With the rapid development of quantum technology, one of the leading applications that has been identified is the simulation of chemistry. Interestingly, even before full scale quantum computers are available, quantum computer science has exhibited a remarkable string of results that directly impact what is possible in a chemical simulation with any computer. Some of these results even impact our understanding of chemistry in the real world. In this Perspective, we take the position that direct chemical simulation is best understood as a digital experiment. While on the one hand, this clarifies the power of quantum computers to extend our reach, it also shows us the limitations of taking such an approach too directly. Leveraging results that quantum computers cannot outpace the physical world, we build to the controversial stance that some chemical problems are best viewed as problems for which no algorithm can deliver their solution, in general, known in computer science as undecidable problems. This has implications for the predictive power of thermodynamic models and topics such as the ergodic hypothesis. However, we argue that this Perspective is not defeatist but rather helps shed light on the success of existing chemical models such as transition state theory, molecular orbital theory, and thermodynamics as models that benefit from data. We contextualize recent results, showing that data-augmented models are a more powerful rote simulation. These results help us appreciate the success of traditional chemical theory and anticipate new models learned from experimental data. Not only can quantum computers provide data for such models, but they can also extend the class and power of models that utilize data in fundamental ways. These discussions culminate in speculation on new ways for quantum computing and chemistry to interact and our perspective on the eventual roles of quantum computers in the future of chemistry.

## I. INTRODUCTION

The study of quantum computing in the abstract is an opportunity to ask ourselves what is
possible if we could attain an almost unimaginable level of control of the microscopic
facets of our universe. It was first proposed as a solution to the problem of simulating
physical systems with strongly quantum characteristics, a task that has proven very
challenging for traditional computers.^{1} The
idea was that if, like the puppet of a marionette puppeteer, one could make a precisely
controllable quantum system act enough like a more interesting system, then the puppet could
answer previously unknown questions about the true system. This core concept of quantum
simulation eventually merged with modern computer science to form the fields of quantum
computer science and quantum computing.^{2} This merger allowed these concepts to be made more precise and expanded applications beyond
physical systems into abstract ones such as breaking cryptography.^{3,4}

Despite the expansion into other applications, the simulation of quantum systems and,
especially, strongly correlated chemistry, has remained a primary application of interest.
Chemistry represents a sweet spot of quantum effects strong enough to make them challenging
for classical computers while still having a well-known application space to motivate
development. Since the original proposal of Aspuru-Guzik *et al.*,^{5} there have been a number of developments in
quantum algorithms for the direct simulation of chemistry bringing costs down from
astronomical numbers to routine calculations for even very challenging systems.^{6–10}

In parallel to algorithmic developments, quantum technology has advanced at a rapid pace.
Recent demonstrations by the Google group have experimentally shown that these devices are
capable of tasks that are incredibly challenging on a classical computer, with a gap that
will only grow with the quality of modern quantum computers.^{11} In addition, a number of prototype chemistry experiments have now
been experimentally demonstrated on quantum devices.^{10,12–14} At present, error rates in quantum
devices are too great to make them competitive with the best classical algorithms for
chemistry, but quantum error correction and mitigation offer paths forward.^{15–19}

However, even without an actual quantum computer running an algorithm, the study of quantum
computer science in the abstract has led to insights about the limitations and possibilities
of chemical simulation, and, indeed, under modest assumptions, about the universe and
chemistry itself. Recent results tying quantum computer science directly to chemistry
include showing that the electronic structure problem is expected to be hard in the worst
case even for quantum computers.^{20–22} In this work, we highlight some of the yet unexplored connections
leading up to a perspective on the role both traditional and quantum computers may play in
the future of chemistry. To build to this perspective, we begin by framing direct chemical
simulation as a digital experiment. In this framework, we exploit known results that
restrict the power of any computer, even a quantum one, to understand limits of such digital
experiments in chemistry. Along the way, we encounter surprising results about the
impossibility of an algorithm for determining if a system ever thermalizes, flying in the
face of conventional thermodynamic analysis. However, this construction guides us to new
ground, where we will see that learning from data can be fundamentally more powerful than
traditional computation. This sheds new light on the success of many existing chemical
theories and the way in which they exceed rote simulation by leveraging the additional power
provided by data.

The relationship between traditional simulation and learning models in chemistry is cartooned in Fig. 1, where classical models with data from nature can exceed the physical prediction horizons of rote simulation. We consider learning models to be models where some amount of high quality training data for the task is available, from either a physical experiment or a simulation in addition to just the specification of the problem. As we will argue, this is not just a reference to modern machine learning methods but rather encompasses the foundations of chemical theory such as transition state theory, molecular orbital theory, and thermodynamically controlled reactions. In addition, we take the opportunity to highlight previously unused technology from quantum information science that may bolster the development of chemistry even before the arrival of full fault tolerant quantum computers. The links from chemistry to results in quantum computer science all build to our ultimate perspective that the eventual role of quantum computers in chemistry will be to aid in the construction of learning models for chemistry. This includes providing data to classical models, constructing quantum models that can make accurate predictions with far less data, and eventually interfacing directly with quantum data from chemical systems. To close, we wrap this Perspective into an outlook for the interplay between these two exciting areas.

## II. DIGITAL CHEMISTRY EXPERIMENTS WITH QUANTUM COMPUTERS

The idea to draw a distinction between theory and computation or simulation is not new,
with suggestions of considering computation as the “third pillar of science,” alongside
experiment and theory.^{23} In this
framework, it can sometimes be more accurate to view simulations as closer to experiments
than to theory. Quantum computing, and especially quantum simulation of physical systems,
helps to make this even more clear by constructing simulations of the physical world at the
quantum level with efficiently refineable (and bounded) accuracy. In addition, limitations
on precision for quantum computers^{24} to
learn physical properties sometimes place them even closer to experiments than classical
digital simulations of the same systems. While a discussion on the precise distinctions
between simulation and theory deserves more exposition than we can provide here with many
gray areas of overlap, we will attempt to provide a working distinction. In particular,
simulation will refer to a rote input and output relationship, where given some
computational problem and inputs, a fixed output is reached. In contrast, theory will refer
to the broader activity of developing and refining models that may go into a simulation,
where those refinements could be based on data from physical experiments, simulations, or
abstract mathematical constructs. In addition, it is worth distinguishing between data that
can be considered as input, such as initial conditions of a dynamical system, and data used
to develop a theory or model, which we will refer to as training data in connection with the
growing literature on learning theory.

The most natural setting for a quantum simulation on a quantum computer is that of watching
the system step forward in time or quantum time dynamics. Within the field of quantum
computing, the term “quantum simulation” is sometimes reserved for time dynamics simulation
specifically, and when we refer to rote simulation, it will typically apply to this case.
That is, one sets an initial state, a physical system defined by its interactions, and some
time to simulate, and the quantum computer performs a computational experiment in mapping
the initial state to the final time, where any reasonable observable of the system can be
probed. This time dynamics obviously mirrors the most natural settings of the real world.
However, despite this natural setting, one will note that a large number of quantum
algorithms and theories closely follow that of the electronic structure performed on
classical computers, where the focus is on low energy eigenstates of the electronic
system,^{5,25} rather than exclusively
the final state after some fixed evolution. This important deviation already wraps in data
from the natural world that tells us in many systems, thermal states are an apt description
of a system, and at low temperatures, low energy states are most commonly observed. As we
will discuss later, the use of such observational knowledge is strictly more powerful than
time dynamics simulation and underpins the great success of many modern chemical theory or
simulation methods. A detailed discussion of this point will follow, but for now, we lump
both static and dynamic experiments into the category of direct simulation.

By direct simulation, we mean a simulation where the system of interest (often processes
such as a reaction) is known, and we seek to use the computational experiment to gain
insights into the process that is challenging to access through the experiment, for example,
details of a reaction mechanism, transition states, or migration of charge through a system.
They represent the bulk of computational experiments, both on classical computers and those
proposed on quantum computers, and we draw a distinction between direct simulation and
design, often characterized as inverse problems.^{26}

For the task of direct simulation of chemical systems, quantum computers have been shown to
demonstrate an exponential advantage over their classical counterparts under modest
assumptions.^{5,27–29} An
exponential speedup has the practical implication that some simulations that might have
taken longer than the age of the universe could be done in mere seconds. More specifically,
full quantum dynamics with no approximations beyond discretization error can be performed on
a quantum computer in a time that scales only polynomially in *t*, *N*, and *M*, where *t* is the simulated
time, *M* is the number of basis functions, and *N* is the
number of electrons.^{9,19,30} Interestingly, along with these speedups, it comes with some of the same limitations
possessed by a physical experiment. For example, in contrast to classical simulations where
more precision in a simulated quantity is often relatively easily obtained, the Heisenberg
limit applies to any measurements one might perform to extract information. These approaches
have also included exploring the ansatz for electronic systems that are inaccessible to
classical computers.^{13,31} Perhaps
surprisingly, recent results have shown that quantum computers can even achieve a scaling
for exact computations that is sub-linear in the basis set size by taking advantage of a
first quantized representation in a way that has no current classical counterpart.^{9}

However, quantum computers do not vastly extend their capabilities into the realm of
discovery or design, which we identify as distinct from direct simulation. That is, while
many of the proposed simulations would be faster and more accurate representations of the
same systems than would be achievable classically, they are much the same type of the
experiment without reaching into the design space. While there has been some notion of how
quantum search may assist in design, most results promise at most a quadratic speedup in
contrast to exponential speedups in direct simulation,^{32} although we note that the direct simulation subroutines used in a
potential search would still benefit from improved speed or accuracy. Due to the immense
size of the design space, this quadratic speedup is not terribly compelling if it is not
combined with structured strategies. That is, any improved search must start not from naive
global search but a search enriched by knowledge of the design space, for example, one that
starts in the chemical neighborhood of known, synthesizable compounds. In addition,
practical issues arising in real quantum computers may prevent quadratic speedups from being
advantageous for some time.^{33} We will
argue later that perhaps it is even more apt and useful to consider an infinite chemical
space as opposed to one which is merely combinatorially large.

That said, direct simulation, of course, has a great value for design. For example, if a
key reaction or catalyst is known to work to catalyze a reaction of interest, the
microscopic detail of a computational experiment that is inaccessible in the lab can offer
insights into the causal mechanism that enable the design of analogs.^{34–36} In addition, a now popular approach is the use
of screening of many candidate compounds through direct simulation in order to identify new
candidate molecules or learn properties, which are predictive for a desired
design task.^{37–39} This type of
approach is also used for selection of candidates in protein to small-molecule docking.^{40} Direct simulation can also be used for
novel discovery of processes through direct time evolution, as has been done with the
molecular nano-reactor.^{41} Quantum
computers serve to strictly improve this type of discovery by expanding the set of systems
that can be accurately treated in a given amount of time by such methods. Advantage in
improving the accuracy in direct simulation has been the primary focus in much of the
literature of quantum computation for chemistry, with the hope that it enables for an
accurate study of important systems inaccessible to current methods. The example receiving
perhaps the most attention is FeMoCo,^{19,28,30,42–44} where the hope is that direct simulation will offer a
mechanistic insight that leads to the design of new catalysts for nitrogen fixation.
However, when leveraging direct simulations results for an improved design, the simulations
are typically tightly coupled to a theory already rooted in data. For example, the energy of
a transition state might be used to justify a reaction rate, but this is inherently coupled
to the empirical success of transition state theory, rather than a result that emerges
directly from simulation.

In contrast, comparatively little attention has been paid to the potential advantages of
improved dynamics simulations, perhaps due to the larger foundation of work available
utilizing static eigenstates in the classical case. In addition, it is perhaps surprising
that dynamics simulations on quantum computers that fully relax the Born–Oppenheimer
approximation, using fully quantum nuclei and exact electronic representations, are about as
costly as those that live within such approximations. However, with each step toward this
nuclear-electron soup, what is gained in accuracy might be paid in ease of interpretation as
the added overhead of re-discovering the notion of a molecule through observable
measurements remains a largely uncharted problem in the context of quantum computing.^{27,45} Taking advantage of these departures
from the tried and true anchors of chemical space and embracing the idea of a fully *ab initio* experiment will require more development in how this type of
simulation can be most beneficial. However, if it is unlocked, it may offer new insights
into complex dynamic processes in spectroscopic or electrochemical processes where both
excited electronic and quantum nuclear states can play a role.^{41,46} It may also offer new insights into the long studied
challenges surrounding proton coupled electron transfer.^{47}

Given the immense power of quantum computers in speeding up these types of direct simulation, one is often left wondering how far these advantages extend. This question is one that has been studied in great detail in computer science, and we now wish to turn to our understanding of the limits of power in quantum computation. The similarity between quantum computations and the physical world implies that the results also tell us something about nature and chemistry itself, and we now turn to this Perspective.

## III. LIMITS OF EVEN QUANTUM COMPUTATIONS IN CHEMISTRY

Having set up the perspective of quantum computation as digital experiments, we now turn to the implications this may have for the physical systems themselves. Due, in part, to the challenge of building scalable quantum hardware, for a good part of its history, quantum computing was a largely theoretical endeavor that aimed at understanding the power of quantum devices and, by extension, physical systems. Considerable progress has been made on this problem, and we wish to highlight those results we believe have the most bearing on chemistry.

Before discussing how they pertain to chemistry, it is worth mentioning a few aspects of the assumptions made in stating the results below. In particular, in order to talk about scaling and cost, it is necessary to have some model of computation where cost can be quantified. We will generally be assuming that this means we use a finite, often local, set of operations called quantum gates, and we work within a digital gate model of quantum computation. This assumption is in some ways analogous to assuming that nature favors local interactions or that computers are built from modular components. Moreover, some of the results below most precisely pertain to the number of times some information, such as the Hamiltonian of a system, must be accessed. This is known as the query complexity, and while it is often closely related to time complexity, these can be different in some circumstances, such as when parallelization is or is not possible and we attempt to be clear as to when this distinction is important.

### A. Quantum computers cannot fast-forward time either

Starting with the most direct form of quantum simulation, time dynamics of a quantum system, we have already noted the exponential advantage over classical simulation achieved by current quantum algorithms. This begs the question how far these quantum algorithms can go in accelerating simulation of physical systems. For example, can a quantum computer simulate a system faster than nature itself evolves the same system, that is, sub-linear in the time being simulated? It turns out that, assuming one only has the available space to represent the system serially, the answer to this is a resounding no. It is also worth noting that if a system is finite and the time to simulate is larger than the number of possible system states, the cost of diagonalization, finite state enumeration, or some equivalent can be paid to fast-forward a system. However, in most cases, the time scale of interest is much less than this, and indeed, as we argue later, infinite perspectives on chemical systems are sometimes more natural.

While there are some special systems that can be fast-forwarded,^{48} it cannot be done in the most general case. This result
can be understood at a high level by imagining if this were true and realizing one could
build a recursive simulation that does any computation in vanishingly small time, by
nesting simulations within simulations with no space overhead. If one allows the space on
the quantum computer to expand polynomially with the simulation time, this result is more
unclear and related to the classical debate of if any computation can be parallelized
arbitrarily well, encapsulated by the theoretical question of if P = NC.^{49} This is an open question that currently
holds a similar status to the more famous question of if P = NP. Some progress has been
made on the question if all computations can be parallelized arbitrarily for specifically
quantum computers, showing that more simulation time is strictly more powerful relative to
an oracle.^{50} To simplify the message
here, we assume that quantum space resources are the minimum required to perform the
simulation and that as a result, oracle queries must be done in serial. More formally, the
limitations of simulation are encapsulated in the no-fast-forwarding theorem.^{51,52} This theorem importantly shows
that not only is universal computation not fast-forwardable but surprisingly even some
Hamiltonians that are incapable of universal computation cannot be fast-forwarded
either.

Under the stated assumptions, the no-fast-forwarding result practically implies that no simulation that retains a complete level of detail at the quantum level can simulate physical systems faster than nature, and likely, there will be some large factor disadvantage between them. This result is in some ways not unexpected from classical simulations of classical chemical models such as molecular dynamics, where for a fully detailed tracking of the model in the most general case, one expects to lag behind the physical time scale of the system unless approximations or reduced models are utilized. The emphasis here is that this limitation provably carries into the case of using a quantum computer as well. Taking this result at a face value along with the knowledge that important chemical reactions such as rusting can take months to occur, using a single molecular realization in a simulation could take comparable times to witness a reaction event, making discovery in this fashion totally impractical. However, most chemists would know that is not the end of the story.

Specifically, methods such as transition state theory based on static calculations of
free energies or transition path sampling^{53} can allow access to rate constants on time scales much faster than
direct physical simulations.^{54} In
addition, the general class of methods related to rare event sampling are often successful
at reaching beyond time scales accessible in direct simulation.^{55} Such methods have been successful at dramatically
increasing the reach of molecular dynamics simulations, often via the observation that
rare events are dominated via relatively few pathways. Hence, while the above theorem is
true for precise simulations of the full quantum state, we see from practice that physical
theories grounded in observation can circumvent such wild overheads by eliminating the
complexity- of the full quantum state and focusing on a reduced set of observables,
reduced precision, or approximate models amenable to fast-forwarding. Such reductions in
the number of important degrees of freedom have been studied in various contexts to try to
understand “what makes science possible.”^{56,57} It should also be noted that these results are about the most
general system, and specific systems may exhibit physics that allow faster than physical
simulation when that data are utilized by validating approximate models. In other cases,
particular models are amenable to classical fast-forwarding, such as non-interacting
electron models, but they are not general enough to encompass all quantum systems with
interactions.

The observation that chemical methods have been so successful in making predictions
beyond physical time scales leaves one believing that perhaps physical systems always
admit some form of reduction that will allow us to forecast results that we actually care
about (as opposed to a full quantum state), well in advance of simply stepping them
forward in time. A recent string of results in quantum computer science has proven that,
in fact, even in some very simple systems by construction, there must exist some questions
that are fundamentally unanswerable ahead of time. This list of problems surprisingly
includes whether simple one-dimensional (1D) systems thermalize,^{58} which has implications for the predictive power of
thermodynamics in general systems. These results are much stronger than simple chaos of
dynamic systems and require some exposition, but we believe they are a value lens to view
chemical problems.

### B. Questions no algorithm can answer and undecidability

A direct reading of the no-fast-forwarding theorem contrasted with the unassailable predictive success of simplified chemical models raises a number of interesting questions. On the one hand, the no-fast-forwarding theorem tells us that even with an exponential advantage over classical simulation of quantum dynamics, a quantum computer aiming to simulate the dynamics of a full wavefunction to a fixed accuracy is bound by some multiplier of the physical time of the actual process. On the other hand, we see that reduced models, for example, mean-field molecular orbital models, reductions based on evaluating the energy of stationary states such as ground and excited states, or even arrow pushing in Lewis dot structures can sometimes be powerfully predictive for a chemical phenomenon of interest at a relative time cost much less than the time scale of a process of interest. This leads one to ask if there is always a predictive model or algorithm for simplified questions. Surprisingly, results in computer science have shown us that there are, indeed, even relatively simple questions related to physical systems for which it can be proven that no single algorithm can answer for all such systems in finite time. Here, while we will certainly not claim that this should cause one to abandon current methods, we will argue that these results should have implications for how one approaches some chemical problems.

To make our case, we will start with a motivating example before we move to the more
general concepts that will allow us to address some of the tensions with this
interpretation. Consider the question of “Given unbounded time, does this physical system
reach thermal equilibrium?” The basic assumptions of conventional thermodynamic theory,
leaning on the ergodic hypothesis,^{59–61} assert that the answer is a resounding “yes” under assumptions
that the system is ergodic. Indeed, many standard molecular dynamics simulations exploit
both possible directions, either determining free energies through proxies derived from
long time simulations^{54} or predicting
long time behavior through models for free energies. However, recent results have shown
that there are 1D translationally invariant systems for which no algorithm that can answer
the question of thermalization or even simple questions about energy gaps in finite
time.^{58,62–64}

To make such a problem even more concrete, let us take the question of a finite gap as an
example. The problem statement can be set up briefly as follows. One is provided as input
of a finite Hamiltonian for a chunk of a system that will be repeated indefinitely. While
any finite chunk of this system could be diagonalized, determining the existence of a gap
for that chunk, this proof implies that no finite number of chunks taken together could
ever tell you about the status of the gap for the infinite system in the general case.
Moreover, any extrapolation on those chunks to a large size limit cannot be guaranteed to
converge. The statement is so strong that we can guarantee that not only will no simple
algorithm work exist, but there cannot exist such an algorithm of any finite time
complexity. This type of problem is known as an undecidable problem, the first example of
which was the famous halting problem of Turing.^{65} Imagining the setup as a physical experiment, one can
characterize a small chunk of material experimentally, determining if it has a gap.
Assuming that one finds no gap, the size of that material could then be extended,
gradually adding more material, and one could again measure at any juncture if the
material has a gap. At any fixed size of the problem, one can easily measure the gap, as
the undecidable problem does not imply that it is difficult to see that the system is
gapless but rather that, in general, one cannot predict whether it will ever exhibit a gap
upon measurement at larger sizes in the most general case. This class of problems is much
more challenging than even the perhaps more familiar class of NP-complete problems, as
even when provided with a supposed answer, a mortal prover cannot verify if it is correct.
This implies that for a number such as an energy or free energy representing a system,
either that number cannot be generally predictive of the behavior of that system or the
number itself must be uncomputable. Said another way, there will be instances where
computing the free energy of a chunk of a system fails to capture the complexity of its
real environment, and qualitatively different behavior than predicted by thermodynamics or
any finite time computation will be observed. By an uncomputable number, we refer to the
idea that if there were such a number that could accurately predict the behavior of any
system for arbitrary times, its computation would involve the solution of undecidable
problems. Showing that this is true, even for a quantum computer, which can perform
classical computation as a special case, underscores the weight of these results.

To draw a connection with recent results in worst-case complexity for chemistry, it has
been shown that the electronic structure problem in a finite basis and some related
problems are in the quantum analog of NP-complete or QMA-complete.^{21,22} Such problems live in a finite state space, and
hence, the implication is that at worst an exhaustive examination of all possible states
will suffice to determine if something will happen akin to a program halting. Despite
these complexity results, practicing chemists have obviously not stopped using simulation
methods. It is important to remember that most of these results apply for worst-case
constructions, and indeed, physical systems may come from a special set of all possible
systems that makes them more amenable to study. For a system with an infinite number of
possible states, total enumeration obviously becomes untenable, and some problems graduate
to become undecidable. Similar to the case of QMA-completeness, however, this does not
imply that every instance of a problem should become impossibly hard.

The idea that bare thermodynamic quantities may not be predictive due to lack of
thermalization is certainly not new in the study of non-ergodic systems, with preeminent
examples such as spontaneous symmetry breaking, spin-glasses, and the general idea of a
Gibbs measure.^{66–69} These
results strengthen these concepts to tell us that in some non-ergodic systems, there is no
computable replacement for something like free energy that is predictive of the behavior
of a system. Hastily defined, the halting problem is the problem of finding an algorithm
that, given a finite program input, can determine in finite time if the program halts or
runs forever. If one views a physical evolution as a computation where halting is defined
as some qualitative change,^{70} for
example, the fraction of random systems that thermalize should be closely related to the
fraction of random programs that halt, a number known as Chaitin’s constant.^{71} Indeed, it is easy to see that such a
thing requires solving the halting problem for every possible instance to determine the
digits of the fraction of halting programs, which defines it as an uncomputable number.
For such systems, assuming ergodicity is analogous to solving the halting problem by
simply hypothesizing that all programs encoded by the physical evolution halt.

For veteran chemists, another way of viewing this result is that it is a vast
strengthening of the concept of kinetically controlled reaction systems to be contrasted
with thermodynamically controlled systems. Previous work in this area has shown that this
type of unpredictability is distinct and stronger than traditional notion sensitivity
induced chaos in rendering systems unpredictable, even when they are deterministic. When
considering a computational view of the universe,^{2} undecidability does not imply that something cannot happen in the
universe. Rather, it implies that in the worst case, some events cannot be predicted using
a finite algorithm, even if those events are relatively ordinary and the outcomes are easy
to describe once they have occurred. Phrased in a physical point of view, for a system
exhibiting undecidable dynamics, there are sudden, qualitative changes that cannot be
predicted in any way other than simply stepping the system forward in time.^{70} For chemical practitioners, such a
description is certainly reminiscent of the emergent phenomena that transition us from
simple molecules to proteins, RNA, DNA, complex catalytic networks, and even living
systems.

Before continuing, we must address a tension that arises with the mapping of physical processes to halting problems in the form of finite vs infinite systems. In order to map to a halting problem rigorously, one typically maps the operations of a physical system to operations of a Turing machine, with an infinite sized tape. If the tape were finite, one can spend a very long, but not infinite, time examining all possible states of the system to arrive at an answer to any question related to the state of the system. For example, in studying the dynamics of a finite physical system, one could discretize the possible states and expend an exponential amount of time in the system size determining an answer. Hence, one might want to argue that since the universe is often presumed finite, these conclusions are of little consequence, even though from a practical point of view, the exponential of even a modest sized finite system would take longer than the age of the universe. Despite their somewhat similar predictions, in practice (too long to compute), we argue that embracing the viewpoint of undecidable dynamics allows one to let go of the impossible dream of treating chemical questions by the exponential enumeration of the combinatorial perspective. Whether or not a physical system exhibits truly undecidable behavior, embracing the possibility leads to a more natural description of many chemical processes and points the way toward more apt solutions.

We depict an example of this Perspective in Fig. 2.
Consider a collection of small molecules attached to a reservoir of source chemicals, and
we wish to know if a particular strand of DNA will ever form, given this setup. If one
took the closed system, or combinatorial, viewpoint, it would suggest that perhaps the
most effective way to address this question is to model the reservoir with some enormous,
but finite, set of explicit molecules and check all the possible states of the dynamic
system. Along the way, one can check things such as the change in free energy
(Δ*G*) or simply proceed with short time dynamics. Not only is the
approach untenable due to the number of possible configurations, but given the nature of
self-catalytic reaction networks, one could accidentally include slightly too few
molecules to form the right catalytic process to enable the formation of that strand and
reach the wrong conclusion. In contrast, the open system, or undecidable, interpretation
of this setup, is that the physical system can continue to borrow from the infinite
reservoir such as states of the Turing tape. If it has to temporarily expand to chemical
networks of unforeseen size, it can do so. In this situation, one tends to lean on data
models for what has existed before, and we detail later how doing so changes the power of
the simulation approach. The undecidable interpretation is supported by recent work,
showing that DNA computing is Turing complete and as such can encode the halting
problem.^{72,73} While DNA computing
is a bit of an artificial construction and real DNA that tends to form in live systems is
subject to other constraints that make its activity more predictable, the possibility of
implementing universal computation in the dynamics of a chemical system pushes us to
embrace the wildest consequences of the theory of computation.

This tension between finite and infinite models is not unfamiliar when related to the
study of other non-ergodic systems, such as spin-glasses.^{66} In such cases, models might predict that finite systems have
some chance of exhibiting substantial rearrangements, albeit on exponentially long time
scales. However, it has been seen to be useful to lean on a model that takes a limit where
these different basins qualitatively fracture from each other and are considered separate.
We suggest here that as molecular systems grow in complexity, the run-away reaction
systems that form are best treated by considering them as fracturing into novel parts of
chemical space, without tracking vanishingly unlikely events that transition one between
these parts of space. Indeed, the undecidability of thermalization suggests that for
infinite systems, it is correct to view such fracturing as exact in the most general case.
Even in simpler cases, this model tracks more closely with the experience of laboratory
chemists who have to contend with side reactions, rendering a thermodynamically favorable
synthesis seemingly impossible. By not splitting hairs about the distinction between a
large finite system and a truly infinite one, we can move toward practical alternatives to
the brute-force combinatorial approach of pure computation.

However, if one resigns themselves to accept that many of these problems have no general algorithm, is one left with no option but to plod forward with rote dynamic simulation or give up? At least the combinatorial approach offered a glimmer of hope, even if it required longer than the age of the universe for tiny problems right? In Sec. IV, we argue that, in fact, this realization does offer an alternative, which focuses the scientist on an approach based on learning and reduced chemical model construction.

## IV. LEARNING AS AN ALTERNATIVE TO COMPUTATION

Rather than claim that one must give up hope on problems where the most natural formulation
looks undecidable or hopelessly expensive, we argue that embracing the undecidable
interpretations of these problems frees one from the distraction of the combinatorial
approaches and leads them to embracing the only known resolutions to problems that exist for
halting problems. The first, and not really palatable solution, that has been mentioned, is
running time dynamics forward and hoping for the event of interest to occur. The second, and
more actionable viewpoint, is to understand that systems with advice can formally resolve
such problems. While advice has a precise definition in theoretical computer science,^{74} it was recently shown that data from the
real world can act as a restricted form of advice.^{75} To solve halting problems, one’s advice would need to constitute a
form of infinite time pre-computation; however, much like the argument of finite vs infinite
systems, the pre-computation performed by the physical universe before this point, though
finite, is already quite strong and available to be revealed through experiments.

To see an example of this in chemistry, one can look to the field of natural product
synthesis, where one seeks to synthesize chemical species found in nature from simpler,
readily available components in a finite number of steps. For the most general complex
molecule, it can be quite difficult to say if a product can be reached with a fixed set of
reagents and conditions or at all, perhaps due to some intervening side reaction. In fact,
we conjecture this form of the problem may be undecidable since we could specify the target
molecule as a particular strand of DNA.^{73} By contrast, in the study of natural product synthesis, the challenging question of chemical
synthesizability is answered by nature itself, and practitioners are freed to discover
practical routes toward a product rather than ponder if success was even possible
theoretically. In this sense, empowering ourselves with observations from the physical world
can enable answers to questions that cannot be practically resolved with rote algorithms.
More generally, we believe one could view the models and study of synthesis in organic
chemistry and their predictive success in this light. Hence, taking this point of view
suggests that approaches based on learning are fundamentally different from those that rely
on computation alone.

With the rise of popularity in machine learning, naturally, there is now a flurry of work
applying techniques of machine learning to chemistry.^{76} The applications range from prediction of direct simulation
quantities^{77} to synthesizability^{78} and inverse design.^{26} We believe that the results we highlight here bolster the
motivation for pursuing this line of work in the general sense but also support the long
standing tradition of chemists developing physical models as well. While much practical work
remains to be done in finding the best representation or approaches, it is now understood
that these approaches are fundamentally different, and indeed more powerful in some ways,
from traditional simulation approaches.

In particular, recent work has shown that the power offered by data from a quantum computer
can lift classical models to be competitive with their quantum counterparts in some
cases.^{75} If nature is, indeed, a
universal quantum computer,^{79} this would
suggest that quantum computations could help fill the role of natural experiments in
providing data to empower learning models, but with improved programmability and
flexibility. This might suggest that in the future, a key role of quantum computers, which
may still remain scarce compared to their traditional counterparts, may be to provide data
for learning models primarily run on traditional computers. However, that would be a rather
unexciting fate for a technology that pushes the limits of our understanding of the
universe.

Instead of accepting this fate, we appeal to other recent work that has shown an
overwhelming advantage for quantum computers in how they can process data moved directly
into the computer from a quantum sensor, a process referred to as transduction.^{80,81} A comparison with a traditional data
pipeline is shown in Fig. 3. In such cases, even
performing quantum data processing on even very few copies stored in quantum memory can
allow one to extract properties of quantum systems that would require exponentially more
data in the classical case.^{80} Even early
machines may be able to manipulate few copies for limited times and store these quantum data
for future models. This separation is stronger than a traditional computational separation,
as if limited measurements are available due to the transient nature of a system, no amount
of computation can allow for the traditional measurement and computation to catch up. This
implies that even systems with a number of qubits that can be classically simulated could
prove advantageous. If the quantum computer has quantum memory that can maintain states in
the quantum form, this extends the power of these approaches even further. The relationship
between this hierarchy of data models and traditional simulation is depicted in Fig. 1. Moreover, if one ventures further into the realm of
speculation, recent computer science results have shown that if one can entangle two parties
trying to prove something to another, what they can convincingly prove expands to
unimaginable heights, including halting problems. While this celebrated result, MIP* =
RE,^{82} has not yet been directly
connected with the physical world, it is tantalizing to imagine how it might reflect on the
power of physical experiments empowered by entanglement to reveal the mysteries of the
universe. The question, of course, becomes where to find such useful quantum data and how to
effectively get it into the quantum computer.

Chemistry has long benefited from the power of quantum sensors in nuclear magnetic
resonance experiments. While the most basic forms of these experiments are based on ensemble
measurements at an effectively high temperature, they offer incredible insights and are
fundamentally quantum. Interestingly, the form of these basic experiments can be closely
identified with the so-called one clean qubit model or DQC1 and this has inspired some
recent work in pursuing advantages with nuclear magnetic resonance (NMR).^{83} However, as NMR technology advances with
techniques such as hyper-polarization,^{84} multi-dimensional methods,^{85} spatial
resolution,^{86} and zero-field
techniques,^{87} we appear to be accessing
more and more pure quantum states with increasing control.^{88} In addition, the design of more advanced molecular quantum sensors
is an active research area.^{89} With
sufficient advances in transduction techniques and developments in quantum error correction,
it may become possible to load data from molecules directly into quantum computers and
perform manipulations that are provably challenging for a classical device even with
unbounded compute time to replicate due to the query separations we mention above. This
could enable faster chemical identification, unprecedented accuracy in examining quantum
effects in molecules, or fundamentally new techniques for the control and manipulation of
chemical states.

## V. UNTAPPED RESOURCES IN QUANTUM COMPUTER SCIENCE

The discussion thus far has largely centered on the role of quantum computer science or
quantum computers themselves in new approaches to understanding chemistry. However, it is
also interesting to ask more generally what theoretical tools from quantum information
science with potential to impact computational chemistry remain untapped. For example,
techniques in tensor networks that received much of their theoretical development in quantum
information have now begun to inform the ansatz for correlated wavefunctions. Both matrix
product states and tensor network states coupled with an entanglement perspective on strong
correlation have led to a wealth of theoretical developments.^{90,91}

An area of remarkable theoretical development that remains relatively unknown in the
chemical community is that of digital quantum error correction.^{16,92} Perhaps for a chemist, the best way to understand
these methods is in relation to the existing and well developed body of research of quantum
control of chemical reactions and chemistry.^{93–95} We depict aspects of this relationship in Fig. 4. In quantum control, a signal, often through a laser excitation or
similar, may be used to control and measure the system through feedback; however, these
measurements are often destructive with regard to quantum states in the system, and we refer
to this as analog quantum-classical feedback control. Through the use of a separate quantum
probe or ancilla system, one can make measurements of a system that actually generate or
stabilize entanglement in the quantum system or an analog quantum–quantum scheme. Taking
this one step further, one may use multiple probes and digital algorithms that adaptively
change based on measurement results to pump entropy out much faster than traditional
thermalization would allow. It turns out that the exponentially long suppression of
transport out of collection of desired quantum states is reliant on an abstract digital
perspective of the states. These constructions are formalized in the theory of digital
quantum error correction, where it has been shown that with only spatially local
measurements and classical computation, it can be possible to stabilize exotic quantum
states for essentially arbitrary lengths of time with asymptotically modest resources. While
quantum control has a rich history in quantum chemistry and, indeed, many of the
developments in reducing errors in quantum experiments^{93,96} have relationships to techniques in chemistry, the full digital
perspective remains relatively unexplored. If such techniques could be ported more directly
into chemistry, stabilizing electronic or excitonic states could lead to novel reactive
methods or wildly improved energy transfer efficiencies. While this remains in the realm of
pure speculation perhaps probe systems that can be interacted via future versions of
tunneling microscopes with near single electron resolution coupled to catalytically active
sites on metal surfaces could offer some of the first places to explore these ideas.

The development of quantum error correction was also closely linked to the development of
simulation for a particular class of quantum states, known as stabilizer states. These
states can exhibit essentially maximal entanglement across a system yet are incredibly
efficient to simulate classically, often scaling like *N*^{2} or *N*^{3} in the number of orbitals.^{97,98} While they are discrete in nature, requiring some
novel optimization techniques such as reinforcement learning, if coupled with simple orbital
rotations, these could form yet another powerful ansatz for efficient classical electronic
structure methods informed by quantum information that has yet to be exploited. We depict
this connection in Fig. 5.

While the existence of a powerful ansatz that has not yet been explored is perhaps strong
enough motivation by itself, the link to quantum error correction provides even more fuel
for this interesting avenue of inquiry. Many of the most challenging bio-metallic enzymes,
such as nitrogenase, are characterized by an active site that is difficult to describe with
traditional methods due to static correlation related to quasi-degeneracy of electronic
configurations. Quantum error correction takes advantage of strong entanglement and
engineered degeneracy as a mechanism for protecting information. If nature were to follow a
similar path, perhaps the electronic near-degeneracy and resulting entanglement in these
systems could be an avenue for protecting a coherent reaction pathway similar to the
approach of quantum error correction. In addition, many modern quantum error correcting
codes take advantage of topological protection, and topological effects have been studied in
the context of electronic degeneracies near conical intersection.^{99} While natural systems do not have access to an optimal
decoder to remove excess entropy, the notion of self-correcting memories where systems
exhibit such behavior more naturally draws an enticing connection.^{100} In such a case, the stabilizer formalism for representing
these ground states would be especially apt, efficiently manipulating untold numbers of
determinants through non-obvious symmetries.^{101} This could offer an electronic (non-point group) symmetry
perspective view on important intermediate or catalytic electronic states. Such statements
are presently at the level of conjecture, but it represents another example of a potential
way in which quantum computation and chemistry may unite. We depict this Perspective in Fig. 5.

## VI. OUTLOOK

As quantum technology advances, so does our understanding of what any computational device can accomplish. In this Perspective, we have explored the ways in which results from quantum computer science may impact our view and approach toward computational chemistry. On the one hand, quantum computers are believed to offer an exponential advantage over classical computers in direct simulation of some quantum chemical theories, making certain tasks that previously seemed impossible into ones that should be relatively routine. On the other hand, we saw that even quantum computers have limits, crystallized through the no-fast-forwarding theorem and strong results on the undecidability of physical processes with quantum computers.

In highlighting the ways some of these speed limits are broken, we were led to a framework where learning from natural data is fundamentally different from rote computation. This viewpoint captures the ability of reduced theories to offer meaningful predictions further than the time scales of experiments, and also offers a strategy for dealing with some of the hardest problems in chemical theory.

It has now been decisively shown that classical models empowered by data from quantum computation (including both nature and engineered computers) are more powerful than traditional computation without data, assuming only that quantum computers may perform some tasks faster than classical computers. While data from quantum devices will play an important role in their future interactions with chemistry, the ability to feed quantum data directly into quantum computers offers more power still. The interplay of quantum computers with more advanced quantum sensors may offer untold possibilities. Finally, we expect the interplay between quantum information theory and chemical theory to continue, where import of ideas from domains such as digital quantum error correction may open new avenues of research.

In looking forward, we see a bright future for the ways in which quantum technology may advance the study of chemistry. While it is true that the direct simulation abilities of quantum computers will offer amazing value, we have argued here that this is just the tip of the iceberg. Our ability to address some of the hardest problems in chemistry and nature through data and learning will only be bolstered by stronger quantum technology, and that future is on its way.

## ACKNOWLEDGMENTS

We thank Nathan Wiebe for helpful discussions and feedback on the draft.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

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