# A perspective on protein structure prediction using quantum computers

Hakan Doga  IBM Quantum, Almaden Research Center, San Jose, California 95120,
USA  Bryan Raubenolt  Center for Computational Life Sciences, Lerner Research
Institute, The Cleveland Clinic, Cleveland, Ohio 44106, USA  Fabio Cumbo
Center for Computational Life Sciences, Lerner Research Institute, The
Cleveland Clinic, Cleveland, Ohio 44106, USA  Jayadev Joshi  Center for
Computational Life Sciences, Lerner Research Institute, The Cleveland Clinic,
Cleveland, Ohio 44106, USA  Frank P. DiFilippo  Center for Computational Life
Sciences, Lerner Research Institute, The Cleveland Clinic, Cleveland, Ohio
44106, USA  Jun Qin  Center for Computational Life Sciences, Lerner Research
Institute, The Cleveland Clinic, Cleveland, Ohio 44106, USA  Daniel
Blankenberg  Center for Computational Life Sciences, Lerner Research
Institute, The Cleveland Clinic, Cleveland, Ohio 44106, USA  Omar Shehab  IBM
Quantum, IBM Thomas J Watson Research Center, Yorktown Heights, NY 10598, USA

###### Abstract

Despite the recent advancements by deep learning methods such as AlphaFold2,
in silico protein structure prediction remains a challenging problem in
biomedical research. With the rapid evolution of quantum computing, it is
natural to ask whether quantum computers can offer some meaningful benefits
for approaching this problem. Yet, identifying specific problem instances
amenable to quantum advantage, and estimating quantum resources required are
equally challenging tasks. Here, we share our perspective on how to create a
framework for systematically selecting protein structure prediction problems
that are amenable for quantum advantage, and estimate quantum resources for
such problems on a utility-scale quantum computer. As a proof-of-concept, we
validate our problem selection framework by accurately predicting the
structure of a catalytic loop of the Zika Virus NS3 Helicase, on quantum
hardware.

##  1 Introduction

The intricate dance of life at the molecular level is orchestrated by
proteins, with virtually all biological activity tied to the three-dimensional
conformations they adopt111Corresponding authors: Hakan Doga
<EMAIL_ADDRESS>and Bryan Raubenolt (raubenb@ccf.org).. The phenomenon by
which these structures are predetermined from their primary amino acid
sequence is known as the “protein folding problem”, and it is inherently
central to all life and its myriad of diseases. In nature, protein folding is
a path-dependent process, meaning that the optimal path is usually taken [1,
2]. For most biomedical research applications, predicting the optimal
structure itself, without necessarily reproducing the optimal path, is
arguably most important and more attainable. This is known as protein
structure prediction (PSP) (see Figure 1). Decades of scientific inquiry have
sought to unravel the mysteries of how and why proteins assume their complex
structures, often driven by the quest for understanding diseases at the
molecular level. To comprehend the function of a protein and devise effective
drugs targeting them, an accurate depiction of their physiologically active
structure is indispensable. Traditionally, this has been achieved through
laborious wet lab experiments involving genetic modifications, protein
isolation, and purification. Techniques like X-ray crystallography, NMR, and
CryoEM have been instrumental in solving protein structures, revolutionizing
our grasp of diseases. However, these methods are time-consuming, expensive,
and not without limitations. Recognizing the need for alternatives,
researchers turned to machine learning, exemplified by AlphaFold2 [3],
RoseTTaFold [4], and ITASSER [5], which leverage experimentally determined
structures. While transformative, these methods may lack a nuanced
understanding of the underlying physics, potentially hindering predictions of
novel protein structures [6]. Physics-based methods on the other had, such as
molecular dynamics (MD) simulations, face challenges in scalability and
practicality. Interested readers may refer to [7, 8] for comparative analyses
of the above-mentioned methods.

Figure 1:  Overview of the PSP pipeline. Following genomic sequencing, the
primary amino acid sequence is determined. The experimental method then starts
with expressing this protein by genetically modifying another organism with
this new sequence. This organism will then translate these proteins, and the
new protein of interest can be isolated, purified, and then solved using X-ray
crystallography, NMR, or CryoEM. The in silico methods on the other hand,
simply take the primary amino acid sequence as input and the structure is
predicted by either a physics-based method (where the underlying biophysics is
somehow simulated) or a template-based method (where machine learning
algorithms predict structures based on patterns found in a training set of
experimental templates). The method we adopt in this work falls under the
category of physics-based algorithms. As an illustrated example, an in silico
model and X-ray crystal structure of the SARS-CoV2 NSP13 helicase (PDB: 7NN0)
are superimposed, along with a docked known inhibitor (colored in magenta).

Fundamentally, protein folding is a physics problem. A computer algorithm
armed with a realistic thermodynamic description and mathematical framework
for modeling interactions between amino acids (residues) could theoretically
navigate the vast conformational space, ultimately arriving at an optimal
solution [9]. Yet, the computational complexity of this challenge has made it
a persistent hurdle in the life sciences. In this landscape, quantum computing
holds the potential to provide meaningful utility for this problem. While
there have been major advances in quantum hardware and algorithm development
[10], finding the appropriate class of problems amenable to quantum advantage
is still an open question for all areas of practical applications. Estimating
the resources needed to enable quantum advantage is equally challenging, with
sporadic progresses. In this work, we provide our perspective, a scalable
framework to identify PSP problem instances that may be amenable to quantum
advantage. Narrowing the problem space through a combination of protein
sequence length, disorder, number of mutations and multiple sequence alignment
(MSA) depth, we opine on how to select problems at different scales that have
been found challenging for state-of-the-art classical methods. Furthermore,
this approach makes our problem selection framework explainable.

Section 2 introduces quantum computing fundamentals, laying the groundwork for
understanding its application in solving the protein folding problem. Section
3 describes our perspective. It delves into the complexities of PSP,
highlighting challenges related to protein size, mutations, and the role of
MSAs. The focus is on identifying the problem subspace where quantum computing
might outperform deep learning methods reliant on MSAs. Experimental results
in Section 4 showcase our quantum-classical hybrid workflow predicting the
structure of the seven amino acid catalytic P-loop from a vital Zika virus
protein (NS3 Helicase) using the IBM_Cleveland quantum computer, validating
our perspective with a concrete example. Section 5 discusses the scalability
of the quantum algorithm and provides high level quantum resource estimation
for PSP problems. The conclusion reflects on the significance of the work,
opening avenues for future exploration in leveraging quantum computing for
protein folding challenges.

##  2 Quantum computing: A brief introduction

Quantum Computing is a new model of high performance computing where the
traditional foundation of computing, i.e. binary logic, has been replaced by
theories of quantum mechanics [11, 12]. This section aims to provide a very
concise introduction to the topic, while the readers are encouraged to review
[13] for a comprehensive overview, and Section 2 of [14] for a primer from the
perspective of healthcare and life sciences. The power of quantum computing
comes from quantum mechanical effects such as superposition, entanglement,
negative state probability i.e. interference, and probabilistic measurement.
These phenomena sometimes allow a quantum algorithm to naturally map the
degrees of freedom of quantum hardware to those of a target quantum system and
simulate in an efficient manner. For some suitably structured problems, it is
possible to exploit these phenomena to design quantum algorithms capable of
traversing a search space or optimization cost landscape in a more efficient
probabilistic manner. Over the last few decades, quantum advantages have been
theoretically demonstrated for prime factorization [15], unstructured search
[16], network flows [17], quantum simulation [18], topological invariants
[19], partition functions [20], semidefinite programming [21], linear systems
[22], differential equations [23], dynamic programming [24], bilinear
functions [25], etc. A few recent works have also reported quantum utility
[26] or empirical quantum advantage [27, 28, 25, 29, 30, 31, 32], as defined
in [31]. Several architectures have been proposed for scalable quantum
computers including neutral atom qubits [33], spin qubits [34], topological
qubits [35], trapped-ion qubits [36], and superconducting qubits [37]. With
30+ quantum computers, IBM Quantum Experience is the first ever and largest
cloud-based quantum computing service. Quantum software development kits like
Qiskit [38], CUDA Quantum [39], Forest [40], PennyLane [41], CirQ [42] and
Braket [43] are also available with increasingly more application verticals in
each release. Several vendors have also announced their developmental road
maps [44, 45, 46, 47] with IBM leading the way to achieve one hundred thousand
qubits with increased gate quality and speed within a decade [10, 48]. In
parallel with the progress of technologies, the research communities have also
presented their point of views for using quantum computers in several areas of
applications including scientific discovery [49], biological sciences [50],
nuclear physics [51], high-energy physics [52], cell-centric therapeutics
[14], financial engineering [53], climate science [54], etc. Finally, the
responsible usage of quantum computing is also emerging as an area of research
[55].

###  2.1 Quantum search and optimization algorithms

While finding the global minima of the energy landscape of protein
conformation is an optimization problem, one could also imagine it as a search
problem. In that case, the database consists of all possible conformational
energies with the lowest energy as the marked item to search for. The quantum
algorithm for unstructured database search is Grover’s algorithm which
provides quadratic speedup over state-of-the-art classical search algorithms
[16]. The algorithm initializes a quantum state in an equal superposition over
all database entries. It then iterates a step which amplifies the amplitude of
the target entry. After $O(\sqrt{N})$ iterations, a measurement will find the
entry with high probability, and it has been shown as optimal [56]. The
optimality of Grover’s algorithm is also supported by the observation that it
defines a geodesic in quantum Hilbert space [57, 58]. Therefore, any other
quantum search or optimization algorithm may be considered as a parametrized
approximation of Grover’s algorithm. One such algorithm is quantum approximate
optimization algorithm (QAOA) [59]. It works by setting up a cost Hamiltonian
whose ground state encodes the search result. A quantum system is initialized
in an easy-to-prepare state. A series of unitary operations alternately apply
a mixer Hamiltonian and the cost Hamiltonian. At the end, the state of the
system is measured, giving a candidate solution. While low-depth QAOA is not
expected to outperform the state-of-the-art classical algorithm[60, 61] (see
[62, 63] for a more formal study of the limitations of variational quantum
algorithms), at a higher depth the algorithm increasingly becomes an
approximation of the Grover’s algorithm and has the potential to maximize
fractional Grover speedup [59]. See Section LABEL:supp-sec:quantum of the
supplementary materials for a detailed discussion of algorithmic structures
needed for quantum speedup.

###  2.2 The protein folding funnel and the prospect of quantum advantage

The theory of protein folding considers how proteins’ primary amino acid
sequences dictate how they fold rapidly and specifically into their native 3D
structures. The classical view was that folding occurred through discrete
intermediates along a linear pathway. In contrast, the energy landscape theory
views folding as a progressive organization of an ensemble of partially folded
structures funneled towards the native state [64, 65, 66]. Evolution has
shaped proteins to have a rugged, funnel-like landscape biased towards the
functional native structure. The funnel shape implies folding robustness, with
different routes down the funnel possible for the same protein. Just like most
other spontaneous thermodynamic phenomena, protein folding is largely path
dependent - nature takes the optimal path to the optimal solution. The native
state’s stability increases moving down the funnel through local folding
events. Common patterns emerge like topology determining mechanisms, but
details depend on subtle sequence variations. The funnel concept explains how
folding can be fast despite the vast number of possible non-native
conformations. The landscape guides the protein through a directed search.
Figure 2 shows four proposed folding funnels.

Figure 2: This graphic was originally created and released in the public
domain by Ken A. Dill. Original caption: The illustrations of proposed energy
landscape that each demonstrate the degree of freedom a protein possesses in
terms of configurations and the multidimensional routes that a protein can
take to achieve its final configuration. From left to right for proposed
funnel-shaped energy landscape: the idealized smooth funnel, the rugged
funnel, the Moat funnel, and the Champagne Glass funnel [67].

Finding the global minimum energy in the folding funnel can be understood as a
search problem where the database entries are conformational energies. The
funnel shape indicates that despite the overall “easy-to-follow” macro
structure, the ruggedness on the funnel wall and the bottom induces hardness
to conformation prediction.

The ruggedness of the energy landscape at the bottom of the folding funnel
correlates with the complexity and diversity of conformations a protein can
adopt [68, 69, 70, 71, 72, 73, 74, 75, 76, 66, 77]. A smooth, single-minimum
funnel bottom indicates a rigid protein with few accessible conformers. A
rugged funnel bottom with many minima reflects a flexible protein existing as
an ensemble of diverse conformers interconverting between sub-states. The more
rugged the bottom, the more conformational flexibility the protein has to
sample alternate structures. Proteins with rugged, multi-minima landscapes can
bind ligands in a non-specific way by selecting conformers complementary to
each ligand from their ensemble. In contrast, proteins with smooth funnels
have less conformational heterogeneity and exhibit more specific binding to
pre-organized native structures. Increased ruggedness also enables phenomena
like induced fit binding, crystal packing effects, domain swapping, and
misfolding aggregation by allowing access to alternate conformers. Molecular
chaperones may smooth rugged landscapes to promote proper folding over
misfolding. In summary, the complexity and interconversion of a protein’s
conformational sub-states is directly related to the ruggedness of the energy
landscape at the bottom of its folding funnel [73]. Rugged landscapes confer
flexibility and multi-specificity, while smooth funnels restrict accessible
conformers and interactions.

The ruggedness in the folding funnel has a curious parallel in unstructured
search problems when one interprets the problem as traversing the geodesic of
a search space (Grover’s quantum search algorithm) or optimizing a rugged cost
function using a variational algorithm with a parameterized quantum circuit
(QAOA algorithm). Besides, the space of the folding pathways has a natural
tree structure (See Figure 5 of [78]) similar to the search tree structure in
Grover solution space. These algorithms have been briefly introduced in
Section 2.1. It has been observed that the ruggedness of the search or
optimization landscape creates amenability for quantum advantage [79, 80, 81,
82, 83, 84, 85, 86, 87, 88] (see Section LABEL:supp-sec:quantum of the
supplementary materials for a detailed discussion). The quantum advantage
appears when there is a least amount of information or a highest amount of
uncertainty about the search space which is also known as the “worst case” in
algorithmic complexity analysis. In the case of Grover’s algorithm one can
attribute this quantum speed up to the fact that quantum mechanics allow us to
associate negative probabilities to search paths. Therefore, a subset of the
search paths will cancel each other, rendering the search problem smaller and
easier. If one is using the QAOA algorithm, the same phenomena may be viewed
as an effect of quantum tunneling where a quantum search process may cross
energy barriers in a probabilistic manner that does not have any classical
analog. Under these circumstances, we can conjecture that the more rugged a
folding funnel is, the easier it will be for a quantum algorithm to find the
lowest energy conformation relative to the performance of its classical
competitors.

##  3 What makes protein structure prediction hard?

###  3.1 Performance of physics-based methods against increasing sequence
length

Regardless of the physics-based computational method used, sequence length
quickly becomes a major limitation. As a protein sequence becomes larger,
there is an exponential increase in the search space (number of possible
conformations), and a corresponding exponential growth in the required run
time for an exhaustive search (see Figure 3a). Most non-molecular dynamics
(non-MD) PSP methods (also known as ab initio or free modeling methods) have
generally been limited to structures of only a few dozen amino acids in
length. One study by M. Yousef et al [89] compared the performance of four of
these methods. While most of these methods appeared to produce backbone Root
Mean Squared Deviations (RMSDs) within three angstroms when compared to the
experimental results, the investigated structures were all 31 amino acids or
less in length. PEP-FOLD3 [90] is designed to model peptides between 5 and 50
amino acids, and has been successful in doing so in different studies. Quark
[91], an ab initio program developed by the same group who created ITASSER
[92], can predict structures accurately, but was mainly designed for fragments
up to 20 amino acids in length. For small peptides at this scale, these
programs perform well (run times of less than 14 hours for all test cases
reported in [89]) and can offer viable solutions with the right compute
resources. For larger, biologically relevant proteins, physics-based classical
algorithms like these may not be as suitable due to the rapid increase in the
conformational space.

Figure 3: . A graphical representation of the Levinthal paradox a) The data
considers that for a protein with n amino acids, there are n-1 peptide bonds.
For each peptide bond, there are also 2 other bond angles on either side of
the $\alpha$ carbon, $\phi$ and $\psi$, and assumes that each peptide in the
sequence can adopt up to 3 conformations (3 combinations of these bond
angles). b) So, for a protein with n amino acids, there are a total of 32(n-1)
possible conformations. Assuming 1 picosecond is spent to sample each
conformation, the y-axis represents the total exploration time in years to
sample all possible conformations for a protein with n amino acids.

Although there has been remarkable success in some studies where all-atom MD
simulations are applied to predict folded structures, with the most impressive
results arguably being those of D.E. Shaw’s research group [93, 94], sequence
length still remains a limitation. These efforts led to the simulation of
possible folding paths, as well as an accurate prediction of the folded state,
for human ubiquitin [95] and a set of 12 “fast folding” proteins [96]. The
main issue is the amount of sampling time (the length of the simulation in
time, as defined by the product of the total number of discrete time steps and
step size) as well as compute resources it took to achieve these results,
since most researchers may have limited access to either. Although the
majority of the structures in [96] folded to near perfect accuracy, much of
this occurred in the order of hundreds of microseconds to milliseconds of
sampling time (depending on the system size, this generally requires
significant run times ranging from weeks to months even when using
conventional HPC resources) and the investigated proteins were relatively
small, ranging in size from roughly 10-80 residues. Much like the other
physics-based methods, standard (unbiased) all-atom molecular simulations
still present a threshold in sequence size for which they may no longer be a
practical solution for predicting larger protein structures. One alternative
for speed up is of course MD simulations using coarse grain potentials, in
particular those derived from neural networks [97], but among the limitations
is the fact that long time scale all-atom simulations still need to be run for
the neural network to train on, and even then these potentials are still not
generalizable (they cannot be readily used for proteins largely different from
the training set).

This gap between practical sampling times in simulation and the real folding
times in experiments has been known for quite some time, and it is one of the
main reasons enhanced sampling methods (such as replica exchange molecular
dynamics REMD) were developed. REMD is an excellent tool for thermodynamic
sampling of large conformational spaces, with relatively short time scales.
Rather than having a single, long simulation of the system, several “replicas”
of the same system are simulated at different temperatures (usually room
temperature, as well as above and below the melting points) and different
random seeds for the initial velocities. In most implementations, the idea is
that these temperature-dependent simulations can then undergo Monte Carlo
swaps between neighboring replicas, preventing the lower temperature
conformations from becoming “trapped” in a local minima. While the sampling
time can be orders of magnitude less than running a single conventional
simulation at fixed temperature, that is compensated by the number of
simulations one has to run, which can regularly exceed a few dozen replicas
even for a small 20 amino acid protein like Trp-Cage [98, 99]. Because of this
burden, new methods of REMD have been developed which can significantly reduce
the number of replicas required to sample the same conformational space [100].

###  3.2 Mutations and intrinsically disordered proteins

One of the main advantages of the template-based deep learning methods like
AlphaFold2 and RoseTTaFold is the sheer size of the structures they are able
to produce. They are not limited to a few dozen residues as we’ve seen with
the physics-based methods. Both AlphaFold2 and RoseTTaFold can readily produce
models up to a couple thousand residues in length, in part due to the fact
that their data bases (PDB222https://www.rcsb.org/.) include experimentally
determined structures which span across this size range. Obviously the larger
the sequence, the more MSAs and fragments are needed, and thus computing
power. The upperbound is roughly between 1280 and 2700 amino acids, according
to the European Bioinformatics Institute 333https://alphafold.ebi.ac.uk/faq..
Despite this impressive capability, success still depends on how accurate the
models are. Specifically, the models are known to produce discrepancies when
dealing with proteins with a) mutated sequences and b) intrinsically
disordered regions. Here we discuss the effect of these two factors.

####  3.2.1 Mutations

While the ability for these programs to predict wildtype proteins containing
mostly ordered secondary structures, i.e. localized structures that form based
on interactions within the protein backbone, is unquestionable, significant
errors have been reported when attempting to predict mutated variants. The
discrepancies can be observed in both the 3D coordinates of the produced
structure as well as the predicted thermodynamic stability of the mutated
residues relative to their confidence scores [101, 102]. Mutations change the
local chemical space, and thus the physics of the interacting residues. Even
the slightest mutations between similar residues (for example valine to
leucine) can have notable impacts on the local structure. Some mutations can
be drastically different, such as going from a slender, flexible, charged
residue like glutamic acid to a bulky hydrophobic sidechain like tryptophan.
Such cases can have profound effects, such as disrupting charge distributions
beyond the local structure and often breaking relevant hydrogen bond networks
vital to the protein’s activity. In order for PSP methods to be successful,
the algorithm needs to capture the consequences of mutations. From a global
perspective, mutations can change the free energy landscape of a protein. This
can sometimes lead to a more rugged landscape (or more smooth), perhaps
introducing new local minima (see Section 2.2 ). This can be problematic, as
an algorithm’s optimization protocol could get “stuck” in these regions,
producing a conformation it thinks corresponds to the lowest energy,
particularly if heuristic optimizers are involved. Of course exhaustive search
algorithms could naturally avoid this, but they are not scalable beyond a
certain size.

Figure 4: Free energy landscapes of a) TC10b and b) TC5b derived from
molecular dynamics simulations at their melting points of 329K and 315K,
respectively. The folded NMR structures are quite similar (0.8 Å RMSD between
them), but the free energy landscape is dramatically different, with TC10b
demonstrating two possible folding pathways and their minima, as opposed to
the more well defined funnel and obvious global minimum in the case of TC5b.
The three point mutations between these structures give rise to different
physicochemical properties such as their melting temperatures, and accordingly
different free energy landscapes.

The Trp-cage proteins are 20 amino acid engineered “miniproteins” that have
long been studied as ideal benchmarking candidates for physics-based PSP
methods [103, 98, 99, 104, 96]. One of the original variants of this protein
is TC10b (PDB: 2JOF), with sequence DAYAQWLKDGGPSSGRPPPS. A closely related
variant is TC5b (PDB: 1L2Y), with sequence NLYIQWLKDGGPSSGRPPPS. The
structures share a sequence identity of 85%, with mutations occurring at
residues 1, 2, and 4. The mutations at 2 and 4 still preserve the hydrophobic
chemical space, but result in more bulky sidechains (leucine and isoleucine
instead of alanines). The mutation at 1 is more pronounced however, with a
charged aspartate residue being substituted by an uncharged, but still polar,
asparagine residue. Although the NMR structures are quite similar, with an
RMSD of 0.8 Å between the two, the folding pathways and free energy landscapes
can be noticeably different.

Proteins at their melting point are in a state of unfolding and refolding,
much like a physical system transitioning between two states. One way to model
what this looks like is through molecular dynamics simulations. As a proof of
concept, we can explore the free energy landscape of the two variants of Trp-
Cage, TC10b and TC5b [105], derived from conventional molecular dynamics
simulations at their theoretical melting points (329K and 315K, respectively).
These simulations were performed with GROMACS2022 [106] (as implemented in the
Galaxy platform [107]) using the FF99SB force field [108] and the TIP3P water
model [109], for approximately 500 nanoseconds in the isothermal-isobaric
(NPT) ensemble. Time-dependent RMSDs and radii of gyration (Rg) of the alpha
carbon backbone were extracted to measure conformational changes over time,
and then posteriorly analyzed using MD DaVis [110]. Figure 4 illustrates the
resulting data, where the free energy (z-axis) is plotted as a function of
RMSD (x-axis) and Rg (y-axis). The difference between both plots is quite
clear. In Figure 4a, TC10b presents a much more rugged landscape, so much so
that there appears to be almost two separate folding paths, leading to two
distinct optimal conformations (circled in green). Both minima are relatively
close, with the smaller funnel reaching -9.01 kJ/mol and the larger funnel
arriving at -9.4 kJ/mol. Both optimal conformations appear to share nearly the
same Rg (6.8 vs 6.9 Å), while having distinct RMSDs (0.8 vs 5 Å). TC5b on the
other hand, presents a much more conservative free energy landscape (Figure
4b). There is a single global minima (circled in cyan) corresponding to an
energy of -13.5 kJ/mol. This optimal conformation shares the same radius of
gyration as TC10b (6.9 Å) with a slightly different RMSD (0.7 Å). Thus,
assuming the force fields are reproducing the experimental melting points
accurately (which is beyond the scope of this work) this brief qualitative
exercise highlights a central thesis of this perspective: even when mutated
sequences still lead to similar experimentally solved structures, the free
energy landscape and dynamic folding pathways can be quite different.
Computational methods must therefore be aware and capable of capturing these
unique interactions, while enumerating and sampling the entire conformational
space. This is precisely what a quantum algorithm could be capable of doing,
at scale.

####  3.2.2 Intrinsically disordered regions

While most proteins have ordered domains (with $\alpha$-helices,
$\beta$-sheets, and $\beta$-barrels), many intrinsically disordered regions
(IDRs) can exist throughout the structure. In fact, there are many proteins
that are significantly disordered (containing at least 30% disordered
residues), appropriately regarded as intrinsically disordered proteins (IDPs).
The human proteome is estimated to be approximately 32% intrinsically
disordered [111]. These regions serve a purpose as they are naturally
flexible, allowing them to often serve as binding domains for other proteins
or ligands. From a thermodynamic perspective, one can think of these regions
as occupying a series of local minima in the free energy landscape, which are
close in magnitude and thus equiprobable, with the global minimum often
induced by a binding process. Despite their disorder, there is likely an
ensemble average of structures, and it is important for PSP methods (and
subsequent drug design efforts) to capture this accurately. The deep learning
methods often fail at properly predicting these conformations [112]. What
appears to be consistent though, at least in the case of AlphaFold2, is that
these regions are usually predicted with low confidence scores, and for those
IDRs which conditionally fold (when bound to a substrate), a recent study
determined that it usually predicts a conformation closely related to that
folded state [113]. This perhaps presents an unfavorable bias in the neural
network, which could mislead researchers who are assuming the model is
predicting the unbound protein. Another study found interesting correlations
between the AlphaFold2 model’s confidence scores and the observed flexibility
throughout the protein in molecular dynamics simulations [114]. Regions with
lower confidence scores in the model were found to have higher flexibilities
in the simulations, while those with higher scores usually corresponded to
more stable secondary structures. This was quantified by measuring root mean
square fluctuations (RMSFs), a common metric to assess flexibility in MD
simulations and NMR ensembles. While this is an interesting correlation, this
inference does not indicate that the predicted conformation of these IDRs is
accurate, and researchers should be aware of this. To get a proper
representation of these disordered regions, an ensemble average must be
sampled. In order to do this, the physics at play must somehow be simulated,
which is something the template-based methods remain arguably agnostic to [6].

A perfect example of the importance of representing the disordered regions
correctly occurs with transmembrane proteins. As the name implies, these
proteins are embedded in cell membranes. They are key players in promoting
cellular homeostasis, by maintaining membrane integrity, aiding in cell
binding and adhesion, active and passive transport of substrates in and out of
the cell, and many other vital functions. They come in many shapes and forms.
One important superfamily of transmembrane proteins is the G-protein coupled
receptors (GPCRs). The basic anatomy of these proteins is outlined in Figure
5a. There are three main domains: the extracellular domain, the transmembrane
(TM) region, and the intracelluar domain (ICD). The intracellular and
extracellular domains interact with molecules in and out of the cell,
respectively. As such, flexible loop regions tend to be found here, in
addition to some segments of ordered secondary structures. The transmembrane
region on the other hand is almost entirely made up of a bundle (seven in this
case) of $\alpha$-helices forming a channel through the lipid bilayer of the
cell membrane, allowing substrates to enter and leave the cell. The structures
in Figure 5 include the incomplete X-ray crystal structure (b) and three
computational models (c-d, obtained from the SWISS-MODEL repository
444https://swissmodel.expasy.org/repository/uniprot/Q9Y5N1.) of the H3 human
histamine receptor (H3HR). Its primary function is the binding and release of
the neurotransmitter histamine. As such, it is vital for normal neurological
function and its impairment is associated with a number neurodegenerative
disorders, including Multiple Sclerosis (MS), Alzheimer’s disease (AD) and
Parkinson’s disease (PD) [115]. The crystal structure (PDB: 7f61) is missing
the entire intracellular domain, approximately 105 residues. Transmembrane
proteins like H3HR and other GPCRs are generally difficult to determine
experimentally for several reasons. Part of this is due to the fact that these
proteins exist in different media - the intracellular and extracellular
domains exist in aqueous hydrophillic media (cytosol and plasma) while the
transmembrane region is trapped in a hydrophobic environment surrounded by
lipids. Experimentally, it is very difficult to provide a solvent for proteins
that can emulate both environments. Thus, this class of proteins relies
heavily on some sort of homology modeling for researchers to further
investigate them. The ability of the deep learning methods to accurately fill
in the gaps here has been investigated before [116, 117]. Hegedus et al [117]
found that AlphaFold2 performed well, but most of their structures appeared to
mainly involve the ordered, transmembrane alpha helical bundles. Similarly,
Lee et al [116] found that AlphaFold2 and RoseTTaFold performed well in
predicting the TM regions, but there were significant discrepancies when
modelling the surrounding loop regions. This is precisely what we observe in
our analysis of the H3HR models. The AlphaFold model is colored in green, a
SWISS-Model based on isoform 7f61.1.A (essentially modelled without a template
for the ICD) colored in orange, and another SWISS-Model based on isoform
5dsg.2.A (using the M4 muscarinic receptor as a template for all domains)
colored in purple. The TM regions align quite well in all 4 structures as seen
in Figure 5c, and in all cases the RMSDs for this region is under 5.0 Å
(Figure 5d). The differences in the ICD structures were much more pronounced
between the models. The largest RMSD is observed between AlphaFold and both
SWISS-MODELS (averaging 26.5 Å). The RMSD in the ICD between the SWISS-MODELS
is still significant (13.8 Å), although about half as much as AlphaFold2’s,
where there appears to be virtually no order in the structure. In reality, the
ICDs of these proteins are neither fully ordered nor fully intrinsically
disordered, but rather a combination of segments of both, much like what is
observed in the case of the purple isoform in Figure 5d. Another interesting
fact worth pointing out is that, for reasons which remain unclear, the
AlphaFold2 model 555https://alphafold.ebi.ac.uk/entry/Q9Y5N1. is noticeably
larger. The model includes an additional 26 residues at the N-terminus and
another 14 residues at the C-terminus. With this, the model has 445 residues
total, while the FASTA sequence for the H3HR crystal structure (PDB:7f61)
contains only 407 residues, including those missing from the ICD.

The ability to accurately generate full length in silico models for
transmembrane proteins is both challenging and crucial. They remain among the
most experimentally difficult to determine and are accordingly the most
expensive human proteins to solve via crystallographic methods (with an
average cost of $2.5M per protein, at a 10% success rate [118]). Yet, solving
these structures holds tremendous therapeutic value. As a clear example,
approximately 35% of FDA approved drugs target GPCRs [119].

Figure 5: a) The common anatomy of transmembrane proteins like the G-protein
coupled receptor (GPCR) superfamily. b) The incomplete experimentally
determined crystal structure of H3HR, which is essentially missing the entire
ICD. c) Alignment and comparison between the experimental structure and
computational models of H3HR, highlighting the excellent agreement in the TM
region, but the significant discrepancies in the predicted ICD. d) Detailed
comparison of the structures reveals that the TM region is closely modeled
with RMSDs of 0.2 to 4.9 Å, while the ICD varies drastically with RMSDs of
13.8 to 26.8 Å. The least accurate model is arguably AlphaFold2’s, with a
highly disordered ICD and an extended C- and N-terminus.

.

###  3.3 Multiple sequence alignment availability and diversity

Given a single amino acid sequence, an MSA is a set of amino acid sequences
generated from the initially queried sequence with respect to how similar
sequences are to the initial sequence. MSAs capture important evolutionary
information that are widely used in PSP, protein classification and protein
design.

Using MSA-based information is not a new approach in PSP. Earlier works [120,
121]laid out the foundations for how one can use covariance methods to
identify the co-evolving pairs of residues, essentially investigating the
substitution patterns correlating among sequences from different organisms.
Later on, more inferential techniques from this statistical summary have been
applied to study protein-protein interaction from MSAs that can be used for
analyzing spatial proximity of residues in the structure prediction [122,
123]. The underlying idea in these analyses, at its core, is that the residues
that have co-evolved tend to stay in close proximity after the 3D structure
stabilizes and achieves its native state. In recent years, one usually applies
a statistical analysis to extract co-evolutionary information about the target
protein and uses this information in a deep learning architecture to predict
the 3D conformation. Over the years, there have been many works that utilize
some type of neural network based architecture to achieve this [124, 125, 126,
127]. However, the biggest breakthrough happened with the introduction of
AlphaFold2 [126] where a transformer block “Evoformer” was introduced that
translates the structure prediction problem to a graph inference problem. This
architecture is capable of capturing long-range dependencies among residues in
the generated MSAs and successfully derives co-evolutionary information that
produces highly accurate structure predictions.

With these new and more sophisticated deep learning techniques such as self-
attention mechanism emerging, using the co-evolutionary information from the
raw MSAs have become widely viable for PSP. Despite all these advances, access
to diverse and deep MSAs present an important computational bottleneck. In
particular, generating MSAs at a large scale is computationally expensive.
Pre-computed MSA databases that contain millions of sequences, such as the one
used in AlphaFold2, are not publicly available. In fact, a recent work [128]
created an open-source database OpenProteinSet that contains more than 16
million pre-computed MSAs to make it accessible to the larger research
community. This is a diverse and large database that researchers can use to
train models for machine learning based structure prediction tasks. Hence, it
is evident that MSAs play a central role for modern deep learning models for
PSP.

It is also important to mention that in parallel with the rising popularity of
large language models, protein language models (PLMs) are becoming a highly
significant tool for PSP [129, 130, 131]. Most notably, MSA Transformer [132]
is a protein language model with a data set of 26 million MSAs that uses row
and column attention mechanisms in its transformer based architecture. These
language models are unsupervised models that predict the conformation of the
protein based on the patterns recognized in the generated MSAs.

The information learned through MSAs by the deep learning models can
potentially go beyond the co-evolution information as evidenced in [133]. It
is hypothesized that the deep learning models can potentially learn an
approximate energy function through MSAs and use this as a starting point in
the energy landscape in the search for a global minimum. As a result, the
learned energy function informs the search locally, giving an advantage to the
model in this search problem. The authors support this claim by showing this
learned energy function can be used to rank the accuracy of some target
protein structures. This experiment reveals that, even though one uses a deep
learning model that is not directly relevant to energy landscape information,
it is still instrumental to understand the physics of the folding process and
an efficient traversal of the energy landscape is necessary to identify the
lowest energy conformation.

###  3.4 A multidimensional perspective on computational hardness

Protein sequence length does not singularly dictate prediction accuracy [134]
but rather, a combination of factors such as mutations, disorder, and MSA
depth collectively influences the ruggedness of the energy landscape. This
complexity necessitates a broader scope of information beyond sequence length
for a quantum solution to protein conformation problems. The fitness landscape
in protein sequences, a critical metric for mutation analysis, directly
influences biomolecular evolution by determining the Colony-Forming Unit (CFU)
and approximating free energy [135]. As mutations occur, local peaks emerge
within the fitness landscape, leading to changes in the energy landscape, as
depicted in Figure 4, introducing uncertainty regarding its structure and
potentially enabling quantum advantage. In comparison to structured proteins,
Intrinsically Disordered Proteins (IDPs) possess shallow, rugged energy
landscapes housing multiple local minima, especially in hypervariable regions
[136]. This stands in contrast to ordered regions like the guanosine
triphophate (GTP) binding domains, which exhibit a deep energy minimum. IDPs’
sensitivity to environmental changes, including post-translational
modifications, can significantly alter their energy landscapes by manipulating
local minima and barriers, thereby creating uncertainty and ruggedness
conducive to leveraging quantum advantages for conformation predictions.
Additionally, studies on kinase families have shown that unregularized
mutational statistics inference struggles to converge for shallow MSAs,
indicating an increasingly rugged and ill-conditioned energy landscape [137].
The results consistently highlight that reduced MSA depths result in Potts
models reflecting more rugged energy landscapes, affirming the potential for
quantum advantage in solving protein prediction problems affected by such
increased ruggedness and uncertainty.

In this section, we present our view on the hardness of a PSP task in terms of
three main parameters: the protein size (sequence length), the number of point
mutations and the MSA depth. We discuss a general framework for identifying
proteins for a structure prediction task, in particular using a physics-based
quantum algorithm, which have high potential for yielding competitive results
compared to state-of-the-art deep learning, MSA-based methods. We expect that
this will be beneficial for both quantum researchers who plan to benchmark
their novel quantum algorithms against state-of-the-art results, and also for
biomedical researchers who wish to explore quantum capabilities to enhance
their structure prediction results. Our perspective relies on a careful
analysis of the current landscape of the PSP research and the most recent
Critical Assessment of Structure Prediction (CASP) outcomes, a biennial event
where the latest PSP methods are showcased and their performance is
benchmarked [138]. Our perspective is meant to serve as a general guideline
for maximizing the best possible outcome in the regime where the best deep-
learning based methods are known to perform poorly. While we identify this
subspace of the problem space based on a particular quantum algorithm from
[139] and the actual boundaries of the subspace may change depending on the
method adopted, we believe that the general framework provided persists if one
wants to use an ab initio method for PSP.

In the previous sections, we discussed in detail three sources of hardness for
the PSP problem and how they relate to energy landscape, in particular
increasing complexity and ruggedness. Some of the target proteins from the
most recent CASP15 provide further evidence for these hardness metrics. As
discussed earlier, it is known that in the absence of deep and variable MSAs,
DL methods fail to produce accurate predictions. In particular, it is reported
in [126] that if the MSA depth is less than 30 sequences, AlphaFold2
prediction accuracy decreases significantly. Moreover, it is also mentioned
that if the MSA depth exceeds 100 sequences, the algorithm has diminishing
returns. Hence, one can conclude that for any protein sequence for which the
generated MSA has depth less than 30, these deep learning based methods will
yield low accuracy predictions. When measuring the performance of the
predictions, there are two common metrics CASP utilizes; Global Distance Test
- Total Score (GDT_TS score) and Template Modeling score (TM score) 666For
detailed description, see https://predictioncenter.org/casp15/doc/help.html..
Two target proteins T1122 and T1131 in CASP15 are examples of orphan proteins
where a reliable MSA cannot be generated. In particular, none of the
participants were able to produce a good prediction in the absence of MSAs.
The best GDT_TS score is less than 40 and and TM scores are around 0.5. A
GDT_TS score around 50 is interpreted to roughly predict the overall topology
of the structure correctly, and similar interpretations for TM score as well.
One can see for most predictions of T1122, only 60% of residues are under 10
angstroms. The other target orphan protein T1131 has similar results for all
participants with the best GDT_TS score around 25 (this is usually deemed as
good as a random guess) and the best TM score in 0.3s (See Figure 6).

Figure 6: The plots highlight the performance of all groups for the target
proteins, however we further distinguish; a) Performance of versions of
AlphaFold2 that participated in CASP15 for the three targets. b) Performance
of top three ranked groups with respect to overall z-scores in CASP15 for the
three targets. The graphs show the percentage of residues in the target
protein that are under a certain distance cut-off. It is easy to see that no
group for the target protein T1122, no group was able to predict all residues
under 10 angstroms. For T1160 and T1161, which exhibit point mutations, deep
learning-based methods used by the groups listed has 80 percent of residues
under 10 angstroms. In general, sharper increasing curves indicate poorer
predictions. Images generated from CASP15 official website from the results.

Similarly, predicting the impact of the mutation is a difficult task. As
detailed in [140], detecting how the mutations alter the conformation was a
big challenge in CASP15. In particular, two target proteins T1160 and T1161
differ by mutations at five residues. Both proteins were categorized as “easy”
since templates were provided and they are relatively short proteins with 48
amino acids. The top performing groups reached TM scores in 0.3 and 0.5s for
the two target proteins respectively. Best GDT_TS scores are 72.41 and 72.4
for the two proteins, while most groups achieved GDT_TS scores in 50s. This is
another example of how disruptive point mutations can be for a structure
prediction task, in particular using a technique that requires reliable MSA-
based information.

Based on these observations, if one wants to consider the parameters that can
make a protein structure task difficulty, we propose that along with the
protein size, MSA depth and the number of (point) mutations play a significant
role. A detailed analysis is presented in the previous sections in terms of
how they increase hardness and introduce more complexity in the energy
landscape. In particular, our perspective is helpful if one chooses to use a
method that does not rely on MSA-based information, such as physics-based or
free modeling methods. In this work, we use the ab initio quantum algorithm
from [139] to identify the subspace of the problem space where one can expect
to produce predictions that are potentially more accurate than state-of-the-
art, deep learning based methods.

The algorithm from [139], to our knowledge, is the best scaling quantum
algorithm that constructs a 3D backbone of the protein from a single amino
acid sequence. In this tetrahedral lattice model, a variational quantum
algorithm is adopted to find the lowest energy conformation of the protein
based on pre-calculated residue-residue interaction energies. We present a
sequence of nested subspaces based on these three sources of hardness and the
resource estimates for this particular quantum algorithm (See Section 5 for
further details).

Figure 7: Subspace of PSP problems partitioned with different color
rectangular boxes. This is in parallel with the IBM Quantum hardware road map
as system size increases. The nested rectangular boxes represent the subset of
proteins where deep learning based methods are known to perform poorly (data
point markers with no facecolors), hence other ab initio methods, including
quantum algorithms, can potentially yield better predictions. CASP data is
obtained from [141], however sequence length data is added from CASP website
directly. For each protein on the plot, average RMSD from top 10% of the
groups is calculated and added to the data. Point mutation data is obtained
from [142], and the MSA depth ($N_{eff}$) values are calculated using HHblits
and HHpred tools [143]. Average RMSD values are not calculated for point
mutation data set since they are not CASP targets. Clearly, there are many
more proteins within these boxes. Our goal is to show that there are non-
trivial, high-value targets for each of these regimes. The alignment with the
IBM Quantum Roadmap is more of a symbolic representation. While we estimated
qubit costs for the problem instances, one needs to perform a rigorous
resource estimation for a concrete representation. This is beyond the scope of
this work. We assume that for any range of qubit number, the quantum computer
is able to perform a reasonable number of gates under reasonable timeframe.

Using the regular encoding of an amino acid sequence in [139], we observe that
we can target proteins up to 22 amino acids in length using IBM’s 127-qubit
devices and extend this to 41 amino acids on IBM Osprey, a 433-qubit device.
We have also estimated how this subspace gets larger as we have access to
larger devices, aligned with IBM Quantum’s hardware roadmap.777For further
information see https://www.ibm.com/quantum/roadmap. Since we have established
that proteins with shallow MSAs are good candidates for this algorithm [139],
we target the protein with MSA depth less than 60 as shown in Figure 7. MSA
depth is measured by the number of effective sequences $N_{eff}$, similar to
[126], and the previous CASP data in the Figure 7 is obtained from the public
repository [141] and added some extra data curated by us. Since the target
protein can exhibit any number of (point) mutations not exceeding the overall
residue count, the protein size potentially determines the boundaries for the
other axis representing the number of point mutations. However, the data we
curated contains maximum 15 point mutations, so the axis boundaries are
adjusted accordingly for better visualization. As a result, we argue that any
target protein contained in the rectangular box regions in Figure 7 is
potentially a suitable candidate for a physics-based quantum algorithm to
benchmark the performance and yield competitive results compared to state-of-
the-art, deep learning template-based algorithms.

We make a few important remarks and observations before concluding this
section. Most importantly, while the actual numerical boundaries of the
subsets provided in Figure 7 depend on the quantum algorithm we adopt, the
general axes that describe computational hardness and the underlying idea of
“going wide instead of going deep” provide a systematic way of benchmarking ab
initio quantum algorithms to maximize the outcome. As quantum hardware becomes
more capable, we expect that this subspace with scale accordingly and include
more potential target proteins. Moreover, with the current boundaries
provided, this is a non-trivial subset of all proteins. There are 1321 protein
structures in PDB that contains 22 or less residues. Similarly, almost 3000
protein sequences contain 41 or less amino acids in PDB database and if we
query the condition that the sequences exhibit at least one mutation, we see
that there are 509 protein sequences 888All data collected from PDB website,
note that some of these sequences can be part of a larger protein sequence.
Since the proteins from PDB are already being utilized for training deep
learning based algorithms, we note that these numerical values from PDB are
meant to represent the variety of proteins that are available at these smaller
scales..

Benchmarking quantum algorithms and techniques is an important research area
within the quantum community. Our perspective provides a novel and systematic
approach that aims to maximize the utility of adopting a physics-based quantum
algorithm and benchmarking the outcome. As a result, we believe that whether
developing a quantum algorithm or exploring existing quantum algorithms for
PSP tasks, selecting a target protein in the regions provided in Figure 7
maximizes the utility and creates the best possible outcome.

##  4 A quantum-classical hybrid workflow for protein structure prediction

While the promise of general quantum advantage comes with strong theoretical
foundation, quantum computers are not expected to replace classical computers
in every task. A scientific workflow may have a component that is
computationally intensive and could be handled by a quantum algorithm while
the remaining steps are treated on classical hardware. For example, simulating
a catalytic reaction in the active site of an enzyme with a quantum
Hamiltonian while the rest of the domains are modeled using classical
algorithms (such as using conventional molecular mechanics force fields)
[144]. Initially, we envision quantum algorithms for PSP to be designed as
part of hybrid quantum-classical workflows. Lattice models can help simplify
the PSP problem by representing proteins as a chain of beads on a lattice.
Generally speaking, these models employ an energy function to enumerate and
score conformations, with a search algorithm used afterwards to identify the
lowest energy conformation. The energy functions may range in sophistication,
from a basic hydrophobic-polar model (where the alphabet is reduced to a
binary representation, allowing for just three possible interaction types) to
more detailed models considering various physical forces including the
hydrophobic effect, electrostatic interactions, $\pi-\pi$ stacking, hydrogen
bonding, induced dipole moments, and side-chain packing. In any case, the
objective should always be to effectively reproduce the free energy associated
with every possible type of interaction. The more detailed the energy function
is and the more granular the lattice model, the greater the number of
physically reasonable enumerated conformations. The challenge for the search
algorithm then becomes the ability to effectively traverse this enormous space
containing all possible conformations, which remains NP-hard even at a coarse
grain resolution on a 2D lattice.

Figure 8: A schematic representation of the workflow used in this study. The
most computationally demanding part, finding a coarse-grain representation of
the lowest energy conformation of the protein structure, is performed on a
quantum computer. The following steps are handled classically to convert the
output into a desired format and post-process to construct the full structure,
while preserving the quantum algorithm’s originally predicted backbone
geometry in the coarse grain model. A final refinement of the all-atom
structure is then performed through further energy minimization using a
molecular mechanics force field. This last step allows the protein to
potentially reach an even more optimal configuration as the atoms and bonds
are no longer constrained to the four turns of the original lattice structure.

The series and order of steps involved in this proposed workflow (see Figure
8, and LABEL:supp-sec:workflow of the supplementary information for a more
detailed discussion of the workflow) are not uncommon in ab initio PSP
methods. For example, the Rosetta algorithm [145] first creates small coarse
grain fragments using a knowledge-based potential (the Rosetta energy
function), much like what is done in the first step of this workflow (using
the Mizayawa-Jernigan potentials instead [146]). These fragments are then
assembled using a set of reference PDB structures until the full size of the
sequence is built. In the later stages, the Rosetta “all-atom” energy function
[147] is then employed which refines the all-atom models. A clustering
analysis is performed on the resulting models from each iteration traversing
the search space, with the average structure from the largest cluster becoming
the chosen solution. The fact that templates are used in this process is part
of the reason why some members of the community have debated as to whether or
not Rosetta is in fact a true ab initio method, since it is mainly the
fragments are constructed exclusively with an energy function. By now, we are
of course aware that efficiency plummets when trying to do this beyond a few
dozen amino acids. Nonetheless, incorporating the physics this way has clearly
allowed the method to preserve accuracy while handling large sequences, an
advantage over most other template-based methods. With the possible advantage
quantum computing brings in handling large search spaces, fragment assembly
may not be required, allowing for the enumeration and sampling of large
conformers in a single pass of the algorithm. This is what our proposed
workflow attempts to do - build the entire sequence in one step, sample the
energies of all possible conformers, and finally refine the optimal solution.

We tested steps 1-3 of this workflow on a small, but highly relevant seven
amino acid fragment (LHPGAGK) - the “P-loop” of the Zika virus helicase
protein. See LABEL:supp-sec:zika_loop of the Supplementary section for more
discussion about the relevance of this protein in drug discovery. We chose the
P-loop as the “toy model” for this workflow and present our results as an
initial proof of concept. A comparison is made in Figure 9 between the results
from a) the full quantum algorithm executed on IBM_Cleveland, b) the quantum
algorithm’s Ising Hamiltonian (a classical cost function originally defined in
[139]) handled classically (by an exhaustive brute force search as well as
Gurobi [148]), and c) AlphaFold2. In every case, the loop’s experimental
structure (extracted from the full crystal structure, PDB: 5gjb) are colored
in cyan, while the models are colored in orange, yellow, and red. The
backbones were aligned to the experimental structure and RMSDs were measured
in each case using VMD [149]. The best model was in fact produced by the
execution on the IBM_Cleveland quantum device, with an RMSD of 1.781 Å with
respect to the crystal structure. During sampling, the conformational energy
initially rises sharply but begins to continously hover around the basin
shortly after a handful of VQE iterations (Figure 9d). The Ising Hamiltonian
produced by the quantum algorithm contains higher order polynomial terms
(cubic and above), which can often increase the complexity of optimization in
general, and more so on classical computers. We have also solved the same
Hamiltonian classically to compare the performance. While the brute force
search was performed on the native high order Hamiltonian, it was also
quadratized and converted into a QUBO (quadratic unoptimized binary
optimization) Hamiltonian using a method developed by Fujitsu [150], and
posteriorly solved with Gurobi. The Hamiltonian term corresponding to the
second best result was obtained as the optimal solution by both Gurobi and a
brute force search, yielding an RMSD of 1.879 Å (Figure 9b). AlphaFold2
produced the least accurate model (Figure 9c), with an RMSD of 3.53 Å, nearly
twice as much as that predicted by the quantum algorithm. Radius of gyration
was also measured in each case, and the same trend can be observed (Figure
9e). The averaged results from IBM_Cleveland (4.6 Å) and the classical solvers
(4.8 Å) is the same as the experimental structure (4.7 Å), while AlphaFold2’s
prediction is noticeably greater (6.9 Å).

These results highlight a few important things: 1) Both the tetrahedral
lattice model and the Miyazawa-Jernigan potential employed in the algorithm
[139] could be sophisticated enough for applications in ab initio PSP at this
scale. 2) Even though VQE may not always find the absolute lower bound in the
energy, it could be well suited for sampling near optimal low energy
conformations that may even produce more accurate results than an exhaustive
search when the measure of success is based on structural similarity to a
ground truth experimental solution. It is well known that there could be
several local minima that are quite close in magnitude and sign, yet be
structurally distinct. An example of this can be seen in Figure 4a, where the
bottom of both funnels correspond to two distinct conformations similar in
energy, but only one of them is closest to the experimental structure. 3)
Classically rebuilding the all-atom structure can still preserve the
originally predicted geometry by the quantum algorithm. 4) A quantum-classical
hybrid workflow like this could possibly be used as a tool in tandem with
template-based programs to predict binding domains characterized by loops.

Figure 9: An initial validation of the workflow with the Zika virus helicase
P-loop (LHPGAGK). In all cases, the coordinates from the experimental crystal
structure are colored in cyan. The most accurate structure is produced by a)
the quantum algorithm executed on IBM_Cleveland, followed by b) the problem
Hamiltonian solved classically by brute force and classical mixed-integer
linear solver. The least accurate model was produced by c) AlphaFold2. This is
observed by the relative RMSD values in each case, as well as e) the measured
radius of gyration. The conformational energy plot in d) appears to
demonstrate that VQE begins to continuously sample conformations around the
basin after a handful of VQE iterations.

##  5 Resource Estimation

Similar to determining which PSP problems may benefit from quantum
computational advantages, accurately estimating the resources required for
quantum algorithms that solve PSP problems presents its own difficulties. It
is important to note that a rigorous resource estimation of a quantum
algorithm for a given PSP problem is beyond the scope of a perspective paper.
However, we contextualize our perspective with a simplified resource
estimation framework. We only consider a utility-scale quantum algorithm
however a similar framework can also be developed for a fault-tolerant quantum
algorithm.

For the discussion presented in this section, we operate under the following
set of assumptions:

  * •

We use IBM quantum computers as our target devices. The diversity in QPU qubit
connectivity and topology can have an impact on the resource estimation. Our
estimates are based on the 127-qubit Eagle (R3) quantum chip on
IBM_Washington.

  * •

The quantum algorithm employed changes the various resources needed to execute
the experiment. We implement the algorithm from [139], which is a variational
quantum algorithm. As a result, qubit count depends on the choice of lattice
for the coarse-grain model and the interaction model between residues.
Moreover, the ansatz we have used is the RealAmplitudes ansatz from Qiskit’s
circuit library for only one repetition. We point the readers to [151] for
comparative overview of different types of ansatze.

  * •

A more comprehensive resource estimation requires analyzing the entire
workflow in Figure 8, whereas we only focus on the first step of the workflow.
Furthermore, while we analyze some commonly used metrics such as qubit count,
circuit depth, ECR depth, and number of measurements, there are other
parameters one can consider such as overall and gate-level execution time,
overhead from error mitigation, etc.

  * •

In a future work, our goal is to provide a robust resource estimation for PSP,
where we consider the other parameters mentioned above.

For simplicity, we have adopted a simple regression model for extrapolation.
Our approach involves mapping the problem to a quantum circuit, characterizing
the circuit in terms of the number of gates and qubits required, applying
different levels of circuit optimization and analyzing the number of ECR gates
required for execution. We derived our estimates using the sequence of an
experimentally verified protein structure (membrane-proximal cytoplasmic
domain tail of platelet integrin $\alpha$IIb-$\beta$3) 999The PDB entry can be
found here https://www.rcsb.org/structure/1M8O.) The estimates provided in the
Figure 10 were calculated for up to 22 amino acids. Using the regression model
we built, we have established the resource cost in different categories such
as number of qubits, number of ECR gates that can be executed in parallel,
total number of gates, total circuit depth for both optimized and unoptimized
circuits, and upper limit on the number of measurements.

Figure 10: a) The total number of qubits scale quadratically as the protein
size increases. We consider both configuration and interaction qubits to
encode a given amino acid sequence, and analyze the scaling. b) We estimate an
upper bound for the number of measurements needed to predict a protein
structure within a fixed energy error margin using the work from [152]. The
energy unit is converted to kcal/mol from Hartree for consistency with our
workflow. See Section LABEL:supp-sec:measurement-formula of the supplementary
materials for mathematical formula to calculate the upper bound. The three
plots show different upper limits to predict a protein structure within
$\varepsilon=1,5,10$ kcal/mol range of the lowest energy conformation. The
upper bound is not known to be tight. We further expect the empirically
sufficient number of measurements to be significantly lower than this upper
bound. c) Shows the total number of ECR gates in the circuit as the protein
size increases for different optimization levels. Qiskit transpiler allows
four levels of optimization which are defined in [153]. d) Shows the ECR-depth
as a function of protein size.

Our estimates show that there is a perfect quadratic relationship between the
protein sequence length and the number of qubits required. The primary reason
we have based our regression model up to 22 amino acids is the current
capabilities of the quantum hardware. Our estimates show that for a protein
sequence with 22 amino acids, the number of qubits needed is 118 (see Figure
10a). Extrapolated according to these estimates, we expect that we can go up
to 41 amino acids with the release of IBM’s Osprey, 433 qubit quantum
computer, and up to 67 amino acids on IBM’s Condor, a 1121 qubit machine. As a
comparison, similar estimates were conducted when converting the Hamiltonian
into a QUBO format and this can be found in LABEL:supp-sec:hamiltonian of the
supplementary materials. While qubit count is an important constraint for
hardware experiments, we are also assuming that we will be able to run the
number of gates that we project in these estimates. As we have access to more
qubits and higher gate fidelity, we believe that it will be possible to
predict larger and larger protein structures using quantum algorithms. While
in terms of protein size, current deep learning based methods can target much
larger structures, as we discussed in Section 3, we believe that there are
still regimes in smaller proteins where physics-based quantum algorithms can
yield better results.

In conclusion, we have established an initial analysis of the quantum
computational resource cost for protein conformation predictions using the
quantum algorithm from [139], based on the full protein structure of platelet
integrin $\alpha$IIb-$\beta$3 cytoplasmic domain [154]. Our estimates are
crucial for understanding the limits of the subset of the problem space we
want to target. There are various other techniques such as dense encoding,
optimizing logical-to-physical qubit mapping, tensor-based circuit
transpilation methods, circuit cutting and dynamic circuits that can enable us
to predict larger proteins. In our future work, our goal is to explore some of
these methods to enhance the performance of the quantum algorithm.

##  6 General discussion

The protein folding problem and protein structure prediction remain
challenging tasks despite advances. Template-based methods, particularly those
propelled by machine learning, have enabled researchers to obtain realistic
models faster than wet lab experiments, establishing structure-function
relationships that would have otherwise taken a very long time to discover.
Growth of databases like the Protein Data Bank, now exceeding 200,000
experimental structures, underpins the success of these methods. However,
known sequences outpace solved structures, around 300 million versus 200,000
[155]. This highlights expansive unseen biodiversity. Advancing physics-based
PSP methods is crucial to help bridge this gap.

Some argue that most natural folds have already been discovered, and by
extension the template-based methods should be capable of predicting most
structures. Remarkable performances like those of AlphaFold2 support this
claim, given enough homology. However, improvement is still needed when
dealing with mutations (even in conserved regions), orphan proteins, shallow
MSAs, and novel sequences. AlphaMissense may address mutations [156]. Still,
there are many missing pieces in this puzzle according to many in the
biophysics community. Overcoming these limitations fundamentally requires
simulating physics. Molecular dynamics could be a solution, but is constrained
by timescales, system sizes, and the chosen force fields. Ab initio methods
also scale poorly on classical hardware as the search space grows unmanageably
large.

Quantum computing shows promise in both accuracy and scalability, as seen by
the results of our quantum-classical workflow in predicting the Zika virus NS3
helicase P-loop. It produced an accurate model with a backbone RMSD of less
than 2.0 Å compared to the experimental structure, and with superior
scalability versus classical optimization [139]. With 10,000 qubits and
sufficient circuit fidelity, prediction of biomedically relevant proteins and
mutants could be attainable. For instance, our estimates indicate hemoglobin’s
141 residues may be predictable using 4,967 entangled qubits – an extremely
valuable prospect since its structure enables fundamental life functions.
Although this structure has long been determined experimentally, it is at this
size and scale where some of the most biologically relevant proteins exist.

Early quantum algorithms will likely mirror classical physics methods.
Eventually quantum chemical calculations could replace knowledge-based energy
functions, initially with DFT or Hartree-Fock for larger peptides than what is
currently possible on classical hardware. The ultimate goal is perhaps
simulating exact electron dynamics, deriving molecular and electronic
structures in real time, capturing nature in its most absolute form. This
could bring about a new era for molecular force fields as well, perhaps
yielding the ability to not only predict optimal protein structures
irrespective of path, but truly help solve the path-dependent protein folding
problem itself.

The future of structural biology depends profoundly on developing and scaling
up efficient physics-based PSP methods. Quantum computing’s potential to
simulate nature’s fundamental mechanics may help overcome the historical
barriers. This will not have to be mutually exclusive, but could be rather
synergistic, with quantum PSP methods helping complement the strengths of
classical template-based methods, and vice versa. Multidisciplinary
collaboration between biophysics, chemistry, computer science, and structural
biology can help unravel the remaining mysteries. For the first time in
decades, elucidating how and why proteins fold appears within reach,
presenting a milestone deeply advancing our comprehension of the subtle
intricacies enabling life.

## Acknowledgments

Portions of this material is based upon work supported by the Defense Advanced
Research Projects Agency (DARPA) under Contract No. HR001122C0102 to O.S. Any
opinions, findings and conclusions or recommendations expressed in this
material are those of the author(s) and do not necessarily reflect the views
of the Defense Advanced Research Projects Agency. The authors thank Kenneth
Merz, Iris Smith, and William Martin from the Cleveland Clinic, and Emily
Pritchett, Saif Rayyan, Antonio Mezzacapo, Courtney Long, Wiktor Mazin, Niall
Robertson, Michal Rosen-Zvi, Jeannette Garcia, Filippo Utro, Kahn
Rhrissorrakrai, Ryan Sweke, Joseph Morrone, Yoel Shoshan, Ivano Tavernelli,
Sarah Mostame, Sara Capponi, Jannis Born and Cynthia McKinney from IBM, and
Robert Jernigan from Iowa State University of Science and Technology for their
expert opinions and insightful feedback on the manuscript or during the
project. They also acknowledge DARPA, NASA, and the MIT Lincoln Lab team
involved in the Phase 1 of the DARPA Quantum Benchmarking program led by
Joseph Altepeter, for rigorous feedback.

## Competing interests

The authors declare no competing interests.

## References

  * [1] Dill KA, Ozkan SB, Shell MS, and Weikl TR.  The protein folding problem.  Annu Rev Biophys., 37:289–316, 2009.
  * [2] Ken A. Dill and Justin L. MacCallum.  The protein-folding problem, 50 years on.  Science, 338(6110):1042–1046, 2012.
  * [3] John Jumper, Richard Evans, and Alexander Pritzel et al.  Highly accurate protein structure prediction with AlphaFold.  Nature, 596(7873):583–589, July 2021.
  * [4] Minkyung Baek, Frank DiMaio, Ivan Anishchenko, Justas Dauparas, Sergey Ovchinnikov, Gyu Rie Lee, Jue Wang, Qian Cong, Lisa N. Kinch, R. Dustin Schaeffer, Claudia Millán, Hahnbeom Park, Carson Adams, Caleb R. Glassman, Andy DeGiovanni, Jose H. Pereira, Andria V. Rodrigues, Alberdina A. van Dijk, Ana C. Ebrecht, Diederik J. Opperman, Theo Sagmeister, Christoph Buhlheller, Tea Pavkov-Keller, Manoj K. Rathinaswamy, Udit Dalwadi, Calvin K. Yip, John E. Burke, K. Christopher Garcia, Nick V. Grishin, Paul D. Adams, Randy J. Read, and David Baker.  Accurate prediction of protein structures and interactions using a three-track neural network.  Science, 373(6557):871–876, August 2021.
  * [5] Xiaogen Zhou, Wei Zheng, Yang Li, Robin Pearce, Chengxin Zhang, Eric W. Bell, Guijun Zhang, and Yang Zhang.  I-TASSER-MTD: a deep-learning-based platform for multi-domain protein structure and function prediction.  Nature Protocols, 17(10):2326–2353, August 2022.
  * [6] Carlos Outeiral, Daniel A Nissley, and Charlotte M Deane.  Current structure predictors are not learning the physics of protein folding.  Bioinformatics, 38(7):1881–1887, January 2022.
  * [7] Frank Noé, Gianni De Fabritiis, and Cecilia Clementi.  Machine learning for protein folding and dynamics.  Current opinion in structural biology, 60:77–84, 2020.
  * [8] Brian Kuhlman and Philip Bradley.  Advances in protein structure prediction and design.  Nature Reviews Molecular Cell Biology, 20(11):681–697, 2019.
  * [9] Kingshuk Ghosh, S. Banu Ozkan, and Ken A. Dill.  The ultimate speed limit to protein folding is conformational searching.  Journal of the American Chemical Society, 129(39):11920–11927, 2007.  PMID: 17824609.
  * [10] Sergey Bravyi, Oliver Dial, Jay M Gambetta, Darío Gil, and Zaira Nazario.  The future of quantum computing with superconducting qubits.  Journal of Applied Physics, 132(16), 2022.
  * [11] Paul Adrien Maurice Dirac.  The principles of quantum mechanics, 1981.
  * [12] Werner Heisenberg.  Development of concepts in the history of quantum theory.  In The physicist’s conception of nature, pages 264–275. Springer, 1973.
  * [13] Michael A Nielsen and Isaac L Chuang.  Quantum computation and quantum information.  Cambridge university press, 2010.
  * [14] Saugata Basu, Jannis Born, Aritra Bose, Sara Capponi, Dimitra Chalkia, Timothy A Chan, Hakan Doga, Mark Goldsmith, Tanvi Gujarati, Aldo Guzman-Saenz, et al.  Towards quantum-enabled cell-centric therapeutics.  arXiv preprint arXiv:2307.05734, 2023.
  * [15] 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.
  * [16] 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.
  * [17] Andris Ambainis and Robert Špalek.  Quantum algorithms for matching and network flows.  In Annual Symposium on Theoretical Aspects of Computer Science, pages 172–183. Springer, 2006.
  * [18] Seth Lloyd.  Universal quantum simulators.  Science, 273(5278):1073–1078, 1996.
  * [19] Dorit Aharonov, Vaughan Jones, and Zeph Landau.  A polynomial quantum algorithm for approximating the jones polynomial.  In Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, pages 427–436, 2006.
  * [20] Pawel Wocjan, Chen-Fu Chiang, Anura Abeyesinghe, and Daniel Nagaj.  Quantum speed-up for approximating partition functions.  arXiv preprint arXiv:0811.0596, 2008.
  * [21] Amir Kalev, Tongyang Li, Cedric Yen-Yu Lin, Krysta M Svore, Xiaodi Wu, et al.  Quantum sdp solvers: Large speed-ups, optimality, and applications to quantum learning.  Leibniz international proceedings in informatics, 2019.
  * [22] Aram W Harrow, Avinatan Hassidim, and Seth Lloyd.  Quantum algorithm for linear systems of equations.  Physical review letters, 103(15):150502, 2009.
  * [23] Alexander Engel, Graeme Smith, and Scott E Parker.  Quantum algorithm for the vlasov equation.  Physical Review A, 100(6):062315, 2019.
  * [24] Andris Ambainis, Kaspars Balodis, Jānis Iraids, Martins Kokainis, Krišjānis Prūsis, and Jevgēnijs Vihrovs.  Quantum speedups for exponential-time dynamic programming algorithms.  In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1783–1793. SIAM, 2019.
  * [25] Corey O’Meara, Kavitha Yogaraj, Kumar Ghosh, Piergiacomo Sabino, Marina Fernández-Campoamor, Giorgio Cortiana, Juan Bernabé-Moreno, Francesco Tacchino, Antonio Mezzacapo, Omar Shehab, et al.  Quadratic quantum speedup in evaluating bilinear risk functions.  arXiv preprint arXiv:2304.10385, 2023.
  * [26] 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.
  * [27] 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.
  * [28] Diego Ristè, Marcus P Da Silva, Colm A Ryan, Andrew W Cross, Antonio D Córcoles, John A Smolin, Jay M Gambetta, Jerry M Chow, and Blake R Johnson.  Demonstration of quantum advantage in machine learning.  npj Quantum Information, 3(1):16, 2017.
  * [29] Howard Dale, David Jennings, and Terry Rudolph.  Provable quantum advantage in randomness processing.  Nature communications, 6(1):8203, 2015.
  * [30] Nikitas Stamatopoulos, Guglielmo Mazzola, Stefan Woerner, and William J Zeng.  Towards quantum advantage in financial market risk using quantum gradient algorithms.  Quantum, 6:770, 2022.
  * [31] 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.
  * [32] Sau Lan Wu, Shaojun Sun, Wen Guan, Chen Zhou, Jay Chan, Chi Lung Cheng, Tuan Pham, Yan Qian, Alex Zeng Wang, Rui Zhang, et al.  Application of quantum machine learning using the quantum kernel algorithm on high energy physics analysis at the lhc.  Physical Review Research, 3(3):033221, 2021.
  * [33] 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.
  * [34] Guido Burkard, Thaddeus D Ladd, John M Nichol, Andrew Pan, and Jason R Petta.  Semiconductor spin qubits.  arXiv preprint arXiv:2112.08863, 2021.
  * [35] 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.
  * [36] Alexander Kozhanov, Yichao Yu, Liudmila Zhukas, Lei Feng, Debopriyo Biswas, Bahaa Harraz, Keqin Yan, Vivian Zhang, Crystal Noel, and Christopher Monroe.  Next-generation trapped-ion quantum computing system.  In Quantum 2.0, pages QM3A–2. Optica Publishing Group, 2023.
  * [37] Matthias Steffen, David P DiVincenzo, Jerry M Chow, Thomas N Theis, and Mark B Ketchen.  Quantum computing: An ibm perspective.  IBM Journal of Research and Development, 55(5):13–1, 2011.
  * [38] Andrew Cross.  The ibm q experience and qiskit open-source quantum computing software.  In APS March meeting abstracts, volume 2018, pages L58–003, 2018.
  * [39] The CUDA Quantum development team.  Cuda quantum, July 2023.  If you use this software, please cite it as below.
  * [40] Rigetti Computing.  Pyquil documentation.  http://pyquil.readthedocs.io/en/latest, 2019.
  * [41] 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.
  * [42] Cirq Developers.  Cirq, December 2022.  See full list of authors on Github: https://github .com/quantumlib/Cirq/graphs/contributors.
  * [43] Constantin Gonzalez.  Cloud based qc with amazon braket.  Digitale Welt, 5:14–17, 2021.
  * [44] Ionq technology roadmap, 2023.
  * [45] Quantinuum technology roadmap, 2023.
  * [46] J Eli Bourassa, Rafael N Alexander, Michael Vasmer, Ashlesha Patil, Ilan Tzitrin, Takaya Matsuura, Daiqin Su, Ben Q Baragiola, Saikat Guha, Guillaume Dauphinais, et al.  Blueprint for a scalable photonic fault-tolerant quantum computer.  Quantum, 5:392, 2021.
  * [47] Sara Bartolucci, Patrick Birchall, Hector Bombin, Hugo Cable, Chris Dawson, Mercedes Gimeno-Segovia, Eric Johnston, Konrad Kieling, Naomi Nickerson, Mihir Pant, et al.  Fusion-based quantum computation.  Nature Communications, 14(1):912, 2023.
  * [48] Ibm quantum roadmap, Oct 2015.
  * [49] Yuri Alexeev, Dave Bacon, Kenneth R Brown, Robert Calderbank, Lincoln D Carr, Frederic T Chong, Brian DeMarco, Dirk Englund, Edward Farhi, Bill Fefferman, et al.  Quantum computer systems for scientific discovery.  PRX Quantum, 2(1):017001, 2021.
  * [50] Prashant S Emani, Jonathan Warrell, Alan Anticevic, Stefan Bekiranov, Michael Gandal, Michael J McConnell, Guillermo Sapiro, Alán Aspuru-Guzik, Justin T Baker, Matteo Bastiani, et al.  Quantum computing at the frontiers of biological sciences.  Nature Methods, 18(7):701–709, 2021.
  * [51] Douglas Beck, Joseph Carlson, Zohreh Davoudi, Joseph Formaggio, Sofia Quaglioni, Martin Savage, Joao Barata, Tanmoy Bhattacharya, Michael Bishof, Ian Cloet, et al.  Quantum information science and technology for nuclear physics. input into us long-range planning, 2023.  arXiv preprint arXiv:2303.00113, 2023.
  * [52] Alberto Di Meglio, Karl Jansen, Ivano Tavernelli, Constantia Alexandrou, Srinivasan Arunachalam, Christian W Bauer, Kerstin Borras, Stefano Carrazza, Arianna Crippa, Vincent Croft, et al.  Quantum computing for high-energy physics: State of the art and challenges. summary of the qc4hep working group.  arXiv preprint arXiv:2307.03236, 2023.
  * [53] Daniel J Egger, Claudio Gambella, Jakub Marecek, Scott McFaddin, Martin Mevissen, Rudy Raymond, Andrea Simonetto, Stefan Woerner, and Elena Yndurain.  Quantum computing for finance: State-of-the-art and future prospects.  IEEE Transactions on Quantum Engineering, 1:1–24, 2020.
  * [54] Casey Berger, Agustin Di Paolo, Tracey Forrest, Stuart Hadfield, Nicolas Sawaya, Michal Stechly, and Karl Thibault.  Quantum technologies for climate change: Preliminary assessment.  arXiv preprint arXiv:2107.05362, 2021.
  * [55] Mauritz Kop, Mateo Aboy, Eline De Jong, Urs Gasser, Timo Minssen, I Glenn Cohen, Mark Brongersma, Teresa Quintel, Luciano Floridi, and Ray Laflamme.  Towards responsible quantum technology.  Available at SSRN 4393248, 2023.
  * [56] Charles H Bennett, Ethan Bernstein, Gilles Brassard, and Umesh Vazirani.  Strengths and weaknesses of quantum computing.  SIAM journal on Computing, 26(5):1510–1523, 1997.
  * [57] JJ Alvarez and C Gomez.  A comment on fisher information and quantum algorithms.  arXiv preprint quant-ph/9910115, 1999.
  * [58] Carlo Cafaro and Stefano Mancini.  On grover’s search algorithm from a quantum information geometry viewpoint.  Physica A: Statistical Mechanics and its Applications, 391(4):1610–1625, 2012.
  * [59] Edward Farhi, Jeffrey Goldstone, and Sam Gutmann.  A quantum approximate optimization algorithm.  arXiv preprint arXiv:1411.4028, 2014.
  * [60] Sergey Bravyi, Alexander Kliesch, Robert Koenig, and Eugene Tang.  Obstacles to variational quantum optimization from symmetry protection.  Physical review letters, 125(26):260505, 2020.
  * [61] Matthew B Hastings.  Classical and quantum bounded depth approximation algorithms.  arXiv preprint arXiv:1905.07047, 2019.
  * [62] Giacomo De Palma, Milad Marvian, Cambyse Rouzé, and Daniel Stilck França.  Limitations of variational quantum algorithms: a quantum optimal transport approach.  PRX Quantum, 4(1):010309, 2023.
  * [63] Jacob M Clary, Eric B Jones, Derek Vigil-Fowler, Christopher Chang, and Peter Graf.  Exploring the scaling limitations of the variational quantum eigensolver with the bond dissociation of hydride diatomic molecules.  International Journal of Quantum Chemistry, page e27097, 2023.
  * [64] Ken A Dill, Darwin OV Alonso, and Karen Hutchinson.  Thermal stabilities of globular proteins.  Biochemistry, 28(13):5439–5449, 1989.
  * [65] Ken A Dill and Hue Sun Chan.  From levinthal to pathways to funnels.  Nature structural biology, 4(1):10–19, 1997.
  * [66] José Nelson Onuchic and Peter G Wolynes.  Theory of protein folding.  Current opinion in structural biology, 14(1):70–75, 2004.
  * [67] Folding funnel.  https://en.wikipedia.org/wiki/Folding_funnel.  Accessed: 2023-10-06.
  * [68] Buyong Ma, Sandeep Kumar, Chung-Jung Tsai, and Ruth Nussinov.  Folding funnels and binding mechanisms.  Protein engineering, 12(9):713–720, 1999.
  * [69] Patricia L Clark.  Protein folding in the cell: reshaping the folding funnel.  Trends in biochemical sciences, 29(10):527–534, 2004.
  * [70] Peter G Wolynes, Jose N Onuchic, and Dave Thirumalai.  Navigating the folding routes.  Science, 267(5204):1619–1620, 1995.
  * [71] Martin Karplus.  Behind the folding funnel diagram.  Nature chemical biology, 7(7):401–404, 2011.
  * [72] J Nelson Onuchic, Peter G Wolynes, Z Luthey-Schulten, and Nicholas D Socci.  Toward an outline of the topography of a realistic protein-folding funnel.  Proceedings of the National Academy of Sciences, 92(8):3626–3630, 1995.
  * [73] Chung-Jung Tsai, Sandeep Kumar, Buyong Ma, and Ruth Nussinov.  Folding funnels, binding funnels, and protein function.  Protein Science, 8(6):1181–1190, 1999.
  * [74] George Chikenji, Yoshimi Fujitsuka, and Shoji Takada.  Shaping up the protein folding funnel by local interaction: Lesson from a structure prediction study.  Proceedings of the National Academy of Sciences, 103(9):3141–3146, 2006.
  * [75] Wenzhen Jin, Ohki Kambara, Hiroaki Sasakawa, Atsuo Tamura, and Shoji Takada.  De novo design of foldable proteins with smooth folding funnel: automated negative design and experimental verification.  Structure, 11(5):581–590, 2003.
  * [76] Ulrich HE Hansmann, Yuko Okamoto, and Jose N Onuchic.  The folding funnel landscape for the peptide met-enkephalin.  Proteins: Structure, Function, and Bioinformatics, 34(4):472–483, 1999.
  * [77] Nicholas D Socci, Hugh Nymeyer, and JoséNelson Onuchic.  Exploring the protein folding funnel landscape.  Physica D: Nonlinear Phenomena, 107(2-4):366–382, 1997.
  * [78] José Nelson Onuchic, Zaida Luthey-Schulten, and Peter G Wolynes.  Theory of protein folding: the energy landscape perspective.  Annual review of physical chemistry, 48(1):545–600, 1997.
  * [79] Aniruddha Bapat and Stephen Jordan.  Bang-bang control as a design principle for classical and quantum optimization algorithms.  arXiv preprint arXiv:1812.02746, 2018.
  * [80] Re-Bing Wu, Michael A Hsieh, and Herschel Rabitz.  Role of controllability in optimizing quantum dynamics.  Physical Review A, 83(6):062306, 2011.
  * [81] CL Baldwin and CR Laumann.  Quantum algorithm for energy matching in hard optimization problems.  Physical Review B, 97(22):224201, 2018.
  * [82] André Leier and Wolfgang Banzhaf.  Exploring the search space of quantum programs.  In The 2003 Congress on Evolutionary Computation, 2003. CEC’03., volume 1, pages 170–177. IEEE, 2003.
  * [83] Susan Stepney and John A Clark.  Searching for quantum programs and quantum protocols.  Journal of Computational and Theoretical Nanoscience, 5(5):942–969, 2008.
  * [84] Michael Streif and Martin Leib.  Training the quantum approximate optimization algorithm without access to a quantum processing unit.  Quantum Science and Technology, 5(3):034008, 2020.
  * [85] Jarrod R McClean, Matthew P Harrigan, Masoud Mohseni, Nicholas C Rubin, Zhang Jiang, Sergio Boixo, Vadim N Smelyanskiy, Ryan Babbush, and Hartmut Neven.  Low-depth mechanisms for quantum optimization.  PRX Quantum, 2(3):030312, 2021.
  * [86] Zhenduo Wang, Pei-Lin Zheng, Biao Wu, Yi Zhang, et al.  Quantum dropout: On and over the hardness of quantum approximate optimization algorithm.  Physical Review Research, 5(2):023171, 2023.
  * [87] Zhihui Wang, Nicholas C Rubin, Jason M Dominy, and Eleanor G Rieffel.  X y mixers: Analytical and numerical results for the quantum alternating operator ansatz.  Physical Review A, 101(1):012320, 2020.
  * [88] Ruslan Shaydulin, Phillip C Lotshaw, Jeffrey Larson, James Ostrowski, and Travis S Humble.  Parameter transfer for quantum approximate optimization of weighted maxcut.  ACM Transactions on Quantum Computing, 4(3):1–15, 2023.
  * [89] Mohamad Yousef, Tamer Abdelkader, and Khaled El-Bahnasy.  Performance comparison of ab initio protein structure prediction methods.  Ain Shams Engineering Journal, 10(4):713–719, December 2019.
  * [90] Alexis Lamiable, Pierre Thévenet, Julien Rey, Marek Vavrusa, Philippe Derreumaux, and Pierre Tufféry.  Pep-fold3: faster de novo structure prediction for linear peptides in solution and in complex.  Nucleic Acids Research, 44(W1):W449–W454, April 2016.
  * [91] Dong Xu and Yang Zhang.  Ab initio protein structure assembly using continuous structure fragments and optimized knowledge-based force field.  Proteins: Structure, Function, and Bioinformatics, 80(7):1715–1735, April 2012.
  * [92] Yang Zhang.  I-TASSER server for protein 3d structure prediction.  BMC Bioinformatics, 9(1), January 2008.
  * [93] David E. Shaw, Peter J. Adams, Asaph Azaria, Joseph A. Bank, Brannon Batson, Alistair Bell, Michael Bergdorf, Jhanvi Bhatt, J. Adam Butts, Timothy Correia, Robert M. Dirks, Ron O. Dror, Michael P. Eastwood, Bruce Edwards, Amos Even, Peter Feldmann, Michael Fenn, Christopher H. Fenton, Anthony Forte, Joseph Gagliardo, Gennette Gill, Maria Gorlatova, Brian Greskamp, J.P. Grossman, Justin Gullingsrud, Anissa Harper, William Hasenplaugh, Mark Heily, Benjamin Colin Heshmat, Jeremy Hunt, Douglas J. Ierardi, Lev Iserovich, Bryan L. Jackson, Nick P. Johnson, Mollie M. Kirk, John L. Klepeis, Jeffrey S. Kuskin, Kenneth M. Mackenzie, Roy J. Mader, Richard McGowen, Adam McLaughlin, Mark A. Moraes, Mohamed H. Nasr, Lawrence J. Nociolo, Lief O'Donnell, Andrew Parker, Jon L. Peticolas, Goran Pocina, Cristian Predescu, Terry Quan, John K. Salmon, Carl Schwink, Keun Sup Shim, Naseer Siddique, Jochen Spengler, Tamas Szalay, Raymond Tabladillo, Reinhard Tartler, Andrew G. Taube, Michael Theobald, Brian Towles, William Vick, Stanley C. Wang, Michael Wazlowski, Madeleine J. Weingarten, John M. Williams, and Kevin A. Yuh.  Anton 3.  In Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis. ACM, November 2021.
  * [94] Paul Robustelli, Stefano Piana, and David E. Shaw.  Developing a molecular dynamics force field for both folded and disordered protein states.  Proceedings of the National Academy of Sciences, 115(21), May 2018.
  * [95] Kresten Lindorff-Larsen, Paul Maragakis, Stefano Piana, and David E. Shaw.  Picosecond to millisecond structural dynamics in human ubiquitin.  The Journal of Physical Chemistry B, 120(33):8313–8320, May 2016.
  * [96] K. Lindorff-Larsen, S. Piana, R. O. Dror, and D. E. Shaw.  How fast-folding proteins fold.  Science, 334(6055):517–520, October 2011.
  * [97] Maciej Majewski, Adrià Pérez, Philipp Thölke, Stefan Doerr, Nicholas E. Charron, Toni Giorgino, Brooke E. Husic, Cecilia Clementi, Frank Noé, and Gianni De Fabritiis.  Machine learning coarse-grained potentials of protein thermodynamics.  Nature Communications, 14(1), September 2023.
  * [98] Dietmar Paschek, Hugh Nymeyer, and Angel E. García.  Replica exchange simulation of reversible folding/unfolding of the trp-cage miniprotein in explicit solvent: On the structure and possible role of internal water.  Journal of Structural Biology, 157(3):524–533, March 2007.
  * [99] Ryan Day, Dietmar Paschek, and Angel E. Garcia.  Microsecond simulations of the folding/unfolding thermodynamics of the trp-cage miniprotein.  Proteins: Structure, Function, and Bioinformatics, 78(8):1889–1899, February 2010.
  * [100] Steven W. Rick, Gregory J. Schwing, and Christopher M. Summa.  An implementation of replica exchange with dynamical scaling for efficient large-scale simulations.  Journal of Chemical Information and Modeling, 61(2):810–818, January 2021.
  * [101] Gwen R. Buel and Kylie J. Walters.  Can AlphaFold2 predict the impact of missense mutations on structure?  Nature Structural and Molecular Biology, 29(1):1–2, January 2022.
  * [102] Marina A. Pak, Karina A. Markhieva, Mariia S. Novikova, Dmitry S. Petrov, Ilya S. Vorobyev, Ekaterina S. Maksimova, Fyodor A. Kondrashov, and Dmitry N. Ivankov.  Using AlphaFold to predict the impact of single mutations on protein stability and function.  PLOS ONE, 18(3):e0282689, March 2023.
  * [103] Jed W. Pitera and William Swope.  Understanding folding and design: Replica-exchange simulations of ``trp-cage'' miniproteins.  Proceedings of the National Academy of Sciences, 100(13):7587–7592, June 2003.
  * [104] Hao Geng, Fangfang Chen, Jing Ye, and Fan Jiang.  Applications of molecular dynamics simulation in structure prediction of peptides and proteins.  Computational and Structural Biotechnology Journal, 17:1162–1170, 2019.
  * [105] B. Barua and N.H. Andersen.  The trp-cage: Optimizing the stability of a globular miniprotein, March 2008.
  * [106] Mark James Abraham, Teemu Murtola, Roland Schulz, Szilárd Páll, Jeremy C. Smith, Berk Hess, and Erik Lindahl.  GROMACS: High performance molecular simulations through multi-level parallelism from laptops to supercomputers.  SoftwareX, 1-2:19–25, September 2015.
  * [107] Enis Afgan, Anton Nekrutenko, Bjórn A Grüning, Daniel Blankenberg, Jeremy Goecks, Michael C Schatz, Alexander E Ostrovsky, Alexandru Mahmoud, Andrew J Lonie, Anna Syme, Anne Fouilloux, Anthony Bretaudeau, Anton Nekrutenko, Anup Kumar, Arthur C Eschenlauer, Assunta D DeSanto, Aysam Guerler, Beatriz Serrano-Solano, Bérénice Batut, Björn A Grüning, Bradley W Langhorst, Bridget Carr, Bryan A Raubenolt, Cameron J Hyde, Catherine J Bromhead, Christopher B Barnett, Coline Royaux, Cristóbal Gallardo, Daniel Blankenberg, Daniel J Fornika, Dannon Baker, Dave Bouvier, Dave Clements, David A de Lima Morais, David Lopez Tabernero, Delphine Lariviere, Engy Nasr, Enis Afgan, Federico Zambelli, Florian Heyl, Fotis Psomopoulos, Frederik Coppens, Gareth R Price, Gianmauro Cuccuru, Gildas Le Corguillé, Greg Von Kuster, Gulsum Gudukbay Akbulut, Helena Rasche, Hans-Rudolf Hotz, Ignacio Eguinoa, Igor Makunin, Isuru J Ranawaka, James P Taylor, Jayadev Joshi, Jennifer Hillman-Jackson, Jeremy Goecks, John M Chilton, Kaivan Kamali, Keith Suderman, Krzysztof Poterlowicz, Le Bras Yvan, Lucille Lopez-Delisle, Luke Sargent, Madeline E Bassetti, Marco Antonio Tangaro, Marius van den Beek, Martin Čech, Matthias Bernt, Matthias Fahrner, Mehmet Tekman, Melanie C Föll, Michael C Schatz, Michael R Crusoe, Miguel Roncoroni, Natalie Kucher, Nate Coraor, Nicholas Stoler, Nick Rhodes, Nicola Soranzo, Niko Pinter, Nuwan A Goonasekera, Pablo A Moreno, Pavankumar Videm, Petera Melanie, Pietro Mandreoli, Pratik D Jagtap, Qiang Gu, Ralf J M Weber, Ross Lazarus, Ruben H P Vorderman, Saskia Hiltemann, Sergey Golitsynskiy, Shilpa Garg, Simon A Bray, Simon L Gladman, Simone Leo, Subina P Mehta, Timothy J Griffin, Vahid Jalili, Vandenbrouck Yves, Victor Wen, Vijay K Nagampalli, Wendi A Bacon, Willem de Koning, Wolfgang Maier, and Peter J Briggs.  The galaxy platform for accessible, reproducible and collaborative biomedical analyses: 2022 update.  Nucleic Acids Research, 50(W1):W345–W351, April 2022.
  * [108] Viktor Hornak, Robert Abel, Asim Okur, Bentley Strockbine, Adrian Roitberg, and Carlos Simmerling.  Comparison of multiple amber force fields and development of improved protein backbone parameters.  Proteins: Structure, Function, and Bioinformatics, 65(3):712–725, September 2006.
  * [109] Pekka Mark and Lennart Nilsson.  Structure and dynamics of the TIP3p, SPC, and SPC/e water models at 298 k.  The Journal of Physical Chemistry A, 105(43):9954–9960, oct 2001.
  * [110] Dibyajyoti Maity.  djmaity/md-davis: Md davis v0.4.0, February 2022.
  * [111] Antonio Deiana, Sergio Forcelloni, Alessandro Porrello, and Andrea Giansanti.  Intrinsically disordered proteins and structured proteins with intrinsically disordered regions have different functional roles in the cell.  PLOS ONE, 14(8):e0217889, August 2019.
  * [112] Kiersten M. Ruff and Rohit V. Pappu.  AlphaFold and implications for intrinsically disordered proteins.  Journal of Molecular Biology, 433(20):167208, oct 2021.
  * [113] T. Reid Alderson, Iva Pritišanac, Desika Kolarić, Alan M. Moses, and Julie D. Forman-Kay.  Systematic identification of conditionally folded intrinsically disordered regions by AlphaFold2.  BioRxiv, feb 2022.
  * [114] Hao-Bo Guo, Alexander Perminov, Selemon Bekele, Gary Kedziora, Sanaz Farajollahi, Vanessa Varaljay, Kevin Hinkle, Valeria Molinero, Konrad Meister, Chia Hung, Patrick Dennis, Nancy Kelley-Loughnane, and Rajiv Berry.  AlphaFold2 models indicate that protein sequence determines both structure and dynamics.  Scientific Reports, 12(1), jun 2022.
  * [115] Sandra M. Rocha, Joel Pires, Marta Esteves, Baltazar GraÃ§a, and Liliana Bernardino.  Histamine: a new immunomodulatory player in the neuron-glia crosstalk.  Frontiers in Cellular Neuroscience, 8, April 2014.
  * [116] Chien Lee, Bo-Han Su, and Yufeng Jane Tseng.  Comparative studies of AlphaFold, RoseTTAFold and modeller: a case study involving the use of g-protein-coupled receptors.  Briefings in Bioinformatics, 23(5), August 2022.
  * [117] Tamás Hegedűs, Markus Geisler, Gergely László Lukács, and Bianka Farkas.  Ins and outs of AlphaFold2 transmembrane protein structure predictions.  Cellular and Molecular Life Sciences, 79(1), January 2022.
  * [118] Raymond C. Stevens.  The cost and value of three-dimensional protein structure, 2003.
  * [119] Krishna Sriram and Paul A. Insel.  G protein-coupled receptors as targets for approved drugs: How many targets and how many drugs?  Molecular Pharmacology, 93(4):251–258, January 2018.
  * [120] White RA, Szurmant H, Hoch JA, and Hwa T.  Features of protein-protein interactions in two-component signaling deduced from genomic libraries.  BioinformaticsMethods Enzymol., 2007.
  * [121] Burger L and van Nimwegen E.  Accurate prediction of protein-protein interactions from sequence alignments using a Bayesian method.  Mol Syst Biol., 2008.
  * [122] Martin Weigt, Robert A. White, Hendrik Szurmant, James A. Hoch, and Terence Hwa.  Identification of direct residue contacts in protein–protein interaction by message passing.  Proceedings of the National Academy of Sciences, 106(1):67–72, 2009.
  * [123] Jones DT, Buchan DW, Cozzetto D, and Pontil M.  PSICOV: precise structural contact prediction using sparse inverse covariance estimation on large multiple sequence alignments.  Bioinformatics, 2012.
  * [124] Liu Y, Palmedo P, Ye Q, Berger B, and Peng J.  Enhancing Evolutionary Couplings with Deep Convolutional Neural Networks.  Cell Syst., 6(1):65–74, 2018.
  * [125] Xu J, Mcpartlon M, and Li J.  Improved protein structure prediction by deep learning irrespective of co-evolution information.  Nat Mach Intell., pages 601–609, 2021.
  * [126] Jumper J et al.  Highly accurate protein structure prediction with AlphaFold.  Nature, pages 583–589, 2021.
  * [127] Baek M et al.  Accurate prediction of protein structures and interactions using a three-track neural network.  Science, pages 871–876, 2021.
  * [128] Gustaf Ahdritz, Nazim Bouatta, Sachin Kadyan, Lukas Jarosch, Daniel Berenberg, Ian Fisk, Andrew M. Watkins, Stephen Ra, Richard Bonneau, and Mohammed AlQuraishi.  Openproteinset: Training data for structural biology at scale, 2023.
  * [129] E.C. et al. Alley.  Unified rational protein engineering with sequence-based deep representation learning.  Nat Methods, 16, 2019.
  * [130] Rives A, Meier J, Sercu T, Goyal S, Lin Z, Liu J, Guo D, Ott M, Zitnick CL, Ma J, and Fergus R.  Biological structure and function emerge from scaling unsupervised learning to 250 million protein sequences.  Proc Natl Acad Sci, 2021.
  * [131] Xiaomin Fang, Fan Wang, Lihang Liu, Jingzhou He, Dayong Lin, Yingfei Xiang, Xiaonan Zhang, Hua Wu, Hui Li, and Le Song.  Helixfold-single: Msa-free protein structure prediction by using protein language model as an alternative, 2023.
  * [132] Roshan M Rao, Jason Liu, Robert Verkuil, Joshua Meier, John Canny, Pieter Abbeel, Tom Sercu, and Alexander Rives.  Msa transformer.  In Marina Meila and Tong Zhang, editors, Proceedings of the 38th International Conference on Machine Learning, volume 139 of Proceedings of Machine Learning Research, pages 8844–8856. PMLR, 18–24 Jul 2021.
  * [133] James P. Roney and Sergey Ovchinnikov.  State-of-the-art estimation of protein model accuracy using alphafold.  Phys. Rev. Lett., 129:238101, Nov 2022.
  * [134] Amy O Stevens and Yi He.  Benchmarking the accuracy of alphafold 2 in loop structure prediction.  Biomolecules, 12(7):985, 2022.
  * [135] Yuuki Hayashi, Takuyo Aita, Hitoshi Toyota, Yuzuru Husimi, Itaru Urabe, and Tetsuya Yomo.  Experimental rugged fitness landscape in protein sequence space.  PLoS One, 1(1):e96, 2006.
  * [136] Jasmine Cornish, Samuel G Chamberlain, Darerca Owen, and Helen R Mott.  Intrinsically disordered proteins and membranes: a marriage of convenience for cell signalling?  Biochemical Society Transactions, 48(6):2669–2689, 2020.
  * [137] Allan Haldane and Ronald M Levy.  Influence of multiple-sequence-alignment depth on potts statistical models of protein covariation.  Physical Review E, 99(3):032405, 2019.
  * [138] Andriy Kryshtafovych, Torsten Schwede, Maya Topf, Krzysztof Fidelis, and John Moult.  Critical assessment of methods of protein structure prediction (casp)—round xiv.  Proteins: Structure, Function, and Bioinformatics, 89(12):1607–1617, 2021.
  * [139] Anton Robert, Panagiotis Kl. Barkoutsos, Stefan Woerner, and Ivano Tavernelli.  Resource-efficient quantum algorithm for protein folding.  npj Quantum Inf, 7, 2021.
  * [140] Arne Elofsson.  Progress at protein structure prediction, as seen in casp15.  Current Opinion in Structural Biology, 80:102594, 2023.
  * [141] M. Pereira.  Casp15 high accuracy, 2023.
  * [142] Joicymara S. Xavier, Pâmela M. Rezende, João P. L. Velloso, Thanh-Binh Nguyen, Malancha Karmarkar, Stephanie Portelli, David B. Ascher, and Douglas E. V. Pires.  Thermomutdb, 2021.
  * [143] Lukas Zimmermann, Andrew Stephens, Seung-Zin Nam, David Rau, Jonas Kübler, Marko Lozajic, Felix Gabler, Johannes Söding, Andrei N. Lupas, and Vikram Alva.  A completely reimplemented mpi bioinformatics toolkit with a new hhpred server at its core.  Journal of Molecular Biology, 430(15):2237–2243, 2018.  Computation Resources for Molecular Biology.
  * [144] Vivien Marx.  Biology begins to tangle with quantum computing.  Nature Methods, 18(7):715–719, June 2021.
  * [145] Julia Koehler Leman, Brian D. Weitzner, Steven M. Lewis, Jared Adolf-Bryfogle, Nawsad Alam, Rebecca F. Alford, Melanie Aprahamian, David Baker, Kyle A. Barlow, Patrick Barth, Benjamin Basanta, Brian J. Bender, Kristin Blacklock, Jaume Bonet, Scott E. Boyken, Phil Bradley, Chris Bystroff, Patrick Conway, Seth Cooper, Bruno E. Correia, Brian Coventry, Rhiju Das, René M. De Jong, Frank DiMaio, Lorna Dsilva, Roland Dunbrack, Alexander S. Ford, Brandon Frenz, Darwin Y. Fu, Caleb Geniesse, Lukasz Goldschmidt, Ragul Gowthaman, Jeffrey J. Gray, Dominik Gront, Sharon Guffy, Scott Horowitz, Po-Ssu Huang, Thomas Huber, Tim M. Jacobs, Jeliazko R. Jeliazkov, David K. Johnson, Kalli Kappel, John Karanicolas, Hamed Khakzad, Karen R. Khar, Sagar D. Khare, Firas Khatib, Alisa Khramushin, Indigo C. King, Robert Kleffner, Brian Koepnick, Tanja Kortemme, Georg Kuenze, Brian Kuhlman, Daisuke Kuroda, Jason W. Labonte, Jason K. Lai, Gideon Lapidoth, Andrew Leaver-Fay, Steffen Lindert, Thomas Linsky, Nir London, Joseph H. Lubin, Sergey Lyskov, Jack Maguire, Lars Malmström, Enrique Marcos, Orly Marcu, Nicholas A. Marze, Jens Meiler, Rocco Moretti, Vikram Khipple Mulligan, Santrupti Nerli, Christoffer Norn, Shane Ó’Conchúir, Noah Ollikainen, Sergey Ovchinnikov, Michael S. Pacella, Xingjie Pan, Hahnbeom Park, Ryan E. Pavlovicz, Manasi Pethe, Brian G. Pierce, Kala Bharath Pilla, Barak Raveh, P. Douglas Renfrew, Shourya S. Roy Burman, Aliza Rubenstein, Marion F. Sauer, Andreas Scheck, William Schief, Ora Schueler-Furman, Yuval Sedan, Alexander M. Sevy, Nikolaos G. Sgourakis, Lei Shi, Justin B. Siegel, Daniel-Adriano Silva, Shannon Smith, Yifan Song, Amelie Stein, Maria Szegedy, Frank D. Teets, Summer B. Thyme, Ray Yu-Ruei Wang, Andrew Watkins, Lior Zimmerman, and Richard Bonneau.  Macromolecular modeling and design in rosetta: recent methods and frameworks.  Nature Methods, 17(7):665–680, June 2020.
  * [146] Miyazawa S and Jernigan RL.  Residue-residue potentials with a favorable contact pair term and an unfavorable high packing density term, for simulation and threading.  J Mol Biol., 3, 1996.
  * [147] Rebecca F. Alford, Andrew Leaver-Fay, Jeliazko R. Jeliazkov, Matthew J. O’Meara, Frank P. DiMaio, Hahnbeom Park, Maxim V. Shapovalov, P. Douglas Renfrew, Vikram K. Mulligan, Kalli Kappel, Jason W. Labonte, Michael S. Pacella, Richard Bonneau, Philip Bradley, Roland L. Dunbrack, Rhiju Das, David Baker, Brian Kuhlman, Tanja Kortemme, and Jeffrey J. Gray.  The rosetta all-atom energy function for macromolecular modeling and design.  Journal of Chemical Theory and Computation, 13(6):3031–3048, May 2017.
  * [148] Gurobi Optimization, LLC.  Gurobi Optimizer Reference Manual, 2023.
  * [149] Mariano Spivak, John E. Stone, João Ribeiro, Jan Saam, Peter L. Freddolino, Rafael C. Bernardi, and Emad Tajkhorshid.  VMD as a platform for interactive small molecule preparation and visualization in quantum and classical simulations.  Journal of Chemical Information and Modeling, 63(15):4664–4678, July 2023.
  * [150] Avradip Mandal, Arnab Roy, Sarvagya Upadhyay, and Hayato Ushijima-Mwesigwa.  Compressed quadratization of higher order binary optimization problems, 2020.
  * [151] Yuxuan Du, Zhuozhuo Tu, Xiao Yuan, and Dacheng Tao.  Efficient measure for the expressivity of variational quantum algorithms.  Physical Review Letters, 128(8):080506, 2022.
  * [152] A. Kandala, A. Mezzacapo, and K. et al. Temme.  Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets.  Nature, 549:242–246, September 2017.
  * [153] Ibm quantum documentation.
  * [154] Olga Vinogradova, Algirdas Velyvis, Asta Velyviene, Bin Hu, Thomas A. Haas, Edward F. Plow, and Jun Qin.  A structural mechanism of integrin $\alpha$IIb$\beta$3 “inside-out” activation as regulated by its cytoplasmic face.  Cell, 110(5):587–597, September 2002.
  * [155] V. A. Jisna and P. B. Jayaraj.  Protein structure prediction: Conventional and deep learning perspectives.  The Protein Journal, 40(4):522–544, May 2021.
  * [156] Jun Cheng, Guido Novati, Joshua Pan, Clare Bycroft, Akvilė Žemgulytė, Taylor Applebaum, Alexander Pritzel, Lai Hong Wong, Michal Zielinski, Tobias Sargeant, Rosalia G. Schneider, Andrew W. Senior, John Jumper, Demis Hassabis, Pushmeet Kohli, and Žiga Avsec.  Accurate proteome-wide missense variant effect prediction with AlphaMissense.  Science, 381(6664), September 2023.