What the foundations of quantum computer science teach us about chemistry

Jarrod R. McClean,1, ∗ Nicholas C. Rubin,1 Joonho Lee,1, 2 Matthew P. Harrigan,1

Thomas E. O’Brien,1 Ryan Babbush,1 William J. Huggins,1 and Hsin-Yuan Huang3

1Google Quantum AI, 340 Main Street, Venice, CA 90291, USA2Department of Chemistry, Columbia University, New York, NY 10027, USA

3Institute for Quantum Information and Matter andDepartment of Computing and Mathematical Sciences, Caltech, Pasadena, CA, USA

(Dated: June 9, 2021)

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

I. INTRODUCTION

The study of quantum computing in the abstract is anopportunity to ask ourselves what is possible if we couldattain an almost unimaginable level of control of the mi-croscopic facets of our universe. It was first proposedas a solution to the problem of simulating physical sys-tems with strongly quantum characteristics, a task thathas proven very challenging for traditional computers [1].The idea was that if, like the puppet of a marionettepuppeteer, one could make a precisely controllable quan-tum system act enough like a more interesting system,the puppet could answer previously unknown questionsabout the true system. This core concept of quantumsimulation eventually merged with modern computer sci-ence to form the fields of quantum computer science andquantum computing [2]. This merger allowed these con-cepts to be made more precise and expanded applicationsbeyond physical systems into abstract ones like breakingcryptography [3, 4].

Despite the expansion into other applications, the sim-ulation of quantum systems, and especially strongly cor-related chemistry, has remained a primary application ofinterest. Chemistry represents a sweet spot of quantumeffects strong enough to make them challenging for classi-

∗ Corresponding author: jmcclean@google.com

cal computers, while still having a well known applicationspace to motivate development. Since the original pro-posal of Aspuru-Guzik et al. [5], there have been a num-ber of developments in quantum algorithms for the directsimulation of chemistry bringing costs down from astro-nomical numbers to routine calculations for even verychallenging systems [6–10].

In parallel to algorithmic developments, quantum tech-nology has advanced at a rapid pace. Recent demonstra-tions by the Google group have experimentally shownthat these devices are capable of tasks that are incredi-bly challenging on a classical computer, with a gap thatwill only grow with the quality of modern quantum com-puters [11]. In addition, a number of prototype chemistryexperiments have now been experimentally demonstratedon quantum devices [10, 12–14]. At present, error ratesin quantum devices are too great to make them com-petitive with the best classical algorithms for chemistry,but quantum error correction and mitigation offer pathsforward [15–19].

However, even without an actual quantum computerrunning an algorithm, the study of quantum computerscience in the abstract has led to insights about the lim-itations and possibilities of chemical simulation, and in-deed under modest assumptions, about the universe andchemistry itself. In this work, we highlight some of thesedevelopments leading up to a perspective on the role bothtraditional and quantum computers may play in the fu-ture of chemistry. To build to this perspective, we begin

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FIG. 1. Cartoon of the relative power of direct simulationvs learning models for chemistry, where a point on the fig-ure is a question related to a chemical system. The ovalson the left depict questions efficiently accessible via simula-tion of time dynamics, where quantum simulation is widelybelieved to be exponentially stronger than classical simula-tion, but still softly bound (non-exponential) by the no-fast-forwarding (NFF) to lag behind the system in nature evenwith a quantum computer. Learning models, which includeboth traditional theories supported by data like thermody-namics as well as modern machine learned models, strictlyinclude all questions answerable by traditional simulation, asthey are assumed to have access to simulation but also havetheir power enhanced by data that comes either from natureor other quantum simulations. The availability of quantumstates held in memory and full quantum computation extendsthe questions that may be efficiently answerable beyond thatof traditional learning models.

by framing direct chemical simulation as a digital ex-periment. In this framework, we exploit known resultsthat restrict the power of any computer, even a quantumone, to understand limits of such digital experiments inchemistry. Along the way, we encounter surprising re-sults about the impossibility of an algorithm for deter-mining if a system ever thermalizes, flying in the faceof conventional thermodynamic analysis. However, thisconstruction guides us to new ground, where we will seethat learning from data can be fundamentally more pow-erful than traditional computation. This sheds new lighton the success of many existing chemical theories and theway in which they exceed rote simulation by leveragingthe additional power provided by data.

The relationship between traditional simulation andlearning models in chemistry is cartooned in Fig. 1, whereclassical models with data from nature can exceed thephysical prediction horizons of rote simulation. We con-sider learning models to be models where some amountof high quality training data for the task is available,either from a physical experiment or a simulation in ad-

dition to just the specification of the problem. As wewill argue, this is not just a reference to modern machinelearning methods, but rather encompasses the founda-tions of chemical theory such as transition state theory,molecular orbital theory, and thermodynamically con-trolled reactions. In addition, we take the opportunityto highlight previously unused technology from quantuminformation science that may bolster the development ofchemistry, even before the arrival of full fault tolerantquantum computers. The links from chemistry to resultsin quantum computer science all build to our ultimateperspective that the eventual role of quantum computersin chemistry will be to aid in the construction of learn-ing models for chemistry. This includes providing datato classical models, constructing quantum models thatcan make accurate predictions with far less data, andeventually interfacing directly with quantum data fromchemical systems. To close, we wrap this perspective intoan outlook for the interplay between these two excitingareas.

II. DIGITAL CHEMISTRY EXPERIMENTSWITH QUANTUM COMPUTERS

The idea to draw a distinction between theory andcomputation is not new, with suggestions of consideringcomputation as the “third pillar of science”, alongsideexperiment and theory [20]. In this framework, it cansometimes be more accurate to view simulations as closerto experiments than to theory. Quantum computing, andespecially quantum simulation of physical systems, helpsto make this even more clear by constructing simulationsof the physical world at the quantum level with efficientlyrefineable (and bounded) accuracy. In addition, limita-tions on precision for quantum computers [21] to learnphysical properties sometimes place them even closer toexperiments than classical digital simulations of the samesystems.

Indeed, the most natural setting for a quantum sim-ulation on a quantum computer is that of watching thesystem step forward in time, or quantum time dynamics.Within the field of quantum computing, the term “quan-tum simulation” is sometimes reserved for time dynamicssimulation specifically, and when we refer to rote simula-tion, it will typically apply to this case. That is, one setsan initial state, a physical system defined by its interac-tions, and some time to simulate, and the quantum com-puter performs a computational experiment in mappingthe initial state to the final time, where any reasonableobservable of the system can be probed. This time dy-namics obviously mirrors the most natural settings of thereal world. However, despite this natural setting, one willnotice that a large number of quantum algorithms andtheories closely follow that of electronic structure per-formed on classical computers, where the focus is on lowenergy eigenstates of the electronic system [5, 22], ratherthan exclusively the final state after some fixed evolu-

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tion. This important deviation already wraps in datafrom the natural world that tells us in many systems,thermal states are an apt description of a system and atlow temperatures low energy states are most commonlyobserved. As we will discuss later, the use of such obser-vational knowledge is strictly more powerful than timedynamics simulation, and underpins the great success ofmany modern chemical theory or simulation methods. Adetailed discussion of this point will follow, but for nowwe lump both static and dynamic experiments into thecategory of direct simulation.

By direct simulation, we mean a simulation where thesystem of interest (often processes such as a reaction)is known, and we seek to use the computational experi-ment to gain insight into the process that is challengingto access through experiment. For example details ofa reaction mechanism, transition states, or migration ofcharge through a system. They represent the bulk ofcomputational experiments, both on classical computers,and those proposed on quantum computers, and we drawa distinction between direct simulation and design, oftencharacterized as inverse problems [23].

For the task of direct simulation of chemical systems,quantum computers have been shown to demonstrate anexponential advantage over their classical counterpartsunder modest assumptions [5, 24–26]. An exponentialspeedup has the practical implication that some simula-tions that might have taken longer than the age of theuniverse, could be done in mere seconds. More specif-ically, full quantum dynamics with no approximationsbeyond discretization error can be performed on a quan-tum computer in a time that scales only polynomially int, N , and M where t is the simulated time, M is thenumber of basis functions and N is the number of elec-trons [9, 19, 27]. Interestingly, along with these speedupsit comes with some of the same limitations possessed by aphysical experiment. For example, in contrast to classicalsimulations where more precision in a simulated quantityis often relatively easily obtained, the Heisenberg limitapplies to any measurements one might perform to ex-tract information. These approaches have also includedexploring ansatz for electronic systems that are inaccessi-ble to classical computers [13, 28]. Perhaps surprisingly,recent results have shown that quantum computers caneven achieve a scaling for exact computations that is sub-linear in the basis set size, by taking advantage of a firstquantized representation in a way that has no currentclassical counterpart [9].

However, quantum computers do not vastly extendtheir capabilities into the realm of discovery or design,which we identify as distinct from direct simulation.That is, while many of the proposed simulations wouldbe faster and more accurate representations of the samesystems than would be achievable classically, they aremuch the same type of experiment without reaching intothe design space. While there has been some notion ofhow quantum search may assist in design, most resultspromise at most a quadratic speedup in contrast to ex-

ponential speedups in direct simulation [29], though wenote the direct simulation subroutines used in a potentialsearch would still benefit from improved speed or accu-racy. Due to the immense size of the design space, thisquadratic speedup is not terribly compelling if it is notcombined with structured strategies. That is, any im-proved 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 neighbor-hood of known, synthesizable compounds. In addition,practical issues arising in real quantum computers mayprevent quadratic speedups from being advantageous forsome time [30]. We will argue later that perhaps it iseven more apt and useful to consider an infinite chemicalspace as opposed to one which is merely combinatoriallylarge.

That said, direct simulation, of course, has great valuefor design. For example, if a key reaction or catalystis known to work to catalyze a reaction of interest, themicroscopic detail of a computational experiment that isinaccessible in the lab can offer insight into the causalmechanism that enable the design of analogs [31–33]. Inaddition, a now popular approach is the use of screen-ing of many candidate compounds through direct sim-ulation in order to identify new candidate molecules orlearn properties which are predictive for a desired designtask [34–36]. This type of approach is also used for se-lection of candidates in protein to small-molecule dock-ing [37]. Direct simulation can also be used for noveldiscovery of processes through direct time evolution, ashas been done with the molecular Nano-Reactor [38].Quantum computers serve to strictly improve this typeof discovery by expanding the set of systems that canbe accurately treated in a given amount of time by suchmethods. Advantage in improving the accuracy in directsimulation has been the primary focus in much of the lit-erature of quantum computation for chemistry, with thehope that it enables accurate study of important systemsinaccessible to current methods. The example receivingperhaps the most attention is FeMoCo [19, 25, 27, 39–41], where the hope is that direct simulation will offermechanistic insight that leads to the design of new cat-alysts for nitrogen fixation. However, when leveragingdirect simulations results for improved design, the simu-lations are typically tightly coupled to a theory alreadyrooted in data. For example, the energy of a transitionstate might be used to justify a reaction rate, but this isinherently coupled to the empirical success of transitionstate theory, rather than a result which emerges directlyfrom simulation.

In contrast, comparatively little attention has beenpaid to the potential advantages of improved dynamicssimulations, perhaps due to the larger foundation of workavailable utilizing static eigenstates in the classical case.In addition, it is perhaps surprising that dynamics simu-lations on quantum computers that fully relax the Born-Oppenheimer approximation, using fully quantum nucleiand exact electronic representations, are about as costly

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as those that live within such approximations. However,with each step towards this nuclear-electron soup, whatis gained in accuracy might be paid in ease of interpreta-tion as the added overhead of re-discovering the notionof a molecule through observable measurements remainsa largely uncharted problem in the context of quantumcomputing [24, 42]. Taking advantage of these depar-tures from the tried and true anchors of chemical space,and embracing the idea of a fully ab initio experimentwill require more development in how this type of simu-lation can be most beneficial. However, if it is unlocked,it may offer new insights into complex dynamic processesin spectroscopic or electrochemical processes where bothexcited electronic and quantum nuclear states can play arole [38, 43]. It may also offer new insights into the longstudied challenges surrounding proton coupled electrontransfer [44].

Given the immense power of quantum computers inspeeding up these types of direct simulation, one is oftenleft wondering how far these advantages extend. Thisquestion is one that has been studied in great detail incomputer science, and we now wish to turn to our under-standing of the limits of power in quantum computation.The similarity between quantum computations and thephysical world imply that the results also tell us some-thing about nature and chemistry itself, and we now turnto this perspective.

III. LIMITS OF EVEN QUANTUMCOMPUTATIONS IN CHEMISTRY

Having set up the perspective of quantum computationas digital experiments, we now turn to the implicationsthis may have for the physical systems themselves. Due,in part, to the challenge of building scalable quantumhardware, for a good part of its history quantum com-puting was a largely theoretical endeavor that aimed tounderstand the power of quantum devices, and by exten-sion, physical systems. Considerable progress has beenmade on this problem, and we wish to highlight thoseresults we believe have the most bearing on chemistry.

Before discussing how they pertain to chemistry, it isworth mentioning a few aspects of the assumptions madein stating the results below. In particular, in order totalk about scaling and cost, it is necessary to have somemodel of computation where cost can be quantified. Wewill generally be assuming this means we use a finite,often local, set of operations called quantum gates, andwe work within a digital gate model of quantum compu-tation. This assumption is in some ways analogous toassuming nature favors local interactions or that com-puters are built from modular components. Moreover,some of the results below most precisely pertain to thenumber of times some information, such as the Hamilto-nian of a system, must be accessed. This is known as thequery complexity, and while it is often closely related totime complexity, these can be different in some circum-

stances, such as when parallelization is or isn’t possibleand we attempt to be clear as to when this distinction isimportant.

A. Quantum computers can’t fast-forward timeeither

Starting with the most direct form of quantum simu-lation, time dynamics of a quantum system, we have al-ready noted the exponential advantage over classical sim-ulation achieved by current quantum algorithms. Thisbegs the question how far these quantum algorithms cango in accelerating simulation of physical systems. For ex-ample, can a quantum computer simulate a system fasterthan nature itself evolves the same system, that is, sub-linear in the time being simulated? It turns out that, as-suming one only has the available space to represent thesystem serially, the answer to this is a resounding no. Itis also worth noting that if a system is finite and the timeto simulate is larger than the number of possible systemstates, the cost of diagonalization, finite state enumer-ation, or some equivalent can be paid to fast-forward asystem. However, in most cases the time-scale of inter-est is much less than this, and indeed as we argue later,infinite perspectives on chemical systems are sometimesmore natural.

While there are some special systems that can be fast-forwarded [45], it cannot be done in the most generalcase. This result can be understood at a high level byimagining if this were true and realizing one could builda recursive simulation that does any computation in van-ishingly small time, by nesting simulations within simu-lations with no space overhead. If one allows the spaceon the quantum computer to expand polynomially withthe simulation time, this result is more unclear and re-lated to the classical debate of if any computation can beparallelized arbitrarily well, encapsulated by the theoret-ical question of if P=NC [46]. This is an open questionthat currently holds a similar status to the more famousquestion of if P=NP. Some progress has been made onthe question if all computations can be parallelized arbi-trarily for specifically quantum computers, showing thatmore simulation time is strictly more powerful relativeto an oracle [47]. To simplify the message here, we as-sume that quantum space resources are the minimumrequired to perform the simulation, and that as a resultoracle queries must be done in serial. More formally,the limitations of simulation are encapsulated in the no-fast-forwarding theorem [48, 49]. This theorem impor-tantly shows that not only is universal computation notfast-forwardable, but surprisingly even some Hamiltoni-ans that are incapable of universal computation cannotbe fast-forwarded either.

Under the stated assumptions, the no-fast-forwardingresult practically implies that no simulation which retainsa complete level of detail at the quantum level can simu-late physical systems faster than nature, and likely there

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will be some large factor disadvantage between them.This result is in some ways not unexpected from classi-cal simulations of classical chemical models like moleculardynamics, where for a fully detailed tracking of the modelin the most general case, one expects to lag behind thephysical timescale of the system unless approximations orreduced models are utilized. The emphasis here is thatthis limitation provably carries into the case of using aquantum computer as well. Taking this result at facevalue along with the knowledge that important chemicalreactions like rusting can take months to occur, usinga single molecular realization in a simulation could takecomparable times to witness a reaction event, making dis-covery in this fashion totally impractical. However, mostchemists would know that is not the end of the story.

Specifically, methods like transition state theory basedon static calculations of free energies or transition pathsampling [50] can allow access to rate constants on timescales much faster than direct physical simulations [51].Hence, while the above theorem is true for precise sim-ulations of the full quantum state, we see from practicethat physical theories grounded in observation can cir-cumvent such wild overheads by eliminating the com-plexity of the full quantum state and focusing on a re-duced set of observables, reduced precision, or approxi-mate models amenable to fast-forwarding. Such reduc-tions of the number of important degrees of freedom havebeen studied in various contexts to try to understand“what makes science possible” [52, 53]. It should also benoted that these results are about the most general sys-tem, and specific systems may exhibit physics that allowfaster than physical simulation when that data is utilizedby validating approximate models. In other cases, par-ticular models are amenable to classical fast-forwarding,like non-interacting electron models, but they are notgeneral enough to encompass all quantum systems withinteractions.

The observation that chemical methods have been sosuccessful in making predictions beyond physical timescales leaves one believing that perhaps physical systemsalways admit some form of reduction that will allow usto forecast results that we actually care about (as op-posed to a full quantum state), well in advance of simplystepping them forward in time. A recent string of re-sults in quantum computer science has proven that, infact, even in some very simple systems by construction,there must exist some questions that are fundamentallyunanswerable ahead of time. This list of problems sur-prisingly includes whether simple one-dimensional (1D)systems thermalize [54], which has implications for thepredictive power of thermodynamics in general systems.These results are much stronger than simple chaos of dy-namic systems and require some exposition, but we be-lieve they are a value lens through which to view chemicalproblems.

B. Questions no algorithm can answer -undecidability

A direct reading of the no-fast-forwarding theoremcontrasted with the unassailable predictive success ofsimplified chemical models raises a number of interest-ing questions. On the one hand, the no-fast-forwardingtheorem tells us that even with an exponential advan-tage over classical simulation of quantum dynamics, aquantum computer aiming to simulate the dynamics ofa full wavefunction to a fixed accuracy is bound by somemultiplier of the physical time of the actual process. Onthe other hand, we see that reduced models, for examplemean-field molecular orbital models, reductions based onevaluating the energy of stationary states like ground andexcited states, or even arrow pushing in Lewis dot struc-tures can sometimes be powerfully predictive for chemicalphenomenon of interest at a relative time cost much lessthan the time scale of a process of interest. This leadsone to ask if there is always a predictive model or al-gorithm for simplified questions. Surprisingly, results incomputer science have shown us that there are indeedeven relatively simple questions related to physical sys-tems for which it can be proven that no single algorithmcan answer for all such systems in finite time. Here, whilewe will certainly not claim that this should cause one toabandon current methods, we will argue that these re-sults should have implications for how one approachessome chemical problems.

To make our case, we will start with a motivatingexample before we move to the more general conceptsthat will allow us to address some of the tensions withthis interpretation. Consider the question of “Given un-bounded time, does this physical system reach thermalequilibrium?”. The basic assumptions of conventionalthermodynamic theory, leaning on the ergodic hypothe-sis [55–57], assert that the answer is a resounding “yes”under assumptions that the system is ergodic. Indeedmany standard molecular dynamics simulations exploitboth possible directions, either determining free energiesthrough proxies derived from long time simulations [51]or predicting long time behavior through models for freeenergies. However, recent results have shown that thereare 1D translationally invariant systems for which noalgorithm that can answer the question of thermaliza-tion or even simple questions about energy gaps in finitetime [54, 58–60].

To make such a problem even more concrete, let’s takethe question of a finite gap as an example. The prob-lem statement can be setup briefly as follows. One isprovided as input a finite Hamiltonian for a chunk ofa system that will be repeated indefinitely. While anyfinite chunk of this system could be diagonalized, deter-mining the existence of a gap for that chunk, this proofimplies that no finite number of chunks taken togethercould ever tell you about the status of the gap for the in-finite system in the general case. Moreover, any extrap-olation on those chunks to a large size limit cannot be

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guaranteed to converge. The statement is so strong thatwe can guarantee that not only will no simple algorithmwork, but there cannot exist such an algorithm of anyfinite time complexity. This type of problem is known asan undecidable problem, the first example of which wasthe famous halting problem of Turing [61]. This class ofproblems is much more challenging than even the per-haps more familiar class of NP-complete problems, aseven when provided with a supposed answer, a mortalprover cannot verify if it is correct. This implies that fora number like an energy or free-energy representing a sys-tem, either that number cannot be generally predictiveof the behavior of that system, or the number itself mustbe uncomputable. Showing that this is true, even for aquantum computer, which can perform classical compu-tation as a special case, underscores the weight of theseresults.

The idea that bare thermodynamic quantities may notbe predictive due to lack of thermalization is certainlynot new in the study of non-ergodic systems, with pre-eminent examples like spontaneous symmetry breaking,spin-glasses, and the general idea of a Gibbs measure [62–65]. These results strengthen these concepts to tell usthat in some non-ergodic systems, there is no computablereplacement for something like free-energy that is pre-dictive of the behavior of a system. Hastily defined, thehalting problem is the problem of finding an algorithmthat, given a finite program input, can determine in finitetime if the program halts or runs forever. If one viewsa physical evolution as a computation where halting isdefined as some qualitative change [66], for example, thefraction of random systems which thermalize should beclosely related to the fraction of random programs thathalt, a number known as Chaitin’s constant [67]. Indeed,it is easy to see such a thing requires solving the haltingproblem for every possible instance to determine the dig-its of the fraction of halting programs, which defines it asan uncomputable number. For such systems, assumingergodicity is analogous to solving the halting problem bysimply hypothesizing that all programs encoded by thephysical evolution halt.

For veteran chemists, another way of viewing this re-sult is that it is a vast strengthening of the concept ofkinetically controlled reaction systems to be contrastedwith thermodynamically controlled systems. Previouswork in this area has shown that this type of unpre-dictability is distinct and stronger than traditional no-tions sensitivity induced chaos in rendering systems un-predictable, even when they are deterministic. Phrasedin a physical point of view, for a system exhibiting unde-cidable dynamics, there are sudden, qualitative changescan occur that cannot be predicted in any way otherthan simply stepping the system forward in time [66].For chemical practitioners, such a description is certainlyreminiscent of the emergent phenomena that transitionus from simple molecules to proteins, RNA, DNA, com-plex catalytic networks, and even living systems.

Before continuing, we must address a tension that

Synthesis

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FIG. 2. Sketch of two viewpoints for the problem of chemicalsynthesis. In this scenario, one wants to know if a certaintarget compound can be reached with a feedstock of specifiedchemicals under typical conditions. The combinatorial viewtries to address this question by introducing a closed reactionsystem with a large but finite number of the reagents, andexplores potential reaction pathways to determine feasibilitythrough some proxy like free energy (∆G) or a short timesimulation. The undecidable view point embraces an opensystems picture and the relationship to DNA computing toargue that the most feasible approach tries to use data eitherfrom nature or related simulations to determine the answer tothe question. Such an approach can more naturally includethe effects of common side-reactions or strong kinetic control.

arises with the mapping of physical processes to halt-ing problems in the form of finite versus infinite systems.In order to map to a halting problem rigorously, one typ-ically maps the operations of a physical system to opera-tions of a Turing machine, with an infinite sized tape. Ifthe tape were finite, one can spend a very long, but notinfinite, time examining all possible states of the systemto arrive at an answer to any question related to the stateof the system. For example, in studying the dynamics ofa finite physical system, one could discretize the possi-ble states, and expend an exponential amount of time inthe system size determining an answer. Hence one mightwant to argue that since the universe is often presumedfinite, these conclusions are of little consequence, eventhough from a practical point of view, the exponentialof even a modest sized finite system would take longerthan the age of the universe. Despite their somewhatsimilar predictions in practice (too long to compute), weargue that embracing the viewpoint of undecidable dy-namics allows one to let go of the impossible dream oftreating chemical questions by the exponential enumer-ation of the combinatorial perspective. Whether or nota physical system exhibits truly undecidable behaviour,embracing the possibility leads to a more natural descrip-

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tion of many chemical processes and points the way to-wards more apt solutions.

We depict an example of this perspective in Fig. 2.Consider a collection of small molecules attached to areservoir of source chemicals, and we wish to know if aparticular strand of DNA will ever form given this setup.If one took the closed system, or combinatorial, view-point, it would suggest that perhaps the most effectiveway to address this question is to model the reservoir withsome enormous, but finite, set of explicit molecules, andcheck all of the possible states of the dynamic system.Along the way, one can check things like the change infree-energy (∆G) or simply proceed with short time dy-namics. Not only is the approach untenable due to thenumber of possible configurations, but given the natureof self-catalytic reaction networks, one could accidentallyinclude slightly too few molecules to form the right cat-alytic process to enable the formation of that strand andreach the wrong conclusion. In contrast, the open sys-tem, or undecidable, interpretation of this setup, is thatthe physical system can continue to borrow from the in-finite reservoir like states of the Turing tape. If it hasto temporarily expand to chemical networks of unfore-seen size, it can do so. In this situation, one tends tolean on data models for what has existed before, andwe detail later how doing so changes the power of thesimulation approach. The undecidable interpretation issupported by recent work showing that DNA computingis Turing complete and as such can encode the haltingproblem [68, 69]. While DNA computing is a bit of an ar-tificial construction, and real DNA that tends to form inlive systems is subject to other constraints that make itsactivity more predictable, the possibility of implement-ing universal computation in the dynamics of a chemicalsystem pushes us to embrace the wildest consequences ofthe theory of computation.

This tension between finite and infinite models is notunfamiliar when related to the study of other non-ergodicsystems, like spin-glasses [62]. In such cases, modelsmight predict that finite systems have some chance of ex-hibiting substantial rearrangements, albeit on exponen-tially long timescales. However, it has been seen to beuseful to lean on a model that takes a limit where thesedifferent basins qualitative fracture from each other, andare considered separate. We suggest here that as molec-ular systems grow in complexity, the run-away reactionsystems that form are best treated by considering themas fracturing into novel parts of chemical space, withouttracking vanishingly unlikely events that transition onebetween these parts of space. Indeed, the undecidabil-ity of thermalization suggests that for infinite systems,it is correct to view such fracturing as exact in the mostgeneral case. Even in simpler cases, this model tracksmore closely with the experience of laboratory chemistswho have to contend with side reactions rendering a ther-modynamically favorable synthesis seemingly impossible.By not splitting hairs about the distinction between alarge finite system and a truly infinite one, we can move

towards practical alternatives to the brute-force combi-natorial approach of pure computation.

But if one resigns themselves to accept that many ofthese problems have no general algorithm, is one left withno option but to plod forward with rote dynamic simu-lation or give up? At least the combinatorial approachoffered a glimmer of hope, even if it required longer thanthe age of the universe for tiny problems right? In thenext section, we argue that in fact this realization doesoffer an alternative, which focuses the scientist on an ap-proach based on learning and reduced chemical modelconstruction.

IV. LEARNING AS AN ALTERNATIVE TOCOMPUTATION

Rather than claim one must give up hope on problemswhere the most natural formulation looks undecidableor hopelessly expensive, we argue that embracing theundecidable interpretations of these problems frees onefrom the distraction of the combinatorial approaches, andleads them to embracing the only known resolutions toproblems that exist for halting problems. The first, andnot really palatable solution, that has been mentioned, isrunning time dynamics forward and hoping for the eventof interest to occur. The second, and more actionableviewpoint, is to understand that systems with advice canformally resolve such problems. While advice has a pre-cise definition in theoretical computer science [70], it wasrecently shown that data from the real world can act as arestricted form of advice [71]. To solve halting problems,one’s advice would need to constitute a form of infinitetime pre-computation, however much like the argumentof finite versus infinite systems, the pre-computation per-formed by the physical universe before this point, thoughfinite, is already quite strong and available to be revealedthrough experiments.

To see an example of this in chemistry, one can lookto the field of natural product synthesis, where one seeksto synthesize chemical species found in nature from sim-pler, readily available components in a finite number ofsteps. For the most general complex molecule, it can bequite difficult to say if a product can be reached witha fixed set of reagents and conditions or at all, perhapsdue to some intervening side reaction. In fact, we con-jecture this form of the problem may be undecidable,since we could specify the target molecule as a particularstrand of DNA [69]. By contrast, in the study of naturalproduct synthesis, the challenging question of chemicalsynthesizability is answered by nature itself, and practi-tioners are freed to discover practical routes towards aproduct rather than ponder if success was even possibletheoretically. In this sense, empowering ourselves withobservations from the physical world can enable answersto questions that cannot be practically resolved with rotealgorithms. More generally, we believe one could view themodels and study of synthesis in organic chemistry and

8

Chemical system

Quantum Sensor

Classical Computer

Quantum Computer

Mea

sure

men

t Transduction

Predictions

FIG. 3. Comparison of quantum and classical data pipelines.The presence of classical data or traditional measurements,e.g. data from laser spectroscopy or thermodynamic mod-els, already confers an advantage over traditional simulation.Here we contrast that pipeline with one were quantum data iscollected from quantum sensors, which can be stored in quan-tum memory, increasing yet again the power of available data.In the classical case, a sensor measures and reports classicalvalues back to a classical computer. In the quantum case, asensor has its state transferred into a quantum computer, aprocess also known as transduction, where it may be com-puted alongside multiple additional copies of such states toperform tasks that would be exponentially more costly on aclassical computer. In such setups, the quantum transductionroute can require exponentially fewer measurements than theclassical case to learn some quantities or perform some tasks.In cases where data is hard to come by, e.g. transient, short-lived, or rare systems, this can make an impossible character-ization into one that is routine.

their predictive success in this light. Hence, taking thispoint of view suggests that approaches based on learn-ing are fundamentally different than those that rely oncomputation alone.

With the rise of popularity in machine learning, nat-urally there is now a flurry of work applying techniquesof machine learning to chemistry [72]. The applicationsrange from prediction of direct simulation quantities [73]to synthesizability [74] and inverse design [23]. We be-lieve the results we highlight here bolster the motivationfor pursuing this line of work in the general sense, but alsosupports the long standing tradition of chemists develop-ing physical models as well. While much practical workremains to be done in finding the best representation orapproaches, it is now understood that these approachesare fundamentally different, and indeed more powerful insome ways, than traditional simulation approaches.

In particular, recent work has shown that the power of-fered by data from a quantum computer can lift classicalmodels to be competitive with their quantum counter-

parts in some cases [71]. If nature is indeed a universalquantum computer [75], this would suggest that quan-tum computations could help fill the role of natural ex-periments in providing data to empower learning models,but with improved programmability and flexibility. Thismight suggest that in the future, a key role of quantumcomputers, which may still remain scarce compared totheir traditional counterparts, may be to provide datafor learning models primarily run on traditional comput-ers. However, that would be a rather unexciting fate fora technology that pushes the limits of our understandingof the universe.

Instead of accepting this fate, we appeal to other re-cent work that has shown an overwhelming advantage forquantum computers in how they can process data moveddirectly into the computer from a quantum sensor, a pro-cess referred to as transduction [76, 77]. A comparisonwith a traditional data pipeline is shown in Fig. 3. Insuch cases, even performing quantum data processing oneven very few copies stored in quantum memory can al-low one to extract properties of quantum systems thatwould require exponentially more data in the classicalcase [76]. Even early machines may be able to manip-ulate few copies for limited times, and store this quan-tum data for future models. This separation is strongerthan a traditional computational separation, as if limitedmeasurements are available due to the transient natureof a system, no amount of computation can allow tradi-tional measurement and computation to catch up. Thisimplies that even systems with a number of qubits thatcan be classically simulated could prove advantageous.If the quantum computer has quantum memory that canmaintain states in quantum form, this extends the powerof these approaches even further. The relationship be-tween this hierarchy of data models and traditional sim-ulation is depicted in Fig. 1. Moreover, if one venturesfurther into the realm of speculation, recent computerscience results have shown that if one can entangle twoparties trying to prove something to another, what theycan convincingly prove expands to unimaginable heights,including halting problems. While this celebrated result,MIP*=RE [78], has not yet been directly connected withthe physical world, it is tantalizing to imagine how itmight reflect on the power of physical experiments em-powered by entanglement to reveal the mysteries of theuniverse. The question, of course, becomes where to findsuch useful quantum data and how to effectively get itinto the quantum computer.

Chemistry has long benefited from the power of quan-tum sensors in nuclear magnetic resonance experiments.While the most basic forms of these experiments arebased on ensemble measurements at an effectively hightemperature, they offer incredible insight and are funda-mentally quantum. Interestingly, the form of these basicexperiments can be closely identified with the so-calledone clean qubit model, or DQC1 and this has inspiredsome recent work in pursuing advantages with NMR [79].However, as NMR technology advances with techniques

9

like hyper-polarization [80], multi-dimensional meth-ods [81], spatial resolution [82], and zero-field tech-niques [83], we appear to be accessing more and morepure quantum states with increasing control [84]. In ad-dition the design of more advanced molecular quantumsensors is an active research area [85]. With sufficientadvances in transduction techniques and developmentsin quantum error correction, it may become possible toload data from molecules directly into quantum comput-ers, and perform manipulations that are provably chal-lenging for a classical device even with unbounded com-pute time to replicate, due to the query separations wemention above. This could enable faster chemical identi-fication, unprecedented accuracy in examining quantumeffects in molecules, or fundamentally new techniques forthe control and manipulation of chemical states.

V. UNTAPPED RESOURCES IN QUANTUMCOMPUTER SCIENCE

The discussion thus far has largely centered around therole of quantum computer science or quantum comput-ers themselves in new approaches to understanding chem-istry. However, it is also interesting to ask more generally,what theoretical tools from quantum information sciencewith potential to impact computational chemistry remainuntapped. For example, techniques in tensor networksthat received much of their theoretical development inquantum information, have now begun to inform ansatzfor correlated wavefunctions. Both matrix product statesand tensor network states coupled with an entanglementperspective on strong correlation has lead to a wealth oftheoretical developments [86, 87].

An area of remarkable theoretical development that re-mains relatively unknown in the chemical community isthat of quantum error correction [16, 88]. Perhaps fora chemist, the best way to understand these methods isin relation to the existing and well developed body ofresearch of quantum control of chemical reactions andchemistry [89–91]. We depict aspects of this relation-ship in Fig. 5. In quantum control, a signal, oftenthrough a laser excitation or similar, may be used to con-trol and measure the system through feedback, howeverthese measurements are often destructive with regardsto quantum states in the system, and we refer to thisas analog quantum-classical feedback control. Throughthe use of a separate quantum probe or ancilla system,one can make measurements of a system that actuallygenerate or stabilize entanglement in the quantum sys-tem, or an analog quantum-quantum scheme. Takingthis one step further, one may use multiple probes anddigital algorithms that adaptively change based on mea-surement results to pump entropy out much faster thantraditional thermalization would allow. It turns out theexponentially long suppression of transport out of collec-tion of desired quantum states is reliant on an abstractdigital perspective of the states. These constructions are

formalized in the theory of digital quantum error correc-tion, where it has been shown that with only spatiallylocal measurements and classical computation, it can bepossible to stabilize exotic quantum states for essentiallyarbitrary lengths of time with asymptotically modest re-sources. If such techniques could be ported more directlyinto chemistry, stabilizing electronic or excitonic statescould lead to novel reactive methods or wildly improvedenergy transfer efficiencies. While this remains in therealm of pure speculation, perhaps probe systems thatcan be interacted with via future versions of tunnelingmicroscopes with near single electron resolution coupledto catalytically active sites on metal surfaces could offersome of the first places to explore these ideas.

The development of quantum error correction was alsoclosely linked to the development of simulation for aparticular class of quantum states, known as stabilizerstates. These states can exhibit essentially maximal en-tanglement across a system, yet are incredibly efficientto simulate classically, often scaling like N2 or N3 in thenumber of orbitals [92, 93]. While they are discrete innature, requiring some novel optimization techniques likereinforcement learning, if coupled with simple orbital ro-tations, these could form yet another powerful ansatz forefficient classical electronic structure methods informedby quantum information that has yet to be exploited. Wedepict this connection in Fig. 4.

While the existence of a powerful ansatz that has notyet been explored is perhaps strong enough motivation byitself, the link to quantum error correction provides evenmore fuel for this interesting avenue of inquiry. Many ofthe most challenging bio-metallic enzymes, such as nitro-genase, are characterized by an active site that is diffi-cult to describe with traditional methods due to staticcorrelation related to quasi-degeneracy of electronic con-figurations. Quantum error correction takes advantageof strong entanglement and engineered degeneracy as amechanism for protecting information. If nature wereto follow a similar path, perhaps the electronic near-degeneracy and resulting entanglement in these systemscould be an avenue for protecting a coherent reactionpathway similar to the approach of quantum error cor-rection. In addition, many modern quantum error cor-recting codes take advantage of topological protection,and topological effects have been studied in the contextof electronic degeneracies near conical intersection [94].While natural systems do not have access to an opti-mal decoder to remove excess entropy, the notion of self-correcting memories where systems exhibit such behaviormore naturally draws an enticing connection [95]. In sucha case, the stabilizer formalism for representing theseground states would be especially apt, efficiently ma-nipulating untold numbers of determinants through non-obvious symmetries [96]. This could offer an electronic(non-point group) symmetry perspective view on impor-tant intermediate or catalytic electronic states. Suchstatements are presently at the level of conjecture, but itrepresents another example of a potential way in which

10

FIG. 4. Contrasting the state vs stabilizer and rotationviewpoint for quasi-degenerate systems. The top shows arepresentation of an electronic state built from rotations ap-plied to a stabilizer state, while the bottom depicts a tra-ditional configuration based view. Electronic catalysts oftenexhibit quasi-degeneracy, making them challenging to treatwith single reference methods. The stabilizer formalism fromquantum error correction, combined with efficient single par-ticle rotations, may offer a compact way to both simulate andanalyze such situations, as the use of degeneracy to protectquantum states is a core principle of quantum error correc-tion. Graph state representations of stabilizer states may offerconnections to chemical bonding and electron correlation the-ory, all while being computationally efficient to simulate andanalyze classically.

quantum computation and chemistry may unite. We de-pict this perspective in Fig. 4.

VI. OUTLOOK

As quantum technology advances, so does our under-standing of what any computational device can accom-plish. In this perspective, we have explored the waysin which results from quantum computer science mayimpact our view and approach towards computationalchemistry. On one hand, quantum computers are be-lieved to offer an exponential advantage over classicalcomputers in direct simulation of some quantum chemi-cal theories, making certain tasks that previously seemedimpossible into ones that should be relatively routine. Onthe other hand, we saw that even quantum computershave limits, crystallized through the no-fast-forwardingtheorem and strong results on undecidability of physicalprocesses with quantum computers.

In highlighting the ways some of these speed limits arebroken, we were led to a framework where learning from

Chemical system

Quantum probe

Analog quantum-classical feedback

Analog quantum-quantum feedback

Digital quantum-quantum feedback

Digital correction algorithm

FIG. 5. Digital quantum feedback control inspired by quan-tum error correction. Quantum control of chemical reactionsis a well studied field, and many aspects of it are mirrored inthe control of qubit systems. While direct measurement of asystem through typical analog signals destroys entanglementin a quantum system (quantum-classical), the use of a quan-tum probe or ancilla system can allow measurements thatpreserve or create entanglement in the target system reliably(quantum-quantum). While this paradigm is already power-ful, the tools of discrete computer science allow one to usesuch interactions in combinations with algorithms that canbe imagined as sophisticated Maxwell’s demons to pump en-tropy out of the system at incredible rates far exceeding thoseof typical thermalization to stabilize exotic quantum states,like electronic states, for time scales that could in principleextend for years, using only local measurements and feedback.These tools are developed at length in the field of quantumerror correction, and we imagine here how those tools mighteventually integrate with chemical systems. Such construc-tions may pave the wave for novel excitonic or energy trans-port designs.

natural data is fundamentally different from rote com-putation. This viewpoint captures the ability of reducedtheories to offer meaningful predictions further than thetimescales of experiments, and also offers a strategy fordealing with some of the hardest problems in chemicaltheory.

It has now been decisively shown that classical mod-els empowered by data from quantum computation (in-cluding both nature and engineered computers) are morepowerful than traditional computation without data, as-suming only that quantum computers may perform sometasks faster than classical computers. While data fromquantum devices will play an important role in their fu-ture interactions with chemistry, the ability to feed quan-tum data directly into quantum computers offers morepower still. The interplay of quantum computers withmore advanced quantum sensors may offer untold possi-bilities. Finally, we expect the interplay between quan-tum information theory and chemical theory to continue,where import of ideas from domains like digital quantum

11

error correction may open new avenues of research.

In looking forward, we see a bright future for the waysin which quantum technology may advance the study ofchemistry. While it is true that the direct simulationabilities 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 inchemistry and nature through data and learning will only

be bolstered by stronger quantum technology, and thatfuture is on its way.

VII. ACKNOWLEDGEMENTS

We thank Nathan Wiebe for helpful discussions andfeedback on the draft.

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