Quantum Computing for Molecular Biology**

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

In the limit of a dense, complete atomic basis set, the many-electron wave function can be encoded exactly as a linear superposition of Slater determinants.

“In the limit of a dense, complete atomic basis set -a limit called the complete basis set limit -the many-electron wave function can be encoded exactly as a linear superposition of Slater determinants.”

Classified as probably not a substantial claim.

Solving the quantum mechanical equations relevant to molecular biology computationally is very difficult.

“the computational solution of the relevant quantum mechanical equations is very hard.”

Probably not a substantial claim: may not pass S4 (based). May combine two claims.

None of these deficiencies is present in approaches that involve quantum mechanical effects, which are hampered by highly unfavorable scaling with system size.

“By contrast, none of these deficiencies is present in approaches that involve quantum mechanical effects, which are, however, hampered by the highly unfavorable scaling problem with system size.”

May combine two claims.

The separation into components makes the solution of the Schrödinger equation feasible for not too large molecular structures and allows understanding biomolecular machinery in terms of individual energies assigned to states of molecular players.

“While this separation into components makes the solution of the Schrödinger equation then feasible (at least for not too large molecular structures), it also allows one to piece together and eventually understand biomolecular machinery in terms of individual energies assigned to the states of the molecular players.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

These algorithms rely on wave function approximations which scale polynomially in the number of expansion parameters but are tailored to a specific regime of electron correlation.

“Specifically, they rely on wave function approximations which scale polynomially in the number of expansion parameters but are tailored to a specific regime of electron correlation.”

Probably not a substantial claim: may not pass S3 (load-bearing).

It remains hard to assess a priori the accuracy of a given density functional approximation for a specific molecule not included in the training data.

“it remains hard to assess a priori the accuracy of a given density functional approximation for a specific molecule not included in the training data.[40]”

• cites [40] sourceNot checked yet

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May combine two claims.

The curse of dimensionality creates a steep wall that cannot be overcome with traditional computational approaches and limits such attempts to small systems of comparatively few atoms.

“This curse of dimensionality creates a steep wall that cannot be overcome with traditional computational approaches (and therefore limits such attempts to small systems of comparatively few atoms, which represent only segments of biomolecules [23][24][25] ).”

• cites [23][24][25]Not checked yet

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Probably not a substantial claim: may not pass S4 (based).

Weak interactions such as van der Waals forces and hydrogen bonds originate from quantum correlations of electrons.

“They originate from the quantum correlations of electrons.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

Large-amplitude motions and long-range effects may be difficult to describe in embedding settings.

“largeamplitude motions and long-range effects may be difficult to describe in such settings.”

Classified as probably not a substantial claim.

In order to represent the many-electron wave function on hardware, either classical or quantum, it must be discretized.

“In order to represent it on hardware, either classical or quantum, the many-electron wave function must be discretized.”

May combine two claims.

Force-field approximations can be combined with classical-mechanics algorithms that do not suffer from the curse of dimensionality and have low polynomial scaling with the number of atoms.

“Such force-field approximations can be combined with algorithms that, being based on classical mechanics, do not suffer from the curse of dimensionality, but have a low polynomial scaling with the number of atoms.”

Probably not a substantial claim: may not pass S4 (based).

Much of biological molecular function can be described with classical mechanics, in which electron-nucleus interactions are replaced by effective classical potentials between atoms or larger units.

“much of molecular function in biology can be understood in terms of classical mechanics, where the interactions of electrons and nuclei have been mapped onto effective classical surrogate potentials that model the interaction of atoms or even larger entities.”

May combine two claims.

The curse of dimensionality restricts practical full-CI applications to molecules with about 20 electrons, which are very small and far smaller than biomolecules.

“The curse of dimensionality limits practical applications of full-CI to molecules with approximately 20 electrons, [65,70] a size that characterizes very small molecules, far smaller than any biomolecule.”

• cites [65] sourceNot checked yet

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• cites [70] sourceNot checked yet

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Probably not a substantial claim: may not pass S3 (load-bearing).

The basis set expansion of the wave function is often supplemented by factors that depend explicitly on interelectronic distance.

“Accordingly, the basis set expansion of the wave function is often supplemented by factors that depend explicitly on the interelectronic distance [69]”

• cites [69] sourceNot checked yet

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May combine two claims.

Very high accuracy can only be achieved for very small molecules; increasing molecular size generally is accompanied by a loss in accuracy for feasibility reasons due to the high computational cost of exact quantum simulations.

“Very high accuracy can only be achieved for very small molecules.Increasing molecular size generally is accompanied by a loss in accuracy for feasibility reasons, due to the high computational cost of exact quantum simulations.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

Such molecular sizes are far smaller than those of functional biological macromolecules embedded in an aqueous phase.

“Obviously, such molecular sizes are far smaller than those of functional biological macromolecules embedded in an aqueous phase.”

May combine two claims.

The Schrödinger equation can be simplified by invoking the Born-Oppenheimer approximation because nuclear mass is at least three orders of magnitude larger than electron mass, so nuclear dynamics are much slower.

“The Schrödinger equation can be simplified by invoking the Born-Oppenheimer approximation: since the mass of the nuclei is at least three orders of magnitude larger than that of the electrons, the nuclear dynamics are much slower than the electronic ones.”

Probably not a substantial claim: may not pass S3 (load-bearing).

In all cases, the wave function is the solution of a differential equation in which all variables are coupled by the potential energy operator.

“In all cases, the wave function is the solution of a differential equation in which all variables are coupled by the potential energy operator.”

Probably not a substantial claim: may not pass S4 (based).

Detailed molecular dynamics simulations rest almost exclusively on classical Newtonian mechanics.

“detailed molecular dynamics simulations rest almost exclusively on classical Newtonian mechanics.”

May combine two claims.

Dynamical correlation does not change the qualitative structure of the full-CI wave function but contributes non-negligibly to the electronic energy due to many configurations with very small weights.

“Although they do not alter the qualitative structure of the full-CI wave function, they yield a nonnegligible contribution to the electronic energy because of the sheer number of configurations with very small weights.”

May combine two claims.

Full-CI is needed to describe the resulting quantum correlations completely, leading once more to the emergence of the curse of dimensionality.

“Full-CI is, therefore, needed to describe the resulting quantum correlations completely, leading once more to the emergence of the curse of dimensionality.”

Classified as probably not a substantial claim. May combine two claims.

In such cases, one resorts to classical models and accepts a reduced accuracy and a loss in transferability.

“In such cases, one resorts to classical models and accepts a reduced accuracy and a loss in transferability.”

Classified as probably not a substantial claim.

Electronic structure is key for quantitative theoretical description and prediction of chemical reactions in terms of reaction energies and activation barriers through the Born-Oppenheimer potential energy surface.

“The electronic structure of molecules is indeed key for the quantitative theoretical description and prediction of chemical reactions in terms of reaction energies and activation barriers through the Born-Oppenheimer potential energy surface (PES; see Figure 1).”

Probably not a substantial claim: may not pass S4 (based). May combine two claims.

More detailed simulations of the quantum mechanical equations are very difficult and only possible for a very small number of atoms.

“more detailed simulations of the quantum mechanical equations are very difficult and only possible for a very small number of atoms.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Electron correlation effects are those that impede encoding the wave function as a single determinant and require it to be expressed as a linear superposition of configurations.

“Therefore, electron correlation effects are those that impede encoding the wave function as a single determinant and require it to be expressed as a linear superposition of configurations.”

Probably not a substantial claim: may not pass S4 (based).

Molecular motion is governed by the laws of quantum mechanics.

“Molecular motion is governed by the laws of quantum mechanics.”

May combine two claims.

This coupling correlates the motion of quantum particles and renders the solution of the differential equation a daunting computational task for large molecules.

“This coupling correlates the motion of the quantum particles and renders the solution of the differential equation a daunting computational task for large molecules.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

A molecular system of a few hundred atomic nuclei can be considered routinely treatable with approximate traditional quantum chemical approaches.

“we may consider the size of a molecular system of a few hundred atomic nuclei to be routinely treatable with approximate traditional quantum chemical approaches.”

Probably not a substantial claim: may not pass S4 (based). May combine two claims.

The promise of quantum computations is that a quantum description can be extended to larger atomistic structures and to higher accuracy compared to approximate traditional quantum chemical approaches.

“The promise of quantum computations is that a quantum description can be extended to larger atomistic structures and to higher accuracy (compared to approximate traditional quantum chemical approaches).”

Probably not a substantial claim: may not pass S4 (based).

Only detailed quantitative physical simulations together with high-resolution experiments will allow extracting a high-confidence picture of molecular events.

“only detailed quantitative physical simulations -in parallel to detailed experiments with a high spatial and temporal resolution -will allow extracting such a picture with high confidence.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

Research in quantum chemistry has mostly focused on taming the curse of dimensionality with algorithms that trade full-CI accuracy and generality for computational efficiency.

“Research in quantum chemistry has mostly focused on taming the curse of dimensionality with algorithms that trade the full-CI accuracy and generality for computational efficiency.”

May combine two claims.

Already for a molecule with a few dozen atoms, the wave function depends on hundreds of variables, and the Schrödinger equations become extremely hard to solve on classical computers.

“Already for a molecule with a few dozen atoms, the wave function depends on hundreds of variables, and Eqs.(1) and (2) become extremely hard to solve on classical computers.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Short-range dynamical correlation is associated with the short-range electron-electron interaction in the limit of vanishing interelectronic distance.

“Short-range correlations are, instead, associated with the short-range electron-electron interaction, i. e., in the limit of a vanishing interelectronic distance in which the positions of two electrons coalesce.”

Probably not a substantial claim: may not pass S4 (based). May combine two claims.

Reversibility is key to molecular function in biochemistry, causing macromolecules to interact predominantly through non-covalent interactions.

“The fact that reversibility is key to molecular function in biochemistry causes macromolecules to interact predominantly through (many) non-covalent interactions.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.

The interactions in a molecule are very well described by pairwise electrostatic Coulomb potential energies, although they neglect magnetic interactions.

“The interactions are very well described by pairwise electrostatic Coulomb potential energies, although they neglect, for instance, magnetic interactions.”

May combine two claims.

We can describe the quantum mechanical behavior of electrons and nuclei in biomolecular systems in principle, but often not in practice, as such calculation would be too computer-time demanding.

“we can describe the quantum mechanical behavior of electrons and nuclei in biomolecular systems in principle, but often not in practice, as such calculation would be too computer-time demanding.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Classical molecular dynamics simulations provide efficient models and can be rigorously based on quantum mechanics through the Born-Oppenheimer approximation.

“Classical molecular dynamics simulations provide efficient models and can be rigorously based on quantum mechanics (technically speaking, this is possible through the Born-Oppenheimer approximation, which separates electronic and nuclear motions, with the latter being then identified as the atomic motion in classical dynamics).”

Probably not a substantial claim: may not pass S3 (load-bearing).

The complete basis set limit cannot be reached in numerical simulations that must rely on finite, incomplete basis sets.

“The complete basis set limit cannot be reached in numerical simulations that must necessarily rely on finite, incomplete basis sets.”

May combine two claims.

The underlying model potentials require sophisticated parametrizations, generally omit quantum mechanical effects, lack transferability to other molecule classes, and therefore cannot describe well arbitrary chemical reactions and molecular interactions.

“the underlying model potentials require sophisticated parametrizations, generally omit quantum mechanical effects, lack transferability to other molecule classes, and therefore cannot describe well arbitrary chemical reactions and molecular interactions (if at all).”

Probably not a substantial claim: may not pass S3 (load-bearing).

Embedding techniques provide a quantum magnifying glass for a cluster of atomic nuclei and electrons considered decisive for the quantum process within a bio-macromolecule.

“embedding techniques have been devised, which provide a quantum magnifying glass for a cluster of atomic nuclei and electrons considered to be decisive for the quantum process within a bio-macromolecule (such as the active site of an enzyme).”

Probably not a substantial claim: may not pass S3 (load-bearing).

Long-range dynamical correlation is associated with London dispersion forces, which describe induced dipolar interactions between molecular fragments from instantaneous electronic polarization.

“longrange correlations are associated with the London dispersion forces, which, qualitatively speaking, describe the induced dipolar interaction between two molecular fragments that result from instantaneous polarization of the corresponding electronic density.”

May combine two claims.

Analog quantum simulators experience decoherence that cannot be suppressed arbitrarily, making formal guaranteed accuracy statements difficult or impossible.

“Yet, since the constituents of analog quantum simulators also experience decoherence -a quantum form of errors -which cannot be suppressed arbitrarily (this is true at present, and may remain so even in the far future), formal guaranteed statements about the overall accuracy of the achieved simulation are difficult if not impossible.”

May combine two claims.

Any outer approximation to the 2-RDM set yields a lower bound on the energy, unlike variational methods.

“we note that any outer approximation to the set will result in a lower bound on the energy, in stark contrast to the situation discussed above for the variational methods.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Coupled cluster (CC) is the established canonical solution to the size-extensivity problem.

“The established canonical solution to this problem is the coupled cluster (CC) method.[72]”

• cites [72] sourceNot checked yet

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Probably not a substantial claim: may not pass S3 (load-bearing).

CC uses an exponential wave function ansatz with parameters scaling polynomially with system size.

“CC relies on an exponential wave function ansatz that contains a number of parameters that scales polynomially with system size.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Time evolution after an external perturbation may lead to continuous growth of entanglement.

“It is known that the time evolution of a wave function following an external perturbation (usually known as quench in quantum physics, which corresponds most often to an electromagnetic perturbation in chemistry) may lead to a continuous growth of its entanglement.”

May combine two claims.

Coulson noted in 1960 that rigorous lower bounds can be obtained because molecular Hamiltonians involve only pairwise interactions, allowing the ground-state energy to be cast as an optimization over the two-particle reduced density matrix (2-RDM).

“It was already noted in 1960 by Coulson [100] that rigorous lower bounds could be obtained by the observation that molecular Hamiltonians only involve pairwise interactions”

• cites [100] sourceNot checked yet

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Probably not a substantial claim: may not pass S3 (load-bearing).

These approximate parametrizations are tailored to two idealized cases: weakly correlated and strongly correlated electronic structures.

“These approximate parametrizations have been tailored to two idealized situations: weakly correlated and strongly correlated electronic structures of molecules.”

May combine two claims.

Protein folding is an optimization problem and a candidate for heuristic quantum optimization algorithms.

“A candidate in the biological context is the protein folding problem, which is an optimization problem, for which heuristic quantum optimization algorithms are being investigated.[109,110]”

• cites [109]Not checked yet

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• cites [110] sourceNot checked yet

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May combine two claims.

Exact characterization of the 2-RDM set is QMA-hard and unlikely to be solved even on a quantum computer.

“Whereas the exact characterization of the set has been shown to be Quantum Merlin Arthur (QMA)-hard (QMA is the quantum analog of the famous class of computational problems known as nondeterministic polynomial time (NP)) and thus unlikely to be solved even on a quantum computer, [103]”

• cites [103] sourceNot checked yet

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May combine two claims.

Quantum information cannot be copied and is therefore fragile and affected by quantum noise (decoherence), so storage and gate operations on qubits are prone to errors needing correction.

“Quantum information cannot be copied and is therefore much more fragile and affected by quantum noise, the previously mentioned decoherence.Consequently, storage and gate operations on quantum bits are prone to errors that need to be corrected.”

May combine two claims.

The hierarchy of truncated CI approximations is ad hoc and does not guarantee inclusion of all truly important configurations.

“We note that the emerging hierarchy of approximations is an ad hoc choice that does not guarantee that all truly important configurations are actually included in the truncated expansion of configurations.”

May combine two claims.

Feynman noted that classical simulation of a quantum system generally incurs exponential overhead due to the curse of dimensionality, while building a simulator from quantum mechanical constituents could circumvent this and achieve efficient simulation.

“It was Feynman [108] who noticed that using a classical system, e. g. a traditional computer, to simulate a quantum system will in general result in an exponential overhead in the time required for the simulation due to the curse of dimensionality, but that building a simulator (or computer) directly from quantum mechanical constituents (henceforth called a quantum simulator or, if very versatile,”

• cites [108] sourceNot checked yet

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Probably not a substantial claim: may not pass S3 (load-bearing).

Designing algorithms to accurately include weak correlation effects on top of DMRG or FCIQMC remains an open challenge in computational chemistry.

“In fact, designing algorithms to include accurately weak correlation effects on top of a DMRG or FCIQMC simulation remains an open challenge of computational chemistry.”

May combine two claims.

Selected CI methods can target up to 100 electrons, a major advance for chemistry but still a severe limitation for large biomolecules.

“In practice, selected CI methods can target up to 100 electrons, which is a huge leap forward for chemical applications, but still represents a severe limitation for generally large biomolecules.”

Probably not a substantial claim: may not pass S3 (load-bearing).

DFT has become the workhorse of computational molecular quantum mechanics.

“Finally, we emphasize that DFT represents a route to solve the electronic structure problem that has become the workhorse of computational molecular quantum mechanics.”

May combine two claims.

The state of n qubits requires 2^n probability amplitudes to describe its superposition, so classical simulation or storage requires more than 2^n bits and exponential overhead.

“hence also of quantum bits, posits that the state of n qubits requires the description with 2 n probability amplitudes (i.e., expansion coefficients; see Figure 3) describing the precise superposition of the strings of n bits (thereby making up the ' state' or ' wave function' of the n qubits).Simulating or even just storing the amplitudes on a standard computer, therefore, requires more than 2 n”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Excited-state solutions are important for describing biomolecular systems after light excitation and provide information related to spectroscopic techniques.

“These solutions are not only important to describe a biomolecular system after light excitation, but they also offer information that can be related to spectroscopic techniques and, hence, to experimental spectroscopies that identify specific states of biomolecular machinery.”

May combine two claims.

The number of physical qubits required for first viable algorithms is extremely large and not expected in the near future, only medium or long term.

“the number of physical qubits required for the first viable algorithms to run is extremely large and therefore not expected to be reached in the near future, but only in the medium or long term.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Because the wave function is normalized, all other determinants have comparatively small weight.

“Since the wave function is normalized to a fixed value (usually one), all other determinants will have a comparatively small weight.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Hybrid quantum-classical algorithms partition workload between classical and quantum computers and are important for NISQ hardware.

“An important class of algorithms for NISQ hardware are socalled hybrid quantum-classical algorithms that partition the computational workload between a classical and a quantum computer.[124,125]”

• cites [124] sourceNot checked yet

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• cites [125] sourceNot checked yet

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May combine two claims.

Matrix product states, a class of TNSs, reliably approximate the electronic Schrödinger equation in many numerical studies, and their parameters can be efficiently optimized with DMRG.

“In many numerical studies it turned out that a specific class of TNSs, so-called matrix product states, reliably approximate the electronic Schrödinger equation, [65,[81][82][83][84][85][86][87][88][89][90] whose parameters can be very efficiently optimized with an iterative algorithm known as density matrix renormalization group (DMRG) approach.[91,92]”

• cites [65] sourceNot checked yet

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• cites [81] sourceNot checked yet

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• cites [91] sourceNot checked yet

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May combine two claims.

Recent work shows only constant overhead is necessary, with logical qubits proportional to physical ones, suggesting no fundamental obstacle to a perfect digital quantum simulator.

“Whereas recently, it has been shown that only a constant overhead is necessary (i.e. the number of logical qubits is proportional to the number of physical ones) [117] and one may therefore generally see no fundamental obstacle to building a perfect digital quantum simulator in the future”

• cites [117] sourceNot checked yet

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May combine two claims.

There can be no rigorous closeness estimate for variational methods because the infinite basis set limit cannot be reached and the ansatz samples only a subset of states.

“there can ultimately not be any rigorous closeness estimate of such methods (as the infinite basis set limit cannot be reached in practice) or the ansatz within a given basis set only samples a subset of the states”

May combine two claims.

Truncated CI is not size extensive: its energy does not scale linearly with electron number, making reliable comparison of molecules with different electron counts impossible.

“The key limitation of truncated CI (by contrast to full-CI), however, is that it is not size extensive, i. e. the corresponding energy does not scale linearly with the number of electrons, [71] which makes it impossible to reliably compare the energy of molecules with different numbers of electrons.”

• cites [71] sourceNot checked yet

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May combine two claims.

Hamiltonians in quantum chemical and quantum biological contexts have drastically more irregular discretized interactions than those in quantum physical problems, so tunability is a key requirement for analog simulators.

“The key difference between Hamiltonians relevant in quantum physical problems and those appearing in quantum chemical and quantum biological contexts is that the (discretized) interaction is drastically more irregular in the latter case than in the former one.Tunability is, therefore, a key requirement of an analog quantum simulator for quantum chemical and quantum biological simulations.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

It is reasonable to expect that the weight of other configurations decreases as their departure from the Hartree-Fock determinant increases.

“Moreover, it is reasonable to expect that the weight of the other configurations will decrease as their departure from the Hartree-Fock determinant increases.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

CCSD(T) has become the de facto reference method for weakly correlated electronic wave functions, reaching chemical accuracy of about 1 kcal/mol.

“Specifically, CC with singles, double, and perturbative triple excitations, abbreviated CCSD(T), has become the de facto reference method for calculating weakly correlated electronic wave functions in order to reach chemical accuracy of about 1 kcal/mol.”

Probably not a substantial claim: may not pass S4 (based). May combine two claims.

The biochemical problem of a local reaction can be reformulated as a purely chemical problem, so solution approaches from chemistry transfer to biochemistry.

“the biochemical problem has been reformulated as a purely chemical problem, and all developed solution approaches in the chemical context automatically transfer to the biochemical application.”

• AssumptionRests on an assumption: Chemical reactions of interest involve only a small number of atoms, with the rest of the biomolecular environment acting as a spectator.

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

VQE is the most prominent hybrid algorithm for electronic structure, providing a variational heuristic for upper-bounding ground-state energy.

“The most prominent example of this class of algorithms relevant for electronic structure calculations is the variational quantum eigensolver (VQE), which presents a variational (and thus heuristic) algorithm to upper bounding the ground state energy problem.[126][127][128]”

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Classified as probably not a substantial claim. May combine two claims.

Some biological processes are driven by absorption of electromagnetic radiation and involve electronically excited state surfaces.

“However, some biological processes are driven by the absorption of electromagnetic radiation.In this case, the reaction involves the surface of an electronically excited state”

May combine two claims.

Building a simulator directly from quantum mechanical constituents remedies this because they adhere to the same quantum mechanical composition rule and are suited to simulating quantum systems.

“This is precisely remedied by building a simulator directly from quantum mechanical constituents, since those evidently adhere to the same quantum mechanical composition rule and, therefore, offer a computational platform perfectly suited to the task of simulating a quantum system.”

May combine two claims.

The ground-state energy problem is QMA-hard, so no general exponential speedup is expected, making the approach heuristic.

“Note, however, that the ground state energy problem is known to be QMA-hard and it might, therefore, be expected that no general exponential speedup will be obtained, rendering this again a heuristic algorithm.”

May combine two claims.

DFT can target systems two orders of magnitude larger than selected CI, but has uncontrollable error for a specific biomolecule.

“Surely, other approximate quantum approaches that do not rely on expansions of configurations -most importantly contemporary methods of Density Functional Theory (DFT; see below) -can target systems that are two orders of magnitude larger, but they also come with an uncontrollable error for a specific biomolecule under consideration.[40]”

• cites [40] sourceNot checked yet

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Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Analog quantum simulators are purpose-built machines that mimic interactions and behavior of the system to be simulated, and are not meant to be universal computing machines.

“Analog quantum simulators are more akin to purpose-built machines that through their build-up mimic the interactions and, therefore, the behavior of the system to be simulated.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Single-reference methods such as truncated CI or CC can become inaccurate when the full-CI wave function is not dominated by a single configuration.

“Approaches such as truncated CI or CC -usually referred to as single-reference methods, because they rely on a single configuration, the Hartree-Fock one, as a reference -can become inaccurate as soon as the full-CI wave function is not dominated by a single configuration.”

May combine two claims.

Variational methods such as Hartree-Fock, full-CI, and DFT restrict the minimization and yield only an upper bound on the true ground-state energy.

“and that methods that optimize over an ansatz (variational methods) such as Hartree-Fock, full-CI, DFT, restrict the minimization and always lead only to an upper bound on the true ground state energy.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Quantum algorithm research splits into fault-tolerant algorithms and NISQ algorithms for devices with dozens to thousands of physical qubits lacking complete fault tolerance.

“Therefore, research on quantum algorithms splits, on the one hand, into algorithms for fault-tolerant quantum computers, which we may think of as a future perfect quantum computer, and, on the other hand, into algorithms for nearterm or noisy-intermediate stage quantum (NISQ) [120] devices of dozens to thousands of physical qubits, that are characterized by the absence of complete fault-tolerance.”

• cites [120] sourceNot checked yet

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Probably not a substantial claim: may not pass S3 (load-bearing).

Modern full-CI-type methods have extended full-CI to systems with about 100 electrons.

“For such multi-reference cases, modern full-CI-type methods [65,73] have made it possible to push the limits of full-CI to systems with approximately 100 electrons.”

• cites [65] sourceNot checked yet

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May combine two claims.

Estimating the accuracy of approximate functionals is often impossible, preventing quantification and systematic improvement of DFT accuracy, which is the major drawback of contemporary DFT.

“Estimating the accuracy of approximate functionals is often impossible, [40] which prevents quantifying and systematically improving the accuracy of a DFT calculation, which is the major drawback of contemporary DFT.”

• cites [40] sourceNot checked yet

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May combine two claims.

The classical composition rule is weaker than the quantum mechanical one, so only a subset of possible quantum states of N qubits can be represented by N classical counterparts.

“In fact, the classical composition rule is weaker than the quantum mechanical.This means, in practice, that only a subset of the possible quantum mechanical states of N qubits can be represented by N of their classical counterparts.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Many approximate methods aim to find wave function parametrizations that approximate full-CI while using a number of parameters scaling polynomially with system size.

“The key idea behind many of these approximations is to identify wave function parametrizations that can approximate full-CI while being defined by a number of parameters that scales polynomially with the system size.”

May combine two claims.

Any biological process cannot be considered isolated; it occurs at about room temperature and involves continuous energy exchange with the environment.

“At this point, it is important to highlight that any biological process cannot be considered an isolated process.In fact, it happens at about room temperature and involves a continuous energy exchange with the environment (for instance, the cell in which a protein is located).”

May combine two claims.

For both light harvesting and magnetoreception, quantum mechanical effects affect the interaction between molecules (chromophores or unpaired electrons).

“For both phenomena, quantum mechanical effects affect the interaction between molecules -the chromophores, for light-harvesting, and the unpaired electrons, for magnetoreception.”

May combine two claims.

The three-dimensional structure of a protein can be reliably predicted with force fields because it is mostly determined by non-covalent interactions between amino acid residues.

“The three-dimensional structure of a protein can be reliably predicted with force fields because it is mostly determined by the non-covalent interactions between the amino acid residues of the protein.”

Probably not a substantial claim: may not pass S3 (load-bearing).

All these optimization algorithms can in principle be applied to biochemical problems.

“All these classes of optimization algorithms can be, in principle, applied to biochemical problems such as those mentioned above.”

Probably not a substantial claim: may not pass S1 (asserted), S3 (load-bearing).

Molecular vibrations, through excitonic coupling, may enhance the coherence lifetime.

“it has been argued that molecular vibrations, through their coupling with the electronic excitation (usually referred to as excitonic coupling), may enhance the coherence lifetime by quantum coherences.[166][167][168]”

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Probably not a substantial claim: may not pass S3 (load-bearing).

For sufficiently long propagation times, the protein will evolve into its most stable conformation, yielding the solution to the protein folding problem.

“For sufficiently long propagation times, the protein will evolve into its most stable conformation, thereby yielding the solution to the protein folding problem.”

May combine two claims.

Quantum computing realizations of the Monte-Carlo algorithm exist and may be applied to calculate thermal averages via sampling.

“Quantum computing realizations of the Monte-Carlo algorithm exist, [183] and so it may be argued that they can be applied to calculate thermal averages via sampling techniques.[184][185][186]”

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Probably not a substantial claim: may not pass S3 (load-bearing). The quote only partly states this claim.

The discrepancy between semiclassical and quantum-mechanical radical pair lifetimes has been taken as an indication that quantum mechanical effects play a key role in magnetoreception.

“As for light harvesting, this has been taken as an indication that quantum mechanical effects play a key role in magnetoreception.”

May combine two claims.

Describing non-covalent interactions with quantum approaches is impractical and superfluous.

“Although these interactions can be described with the quantum approaches presented in Section 5, such a route is both impractical and superfluous.”

May combine two claims.

Condensing all electrons of a chromophore into a two-level quantum system drastically reduces Hamiltonian dimensionality and complexity, but at a huge loss of detail and resolution.

“Condensing all the electrons of a chromophore into a two-level quantum system drastically reduces the Hamiltonian dimensionality and, consequently, the complexity of solving the corresponding quantum equations, but at a huge loss in terms of detail and resolution.”

• AssumptionThe authors state this as an assumption: Chromophores can be approximated as two-level quantum systems (excitons) for modeling EET.

May combine two claims.

Pilot experimental realizations of quantum simulators for light-harvesting systems have been proposed, but they are limited to very simple excitonic Hamiltonians.

“As a matter of fact, some pilot experimental realizations of quantum simulators for light-harvesting systems have been proposed in recent years, [175,176] although they are limited to very simple excitonic Hamiltonians.”

• cites [175] sourceNot checked yet

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• cites [176] sourceNot checked yet

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May combine two claims. The quote only partly states this claim.

In light harvesting and magnetoreception, quantum effects span the whole molecule.

“In contrast to the chemical reactions described in the previous section, quantum effects span, in this case, the whole molecule.”

Classified as probably not a substantial claim.

These algorithms assume that the distribution function can be encoded as the probability distribution of a many-body wave function.

“However, these algorithms assume that the distribution function can be encoded as the probability distribution of a many-body wave function (see Section 3).”

Classified as probably not a substantial claim. May combine two claims.

Quantum tunneling is thought to explore the energy landscape more efficiently and avoid local minima.

“The idea is that quantum tunneling will explore the complicated energy landscape of this model more efficiently and, therefore, avoid converging into local minima.[203,204]”

• cites [203] sourceNot checked yet

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• cites [204] sourceNot checked yet

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May combine two claims.

Classical simulations can target much larger molecules than quantum mechanical simulations, making folding simulations of small fast-folding proteins possible.

“Not being affected by the curse of dimensionality, classical simulations can target much larger molecules compared to quantum mechanical simulations so that folding simulations of small fast-folding proteins are possible.[187][188][189]”

• cites [187][188][189]Not checked yet

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May combine two claims.

Polynomial-time quantum algorithms are not expected for NP-hard problems, but more modest speedups might be achievable.

“Whereas polynomial-time quantum algorithms are not expected for NPhard problems, [196] like general optimization problems, more modest speedups, still significant in practice, might be achievable.”

• cites [196] sourceNot checked yet

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Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing).

This limited exploration may indicate that these quantum algorithms are promising candidates for molecular biology applications.

“On the one hand, this may be taken as an indication that these quantum algorithms may be promising candidates for molecular biology applications.”

Probably not a substantial claim: may not pass S3 (load-bearing).

It has been proposed to reformulate finding the most favourable reaction pathways as an optimization problem and solve it with quantum annealing.

“For instance, it was proposed in Ref. [204] to reformulate the problem finding the most favourable reaction pathways as an optimization problem, and to solve it with quantum-annealing algorithms.”

• cites [204] sourceNot checked yet

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May combine two claims.

In both light harvesting and magnetoreception, the many-body Hamiltonians depend on more than 100 particles, which is out of reach of classical simulation strategies.

“In both cases, this results in many-body Hamiltonians depending on more than 100 particles, which is out of the reach of the classical simulation strategies described in Section 3.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The potential of mean force generalizes the PES to thermal systems.

“Typically, the potential of mean forces [181] is then considered, which generalizes the concept of PES to thermal systems.”

• cites [181] sourceNot checked yet

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May combine two claims.

Solving folding as an optimization problem yields only the final structure (lowest free energy) and does not provide insights into the physical dynamics of folding.

“Solving folding as an optimization problem yields only the final structure, corresponding to the lowest (free) energy, but this procedure does not provide any insights into the physical dynamics of the folding process itself.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

QAOA is noteworthy and currently under in-depth investigation.

“In the latter category, the quantum approximate optimization algorithm (QAOA) [198] is noteworthy and is currently under indepth investigation.[199][200][201][202]”

• cites [198][199][200]Not checked yet

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May combine two claims.

A small polypeptide may contain a few dozen atoms and a few hundred electrons, making it too large for classical quantum chemical calculations.

“A small polypeptide may contain a few dozen atoms and a few hundred electrons and is, therefore, too large to be targeted by classical quantum chemical calculations.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Quantum annealing solves quadratic binary optimization by mapping it to finding the ground state of an Ising Hamiltonian.

“Another example of heuristic algorithms is quantum annealing, which solves a quadratic binary optimization by mapping it onto the equivalent problem of calculating the ground state configuration of an Ising Hamiltonian.”

May combine two claims.

Algorithms to calculate the partition function of quantum systems on quantum hardware have been proposed, but extensions to compute classical thermal averages are still scarce.

“While algorithms to calculate the partition function of quantum systems on quantum hardware have been proposed, [182] which is not surprising in view of the quantum simulates quantum principle introduced above, extensions of these schemes to compute classical thermal averages are still scarce.”

• cites [182] sourceNot checked yet

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Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

Large amplitude motions of solvated proteins are governed by dynamic electron correlation effects.

“large amplitude motions of solvated proteins are, at their very heart, governed by dynamic electron correlation effects.”

Probably not a substantial claim: may not pass S4 (based).

Achieving such stability is particularly hard in digital quantum computing.

“As we display Figure 5, achieving such stability is particularly hard in digital quantum computing.”

May combine two claims.

Classical surrogate potentials can often model large amplitude motions of solvated proteins, but not always.

“While these can often be well modelled by classical surrogate potentials, this is not always the case.”

May combine two claims.

Both the efficiency of EET in pigment-protein complexes and the radical pair lifetime can be predicted by solving the time-dependent Schrödinger equation for the corresponding Hamiltonian.

“In fact, both the efficiency of the EET process in pigmentprotein complexes and the lifetime of the radical pair can be predicted by solving the time-dependent Schrödinger equation for the corresponding Hamiltonian.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The lifetime of electronic coherences may not be long enough to affect the EET rate.

“Although it was later observed that the lifetime of electronic coherences may not be long enough to have an effect on the EET rate, [164,165]”

• cites [164] sourceNot checked yet

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• cites [165] sourceNot checked yet

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May combine two claims.

The electronic wave functions of light-harvesting and magnetoreception systems depend on thousands of electrons, making simulation extremely challenging for both classical and quantum computers.

“The electronic wave functions of these systems depend on thousands of electrons, and this makes applying the theory described in Section 3 extremely challenging for both classical and quantum computations.”

Probably not a substantial claim: may not pass S4 (based). May combine two claims.

Analog quantum simulation does not admit rigorous approximation bounds yet aims to give an accurate qualitatively reliable simulation for a long propagation time.

“The analog quantum simulation does not admit rigorous approximation bounds yet aims to give an accurate qualitatively reliable simulation for a long propagation time.”

Probably not a substantial claim: may not pass S4 (based). May combine two claims.

Moving to larger structures with an effectively cured curse of dimensionality, biocomputations face solving a hard classical computational problem, often an optimization problem such as protein folding.

“Moving to even larger structures with an effectively cured curse of dimensionality, biocomputations are now faced with solving a still hard, but now entirely classical, computational problem, often in the form of an optimization problem, e. g. in protein folding.”

• AssumptionRests on an assumption: Classicality emerges via decoherence due to the large size of most biomolecules and the relatively high temperatures of living organisms.

Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing), S4 (based).

It seems possible that quantum computers might become a supplement and accelerator of machine learning methods.

“Here, it seems possible that quantum computers might become a supplement and accelerator of machine learning methods.”

May combine two claims.

Intrinsically disordered proteins have only a small fraction of their amino acid sequence organized into a well-defined three-dimensional structure, while the rest is inherently dynamical and disordered.

“intrinsically disordered proteins, [215] in which only a small fraction of the amino acid sequence is organized into a well-defined three-dimensional structure, while the rest of the protein is inherently dynamical and disordered.”

• cites [215] sourceNot checked yet

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May combine two claims.

Formal guarantees of quantum speedup over classical counterparts might be available for these algorithms, but they require preparing qubits in a linear superposition with coefficients depending on the input data set.

“Formal guarantees of their quantum speedup over their classical counterpart might be available for these algorithms, but they require, as the initial step, preparing the qubits in a linear superposition with coefficients that depend on the input data set.”

Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing).

The same requirement applies more generally to important classical computational problems in the life sciences, ranging from genomics to patient data from imaging or sensory data.

“The same is true more generally for important classical computational problems in the life sciences, ranging from genomics to patient data from, e. g., imaging or sensory data such as from wearable devices.”

May combine two claims.

Physical modeling such as first-principles simulations does not suffer from these limitations by construction, but many practical challenges of simulating intrinsically disordered proteins with classical algorithms remain.

“Physical modeling such as first-principles simulations does not suffer from these limitations by construction, but many practical challenges of simulating intrinsically disordered proteins with classical algorithms remain.[216]”

• cites [216] sourceNot checked yet

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Probably not a substantial claim: may not pass S3 (load-bearing).

There is a long list of quantum computing approaches for optimization problems, from proven quadratic Grover speedup for exhaustive search to tailored algorithms for fault-tolerant computers to heuristic approaches for near-term quantum devices.

“From a proven quadratic Grover speedup for exhaustive search strategies to tailored algorithms to be run on fault-tolerant computers, to the heuristic approaches for near-term quantum devices, there is a long list of approaches for this class of problems”

Classified as probably not a substantial claim.

Genome sequencing requires reconstructing the nucleobase sequence of an organism from small portions of the full DNA sequence.

“For instance, genome sequencing requires reconstructing the nucleobase sequence of an organism based on small portions of the full DNA sequence.[232]”

• cites [232] sourceNot checked yet

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Probably not a substantial claim: may not pass S3 (load-bearing).

Approaches for protein simulation will also be applicable in the context of polymer physics.

“For instance, approaches for protein simulation will also be applicable in the context of polymer physics.”

Probably not a substantial claim: may not pass S2 (checkable), S4 (based).

The specifics of optimization problems in biomolecular sciences require a targeted and in-depth understanding of the type of optimization problem and a tailored solution.

“Nevertheless, the specifics of the optimization problems in biomolecular sciences require a targeted and in-depth understanding of the type of optimization problem as well as a tailored solution.”

Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing).

Other applications of quantum machine learning algorithms may yield a practical quantum advantage in the future.

“Other applications of quantum machine learning algorithms may, in principle, yield a practical quantum advantage in the future.”

Probably not a substantial claim: may not pass S4 (based).

Quantum mechanical simulations are essential when simulating biochemical processes that are accompanied by the creation of new chemical bonds.

“Quantum mechanical simulations are essential when simulating biochemical processes that are accompanied by the creation of new chemical bonds.”

• AssumptionRests on an assumption: The living world is governed by quantum physics.

Probably not a substantial claim: may not pass S3 (load-bearing).

The accuracy of machine learning for genuinely new instances depends on its transferability to unseen data (generalization power), which is extremely hard to estimate a priori.

“However, since the former make predictions based on previously learned data, the accuracy of machine learning for genuinely new instances depends on its transferability to these unseen input data -a property that is usually referred to as generalization power, which is extremely hard to estimate a priori.”

Probably not a substantial claim: may not pass S2 (checkable), S4 (based). May combine two claims.

Quantum effects may be crucial also in larger functional biomolecules, where they shape classical effective descriptions and require a more targeted use of quantum computing at the quantum-classical barrier.

“However, we also discussed how quantum effects may be crucial also in larger functional biomolecules, where quantum effects shape classical effective descriptions and require a more targeted use of quantum computing at this quantum-classical barrier.”

• AssumptionRests on an assumption: The living world is governed by quantum physics.
• AssumptionRests on an assumption: Classicality emerges via decoherence due to the large size of most biomolecules and the relatively high temperatures of living organisms.

May combine two claims.

Quantum computing may boost machine learning in two respects: speeding up training and making algorithms more expressive to predict more complex functions.

“Quantum computing may boost machine learning in two respects.First, it can speed up the training process.Second, it can make the algorithm more expressive in order to be able to predict more complex functions.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Local interactions between amino acids drive the protein toward the lowest-energy conformation during folding, as highlighted by the folding funnel metaphor.

“The local interactions between amino acids drive, while the protein folds, the protein toward the lowestenergy conformation, as highlighted by the famous”

May combine two claims.

This makes the training process computationally hard and eradicates a quantum speedup.

“This makes the training process itself a computationally hard problem and therefore eradicates a quantum speedup.”

Classified as probably not a substantial claim.

The first-principles nature of the quantum mechanical basis of the physical modelling approach makes the methods discussed applicable to other branches of molecular and materials science.

“We emphasize that the first-principles nature of the quantum mechanical basis of the physical modelling approach makes the methods discussed in this work also applicable to other branches of molecular and materials science.”

• AssumptionRests on an assumption: The living world is governed by quantum physics.

Probably not a substantial claim: may not pass S2 (checkable).

Quantum computing might supplement classical computers for these optimization problems.

“Also here, quantum computing might come to supplement classical computers.”

May combine two claims.

Otherwise, either loading classical data onto quantum hardware or carrying out classical optimization will become the main bottleneck, nullifying any quantum advantage.

“Otherwise, either loading the classical data on quantum hardware or carrying out the classical optimization will become the main bottleneck, nullifying any quantum advantage.”

May combine two claims.

Digital simulation tracks a property with high accuracy, but only for a shorter period of time due to stringent hardware requirements of fault-tolerant quantum computation.

“The digital simulation, by contrast, will track the property with high accuracy, yet only for a shorter period of time due to the stringent hardware requirements of fault-tolerant quantum computation.”

May combine two claims.

Recently proposed quantum algorithms for DNA sequencing and for identifying the most probable binding site of a transcription factor to DNA exist.

“This is the case of the recently proposed quantum algorithm for DNA sequencing [240,241] and for identifying the most probable binding site of a transcription factor to DNA.[242]”

• cites [240]Not checked yet

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• cites [241] sourceNot checked yet

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• cites [242] sourceNot checked yet

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May combine two claims.

Quantum technologies such as quantum sensing might enhance the quality of imaging data of biomolecular structures and improve the quality of data for further classical or quantum computations.

“More generally, quantum technologies, such as quantum sensing, [243] might enhance the quality of imaging data of biomolecular structures (see e. g.Ref. [244]) and, therefore, improve the quality of the data on which further classical or quantum computations would be based.”

• cites [243] sourceNot checked yet

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• cites [244] sourceNot checked yet

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Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Finding a Hamiltonian cycle is NP-hard classically, though efficient heuristic classical solutions have been developed.

“This problem is known to be NP-hard classically, [233,234] although efficient heuristic classical solutions to it have been developed over the years.[235,236]”

• cites [233] sourceNot checked yet

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• cites [234] sourceNot checked yet

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• cites [235] sourceNot checked yet

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• cites [236] sourceNot checked yet

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May combine two claims.

Quantum realizations of principal components analysis and support vector machine have been derived.

“The first advantage has been leveraged in deriving the quantum realization of the principal components analysis algorithm [221] and of a support vector machine.[222]”

• cites [221] sourceNot checked yet

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• cites [222] sourceNot checked yet

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