapeutics

Therapeutic design and discovery has traditionally focused on drug-target
identification and interaction optimization, and there exist many classical
approaches to perform such analyses as well as clear quantum applicable
algorithms. This target-centric approach has been the dominant paradigm in
therapeutic design and led to the successful approval of many novel
therapeutics (e.g., small molecule inhibitors, chemotherapeutic, antibody
therapies) across a multitude of diseases. However, the cost in research &
development per new approved drug has been doubling $\sim$ nine years since
the 1950s [161] and, for many diseases (e.g., rare diseases or particularly
aggressive cancers like pancreatic), effective therapies are still far away.
While typically still considered the gold standard, the target-centric
approach to drug discovery appears to be falling short on delivering
significant numbers of therapeutic advances [162], which may be attributable
to the complexity and uncertainty in target validation [163]. The validity of
target-centric approaches may be reaching the point of diminishing returns as
evidenced by the observation that many anticancer drugs in clinical trials
exert their efficacy not through their ostensible mechanism of action but via
some off-target cytotoxicity [164]. Moreover, medicine itself is transforming.
As we progress towards precision medicine, treatments at the level of an
individual, and even better proactive interventions to keep people healthy,
are needed. Today, many standardized therapies fail to achieve their intended
outcomes, a case in point being that less than half of cancer patients respond
to immunotherapy [165], thus necessitating more tailored methods [166].

Figure 2: Overview of quantum-enabled cell-centric therapeutics.
Spatiotemporal single-cell, cell-line, imaging, drug profile, and clinical
data are analyzed with four quantum computing technologies to capture varying
aspects of cellular behavior. These technologies include: (top left) QCNNs to
learn optimal CAR T-cell intracellular signalling domain design from limited
experiment data; (center) hybrid classical-quantum GNNs to model tumor
microenvironments from single cell spatial data; (top right) single cell
perturbation response using Quantum Conditional OT; and (bottom) QTDA to
identify topological signatures of single cell perturbation response. Using
insights from each research area and by combining the efforts between them, we
aim to provide treatment options that optimize the cellular context to improve
therapeutic response.

With the advent of single-cell and spatial single-cell technologies there is
now an opportunity to more precisely understand the interactions between
disease cells, microenvironment, and therapeutics to accelerate cell-centric
therapeutic approaches and precision medicine. These technologies provide a
detailed accounting of the activities occurring within each cell and of how
those cells are interacting in a tissue. In addition, they enable the capture
of a holistic perspective of the cellular and metabolic activities of
malfunctioning tissue with single-cell precision, allowing the study of
disease dynamics in highly heterogeneous tissues with unprecedented
spatiotemporal resolution. The trajectory of these technological advancements
points towards a transformation of therapeutic design. This is made more
relevant as we seek to characterize complex diseases with heterogeneous
disease pathologies that cannot be described through single-protein effects,
and where a conventional target-centric approach cannot be successful without
the involvement of serendipity [167].

By using the data obtained from these new technologies to examine the cellular
state with respect to its cellular context and in combination with the
potential of quantum computing to complement classical computing techniques, a
cell-centric therapeutic design can be realized. Under this regime, the goal
is to identify therapeutics, biologics, or other interventions that modify the
cellular disease ecosystem to make it more responsive to therapy or shift it
to a quiescent or even dying state. This is achieved by modeling the disease
microenvironment at single-cell resolution to understand the cellular
interactions, feature space, and the needed changes that shift the environment
from a non-responsive to a responsive state.

Though classical AI methods have shown themselves to be critical in analysing
single cell and spatial single cell data, there remain challenges where
quantum computing approaches may offer significant advantages. Single cell
omics data is often high dimensional and with high sparsity where most genes
are not altered or expressed in any given single cell. These cells are
typically not labeled and thus semi-supervised approaches are needed to learn
from labeled and unlabeled single cell data. These challenges in single cell
analysis are being addressed with breakthrough advancements in transfer
learning and transformer models [168, 169, 170, 171]. Yet there remains
important limitations with these classical computing technologies, including
requiring large amount of training data, poor explainability or
interpretability, limitations in capturing global contextual signals,
inability to control attention, and quadratic space and time complexity. These
challenges are exacerbated when data sets from related techniques are
combined, for example from flow cytometry and single-cell sequencing [172];
classical methods struggle with leveraging such heterogeneous data to
effectively classify cells based on their physical and biochemical
characteristics.

A cell-centric approach to understand perturbation response and disease
behaviors presents an unique opportunity for multiple quantum and quantum-
classical optimization and machine learning techniques to be brought together
to address these challenges and further advance cell-centric therapeutic
design strategies. In the following, we will describe several avenues of
research capturing varying aspects of this cell-centric approach and how they
may come together by focusing on a cancer use case (Fig. 2). These include
approaches for optimizing CAR T-Cell engineering; representing and analyzing
spatial, single-cell data; developing predictive models of single-cell
perturbation; and extracting n-th order feature interactions that inform
cellular behavior.

###  3.1 Cell engineering for immunotherapy

Figure 3: Quantum Convolutional Neural Network for cellular engineering.
A: Cartoon of a Chimeric antigen receptor (CAR) T-cells (brown colored)
surrounding a tumor with its microenvironment. CAR T-cells with both the
engineered extracellular and intracellular domains of the receptor are shown.
B: Schematic representation of the combinatorial motif domain derived from a
library of signal motif domains. These domains can be arranged in up to three
positions within the CAR T-cell intracellular region. Each combination gives
rise to specific CAR T-cell phenotypes. The combinatorial space (cyan) is
typically much larger than the experimentally generated CAR T-cells used in
development (green). The optimal motif combination giving rise to optimal CAR
T cell phenotype may fall outside of what was experimentally generated (red).
C: Schematic representation of the learning of the optimal combination of the
signalling motifs using convolutional neural networks, either current
classical CNNs (top) or quantum CNNs (below). For the learning process,
experimental data are used as input data.

Cell therapies are powerful new medicines in which a class of human cells are
reprogrammed to carry out specific functions such as killing cancer cells (an
example is represented schematically in Fig. 3A). These therapies offer novel
approaches that eventually will lead to treatments of other diseases such as
autoimmune disorders, inflammation, and neuro-degeneration, by overcoming
immunosuppressive tumor microenvironments, reducing toxicities, and preventing
antigen escape. The great advantage of cell therapies lies in the possibility
of engineering each cell modular component, each characterized by specific
features, and building a synthetic molecule or cell with desired functions. A
cell’s phenotype can be reprogrammed by engineering cells at different scales,
whether single point amino acid mutations [173], designed peptides (e.g.
antimicrobial peptides) [174], protein receptors [175, 176, 177, 178], or
cells and tissues [179, 180, 181]. The downside of such reprogramming is that
the possibility of engineering and combining different cell modular components
generates a vast, complex combinatorial design space that is difficult to
explore experimentally. For this reason, AI/ML models are powerful tools to
address this challenge because such models can learn complex patterns and
features from a given dataset and provide predictions on different phenotypes
of interest.

Chimeric antigen receptors (CARs) are genetically modified T cell receptors
designed to repurpose the phenotypic output of natural T cells. The
extracellular domain of CAR T cells is engineered to identify specific tumor-
associated antigen and the intracellular domain is engineered to include
intracellular motifs that enhance the T cell activation and function (Fig.
3A). These motifs are part of the co-stimulatory domains and are responsible
for the antitumor efficacy of CAR T cells. Combinations of these domains
generate CAR T cells with different features, and currently, six CAR T cell
therapies for 12 applications are approved by the US Food and Drug
Administration (FDA) [182]. Hence, given the proven efficacy of this new
technology, the outstanding question is how can we efficiently explore all the
possible costimulatory domain combinations to design optimal CAR T cell for a
given patient? Since there are many potential combinations of associated sets
of signalling and activation motifs for an engineered T cell, novel approaches
based on screening pooled CAR signaling domain libraries have been proposed
[183, 184]. Specifically, KG Daniels et al. [22] seeking to optimize the
phenotypic response of CAR T cells, the authors defined a combinatorial
library of 13 motifs located in 3 different positions of the receptor
intracellular domain (Fig. 3B). With $\approx 2350$ potential combinations,
only $\approx 250$ were tested experimentally (Fig. 3B) and a ML model was
used to predict the cytotoxicity of the other remaining combinations. The ML
algorithm, based on a Convolutional Neural Network (CNN) (3C) with long-short
term memory (LSTM), was able to reach $\approx 70$% accuracy at predicting CAR
T cell phenotype.

Given the highly data-constrained problem described above, we have identified
this study as an instance where quantum neural networks (QNNs) would
potentially be able to provide advancements. Adding to the power of classical
neural network models, QNNs utilize quantum mechanical effects such as
superposition, entanglement, and interference to represent complex relations
among data. As such, certain QNN architectures have been shown to have greater
expressivity than some of their classical counterparts, allowing them to
capture more complex probability distributions than classical models, and
additionally indicating that there might be potential speed-ups in training
time. In general, identifying problems where QNNs are more advantageous than
classical models in training or model accuracy remains an open question with
great room for improvement and requires heuristic experimentation with the
datasets considered. As such the use of Quantum Convolutional Neural Networks
(QCNNs) [185] (Fig. 3C) can be employed to improve the aforementioned 70%
accuracy. QCNNs have several useful properties, including the number of
variational parameters that scale logarithmically with the number of qubits
and the absence of barren plateaus during training that can affect other types
of QNNs [186]. Finally, it has been shown that some quantum ML models, such as
QCNNs, can reach low generalization errors even in the case of limited
training data, which further motivates the use of QCNNs for this problem
domain.

###  3.2 Modeling tumor microenvironments with hybrid classical-quantum GNNs

Figure 4: Quantum Geometric Machine Learning for spatial tissue modeling.
A: Spatial distribution of cells in a tissue microenvironment is modeled as a
cell-graph, where nodes are cells, edges are cell-to-cell interactions, and
nodes encode cell-level multi-omics information. B: Learning an effective
representation of a cell-graph can address immediate clinical applications
(diagnosis, prognosis, treatment selection) as well as discover crucial
patterns driving the disease. To this end, we propose a novel hybrid
classical-quantum Graph Neural Network (GNN) by integrating the capabilities
of GNNs and quantum machine learning. C: Though a classical GNN operating on a
cell-graph can tackle the aforementioned applications, it poses a number of
crucial challenges (e.g., a sub-optimal embedding space, ineffective in
addressing $NP$-hard combinatorial tasks) that can benefit from Quantum
Computing. D: The proposed algorithm extends the classical embedding space to
exponentially large qubit Hilbert space and captures hidden correlations among
the cells to improve class separability. Showcased on spatial tissue modeling
for various downstream applications, this algorithm finds general utility in
other graph-structured data.

Spatial single-cell omics are currently revolutionizing and paving way for
advancing cancer biology by enabling deep phenotypic profiling of each
individual cell within the tumor ecosystem while preserving its topology
[187]. From multiplexed imaging to spatial single-cell transcriptomics, such
spatial data can be elegantly modeled using cell-graph representations (Fig.
4A), where cells are the nodes, including cell-specific information, and edges
denote cell-to-cell interactions. Consequently, Graph Neural Networks (GNNs)
have found initial applications in learning on spatial single-cell data (Fig.
4C). Modeling and learning on spatial cell-graphs exhibit several attractive
properties: GNNs explicitly learn on cells instead of encoding pixels, and can
elegantly integrate single-cell information with disease tissue (e.g. tumor)
morphology, topology, and interactions among cells and/or tissue structures
[188, 189]. The cell-graphs bestow an interpretable input space, which enables
incorporation of prior domain knowledge from medical experts. Furthermore,
GNNs are not limited by variations in resolution and can be easily coupled
with explainability techniques to provide valuable insights on which cells or
cell neighborhoods drive the decision [190]. Initial attempts to leverage the
advantages of GNNs on spatial single-cell data have started to emerge, with
applications in cell phenotyping [191], learning cell-cell communication [192,
193] and, recently, modeling tumor microenvironments [194]. Yet, the
complexity of tumor graphs and the entangled cell neighborhoods lead to
suboptimal embedding spaces of GNNs, which in turn struggle with learning
clinically meaningful patterns from the data. At the same time, searching for
relatively small query subgraphs over large, complex graphs is an $NP$-hard
problem where classical computing approaches do not suffice.

These limitations present an interesting opportunity for quantum computing.
Mapping spatial data to the exponentially large Hilbert spaces can potentially
solve the sub-optimal embedding of cell neighborhoods, searching for small
query sub-graphs within large graphs can be improved, and leveraging the
state-space available in quantum can lead to higher predictive ability of QML
models. It is also possible for quantum computing to implement biases and
symmetries as well as capture hidden correlations more efficiently (Fig. 4D).
Currently, hybrid quantum-classical solutions are implemented that combine
GNNs for data pre-processing with QML algorithms, such as Variational Quantum
Classifiers (VQC), Quantum Neural Networks (QNN), and Quantum Support Vector
Machines (QSVM), and validated for downstream tasks such as tumor subtyping
[195, 196] (Fig. 4B). Quantum versions of GNNs can also be created and
researched upon for these inherent graph problems to study the possible
advantages quantum can provide as new research directions and advance the
state-of-art in the spatial single-cell omics.

###  3.3 Inferring single-cell drug perturbations with quantum conditional
optimal transport

Figure 5: Quantum Conditional Optimal Transport for single-cell perturbation
modeling.
A: In a single cell perturbation atlas with a given initial cell state
(center), different perturbations (in this case drug administrations) result
in different cell states with varying favorability. B: Modeling perturbational
effects of biological systems with single cell resolution may impact immediate
clinical applications (personalized treatment recommendation) as well as
discovery of novel therapeutics. C: Rooting in optimization and transportation
theory, Conditional Optimal Transport recently emerged as a parametric
approach to estimate transportation plans conditioned on a context variable.
D: Building upon OT principles, we propose a novel Quantum algorithm for
parameterizable transportation map estimation that leverages unitarity as well
as Sinkhorn rescaling. Showcased on single cell perturbation tasks, this
algorithm finds general application in simulating effects of interventions on
a given system.

At the core of finding effective, novel therapeutics lies understanding the
tissue response to specific therapeutic interventions, such as drug
administrations. While cell line perturbation studies have long been
successfully used for pre-clinical validation of targeted cancer drug
candidates, recent technological developments have allowed for such
perturbation studies at the single-cell level [197]. In these studies, the
multiomic states for each individual cell are measured before and after drug
or genetic perturbations resulting in perturbation atlases that also capture
the underlying heterogeneity of drug response. These highly informative
atlases facilitate prediction of perturbation response on the tumour tissue at
the resolution of single-cells, in the near future even for entire tumors
(Fig. 5A). The overarching long term objective of computational models in this
field will be to simulate responses of tumor microenvironments to therapeutic
interventions, e.g. drug administrations, and, ultimately, to develop
trajectories that capture tumor growth patterns within its native
microenvironment. Early attempts in this direction used autoencoders with
linear latent arithmetics and achieved reasonable success at approximating the
effect of such interventions [198]. Notably however, the problem formulation
is deeply rooted in optimal transport theory, which in simple intuitive terms
can be illustrated by minimization of the earth mover distance when moving one
distribution of earth to another [199]. Thanks to approximation techniques,
such as entropic regularizations [200], Optimal Transport (OT) recently gained
popularity across machine learning applications [201]. Contemporary work
suggests that the posed task can be approached best with conditional OT [202]
– which leverages the OT principles, thus benefits from strong theoretical
support and explainability while allowing to condition the OT on a desired
perturbation (Fig. 5C).

We propose a novel hybrid quantum optimization algorithm for single-cell
perturbation modeling that is able to predict transportation maps tailored to
individual patients and therapeutic hypotheses. While the theoretical support
for such a quantum OT algorithm is in development, this formulation exploits
some natural links between unitary operators and the structure of the OT maps
(Fig. 5D). The proposed system is currently validated on several public
datasets of single-cell perturbations of different drugs, drug dosages, and
CRISPR gene knockouts [197, 203, 204]; with the objective to achieve a
significant performance improvement in capturing therapeutic effects on a
tumor level. Such a methodology naturally integrates with the previously
discussed activities, in particular the spatial tumor modeling work which
facilitates downstream clinical applications, e.g. recommendation of
therapeutic interventions, based on the simulated tumors (Fig. 5B). The method
is built largely in anticipation of novel single-cell data acquisition
techniques that will arise in the near future, such as spatiotemporal
perturbation data or combinatorial perturbations. Future extensions of this
quantum framework will focus on modeling cell system evolution over time,
e.g., reconstructing trajectories of cell differentiation. Remarkably, the
devised conditional OT methodology is generic and finds immediate application
across a rich set of problems in the healthcare and life sciences –
essentially all scenarios where the state of a system is captured before and
after a certain intervention and where the goal is to understand how the
intervention alters the before state to a hypothetical after state.

###  3.4 BioTopology for cellular behavior

Combining insights from both a spatial cell model and cell perturbation model
enabled with quantum would provide valuable insights into how tumors evade
therapeutic response. Yet it is known that despite the incredible amount of
information contained in imagery and omics data, this information is
incomplete. This incompleteness may be due to throughput constraints on
technology, constraints on funding, measurement variances between technology
platforms, or lacking the technology to measure every relevant molecular
entity within a cell at once. Furthermore, cell behaviors and hence disease
phenotypes are the result of complex relationships dynamically occurring over
time that can not be captured at scale. Hence novel methods are needed to
discover those cryptic relationships, likely residing in connections among
significant higher order interactions between the dimensions of a given
feature space.

Figure 6: Computational Topology for learning cellular behavior.
A: Examples of simplicial complexes detected in multidimensional data with
each increase in the filtration distance (left to right) represented as
increasing radii from the data points (top row) or new edges (red) drawn
between points (bottom row). B: For a multinomal dataset, n-th order
interactions and topological signatures can be discovered that define
different phenotypes. The association between features or cumulants with some
distance measure, e.g. pairwise Jaccard distances between features (heatmap).
Applying TDA to the projection of data in that defined distance space
generates topological features such as barcode signatures or cycles containing
n-th order interactions that describe known and novel phenotypes in the data.
C: Phase shifts in the computational time (z-axis) for classical TDA when
testing combinations of density threshold (x-axis) and maximum distance
(y-axis) when computing Vietoris-Rips complexes indicates there are specific
problems spaces where quantum computing has opportunity to provide significant
speed ups over classical TDA.

Topological data analysis and cumulants can capture these n-th order
interactions that would represent learned complex associations leading to
logical relationships explaining a given phenotype, like interactions between
the community of cells/features/alterations that lead to therapeutic response
(Fig. 6). As described earlier (see Section 2.5.3), TDA discovers hidden
structures in the shape of data to gain novel insights (Fig. 6A). TDA is
already being used in computer vision and atomistic models, and cumulants of
order $d>3$ are playing important roles in financial data analyses, economics,
hyper-spectral image analyses, etc. Yet, they are classically constrained as
to how far they can scale in terms of dimension ($d<5$) and feature space
($<100$). Cumulants are a useful tool for integrating and analysing multi-
omics data, whether single-cell or patient omics data, where they can provide
a unique measurement of liability or risk of a given disease or trait, for
each cell or individual (Fig. 6B). The association, i.e. distance, between
features or cumulants can be calculated and TDA applied to generate the
topological features, e.g., barcode signatures or cycles, that characterize
the clusters of features. The utility of this type of mining was demonstrated
by constructing aggregated phenotypes representing distinct pathways to
identify genetic variants relevant to that pathway via genome-wide association
studies [205], exploring metabolic syndrome with coronary artery disease, and
metabolic interactions leading to COVID-19 severity, where it was found that
renin angiotensin aldosterone system (RAAS) drugs mediated the risk of severe
COVID-19 due to hypertension, and that lipids mediated several other metabolic
syndrome risk factors for severity [153, 152].

Overall, TDA and cumulant calculation are powerful tools for analyzing
healthcare data, allowing researchers and healthcare professionals to gain a
deeper understanding of complex medical conditions and improve patient
outcomes. However there are classical bottlenecks in cumulant computation
owing to the combinatorial expansions of the multinomial feature combinations,
and in TDA computations where calculating exact and approximate Betti numbers
is $\\#P$ and $NP$-hard[147], respectively. For instance, in most analyses of
persistent Betti number calculations, there is a phase shift in the
computational time needed to perform the calculation (Fig. 6C). This shift
represents areas where quantum may offer particularly significant advantages
over classical TDA. These classical constraints can be overcome using QTDA and
quantum computing for cumulant computation.

Quantum computing offers the potential for moving past the classical
limitations for these techniques enabling more sophisticated applications of
TDA and cumulant analysis, and other mathematical techniques to improve
biomedical data analysis. Hereto, quantum walks may also prove useful on the
discovered knowledge graphs, as they have already been shown to be able to
infer missing links in protein-protein interaction networks [160]. By applying
these techniques to the spatial single cell omics data, it would be possible
to capture those cryptic interactions that would have been hidden even with
complete biological measurements let alone incomplete ones encountered in
practice to better inform spatial and perturbation models described above.

##  4 Conclusion

A cell-centric therapeutic design strategy can provide clinicians and patients
with much needed additional treatment options, bringing us significantly
closer to precision medicine. By developing a deeper understanding of and
modeling how cancer cells behave individually and in aggregate, treatment
plans can be developed to manipulate the cancer and its tumor microenvironment
into a more therapeutically responsive state or shift the tumor into an
indolent phase transforming the disease into a more manageable, chronic
condition. Quantum computing is a powerful enabling technology to help push
this approach to design therapeutics forward, and this case study may serve as
an exemplar of how quantum computing can meaningfully contribute to HCLS.

##  5 Acknowledgments

Portions of this material is based upon work supported by NSF Center for
Cellular Construction DBI-1548297 grant to S.C.

## References

  * [1] Ned Stafford.  Science in the digital age.  Nature, 467(7317):S19–S21, 2010.
  * [2] Bruce M Boghosian and Washington Taylor IV.  Simulating quantum mechanics on a quantum computer.  Physica D: Nonlinear Phenomena, 120(1-2):30–42, 1998.
  * [3] Peter W Shor.  Algorithms for quantum computation: discrete logarithms and factoring.  In Proceedings 35th annual symposium on foundations of computer science, pages 124–134. Ieee, 1994.
  * [4] Alberto Di Meglio, Karl Jansen, Ivano Tavernelli, Constantia Alexandrou, Srinivasan Arunachalam, Christian W. Bauer, Kerstin Borras, Stefano Carrazza, Arianna Crippa, Vincent Croft, Roland de Putter, Andrea Delgado, Vedran Dunjko, Daniel J. Egger, Elias Fernandez-Combarro, Elina Fuchs, Lena Funcke, Daniel Gonzalez-Cuadra, Michele Grossi, Jad C. Halimeh, Zoe Holmes, Stefan Kuhn, Denis Lacroix, Randy Lewis, Donatella Lucchesi, Miriam Lucio Martinez, Federico Meloni, Antonio Mezzacapo, Simone Montangero, Lento Nagano, Voica Radescu, Enrique Rico Ortega, Alessandro Roggero, Julian Schuhmacher, Joao Seixas, Pietro Silvi, Panagiotis Spentzouris, Francesco Tacchino, Kristan Temme, Koji Terashi, Jordi Tura, Cenk Tuysuz, Sofia Vallecorsa, Uwe-Jens Wiese, Shinjae Yoo, and Jinglei Zhang.  Quantum computing for high-energy physics: State of the art and challenges. summary of the qc4hep working group.  2023\.
  * [5] Frederik F. Flöther.  The state of quantum computing applications in health and medicine.  Research Directions: Quantum Technologies, page 1–21, 2023.
  * [6] Yudong Cao, Jhonathan Romero, and Alán Aspuru-Guzik.  Potential of quantum computing for drug discovery.  IBM Journal of Research and Development, 62(6):6–1, 2018.
  * [7] Lothar Schermelleh, Alexia Ferrand, Thomas Huser, Christian Eggeling, Markus Sauer, Oliver Biehlmaier, and Gregor PC Drummen.  Super-resolution microscopy demystified.  Nature cell biology, 21(1):72–84, 2019.
  * [8] Hans Clevers.  Modeling development and disease with organoids.  Cell, 165(7):1586–1597, 2016.
  * [9] Laura Kuett, Raúl Catena, Alaz Özcan, Alex Plüss, Peter Schraml, Holger Moch, Natalie de Souza, and Bernd Bodenmiller.  Three-dimensional imaging mass cytometry for highly multiplexed molecular and cellular mapping of tissues and the tumor microenvironment.  Nature cancer, 3(1):122–133, 2022.
  * [10] Orit Rozenblatt-Rosen, Michael JT Stubbington, Aviv Regev, and Sarah A Teichmann.  The human cell atlas: from vision to reality.  Nature, 550(7677):451–453, 2017.
  * [11] Anoop P Patel, Itay Tirosh, John J Trombetta, Alex K Shalek, Shawn M Gillespie, Hiroaki Wakimoto, Daniel P Cahill, Brian V Nahed, William T Curry, Robert L Martuza, et al.  Single-cell rna-seq highlights intratumoral heterogeneity in primary glioblastoma.  Science, 344(6190):1396–1401, 2014.
  * [12] Johanna Wagner, Maria Anna Rapsomaniki, Stéphane Chevrier, Tobias Anzeneder, Claus Langwieder, August Dykgers, Martin Rees, Annette Ramaswamy, Simone Muenst, Savas Deniz Soysal, et al.  A single-cell atlas of the tumor and immune ecosystem of human breast cancer.  Cell, 177(5):1330–1345, 2019.
  * [13] Itay Tirosh, Benjamin Izar, Sanjay M Prakadan, Marc H Wadsworth, Daniel Treacy, John J Trombetta, Asaf Rotem, Christopher Rodman, Christine Lian, George Murphy, et al.  Dissecting the multicellular ecosystem of metastatic melanoma by single-cell rna-seq.  Science, 352(6282):189–196, 2016.
  * [14] David T Paik, Sangkyun Cho, Lei Tian, Howard Y Chang, and Joseph C Wu.  Single-cell rna sequencing in cardiovascular development, disease and medicine.  Nature Reviews Cardiology, 17(8):457–473, 2020.
  * [15] Åsa Segerstolpe, Athanasia Palasantza, Pernilla Eliasson, Eva-Marie Andersson, Anne-Christine Andréasson, Xiaoyan Sun, Simone Picelli, Alan Sabirsh, Maryam Clausen, Magnus K Bjursell, et al.  Single-cell transcriptome profiling of human pancreatic islets in health and type 2 diabetes.  Cell metabolism, 24(4):593–607, 2016.
  * [16] Wendell A Lim.  Designing customized cell signalling circuits.  Nature reviews Molecular cell biology, 11(6):393–403, 2010.
  * [17] Michael A Fischbach, Jeffrey A Bluestone, and Wendell A Lim.  Cell-based therapeutics: the next pillar of medicine.  Science translational medicine, 5(179):179ps7–179ps7, 2013.
  * [18] Wendell A. Lim.  The emerging era of cell engineering: Harnessing the modularity of cells to program complex biological function.  Science, 378(6622):848–852, 2022.
  * [19] Carl H June and Michel Sadelain.  Chimeric antigen receptor therapy.  New England Journal of Medicine, 379(1):64–73, 2018.
  * [20] Evan W Weber, Marcela V Maus, and Crystal L Mackall.  The emerging landscape of immune cell therapies.  Cell, 181(1):46–62, 2020.
  * [21] Mihe Hong, Justin D Clubb, and Yvonne Y Chen.  Engineering car-t cells for next-generation cancer therapy.  Cancer cell, 38(4):473–488, 2020.
  * [22] Kyle G Daniels, Shangying Wang, Milos S Simic, Hersh K Bhargava, Sara Capponi, Yurie Tonai, Wei Yu, Simone Bianco, and Wendell A Lim.  Decoding car t cell phenotype using combinatorial signaling motif libraries and machine learning.  Science, 378(6625):1194–1200, 2022.
  * [23] Katharina Reinhard, Benjamin Rengstl, Petra Oehm, Kristina Michel, Arne Billmeier, Nina Hayduk, Oliver Klein, Kathrin Kuna, Yasmina Ouchan, Stefan Wöll, et al.  An rna vaccine drives expansion and efficacy of claudin-car-t cells against solid tumors.  Science, 367(6476):446–453, 2020.
  * [24] Jolanda De Vries and Carl Figdor.  Cancer vaccine triggers antiviral-type defences.  Nature, 534(7607):329–331, 2016.
  * [25] Lena M Kranz, Mustafa Diken, Heinrich Haas, Sebastian Kreiter, Carmen Loquai, Kerstin C Reuter, Martin Meng, Daniel Fritz, Fulvia Vascotto, Hossam Hefesha, et al.  Systemic rna delivery to dendritic cells exploits antiviral defence for cancer immunotherapy.  Nature, 534(7607):396–401, 2016.
  * [26] Andreas Mackensen, Fabian Müller, Dimitrios Mougiakakos, Sebastian Böltz, Artur Wilhelm, Michael Aigner, Simon Völkl, David Simon, Arnd Kleyer, Luis Munoz, et al.  Anti-cd19 car t cell therapy for refractory systemic lupus erythematosus.  Nature medicine, 28(10):2124–2132, 2022.
  * [27] Joel G Rurik, István Tombácz, Amir Yadegari, Pedro O Méndez Fernández, Swapnil V Shewale, Li Li, Toru Kimura, Ousamah Younoss Soliman, Tyler E Papp, Ying K Tam, et al.  Car t cells produced in vivo to treat cardiac injury.  Science, 375(6576):91–96, 2022.
  * [28] Corina Amor, Judith Feucht, Josef Leibold, Yu-Jui Ho, Changyu Zhu, Direna Alonso-Curbelo, Jorge Mansilla-Soto, Jacob A Boyer, Xiang Li, Theodoros Giavridis, et al.  Senolytic car t cells reverse senescence-associated pathologies.  Nature, 583(7814):127–132, 2020.
  * [29] Caroline Raffin, Linda T Vo, and Jeffrey A Bluestone.  Treg cell-based therapies: challenges and perspectives.  Nature Reviews Immunology, 20(3):158–172, 2020.
  * [30] Norbert Pardi, Michael J Hogan, Frederick W Porter, and Drew Weissman.  mrna vaccines—a new era in vaccinology.  Nature reviews Drug discovery, 17(4):261–279, 2018.
  * [31] Norbert Pardi, Michael J Hogan, and Drew Weissman.  Recent advances in mrna vaccine technology.  Current opinion in immunology, 65:14–20, 2020.
  * [32] Ken A Dill and Justin L MacCallum.  The protein-folding problem, 50 years on.  Science, 338(6110):1042–1046, 2012.
  * [33] John Jumper, Richard Evans, Alexander Pritzel, Tim Green, Michael Figurnov, Olaf Ronneberger, Kathryn Tunyasuvunakool, Russ Bates, Augustin Žídek, Anna Potapenko, et al.  Highly accurate protein structure prediction with alphafold.  Nature, 596(7873):583–589, 2021.
  * [34] Jung-Eun Shin, Adam J Riesselman, Aaron W Kollasch, Conor McMahon, Elana Simon, Chris Sander, Aashish Manglik, Andrew C Kruse, and Debora S Marks.  Protein design and variant prediction using autoregressive generative models.  Nature communications, 12(1):2403, 2021.
  * [35] Ian R Humphreys, Jimin Pei, Minkyung Baek, Aditya Krishnakumar, Ivan Anishchenko, Sergey Ovchinnikov, Jing Zhang, Travis J Ness, Sudeep Banjade, Saket R Bagde, et al.  Computed structures of core eukaryotic protein complexes.  Science, 374(6573):eabm4805, 2021.
  * [36] BIM Wicky, LF Milles, A Courbet, RJ Ragotte, J Dauparas, E Kinfu, S Tipps, RD Kibler, M Baek, F DiMaio, et al.  Hallucinating symmetric protein assemblies.  Science, 378(6615):56–61, 2022.
  * [37] Patrick Bryant, Gabriele Pozzati, Wensi Zhu, Aditi Shenoy, Petras Kundrotas, and Arne Elofsson.  Predicting the structure of large protein complexes using alphafold and monte carlo tree search.  Nature Communications, 13(1):6028, 2022.
  * [38] Ivan Anishchenko, Samuel J Pellock, Tamuka M Chidyausiku, Theresa A Ramelot, Sergey Ovchinnikov, Jingzhou Hao, Khushboo Bafna, Christoffer Norn, Alex Kang, Asim K Bera, et al.  De novo protein design by deep network hallucination.  Nature, 600(7889):547–552, 2021.
  * [39] Justas Dauparas, Ivan Anishchenko, Nathaniel Bennett, Hua Bai, Robert J Ragotte, Lukas F Milles, Basile IM Wicky, Alexis Courbet, Rob J de Haas, Neville Bethel, et al.  Robust deep learning–based protein sequence design using proteinmpnn.  Science, 378(6615):49–56, 2022.
  * [40] Jue Wang, Sidney Lisanza, David Juergens, Doug Tischer, Joseph L Watson, Karla M Castro, Robert Ragotte, Amijai Saragovi, Lukas F Milles, Minkyung Baek, et al.  Scaffolding protein functional sites using deep learning.  Science, 377(6604):387–394, 2022.
  * [41] Stanislav Mazurenko, Zbynek Prokop, and Jiri Damborsky.  Machine learning in enzyme engineering.  ACS Catalysis, 10(2):1210–1223, 2019.
  * [42] Kevin K Yang, Zachary Wu, and Frances H Arnold.  Machine-learning-guided directed evolution for protein engineering.  Nature methods, 16(8):687–694, 2019.
  * [43] Jian Zhou and Olga G Troyanskaya.  Predicting effects of noncoding variants with deep learning–based sequence model.  Nature methods, 12(10):931–934, 2015.
  * [44] Noah F Greenwald, Geneva Miller, Erick Moen, Alex Kong, Adam Kagel, Thomas Dougherty, Christine Camacho Fullaway, Brianna J McIntosh, Ke Xuan Leow, Morgan Sarah Schwartz, et al.  Whole-cell segmentation of tissue images with human-level performance using large-scale data annotation and deep learning.  Nature biotechnology, 40(4):555–565, 2022.
  * [45] Diego Ardila, Atilla P Kiraly, Sujeeth Bharadwaj, Bokyung Choi, Joshua J Reicher, Lily Peng, Daniel Tse, Mozziyar Etemadi, Wenxing Ye, Greg Corrado, et al.  End-to-end lung cancer screening with three-dimensional deep learning on low-dose chest computed tomography.  Nature medicine, 25(6):954–961, 2019.
  * [46] Andre Esteva, Brett Kuprel, Roberto A Novoa, Justin Ko, Susan M Swetter, Helen M Blau, and Sebastian Thrun.  Dermatologist-level classification of skin cancer with deep neural networks.  Nature, 542(7639):115–118, 2017.
  * [47] Adam Yala, Constance Lehman, Tal Schuster, Tally Portnoi, and Regina Barzilay.  A deep learning mammography-based model for improved breast cancer risk prediction.  Radiology, 292(1):60–66, 2019.
  * [48] Rishi Bommasani, Drew A Hudson, Ehsan Adeli, Russ Altman, Simran Arora, Sydney von Arx, Michael S Bernstein, Jeannette Bohg, Antoine Bosselut, Emma Brunskill, et al.  On the opportunities and risks of foundation models.  arXiv preprint arXiv:2108.07258, 2021.
  * [49] Michael Moor, Oishi Banerjee, Zahra Shakeri Hossein Abad, Harlan M Krumholz, Jure Leskovec, Eric J Topol, and Pranav Rajpurkar.  Foundation models for generalist medical artificial intelligence.  Nature, 616(7956):259–265, 2023.
  * [50] Christopher J Kelly, Alan Karthikesalingam, Mustafa Suleyman, Greg Corrado, and Dominic King.  Key challenges for delivering clinical impact with artificial intelligence.  BMC medicine, 17:1–9, 2019.
  * [51] Benjamin A Cordier, Nicolas PD Sawaya, Gian Giacomo Guerreschi, and Shannon K McWeeney.  Biology and medicine in the landscape of quantum advantages.  Journal of the Royal Society Interface, 19(196):20220541, 2022.
  * [52] Simon Ott, Adriano Barbosa-Silva, Kathrin Blagec, Jan Brauner, and Matthias Samwald.  Mapping global dynamics of benchmark creation and saturation in artificial intelligence.  Nature Communications, 13(1):6793, 2022.
  * [53] Andrew J Daley, Immanuel Bloch, Christian Kokail, Stuart Flannigan, Natalie Pearson, Matthias Troyer, and Peter Zoller.  Practical quantum advantage in quantum simulation.  Nature, 607(7920):667–676, 2022.
  * [54] Francesco Tacchino, Alessandro Chiesa, Stefano Carretta, and Dario Gerace.  Quantum computers as universal quantum simulators: state-of-the-art and perspectives.  Advanced Quantum Technologies, 3(3):1900052, 2020.
  * [55] Ehud Altman, Kenneth R Brown, Giuseppe Carleo, Lincoln D Carr, Eugene Demler, Cheng Chin, Brian DeMarco, Sophia E Economou, Mark A Eriksson, Kai-Mei C Fu, et al.  Quantum simulators: Architectures and opportunities.  PRX Quantum, 2(1):017003, 2021.
  * [56] Frederik F. Flöther and Paul F. Griffin.  How can quantum technologies be applied in healthcare, medicine and the life sciences?  Research Directions: Quantum Technologies, 1:e7, 2023.
  * [57] Jerry Chow, Oliver Dial, and Jay Gambetta.  IBM quantum breaks the 100‑qubit processor barrier.  https://research.ibm.com/blog/127-qubit-quantum-processor-eagle, Feb 2022\.
  * [58] Sophia Fuhui Lin, Joshua Viszlai, Kaitlin N Smith, Gokul Subramanian Ravi, Charles Yuan, Frederic T Chong, and Benjamin J Brown.  Empirical overhead of the adapted surface code on defective qubit arrays.  arXiv preprint arXiv:2305.00138, 2023.
  * [59] Google A. I. Quantum.  Suppressing quantum errors by scaling a surface code logical qubit.  Nature, 614(7949):676–681, 2023.
  * [60] Laird Egan, Dripto M Debroy, Crystal Noel, Andrew Risinger, Daiwei Zhu, Debopriyo Biswas, Michael Newman, Muyuan Li, Kenneth R Brown, Marko Cetina, et al.  Fault-tolerant control of an error-corrected qubit.  Nature, 598(7880):281–286, 2021.
  * [61] Andrew Cross, Ali Javadi-Abhari, Thomas Alexander, Niel De Beaudrap, Lev S Bishop, Steven Heidel, Colm A Ryan, Prasahnt Sivarajah, John Smolin, Jay M Gambetta, et al.  Openqasm 3: A broader and deeper quantum assembly language.  ACM Transactions on Quantum Computing, 3(3):1–50, 2022.
  * [62] Petar Jurcevic, Ali Javadi-Abhari, Lev S Bishop, Isaac Lauer, Daniela F Bogorin, Markus Brink, Lauren Capelluto, Oktay Günlük, Toshinari Itoko, Naoki Kanazawa, et al.  Demonstration of quantum volume 64 on a superconducting quantum computing system.  Quantum Science and Technology, 6(2):025020, 2021.
  * [63] Andrew Wack, Hanhee Paik, Ali Javadi-Abhari, Petar Jurcevic, Ismael Faro, Jay M. Gambetta, and Blake R. Johnson.  Quality, speed, and scale: three key attributes to measure the performance of near-term quantum computers.  arXiv preprint arXiv:2110.14108, 2021.
  * [64] Christophe Piveteau and David Sutter.  Circuit knitting with classical communication.  arXiv preprint arXiv:2205.00016, 2022.
  * [65] Omar Shehab, Isaac H Kim, Nhung H Nguyen, Kevin Landsman, Cinthia H Alderete, Daiwei Zhu, C Monroe, and Norbert M Linke.  Noise reduction using past causal cones in variational quantum algorithms.  arXiv preprint arXiv:1906.00476, 2019.
  * [66] Taylor L Patti, Omar Shehab, Khadijeh Najafi, and Susanne F Yelin.  Markov chain monte carlo enhanced variational quantum algorithms.  Quantum Science and Technology, 8(1):015019, 2022.
  * [67] Kaitlin N Smith, Michael A Perlin, Pranav Gokhale, Paige Frederick, David Owusu-Antwi, Richard Rines, Victory Omole, and Frederic T Chong.  Clifford-based circuit cutting for quantum simulation.  arXiv preprint arXiv:2303.10788, 2023.
  * [68] Max Rossmannek, Panagiotis Kl. Barkoutsos, Pauline J. Ollitrault, and Ivano Tavernelli.  Quantum HF/DFT-embedding algorithms for electronic structure calculations: Scaling up to complex molecular systems.  J. Chem. Phys., 154(11), 2021.
  * [69] Max Rossmannek, Fabijan Pavošević, Angel Rubio, and Ivano Tavernelli.  Quantum embedding method for the simulation of strongly correlated systems on quantum computers.  J. Chem. Phys. Lett., 14(14):3491–3497, 2023.
  * [70] Youngseok Kim, Christopher J Wood, Theodore J Yoder, Seth T Merkel, Jay M Gambetta, Kristan Temme, and Abhinav Kandala.  Scalable error mitigation for noisy quantum circuits produces competitive expectation values.  Nature Physics, pages 1–8, 2023.
  * [71] Ewout Van Den Berg, Zlatko K Minev, Abhinav Kandala, and Kristan Temme.  Probabilistic error cancellation with sparse pauli–lindblad models on noisy quantum processors.  Nature Physics, pages 1–6, 2023.
  * [72] Sergey Bravyi, Oliver Dial, Jay M Gambetta, Dario Gil, and Zaira Nazario.  The future of quantum computing with superconducting qubits.  Journal of Applied Physics, 132(16):160902, 2022.
  * [73] M Cerezo, Guillaume Verdon, Hsin-Yuan Huang, Lukasz Cincio, and Patrick J Coles.  Challenges and opportunities in quantum machine learning.  Nature Computational Science, 2(9):567–576, 2022.
  * [74] Zoran Krunic, Frederik F Flöther, George Seegan, Nathan D Earnest-Noble, and Omar Shehab.  Quantum kernels for real-world predictions based on electronic health records.  IEEE Transactions on Quantum Engineering, 3:1–11, 2022.
  * [75] IBM Research.  The Next Wave of Quantum-Centric Supercomputing.  https://research.ibm.com/blog/next-wave-quantum-centric-supercomputing, March 2021.  Accessed: May 3, 2023.
  * [76] Stefano Mensa, Emre Sahin, Francesco Tacchino, Panagiotis Kl Barkoutsos, and Ivano Tavernelli.  Quantum machine learning framework for virtual screening in drug discovery: a prospective quantum advantage.  Machine Learning: Science and Technology, 4(1):015023, feb 2023\.
  * [77] P. K. Barkoutsos, F. Gkritsis, P. J. Ollitrault, I. O. Sokolov, S. Woerner, and I. Tavernelli.  Quantum algorithm for alchemical optimization in material design.  Chem Sci, 12(12):4345–4352, 2021.
  * [78] Oriel Kiss, Francesco Tacchino, Sofia Vallecorsa, and Ivano Tavernelli.  Quantum neural networks force fields generation.  arXiv:2203.04666 [physics, physics:quant-ph], March 2022.  arXiv: 2203.04666.
  * [79] Andrew W Cross, Lev S Bishop, Sarah Sheldon, Paul D Nation, and Jay M Gambetta.  Validating quantum computers using randomized model circuits.  Physical Review A, 100(3):032328, 2019.
  * [80] IBM Quantum Experience.  Qiskit Runtime overview.  https://quantum-computing.ibm.com/lab/docs/iql/runtime/, 2021.  Accessed: May 3, 2023.
  * [81] David J Griffiths and Darrell F Schroeter.  Introduction to quantum mechanics.  Cambridge university press, 2018.
  * [82] Harry Levine, Alexander Keesling, Giulia Semeghini, Ahmed Omran, Tout T Wang, Sepehr Ebadi, Hannes Bernien, Markus Greiner, Vladan Vuletić, Hannes Pichler, et al.  Parallel implementation of high-fidelity multiqubit gates with neutral atoms.  Physical review letters, 123(17):170503, 2019.
  * [83] Guido Burkard, Thaddeus D Ladd, John M Nichol, Andrew Pan, and Jason R Petta.  Semiconductor spin qubits.  arXiv preprint arXiv:2112.08863, 2021.
  * [84] Aaron Conlon, Domenico Pellegrino, Johannes K Slingerland, Shane Dooley, and Graham Kells.  Error generation and propagation in majorana-based topological qubits.  Physical Review B, 100(13):134307, 2019.
  * [85] John M Martinis.  Optimal design of a superconducting transmon qubit with tapered wiring.  arXiv preprint arXiv:2104.01544, 2021.
  * [86] Christopher Chamberland, Kyungjoo Noh, Patricio Arrangoiz-Arriola, Earl T Campbell, Connor T Hann, Joseph Iverson, Harald Putterman, Thomas C Bohdanowicz, Steven T Flammia, Andrew Keller, et al.  Building a fault-tolerant quantum computer using concatenated cat codes.  PRX Quantum, 3(1):010329, 2022.
  * [87] Zhenyu Cai, Ryan Babbush, Simon C Benjamin, Suguru Endo, William J Huggins, Ying Li, Jarrod R McClean, and Thomas E O’Brien.  Quantum error mitigation.  arXiv preprint arXiv:2210.00921, 2022.
  * [88] Youngseok Kim, Andrew Eddins, Ewout Berg, Ken Wei, Sergey Bravyy, Kristan Temme, and Abhinav Kandala.  Recent progress and challenges in error mitigation with fixed-frequency superconducting quantum processors.  Bulletin of the American Physical Society, 2023.
  * [89] Liam Madden and Andrea Simonetto.  Best approximate quantum compiling problems.  ACM Transactions on Quantum Computing, 3(2):1–29, 2022.
  * [90] Youngseok Kim, Andrew Eddins, Sajant Anand, Ken Xuan Wei, Ewout Van Den Berg, Sami Rosenblatt, Hasan Nayfeh, Yantao Wu, Michael Zaletel, Kristan Temme, et al.  Evidence for the utility of quantum computing before fault tolerance.  Nature, 618(7965):500–505, 2023.
  * [91] Daniel Gottesman.  Opportunities and challenges in fault-tolerant quantum computation.  arXiv preprint arXiv:2210.15844, 2022.
  * [92] Craig Gidney and Martin Ekerå.  How to factor 2048 bit rsa integers in 8 hours using 20 million noisy qubits.  Quantum, 5:433, 2021.
  * [93] Andrew Cross.  The ibm q experience and qiskit open-source quantum computing software.  In APS March meeting abstracts, volume 2018, pages L58–003, 2018\.
  * [94] Rigetti Computing.  Pyquil documentation.  http://pyquil.readthedocs.io/en/latest, 2019.
  * [95] Ville Bergholm, Josh Izaac, Maria Schuld, Christian Gogolin, Shahnawaz Ahmed, Vishnu Ajith, M Sohaib Alam, Guillermo Alonso-Linaje, B AkashNarayanan, Ali Asadi, et al.  Pennylane: Automatic differentiation of hybrid quantum-classical computations.  arXiv preprint arXiv:1811.04968, 2018.
  * [96] Cirq Developers.  Cirq, December 2022.  See full list of authors on Github: https://github .com/quantumlib/Cirq/graphs/contributors.
  * [97] Constantin Gonzalez.  Cloud based qc with amazon braket.  Digitale Welt, 5:14–17, 2021.
  * [98] Algorithmiq’s platform for drug discovery with quantum computers.  https://algorithmiq.fi/news/quantum-computing-aurora-launch.  Accessed: 2023-17-05.
  * [99] IBM Research.  Charting the course to 100,000 qubits.  https://research.ibm.com/blog/100k-qubit-supercomputer, 2023.  Accessed: Jun 21, 2023.
  * [100] Richard P Feynman.  Quantum mechanical computers.  Optics news, 11(2):11–20, 1985.
  * [101] Yuri I Manin.  Mathematics as metaphor: Selected essays of Yuri I. manin.  American Mathematical Society, 2023.
  * [102] Lov K Grover.  A fast quantum mechanical algorithm for database search.  In Proceedings of the twenty-eighth annual ACM symposium on Theory of computing, pages 212–219, 1996.
  * [103] Seth Lloyd.  Universal quantum simulators.  Science, 273(5278):1073–1078, 1996.
  * [104] Seth Lloyd.  Quantum algorithm for solving linear systems of equations.  In APS March Meeting Abstracts, volume 2010, pages D4–002, 2010\.
  * [105] Vojtěch Havlíček, Antonio D Córcoles, Kristan Temme, Aram W Harrow, Abhinav Kandala, Jerry M Chow, and Jay M Gambetta.  Supervised learning with quantum-enhanced feature spaces.  Nature, 567(7747):209–212, 2019.
  * [106] Edward Farhi, Jeffrey Goldstone, and Sam Gutmann.  A quantum approximate optimization algorithm.  arXiv preprint arXiv:1411.4028, 2014.
  * [107] Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J Love, Alán Aspuru-Guzik, and Jeremy L O’brien.  A variational eigenvalue solver on a photonic quantum processor.  Nature communications, 5:4213, 2014.
  * [108] Mario Motta and Julia E Rice.  Emerging quantum computing algorithms for quantum chemistry.  Wiley Interdisciplinary Reviews: Computational Molecular Science, 12(3):e1580, 2022.
  * [109] Sam McArdle, Suguru Endo, Alán Aspuru-Guzik, Simon C Benjamin, and Xiao Yuan.  Quantum computational chemistry.  Reviews of Modern Physics, 92(1):015003, 2020.
  * [110] Yunseong Nam, Jwo-Sy Chen, Neal C Pisenti, Kenneth Wright, Conor Delaney, Dmitri Maslov, Kenneth R Brown, Stewart Allen, Jason M Amini, Joel Apisdorf, et al.  Ground-state energy estimation of the water molecule on a trapped-ion quantum computer.  npj Quantum Information, 6(1):33, 2020.
  * [111] Pranav Gokhale, Olivia Angiuli, Yongshan Ding, Kaiwen Gui, Teague Tomesh, Martin Suchara, Margaret Martonosi, and Frederic T Chong.  Optimization of simultaneous measurement for variational quantum eigensolver applications.  In 2020 IEEE International Conference on Quantum Computing and Engineering (QCE), pages 379–390. IEEE, 2020.
  * [112] Eugene F Dumitrescu, Alex J McCaskey, Gaute Hagen, Gustav R Jansen, Titus D Morris, T Papenbrock, Raphael C Pooser, David Jarvis Dean, and Pavel Lougovski.  Cloud quantum computing of an atomic nucleus.  Physical review letters, 120(21):210501, 2018.
  * [113] Omar Shehab, K Landsman, Yunseong Nam, Daiwei Zhu, Norbert M Linke, M Keesan, Raphael C Pooser, and C Monroe.  Toward convergence of effective-field-theory simulations on digital quantum computers.  Physical Review A, 100(6):062319, 2019.
  * [114] Aaron Fitzpatrick, Anton Nykänen, N. Walter Talarico, Alessandro Lunghi, Sabrina Maniscalco, Guillermo García-Pérez, and Stefan Knecht.  A self-consistent field approach for the variational quantum eigensolver: orbital optimization goes adaptive.  arXiv preprint arXiv:2212.11405, 2022.
  * [115] Oscar Higgott, Daochen Wang, and Stephen Brierley.  Variational quantum computation of excited states.  Quantum, 3:156, 2019.
  * [116] Pauline J. Ollitrault, Abhinav Kandala, Chun-Fu Chen, Panagiotis Kl. Barkoutsos, Antonio Mezzacapo, Marco Pistoia, Sarah Sheldon, Stefan Woerner, Jay M. Gambetta, and Ivano Tavernelli.  Quantum equation of motion for computing molecular excitation energies on a noisy quantum processor.  Phys. Rev. Res., 2:043140, Oct 2020.
  * [117] Mario Motta, Gavin O. Jones, Julia E. Rice, Tanvi P. Gujarati, Rei Sakuma, Ieva Liepuoniute, Jeannette M. Garcia, and Yu-ya Ohnishi.  Quantum chemistry simulation of ground- and excited-state properties of the sulfonium cation on a superconducting quantum processor.  Chemical Science, 14(11):2915–2927, March 2023.  Publisher: The Royal Society of Chemistry.
  * [118] Jarrod R McClean, Mollie E Kimchi-Schwartz, Jonathan Carter, and Wibe A de Jong.  Hybrid quantum-classical hierarchy for mitigation of decoherence and determination of excited states.  Physical Review A, 95(4):042308, 2017.
  * [119] William Kirby, Mario Motta, and Antonio Mezzacapo.  Exact and efficient lanczos method on a quantum computer.  arXiv preprint arXiv:2208.00567, 2022.
  * [120] Claudio Gambella and Andrea Simonetto.  Multiblock admm heuristics for mixed-binary optimization on classical and quantum computers.  IEEE Transactions on Quantum Engineering, 1:1–22, 2020.
  * [121] Mario Szegedy.  Quantum speed-up of markov chain based algorithms.  In 45th Annual IEEE symposium on foundations of computer science, pages 32–41. IEEE, 2004.
  * [122] Pawel Wocjan and Kristan Temme.  Szegedy walk unitaries for quantum maps.  arXiv preprint arXiv:2107.07365, 2021.
  * [123] Andrew M Childs.  Universal computation by quantum walk.  Physical review letters, 102(18):180501, 2009.
  * [124] Michal Stechly, Lanruo Gao, Boniface Yogendran, Enrico Fontana, and Manuel Rudolph.  Connecting the hamiltonian structure to the qaoa energy and fourier landscape structure.  arXiv preprint arXiv:2305.13594, 2023.
  * [125] Jacob Biamonte, Peter Wittek, Nicola Pancotti, Patrick Rebentrost, Nathan Wiebe, and Seth Lloyd.  Quantum machine learning.  Nature, 549(7671):195–202, 2017.
  * [126] Maria Schuld, Ilya Sinayskiy, and Francesco Petruccione.  An introduction to quantum machine learning.  Contemporary Physics, 56(2):172–185, 2015.
  * [127] Yunchao Liu, Srinivasan Arunachalam, and Kristan Temme.  A rigorous and robust quantum speed-up in supervised machine learning.  Nature Physics, 17(9):1013–1017, 2021.
  * [128] Patrick Rebentrost, Masoud Mohseni, and Seth Lloyd.  Quantum support vector machine for big data classification.  Physical Review Letters, 113(13), sep 2014.
  * [129] S. S. Kavitha and N. Kaulgud.  Quantum k-means clustering method for detecting heart disease using quantum circuit approach.  Soft comput, pages 1–14, 2022.
  * [130] Amira Abbas, David Sutter, Christa Zoufal, Aurélien Lucchi, Alessio Figalli, and Stefan Woerner.  The power of quantum neural networks.  Nature Computational Science, 1(6):403–409, 2021.
  * [131] Gunnar Carlsson, Afra Zomorodian, Anne Collins, and Leonidas J. Guibas.  Persistence barcodes for shapes.  International Journal of Shape Modeling, 11(02):149–187, 2005.
  * [132] H. Edelsbrunner, D. Letscher, and A. Zomorodian.  Topological persistence and simplification.  In Proceedings 41st Annual Symposium on Foundations of Computer Science, pages 454–463, 2000.
  * [133] Yuan Yao, Jian Sun, Xuhui Huang, Gregory R. Bowman, Gurjeet Singh, Michael Lesnick, Leonidas J. Guibas, Vijay S. Pande, and Gunnar Carlsson.  Topological methods for exploring low-density states in biomolecular folding pathways.  The Journal of Chemical Physics, 130(14), 04 2009.
  * [134] Meryll Dindin, Yuhei Umeda, and Frederic Chazal.  Topological data analysis for arrhythmia detection through modular neural networks.  In Cyril Goutte and Xiaodan Zhu, editors, Advances in Artificial Intelligence, pages 177–188, Cham, 2020. Springer International Publishing.
  * [135] Bastian Rieck, Matteo Togninalli, Christian Bock, Michael Moor, Max Horn, Thomas Gumbsch, and Karsten Borgwardt.  Neural persistence: A complexity measure for deep neural networks using algebraic topology.  In International Conference on Learning Representations, 2019.
  * [136] Yuhei Umeda.  Time series classification via topological data analysis.  Information and Media Technologies, 12:228–239, 2017.
  * [137] T. K. Dey, S. Mandal, and S. Mukherjee.  Gene expression data classification using topology and machine learning models.  BMC Bioinformatics, 22(Suppl 10):627, 2022.
  * [138] H. Masoomy, B. Askari, S. Tajik, A. K. Rizi, and G. R. Jafari.  Topological analysis of interaction patterns in cancer-specific gene regulatory network: persistent homology approach.  Scientific Reports, 11(1):16414, 2021.
  * [139] C. F. Loughrey, P. Fitzpatrick, N. Orr, and A. Jurek-Loughrey.  The topology of data: opportunities for cancer research.  Bioinformatics, 37(19):3091–3098, 2021.
  * [140] R. Rabadan, Y. Mohamedi, U. Rubin, T. Chu, A. N. Alghalith, O. Elliott, L. Arnes, S. Cal, A. J. Obaya, A. J. Levine, and P. G. Camara.  Identification of relevant genetic alterations in cancer using topological data analysis.  Nature Communications, 11(1):3808, 2020.
  * [141] P. G. Camara.  Topological methods for genomics: present and future directions.  Curr Opin Syst Biol, 1:95–101, 2017.
  * [142] M. Nicolau, A. J. Levine, and G. Carlsson.  Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival.  Proc Natl Acad Sci U S A, 108(17):7265–70, 2011.
  * [143] Seth Lloyd, Silvano Garnerone, and Paolo Zanardi.  Quantum algorithms for topological and geometric analysis of data.  Nature Communications, 7, 2014.
  * [144] Sam Gunn and Niels Kornerup.  Review of a quantum algorithm for betti numbers.  arXiv preprint arXiv:1906.07673, 2019.
  * [145] Casper Gyurik, Chris Cade, and Vedran Dunjko.  Towards quantum advantage via topological data analysis.  Quantum, 6:855, November 2022.
  * [146] Ryu Hayakawa.  Quantum algorithm for persistent Betti numbers and topological data analysis.  Quantum, 6:873, December 2022.
  * [147] Alexander Schmidhuber and Seth Lloyd.  Complexity-theoretic limitations on quantum algorithms for topological data analysis.  arXiv preprint arXiv:2209.14286, 2022.
  * [148] Shashanka Ubaru, Ismail Yunus Akhalwaya, Mark S. Squillante, Kenneth L. Clarkson, and Lior Horesh.  Quantum topological data analysis with linear depth and exponential speedup.  arXiv preprint arXiv:2108.02811, 2021.
  * [149] Ismail Yunus Akhalwaya, Shashanka Ubaru, Kenneth L. Clarkson, Mark S. Squillante, Vishnu Jejjala, Yang-Hui He, Kugendran Naidoo, Vasileios Kalantzis, and Lior Horesh.  Towards quantum advantage on noisy quantum computers.  arXiv preprint arXiv:2209.09371, 2022.
  * [150] Laxmi Parida and Naren Ramakrishnan.  Redescription mining: Structure theory and algorithms.  Proceedings of the 20th national conference on Artificial intelligence, 2, 2005.
  * [151] D. E. Platt, S. Basu, P. A. Zalloua, and L. Parida.  Characterizing redescriptions using persistent homology to isolate genetic pathways contributing to pathogenesis.  BMC Syst Biol, 10 Suppl 1(Suppl 1):10, 2016.
  * [152] Negin Karisani, Daniel E Platt, Saugata Basu, and Laxmi Parida.  Topology and redescriptions detect multiple alternative biological pathways from clinical phenotypes.  Experimental Biology and Medicine, 247(22):2015–2024, November 2022\.  Publisher: SAGE Publications.
  * [153] Daniel Platt, Aritra Bose, Chaya Levovitz, Kahn Rhrissorrakrai, and Laxmi Parida.  Epidemiological topology data analysis links severe covid-19 to raas and hyperlipidemia associated metabolic syndrome conditions.  medRxiv, 2022.
  * [154] Aritra Bose, Daniel E. Platt, Niina Haiminen, and Laxmi Parida.  Cuna: Cumulant-based network analysis of genotype-phenotype associations in parkinson’s disease.  medRxiv, 2021.
  * [155] J. K. Percus.  Correlation inequalities for Ising spin lattices.  Communications in Mathematical Physics, 40(3):283–308, October 1975\.
  * [156] T. D. Lee and C. N. Yang.  Statistical Theory of Equations of State and Phase Transitions. II. Lattice Gas and Ising Model.  Physical Review, 87(3):410–419, August 1952.  Publisher: American Physical Society.
  * [157] A. L. Barabasi, N. Gulbahce, and J. Loscalzo.  Network medicine: a network-based approach to human disease.  Nat Rev Genet, 12(1):56–68, 2011.
  * [158] K. Luck, D. K. Kim, L. Lambourne, K. Spirohn, B. E. Begg, W. Bian, R. Brignall, T. Cafarelli, F. J. Campos-Laborie, B. Charloteaux, D. Choi, A. G. Cote, M. Daley, S. Deimling, A. Desbuleux, A. Dricot, M. Gebbia, M. F. Hardy, N. Kishore, J. J. Knapp, I. A. Kovacs, I. Lemmens, M. W. Mee, J. C. Mellor, C. Pollis, C. Pons, A. D. Richardson, S. Schlabach, B. Teeking, A. Yadav, M. Babor, D. Balcha, O. Basha, C. Bowman-Colin, S. F. Chin, S. G. Choi, C. Colabella, G. Coppin, C. D’Amata, D. De Ridder, S. De Rouck, M. Duran-Frigola, H. Ennajdaoui, F. Goebels, L. Goehring, A. Gopal, G. Haddad, E. Hatchi, M. Helmy, Y. Jacob, Y. Kassa, S. Landini, R. Li, N. van Lieshout, A. MacWilliams, D. Markey, J. N. Paulson, S. Rangarajan, J. Rasla, A. Rayhan, T. Rolland, A. San-Miguel, Y. Shen, D. Sheykhkarimli, G. M. Sheynkman, E. Simonovsky, M. Tasan, A. Tejeda, V. Tropepe, J. C. Twizere, Y. Wang, R. J. Weatheritt, J. Weile, Y. Xia, X. Yang, E. Yeger-Lotem, Q. Zhong, P. Aloy, G. D. Bader, J. De Las Rivas, S. Gaudet, T. Hao, J. Rak, J. Tavernier, D. E. Hill, M. Vidal, F. P. Roth, and M. A. Calderwood.  A reference map of the human binary protein interactome.  Nature, 580(7803):402–408, 2020.
  * [159] Sabrina Maniscalco, Elsi-Mari Borrelli, Daniel Cavalcanti, Caterina Foti, Adam Glos, Mark Goldsmith, Stefan Knecht, Keijo Korhonen, Joonas Malmi, Anton Nykänen, Matteo A. C. Rossi, Harto Saarinen, Boris Sokolov, N. Walter Talarico, Jussi Westergren, Zoltán Zimborás, and Guillermo García-Pérez.  Quantum network medicine: rethinking medicine with network science and quantum algorithms.  arXiv preprint arXiv:2206.12405, 2022.
  * [160] M. Goldsmith, H. Saarinen, G. Garcia-Perez, J. Malmi, M. A. C. Rossi, and S. Maniscalco.  Link prediction with continuous-time classical and quantum walks.  Entropy (Basel), 25(5), 2023.
  * [161] Jack W Scannell, Alex Blanckley, Helen Boldon, and Brian Warrington.  Diagnosing the decline in pharmaceutical r&d efficiency.  Nature reviews Drug discovery, 11(3):191–200, 2012.
  * [162] John G Moffat, Fabien Vincent, Jonathan A Lee, Jörg Eder, and Marco Prunotto.  Opportunities and challenges in phenotypic drug discovery: an industry perspective.  Nature reviews Drug discovery, 16(8):531–543, 2017.
  * [163] Frank Sams-Dodd.  Target-based drug discovery: is something wrong?  Drug discovery today, 10(2):139–147, 2005.
  * [164] Ann Lin, Christopher J Giuliano, Ann Palladino, Kristen M John, Connor Abramowicz, Monet Lou Yuan, Erin L Sausville, Devon A Lukow, Luwei Liu, Alexander R Chait, et al.  Off-target toxicity is a common mechanism of action of cancer drugs undergoing clinical trials.  Science translational medicine, 11(509):eaaw8412, 2019.
  * [165] Padmanee Sharma, Siwen Hu-Lieskovan, Jennifer A Wargo, and Antoni Ribas.  Primary, adaptive, and acquired resistance to cancer immunotherapy.  Cell, 168(4):707–723, 2017.
  * [166] Alison Abbott.  Quantum computers to explore precision oncology.  Nature Biotechnology, 39(11):1324–1325, 2021.
  * [167] Emily Hargrave-Thomas, Bo Yu, and Jóhannes Reynisson.  Serendipity in anticancer drug discovery.  World journal of clinical oncology, 3(1):1, 2012.
  * [168] J. Chen, H. Xu, W. Tao, Z. Chen, Y. Zhao, and J. J. Han.  Transformer for one stop interpretable cell type annotation.  Nature Communications, 14(1):223, 2023.
  * [169] C. V. Theodoris, L. Xiao, A. Chopra, M. D. Chaffin, Z. R. Al Sayed, M. C. Hill, H. Mantineo, E. M. Brydon, Z. Zeng, X. S. Liu, and P. T. Ellinor.  Transfer learning enables predictions in network biology.  Nature, 2023.
  * [170] Haotian Cui, Chloe Wang, Hassaan Maan, and Bo Wang.  scgpt: Towards building a foundation model for single-cell multi-omics using generative ai.  bioRxiv, 2023.
  * [171] Minsheng Hao, Jing Gong, Xin Zeng, Chiming Liu, Yucheng Guo, Xingyi Cheng, Taifeng Wang, Jianzhu Ma, Le Song, and Xuegong Zhang.  Large scale foundation model on single-cell transcriptomics.  bioRxiv, 2023.
  * [172] Dmitry S Andreyev and Boris L Zybailov.  Integration of flow cytometry and single cell sequencing.  Trends in Biotechnology, 38(2):133–136, 2020.
  * [173] Sonia Guedan, Aviv Madar, Victoria Casado-Medrano, Carolyn Shaw, Anna Wing, Fang Liu, Regina M Young, Carl H June, Avery D Posey, et al.  Single residue in cd28-costimulated car-t cells limits long-term persistence and antitumor durability.  The Journal of clinical investigation, 130(6):3087–3097, 2020.
  * [174] Payel Das, Tom Sercu, Kahini Wadhawan, Inkit Padhi, Sebastian Gehrmann, Flaviu Cipcigan, Vijil Chenthamarakshan, Hendrik Strobelt, Cicero Dos Santos, Pin-Yu Chen, et al.  Accelerated antimicrobial discovery via deep generative models and molecular dynamics simulations.  Nature Biomedical Engineering, 5(6):613–623, 2021.
  * [175] Anja Krause, Hong-Fen Guo, Jean-Baptiste Latouche, Cuiwen Tan, Nai-Kong V Cheung, and Michel Sadelain.  Antigen-dependent cd28 signaling selectively enhances survival and proliferation in genetically modified activated human primary t lymphocytes.  The Journal of experimental medicine, 188(4):619–626, 1998.
  * [176] Kole T Roybal, Jasper Z Williams, Leonardo Morsut, Levi J Rupp, Isabel Kolinko, Joseph H Choe, Whitney J Walker, Krista A McNally, and Wendell A Lim.  Engineering t cells with customized therapeutic response programs using synthetic notch receptors.  Cell, 167(2):419–432, 2016.
  * [177] Aidan M Tousley, Maria Caterina Rotiroti, Louai Labanieh, Lea Wenting Rysavy, Won-Ju Kim, Caleb Lareau, Elena Sotillo, Evan W Weber, Skyler P Rietberg, Guillermo Nicolas Dalton, et al.  Co-opting signalling molecules enables logic-gated control of car t cells.  Nature, pages 1–10, 2023.
  * [178] Kole T Roybal, Levi J Rupp, Leonardo Morsut, Whitney J Walker, Krista A McNally, Jason S Park, and Wendell A Lim.  Precision tumor recognition by t cells with combinatorial antigen-sensing circuits.  Cell, 164(4):770–779, 2016.
  * [179] Guillermo Martínez-Ara, Kristina S Stapornwongkul, and Miki Ebisuya.  Scaling up complexity in synthetic developmental biology.  Science, 378(6622):864–868, 2022.
  * [180] S. E. Harrison, B. Sozen, N. Christodoulou, C. Kyprianou, and M. Zernicka-Goetz.  Assembly of embryonic and extraembryonic stem cells to mimic embryogenesis in vitro.  Science, 356(6334), 2017.
  * [181] Marta N Shahbazi and Magdalena Zernicka-Goetz.  Deconstructing and reconstructing the mouse and human early embryo.  Nature cell biology, 20(8):878–887, 2018.
  * [182] Louai Labanieh and Crystal L Mackall.  Car immune cells: design principles, resistance and the next generation.  Nature, 614(7949):635–648, 2023.
  * [183] Khloe S Gordon, Taeyoon Kyung, Caleb R Perez, Patrick V Holec, Azucena Ramos, Angela Q Zhang, Yash Agarwal, Yunpeng Liu, Catherine E Koch, Alina Starchenko, et al.  Screening for cd19-specific chimaeric antigen receptors with enhanced signalling via a barcoded library of intracellular domains.  Nature Biomedical Engineering, 6(7):855–866, 2022.
  * [184] Rocío Castellanos-Rueda, Raphaël B Di Roberto, Florian Bieberich, Fabrice S Schlatter, Darya Palianina, Oanh TP Nguyen, Edo Kapetanovic, Heinz Läubli, Andreas Hierlemann, Nina Khanna, et al.  speedingcars: accelerating the engineering of car t cells by signaling domain shuffling and single-cell sequencing.  Nature Communications, 13(1):6555, 2022.
  * [185] Iris Cong, Soonwon Choi, and Mikhail D Lukin.  Quantum convolutional neural networks.  Nature Physics, 15(12):1273–1278, 2019.
  * [186] Arthur Pesah, Marco Cerezo, Samson Wang, Tyler Volkoff, Andrew T Sornborger, and Patrick J Coles.  Absence of barren plateaus in quantum convolutional neural networks.  Physical Review X, 11(4):041011, 2021.
  * [187] Sabrina M Lewis, Marie-Liesse Asselin-Labat, Quan Nguyen, Jean Berthelet, Xiao Tan, Verena C Wimmer, Delphine Merino, Kelly L Rogers, and Shalin H Naik.  Spatial omics and multiplexed imaging to explore cancer biology.  Nature methods, 18(9):997–1012, 2021.
  * [188] Leon Hetzel, David S Fischer, Stephan Günnemann, and Fabian J Theis.  Graph representation learning for single-cell biology.  Current Opinion in Systems Biology, 28:100347, 2021.
  * [189] Michelle M Li, Kexin Huang, and Marinka Zitnik.  Graph representation learning in biomedicine and healthcare.  Nature Biomedical Engineering, pages 1–17, 2022.
  * [190] Guillaume Jaume, Pushpak Pati, Behzad Bozorgtabar, Antonio Foncubierta, Anna Maria Anniciello, Florinda Feroce, Tilman Rau, Jean-Philippe Thiran, Maria Gabrani, and Orcun Goksel.  Quantifying explainers of graph neural networks in computational pathology.  In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 8106–8116, 2021.
  * [191] Maria Brbić, Kaidi Cao, John W Hickey, Yuqi Tan, Michael P Snyder, Garry P Nolan, and Jure Leskovec.  Annotation of spatially resolved single-cell data with stellar.  Nature Methods, pages 1–8, 2022.
  * [192] David S Fischer, Anna C Schaar, and Fabian J Theis.  Modeling intercellular communication in tissues using spatial graphs of cells.  Nature Biotechnology, pages 1–5, 2022.
  * [193] Ye Yuan and Ziv Bar-Joseph.  Gcng: graph convolutional networks for inferring gene interaction from spatial transcriptomics data.  Genome biology, 21(1):1–16, 2020.
  * [194] Zhenqin Wu, Alexandro E Trevino, Eric Wu, Kyle Swanson, Honesty J Kim, H Blaize D’Angio, Ryan Preska, Gregory W Charville, Piero D Dalerba, Ann Marie Egloff, et al.  Graph deep learning for the characterization of tumour microenvironments from spatial protein profiles in tissue specimens.  Nature Biomedical Engineering, pages 1–14, 2022.
  * [195] Nadia Brancati, Anna Maria Anniciello, Pushpak Pati, Daniel Riccio, Giosuè Scognamiglio, Guillaume Jaume, Giuseppe De Pietro, Maurizio Di Bonito, Antonio Foncubierta, Gerardo Botti, et al.  Bracs: A dataset for breast carcinoma subtyping in h&e histology images.  Database, 2022, 2022.
  * [196] W. Bulten, K. Kartasalo, P.C. Chen, P. Ström, H. Pinckaers, K. Nagpal, Y. Cai, D.F. Steiner, H. van Boven, R. Vink, C. Hulsbergen-van de Kaa, J. van der Laak, M.B. Amin, A.J. Evans, T. van der Kwast, R. Allan, P.A. Humphrey, H. Grönberg, H. Samaratunga, B. Delahunt, T. Tsuzuki, T. Häkkinen, L. Egevad, M. Demkin, S. Dane, F. Tan, M. Valkonen, G.S. Corrado, L. Peng, C.H. Mermel, P. Ruusuvuori, G. Litjens, M. Eklund, and the PANDA challenge consortium.  Artificial intelligence for diagnosis and gleason grading of prostate cancer: the panda challenge.  Nature Medicine, 28:154–163, 2022.
  * [197] Sanjay R Srivatsan, José L McFaline-Figueroa, Vijay Ramani, Lauren Saunders, Junyue Cao, Jonathan Packer, Hannah A Pliner, Dana L Jackson, Riza M Daza, Lena Christiansen, et al.  Massively multiplex chemical transcriptomics at single-cell resolution.  Science, 367(6473):45–51, 2020.
  * [198] Mohammad Lotfollahi, F Alexander Wolf, and Fabian J Theis.  scgen predicts single-cell perturbation responses.  Nature methods, 16(8):715–721, 2019.
  * [199] Cédric Villani et al.  Optimal transport: old and new, volume 338.  Springer, 2009.
  * [200] Marco Cuturi.  Sinkhorn distances: Lightspeed computation of optimal transport.  Advances in neural information processing systems, 26, 2013.
  * [201] Gabriel Peyré, Marco Cuturi, et al.  Computational optimal transport: With applications to data science.  Foundations and Trends® in Machine Learning, 11(5-6):355–607, 2019.
  * [202] Charlotte Bunne, Andreas Krause, and marco cuturi.  Supervised training of conditional monge maps.  In Alice H. Oh, Alekh Agarwal, Danielle Belgrave, and Kyunghyun Cho, editors, Advances in Neural Information Processing Systems, 2022.
  * [203] Thomas M Norman, Max A Horlbeck, Joseph M Replogle, Alex Y Ge, Albert Xu, Marco Jost, Luke A Gilbert, and Jonathan S Weissman.  Exploring genetic interaction manifolds constructed from rich single-cell phenotypes.  Science, 365(6455):786–793, 2019.
  * [204] Jase Gehring, Jong Hwee Park, Sisi Chen, Matthew Thomson, and Lior Pachter.  Highly multiplexed single-cell rna-seq by dna oligonucleotide tagging of cellular proteins.  Nature biotechnology, 38(1):35–38, 2020.
  * [205] Daniel E. Platt, Saugata Basu, Pierre A. Zalloua, and Laxmi Parida.  Characterizing redescriptions using persistent homology to isolate genetic pathways contributing to pathogenesis.  BMC Systems Biology, 10(1):S10, January 2016.